Paper Title: Using low-cost Blu-Ray Optical Pickup Units for Measurement of Single Photon Emission from NV-Centers.



Global convergence of a modified BFGS-type method based on function information for nonconvex multiobjective optimization problems
    
		Ying Xue yang 
	
    August 5, 2024
==================================================================================================================================





§ ABSTRACT  
This work presents a cost-effective method for collecting single photons emitted from single nitrogen-vacancy centers in nanodiamonds. Conventional components of a confocal laser-scanning microscope, such as microscope objectives and the piezo translation stages, are replaced by two affordable Blu-ray optical pickup units. A Hanbury Brown and Twiss setup is used to identify single photon emission. The proposed approach is inexpensive and simple and lowers the entry-level to single photon research for quantum technologies. This enables student lab experiments or demonstration experiments at schools and shows that efficient sources of quantum light can be made from standard components compatible with established industry processes.






§ INTRODUCTION

In the developing field of quantum technologies, single photon sources (SPSs) have attracted considerable interest due to their role in various quantum-based applications, including quantum communications, cryptography, computing, and metrology <cit.>. These applications rely significantly on the controlled generation and coupling of single photons. Among the various methods investigated for single photon generation, the use of single nitrogen-vacancy centers (NV-centers) in diamonds has proven to be one of the promising and versatile solutions, since they exhibit efficient photon emission at room temperature, photostability, and offer easy ways to manipulate its spin quantum state <cit.>.

There are different approaches for interfacing single photons, such as using high-NA microscope objectives with piezo positioning systems for accurate sub-micrometer positioning <cit.>, direct coupling to fibers <cit.> or employing special optical elements <cit.>, although these methods are usually expensive to produce or build.
The goal of this work was to develop a cost-effective solution based on a Confocal Laser Scanning Microscope (CLSM) setup for the efficient optical coupling of single quantum emitters based on NV centers. A major challenge is to identify an alternative option for precise and selective control of individual diamonds without relying on expensive piezoelectric technology and replacing expensive optical components like high-NA microscope objectives. 
 In this work, a commercially available optical pickup unit (OPU) (Blu-Ray, PHR-803T, Toshiba Co., Tokyo, Japan), which has already been successfully used in other low-cost applications <cit.>, is used to replace the high-NA microscope objective and piezoelectric positioner. 



§ EXPERIMENTAL SETUPS






 §.§ Reference CLSM Setup


CLSMs have proven to be a powerful technique for detecting individual NV centers making it an ideal choice for investigating individual nanodiamonds on a coverslip. 

To get more detailed information about the diamond samples and to get a comparison for a potential 2D scan with the low-cost setup, a reference CLSM is built (see Fig. <ref>). Therefore, a single frequency diode-pumped laser (Cobolt 08-DPL) with a good beam quality (M^2 < 1.1) is expanded by a Galilean telescope (Thorlabs ACN127-030-A + Thorlabs LA1708-AB) with a magnification of  M=6.6̅  from approx. ≈ 0.6  mm to approx. ≈ 4  mm in diameter to fully illuminate the lens system inside the OPU. The beam is reflected by a dichroic beamsplitter (Thorlabs DMLP550R) and focused on the sample through a microscope objective (Nikon CFI Plan Apo Lambda 60XC) with a high NA of 0.95. To eliminate the laser- and background light, the collected fluorescence is spectrally filtered by the dichroic beam splitter, two long-pass (Thorlabs FELH0550) and one short-pass (Thorlabs FESH0750) filter. Following this, a spatial frequency filter (30   m pinhole,  2  × Edmund Optics  48774  f=100 mm) is used to eliminate out-of-focus light. To perform a 2d confocal scan of a sample, a piezo translation stage (PiezoSystem Jena Tritor 100, open loop) is used to allow movements with nanometer precision and ensure accurate positioning. For the Hanbury Brown and Twiss (HBT) experiment, two actively quenched and cooled SPAD (Laser Components COUNT T-series) and the correlation electronics (TimeTagger 20, Swabian Instruments) are utilized. Furthermore, a scientific CMOS camera (Hamamatsu Orca Fusion BT) for widefield microscopy for pre-aligning the nanodiamond sample slide is integrated into the beam path.



 §.§ Low Cost Single Photon Coupling


NV-centers can exist in the NV^- negative or NV^0 neutral charge state and both can act as a source of single photons. The NV-centers can efficiently be excited using a laser at a wavelength of 532  nm. The fluorescence spectrum for an ensemble of NC-centers inside a 100  m diamond is shown in figure  <ref> (a). Both charge states can be identified in the sample by the zero phonon line at 575  nm <cit.> for the NV^0-center and 638 nm  <cit.> for the NV^--center. At room temperature, the emission is broader ranging from 630   nm to 800  nm. The lifetimes of the two charge states in nanodiamonds differ slightly. The NV^- centers has a lifetime of 12 ns compared to 21  ns for the NV^0 center <cit.>. This makes the NV^- center brighter and more easily to be detected. A spectrum of a bulk diamond with 1 m in diameter containing NV centers in both charge states was captured using a spectrometer (Ocean HDX spectrometer) in the described optical setup and is exemplarily depicted in Figure <ref> (a) (red).

As a low-cost approach, and as a replacement for the microscope objective to collect the photons, the integrated lens from a PHR-803T Blu-Ray pickup is to be used (see Fig. <ref> (a)). As shown in Figure <ref>, it is integrated into the existing setup using a flip mirror. The aspheric lens has a large NA of about 0.6 at 650 nm <cit.>. Transmission measurements for this optic were taken using a spectrophotometer (PerkinElmer Lambda 1050), which shows that the lens has an anti-reflective coating and high transmission, particularly in the spectral range of the NV center emission (see Fig. <ref> (a) (black)). The OPU has a smaller transmission at the pump laser wavelength and can be compensated by increased laser power.





 §.§ Scanning OPU Setup


To be able to perform spatial scans without employing piezo actuators, the internal positioning mechanisms of the PHR-803T are utilized for the XY-scan. The optics of a single OPU can be moved along the horizontal and vertical direction and can also be tilted in the horizontal direction. The operation principle is similar to voice-coil actuators.
Five individual coils are directly fixed to the lens holder.  Two permanent magnets near the coils create an external static magnetic field and allow the optics to be positioned precisely. For the laboratory demonstrator, the OPUs were clamped in optical mounts and placed opposite each other. In order to realize the 2D scan, it is necessary to install two PHR-803T at an angle of 90^ ∘ to each other, as seen in Figure <ref> (b) <cit.>. A solid 3D-printed housing is intended for future setups.
  

Since the lens is mounted in the center position and is held by four springs, it can deflect in two directions.  Therefore, the direction of the currents within coils needs to be controlled. To ensure this, DC-motor control operational amplifiers (TCA0372) were used <cit.>, which provide an output current of up to one ampere. The input voltages for the scan are provided by the 16-bit data digital-to-analog converter (DAC, RedLab 3103 USB), which was also used to provide the input voltages for the piezo driver in the reference setup. In the DC motor circuit used here, the virtual ground is at ≈ 2  V output voltage of the DAC. Here, the lens is in the center position and can be deflected bidirectionally with a control voltage between 1  to  3  V, resulting in coil voltages between -1  to 1  V and coil currents between -200  to  200  mA.  The deflection in the horizontal direction in  Figure <ref> (b) was measured by continuously increasing the current through the coils and tracking it with a manual translation stage. The deflection has a high linearity in both directions around the center position of the lens.  




One of the factors limiting the resolution of a CSLM is the size of the exciting laser spot on the sample. For the nanodiamonds to be excited individually, the distribution on the slide and thus the minimum distance between the individual diamonds needs to agree with the diameter of the laser spot. Since these OPUs are not optimized for the green wavelength of the excitation laser and no information about the focal length is available, an optical approach was chosen to determine the focus size. We positioned a CMOS image sensor (Raspberry Pi High Quality Camera Module) at various distances behind the lens. Gaussian fits through the center of the mode images (an example is shown in Figure <ref> (c)) were used to measure diameters at different positions. Subsequently, the transverse Gaussian beam profile w(z) = w_0 √(1 + ( z λ/π w_0^2)^2) was fitted for all measured diameters, as shown in Figure <ref> (c). From this fit, a beam radius in the focus of w_0 = 1.66 ± 0.74  m can be estimated. The parameters for spin coating should therefore be selected so that the diamonds are at least 2 · w_0 apart. The mode profile pictured in Figure <ref> (c) appears suboptimal, suggesting that the smallest achievable focus diameter may not be reached, possibly due to aberration in the OPUs imaging system, since it is not optimized for our external pump wavelength of 532 nm. Nevertheless, the estimated spot diameter is sufficient for our intended measurements.



 §.§ Samples Preparation


 Fused silica serves as an ideal coverslip substrate for nanodiamond adherence due to its minimal autofluorescence. This reduces unwanted background signals and ensures a better signal-to-noise ratio, which is very important considering the small fluorescence signal of single NV centers.

To achieve a uniform deposition of nanodiamonds (Adamas Nanotechnologies NDNV40nmLw10ml) on the coverslip, a multi-step process is employed. A concentration of 1:100 nanodiamonds in deionized water (DI-water) is explored, with a solution comprising 1 mg/ml diamonds in DI-water and 0.3  wt% PVA to enhance adhesion and distribution. The coverslips undergo pretreatment with a 2% Hellmanex solution in DI-water, as suggested in <cit.>, utilizing ultrasonic bath exposure at 50^∘ C for 10 minutes to enhance cleaning. This step minimizes contamination from residues, oils, and particles while improving nanodiamond adhesion. The ultrasonic bath also aids in dispersing nanodiamonds in DI-water and PVA, ensuring a homogeneous particle distribution. Subsequently, a spin-coating technique is employed, where 30  l of nanodiamond suspension is placed at the coverslip center and rotated at 3500  rpm. This process utilizes centrifugal force to create a thin, uniform nanodiamond film, offering precise control over coating density through spin speed and duration adjustments.



§ CONFOCAL SCAN AND MEASUREMENTS OF THE NV CENTER



The 2D scanning, focusing, and data acquisition are controlled for both setups using a Python script.
The fluorescence intensity plots generated from the 2D scans indicate the quality of the control of the pickup coils. The bright spots in the image of the low-cost system (see Fig. <ref>(b)), are slightly oval with a lower signal-to-noise ratio compared to those in the reference setup (see Fig. <ref> (a)). This may imply that the wire resistance of the pickup coils varies slightly from coil to coil, resulting in differences in the strength of linear displacement with the control voltages.  The reduced contrast observed in the scan obtained from the low-cost setup compared to that obtained from the reference setup can be attributed to the increased autofluorescence emanating at 532 nm excitation wavelength from the polymeric aspheric lens of the OPU (see Fig. <ref> (e)). Unfortunately, the autofluorescence occurs in the same spectral region as the emission spectrum of the NV center, thus reducing the signal-to-noise ratio.

The detection of single photons with low-cost optics can be demonstrated by measuring the anti-bunching behavior and comparing it with the reference setup.
For the red-marked diamonds of the reference and low-cost scans (see Fig. <ref> (a), (b)), the g^2(τ)-functions were measured. Surprisingly, during the measurement, the PHR-803T coils showed significantly less drift than the open-loop piezo stages of the reference setup.  An antibunching effect can be seen with both setups (see Fig. <ref> (c), (d)). 
The fact that the g^(2)(0) function reaches a value below 0.5 is a strong indicator of the presence of a single NV center.




§ CONCLUSION

To summarise, the development of mechanically stable 3D confocal scanning for measuring single photon emission is a significant advance in terms of cost efficiency and applicability in this field. By using affordable components such as two OPUs,  that cost about 30   in sum, this approach stands in strong contrast to conventional setups, like our reference setup presented here, with expensive high-NA microscope objectives and piezo stages costing routinely more than 10k .  
The development of a low-cost single photon collection system based on OPUs holds the potential to advance quantum technologies and promote their widespread adoption in industrial applications. This would also benefit schools and universities, for which quantum experiments are often too expensive and complex. The integrated lenses and voice coil actuators for positioning ensure a compact and alignment-friendly design for further experiments in single photon source designs and laser-scanning confocal microscopes.






§ DISCLOSURES
 The authors declare no conflicts of interest.



§ DATA AVAILABILITY
 The data that support the findings of this study are available from the corresponding author upon reasonable request.


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 Paper Title: Simulating Vibrationally-Resolved X-ray Photoelectron Spectra of Flexible Molecules: Linear Alkanes C$_{n}$H$_{2n+2}$ ($n$=1-8).






 MIIT Key Laboratory of Semiconductor Microstructure and Quantum Sensing, Department of Applied Physics, School of Physics, Nanjing University of Science and Technology, 210094 Nanjing, China
Hefei National Laboratory for Physical Science at the Microscale, University of Science and Technology of China, 230026 Hefei, China
Department of Theoretical Chemistry and Biology, School of Engineering Sciences in Chemistry, Biotechnology and Health, Royal Institute of Technology, SE-106 91, Stockholm, Sweden



MIIT Key Laboratory of Semiconductor Microstructure and Quantum Sensing, Department of Applied Physics, School of Physics, Nanjing University of Science and Technology, 210094 Nanjing, China



Key Laboratory for Microstructural Material Physics of Hebei Province, School of Science, Yanshan University, Qinhuangdao 066004, China








wjhua@njust.edu.cn
 MIIT Key Laboratory of Semiconductor Microstructure and Quantum Sensing, Department of Applied Physics, School of Physics, Nanjing University of Science and Technology, 210094 Nanjing, China


§ ABSTRACT  
We integrated full core-hole density functional theory with Franck-Condon calculations, considering Duschinsky rotation, to simulate vibrationally-resolved C1s X-ray photoelectron spectra (XPS) of eight linear alkanes, from methane to octane (C_nH_2n+2, n=1–8). Results align excellently with experimental absolute binding energies and profiles. The spectrum of ethane serves as a “spectral seed”, with each longer alkane's atom-specific spectrum displaying similar characteristics, albeit with shifts and slight intensity adjustments. Detailed assignments clarify the distinct spectra in short alkanes (n=1–4) and their stabilization in long alkanes (n=5–8). Carbons are classified as central or distal (C_1 and C_2), with central carbons contributing nearly identically to the lowest-energy feature A, while distal carbons contribute to the second lowest-energy feature B (both are 0-0 transitions). Increasing molecule size adds more central carbons, enhancing feature A and weakening feature B. Our analysis identifies two dominant Franck-Condon-active vibrations: C^*–H stretching (∼3400-3500 cm^-1) and bending (∼1400-1500 cm^-1) modes. Structural analysis shows that core ionization minimally affects alkane geometry, less than in ring compounds from previous studies. This work extends our protocol from rigid ring compounds to flexible molecules, contributing to building a high-resolution theoretical XPS library and enhancing the understanding of vibronic coupling.

 

Simulating Vibrationally-Resolved X-ray Photoelectron Spectra of Flexible Molecules: Linear Alkanes C_nH_2n+2 (n=1-8)
    Weijie Hua
    August 5, 2024
=====================================================================================================================



§ INTRODUCTION


X-ray photoelectron spectroscopy (XPS) is a powerful analytical technique that provides detailed information about the chemical composition and electronic structure of molecules and materials.<cit.> High-resolution XPS spectra allow for investigation of fine structures, which predominantly originate from the vibronic coupling effect. The sub-eV fine structures carry crucial insights into the interplay between nuclear and electronic motions, unveiling the fundamental physics underpinning core hole ionization dynamics and alterations in potential energy surfaces (PESs) between ground and core-ionized states. Historically, vibronic fine structures in XPS spectra were observed as early as 1974 by Siegbahn and coworkers<cit.> while studying the spectra of CH_4. The continuous advancements in synchrotron radiation technologies have propelled XPS to higher resolutions, unlocking more comprehensive chemical information about the chemical systems.<cit.> 

XPS databases have been helpful in interpreting the structure-spectroscopy relation. However, existing experimental XPS databases<cit.> primarily focus on binding energy (BE) values, often omitting intensity data, limiting the depth of analysis. Even for BEs, different experiments sometimes report values that differ over 1 eV. An ultimate goal is to create a comprehensive, high-resolution theoretical library for XPS spectra with both accurate BEs and profiles. Preliminary studies were conducted to investigate a family of molecules (azines<cit.>, indoles<cit.>, polycyclic aromatic hydrocarbons (PAHs)<cit.>, and some other five- and six-membered ring molecules<cit.>, biomolecules<cit.>) with similar structures, by combing density functional theory (DFT) with the full core-hole (FCH) approximation and Franck-Condon (FC) simulations which incorporate the Duschinsky rotation (DR) effect<cit.> (denoted as the FCH-DR method for simplicity).<cit.> Grounded on the harmonic oscillation (HO) approximation, the calculations yielded good agreement with experiments.<cit.> The comparative analysis on theoretical and experimental high-resolution spectra offers a more rigorous evaluation, extending beyond solely assessing binding energies, for the accuracy of the chosen theoretical methodology, validating prediction of spectra of related molecules with no experimental data or no high-resolution experimental data. The studies provide  data on equal footing, serving as a reliable stand point for analyzing vibronic properties within a specific family and edge. Extensive investigations were performed including influences on functionals, basis sets, temperature effects, and tautomeric effects, as well as analyses on core-ionization induced structural changes, active vibronal modes, etc. Additionally, the experiment-theory agreement also validated the HO approximation used which is less sensitive for polyatomic molecules than diatomic systems.<cit.>

However, all previous calculations on polyatomic systems have predominantly focused on ring compounds which exhibit relatively rigid structures including PAHs. It would be natural to investigate the performance on flexible molecules. In this study, we  extend our research to a series of straight-chain alkanes (C_nH_2n+2, n=1-8; see Fig. <ref>): methane (CH_4), ethane (C_2H_6), propane (C_3H_8), n-butane (C_4H_10), n-pentane (C_5H_12), n-hexane (C_6H_14), n-heptane (C_7H_16), and n-octane (C_8H_18).  They represent the simplest organic molecule family, composed solely of sp^3-hybridized carbon atoms and hydrogen atoms. Gradually increasing the –CH_2 groups provide an ideal set for investigating size effects. There are access to high-resolution experimental spectra<cit.> (except heptane), facilitating the performance evaluation of the theoretical method. Technically, as linear molecules generally possess greater flexibility than their ring counterparts, the geometry optimization of core-ionized states becomes a more intricate task than rigid molecules. 

Alkanes are naturally occurring hydrocarbons found in crude oil, serving as high-value petrochemical feedstock materials for the petroleum industry<cit.>. They also play a prominent role as a major component of non-methane volatile organic compounds in vehicle emissions and urban atmospheric environments<cit.>. Methane, propane, and butane are clean energy sources for domestic and industrial applications, with methane dominating liquefied natural gas and propane/butane used in liquefied petroleum gas. Ethane, a minor natural gas component, holds significant value as a petrochemical feedstock<cit.>, primarily for ethylene production. Pentane is used as a polystyrene foam blowing agent and nonpolar solvent, while hexane is a common affordable, safe, and rapidly evaporable non-polar organic solvent. Heptane serves in various industrial applications, including paints, coatings, and pharmaceuticals, and is a significant gasoline component. Octane, another vital gasoline component, is often blended with the highly branched isomer iso-octane to establish a standard reference for measuring the fuel's resistance to engine knock. Methane exhibits T_d symmetry, while ethane possesses D_3d symmetry. Longer straight-chain alkanes  have either C_2v (propane, pentane, and heptane) or C_2h (butane, hexane, and octane) symmetry, as labeled in Fig. <ref> together with all non-equivalent carbon atoms.

It is necessary to acknowledge early theoretical and experimental studies on vibrationally-resolved XPS of alkanes, particularly the work by Svensson and coworkers.<cit.> They employed equivalent-core hole (ECH) DFT, replacing the core-ionized carbon (C^*) with a nitrogen atom,<cit.>  and Franck-Condon simulations to compute spectra for methane, ethane, and other small hydrocarbons, achieving good agreement with experiments.<cit.>  Subsequently, they transitioned to another electronic structure method, using a nitrogen basis set for C^* with scaled exponents and an effective core potential (ECP) with adjusted parameters to account for one electron in the C1s shell.<cit.> This approach was applied to simulate larger linear alkanes (n=1-6,8) and extrapolated to alkanes with n=1000, providing valuable insights into the vibronic coupling properties of alkanes in XPS.

However, these calculations have several limitations. The ECH-DFT approach<cit.> does not yield accurate absolute binding energies. While theoretical spectra can be adjusted to align with experimental values, this method is unsuitable for predictions when no corresponding experiments exist (e.g., heptane). Additionally, our earlier calculations<cit.> indicated that ECH produces less accurate vibronic profiles compared to FCH, particularly regarding the intensity ratios of the 0-0 and 0-1 peaks, F_01/F_00. In the alternative approach<cit.> using a nitrogen basis set and ECP, scaling issues arise with the basis sets, ECPs, and vibrational frequencies (with different factors for different modes), leading to empirical adjustments that may not be valid for other systems. Furthermore, the linear coupling method (product of Poisson distributions) does not account for mode mixing effects (i.e., Duschinsky rotation). Lastly, while analyses have been conducted on core hole-induced structural changes and potential energy curves, there is a lack of comprehensive examination of active vibrational modes.

The goal of this study is twofold. First, we aim to evaluate the performance of a more accurate FCH-DR approach<cit.> for alkane molecules. This computational protocol has been validated<cit.> to accurately predict both binding energies (within 1 eV) and vibronic fine structures for XPS of cyclic compounds, including PAHs. The computational procedure employs only standard basis sets and model core potentials (MCPs) without any empirical parameter scaling. Second, with accurate theoretical spectra at hand, we will investigate general vibronic coupling rules in gaseous linear alkanes. This comprehensive analysis of aliphatic hydrocarbons, alongside our study of PAHs,<cit.> will enhance our understanding of hydrocarbon systems and contribute to our ongoing goal of constructing a high-resolution XPS library.  




§ COMPUTATIONAL METHODS


The FCH-DR method<cit.> was employed for the XPS computations in this work. Details of the methodology have been outlined previously. Briefly, DFT-based electronic structure calculations were performed using the B3LYP functional<cit.> and the GAMESS software package<cit.>. A double basis set approach<cit.> was utilized for geometry optimization and frequency calculations, involving distinct basis sets for the ground state (χ _GS) and core-ionized state (χ _FCH). Specifically, the cc-pVTZ basis set was used for χ _GS, while for χ _FCH, the IGLO-III basis set<cit.> was used for core-ionized carbon atoms (C^*) and the MCP/NOSeC-V-TZP basis set<cit.> was used for non-excited carbon atoms, complemented by the corresponding MCPs. This choice of basis sets has been validated for its accuracy in spectral simulations of various cyclic molecules<cit.>. Fine integration grid and a stringent gradient convergence tolerance of 1.0 × 10^-7 Hartree/Bohr were used.  The optimized structures and vibrational modes for both the ground and FCH states were used as input to simulate FCFs using our in-house software package, DynaVib<cit.>,  modified<cit.> to extend its capabilities from the optical to the X-ray regimes. The DR method<cit.> was employed in the computation of FCFs, which characterizes the differences between the normal coordinates of the initial and final states. The ⟨ 0|n ⟩ amplitudes were computed through an iterative procedure<cit.> initiated with ⟨ 0|0 ⟩. Lorentzian broadening with a half width at half maximum (HWHM) γ=0.05 eV (i.e., the C1s core hole lifetime<cit.>) was applied to all C1s ionizations, and a small uniform shift δ_rel=0.2 eV<cit.> was added to account for the scalar relativistic effect. The total XPS profile of each alkane is a weighted combination of the XPS spectrum from its nonequivalent carbon atoms. To better compare with the experimental data,<cit.> theoretical spectrum of each alkane molecules was shifted by δ=-0.27 eV, except that a slightly different value of δ=-0.33 eV was applied for methane.



§ RESULTS




 §.§ Overview and notations for peak labels


Figure <ref> presents the simulated vibrationally-resolved C1s XPS spectra for all alkanes under study, compared to available experimental data<cit.> (note that no experimental data is available for heptane). As shown in Figs. <ref>–<ref>, more detailed analyses–including atom-specific spectra, interpretations of vibronic transitions, and identifications of active modes–are provided for each pair of molecules with the same number of non-equivalent carbons, C_nH_2n+2 and C_n+1H_2n+4 (where n=1, 3, 5, 7).  Figure <ref> consolidates the atomic-specific contributions from the edge carbons C_1 and C_2 to highlight patterns in spectral evolution.


For methane and ethane, each containing only one non-equivalent carbon, we use Roman numerals i–iii and i-v to label their major peaks, respectively. The same numeral indicates the same vibronic assignment and is transferable between the two molecules. For other alkanes with multiple non-equivalent carbons, we use Latin letters A, B, C, etc., to label features in ascending order of energy in the total spectrum. These labels are molecule-specific and not necessarily transferable among all molecules, although they are indeed transferable for long alkanes with n=5–8.



 §.§ Methane and ethane


The T_d symmetry molecule methane is the simplest alkane with a single carbon atom. Its experimental spectrum [Fig. <ref>(a)] exhibits three prominent peaks (i-iii) that are well-reproduced by theory [Fig. <ref>(b)]: peaks i (experimental, 290.70 eV; theory, 290.71 eV ), ii (experimental, 291.10 eV; theory, 291.12 eV), and iii (experimental, 291.51 eV; theory, 291.55 eV). One can see that the experiment-theory deviation in peak energies is less than 0.1 eV. These peaks correspond to the 0-0, 0-1, and 0-2 transitions of the totally-symmetric (A_1) C^*–H stretching mode ν(6) (3383 cm^-1) at the final-state geometry min FCH, which is the only Franck-Condon-active mode [Fig. <ref>(a,c)].  For clarity, we label peaks ii and iii as 0-16 and 0-26, respectively [Fig. <ref>(a)], with the mode index indicated as a subscript.<cit.> Beyond these peaks, no additional features are clearly identifiable in the experimental spectrum [Fig. <ref>(a), bottom], while the 0-3_6 peak is predicted at approximately 292.00 eV [Fig. <ref>(b), bottom; Fig. <ref>(a)].
 
The D_3d symmetry molecule ethane also contains one single non-equivalent carbon atom. Its theoretical spectrum [Fig. <ref>(b)] closely matches experiment [Fig. <ref>(a)]. The main peak i (0-0 peak) exhibits a red shift of ca. 0.15 eV compared to methane. Peaks ii-iii are again observed, assigned to the C^*–H stretching mode ν(16) (3428 cm^-1) [Figs. <ref>(d), right]. This mode is essentially similar to the only active mode of methane, and the vibrational frequency is slightly (45 cm^-1) larger than that of methane. Besides, two additional peaks, iv (experimental, 290.73 eV; theory, 290.76 eV) and v (experimental, 291.36 eV; theory, 291.19 eV), are resolved and primarily attributed to the C^*–H bending (scissoring) mode ν(7) (1473 cm^-1), which also includes contributions from the bending vibrations of the other –CH_3 group of the non-excited carbon [Fig. <ref>(d), left]. Specifically, peaks iv and v are assigned as the 0→1_7 and 0→1_7,1_16 transitions, respectively.



 §.§ Propane and butane


Propane with C_2v symmetry contains two non-equivalent carbons, distal carbon C_1 and central carbon C_2, in a 2:1 ratio. In the total theoretical spectrum [Fig. <ref>(b)], it now contains rich features (labeled by A-H), since C_1 and C_2 curves are similar in shape but with different “phase” [see also Fig. <ref>(a-b)]. Feature A at 290.39 eV and feature B at 290.50 eV are 0-0 transitions by C_1  and  C_2, respectively. Corresponding experimental values  are 290.40 and 290.50 eV, respectively, almost equivalent to our predictions.  Features C and D come from 0-1_12 and 0-1_14 transitions for C^*-H bending modes ν(12) [1412 cm^-1, Fig. <ref>(e)] for C_1 and ν(14) [1488 cm^-1, Fig. <ref>(f)] for C_2, respectively. Meanwhile, peak features E and F come from  0-1_25 and 0-1_26 transitions for C^*-H stretching modes ν(25) [3430 cm^-1, Fig. <ref>(e)] for C_1 and ν(26) [3465 cm^-1, Fig. <ref>(f)] for C_2, respectively. Features G and H are both assigned as 0-2 transitions with contributions of both carbons. Feature G is assigned as mixed 0-1 transitions for both the bending and stretching modes; while feature H is assigned as pure 0-2 transitions of the stretching mode.

Similarly, butane with C_2h symmetry also has two non-equivalent carbons, C_1 and C_2, in a 2:2 ratio. Compared to propane, the fine structure looks simper because in the C_1 and C_2 contributions, all peaks almost coincide with each other [see also Fig. <ref>(c-d)]. The active mode analyses for butane [Fig. <ref>(g-h)] mirror the discussions for propane, identifying a C^*-H stretching and a C^*-H bending modes for each carbon. Peak A is contributed by  0-0 transitions; peaks B and C are 0-1 transitions for the bending and stretching modes, respectively; peaks D and E are 0-2 transitions, either mixed 0-1 transitions for both the bending and stretching modes (feature D) or pure 0-2 transitions of the stretching mode.





 §.§ Pentane and hextane


Pentane (C_2v) and hexane (C_2h) both have three non-equivalent carbon atoms, C_1–C_3, with weight ratios of 2:2:1 and 2:2:2, respectively. Both molecules exhibit similar spectra with six major features A-F. Peak A is assigned as 0-0 transition by the central carbon C_3. Peak B is from 0-0 transitions of C_1 and C_2 as well as 0-1 transition of C_3 for a  C^*-H bending mode (Fig. <ref>). Peaks C and D are assigned as 0-1 transitions, and Peaks E and F as 0-2 transitions. Active modes of the two molecules are analyzed in Fig. <ref> (g)-(i) and Fig. <ref> (j)-(l). Each atom-specific spectrum is identified with a medium-frequency (1400–1500 cm^-1) C^*–H bending mode and a high-frequency (3400-3500 cm^-1) C^*–H stretching mode.



 §.§ Heptane and octane


Heptane (C_2v) and octane (C_2h) have four non-equivalent carbon atoms, with weight ratios of 2:2:2:1 and 2:2:2:2, respectively. Although one additional non-equivalent carbon is present as compared to pentane and hextane, the total spectra do not change much, also exhibiting six features A-F. In other words, when n changes from 5 to 8, the vibronic profile almost converges into a consistent pattern. Concerning binding energies, the spectral series exhibit a slight red shift with increasing n. Despite the lack of experimental spectrum for heptane, our predicted theoretical spectrum matches the overall evolutionary trend.

The underlying reason is that in  heptane and  octane with relatively long length, spectra contributed by the two central carbons C_3 and C_4 almost coincide with each other, indicating their similar local chemical environment. Their 0-0 transitions both contribute to feature A in the total spectrum. 0-0 transitions of distal carbons C_1 and C_2 make contributions to feature B some 0.2 eV beyond. The deviation of 0-0 transition energies for C_1 and C_2 makes peak B broader than peak A. In a even longer linear alkane, it is reasonable to yield that the spectral contributions for two neighboring carbons near the terminal deviates more than two neighboring carbons in the central region.  Based on each atom-specific spectrum  of both molecules [Fig. <ref> (a)-(h)], each was identified with two active modes, including one  medium-frequency (1400–1500 cm^-1) C^*–H bending  mode and a high-frequency (3400-3500 cm^-1) C^*–H stretching mode [Fig. <ref> (i)-(p)].



§ DISCUSSION




 §.§ Red shift of the lowest peak A


In the total theoretical spectra [Fig. <ref>(b)], the lowest-energy peak, peak A, exhibits a red shift, with energies of 290.70 eV (methane), 290.58 eV (ethane), 290.38 eV (propane), 290.32 eV (butane), 290.12 eV (pentane), 290.09 eV (hexane), 290.07 eV (heptane), and 290.05 eV (octane), as indicated by the numbers at the top of this panel. The magnitude of the red shift is relatively large for alkanes with n=1 to 4 but becomes significantly smaller starting from n=5. As discussed earlier, for n=5 to 8, peak A for each molecule shares the same assignment (from central carbons). The energy differences between neighboring alkanes are only 0.02 to 0.03 eV (approximately 1 milli-Hartree), indicating that the values are approaching convergence.




 §.§ Ratio of distal and central carbons controls intensity ratios of peaks A and B


Above analyses have shown stable spectral profiles of six features A-F for linear alkanes at n=5-8. Peak A and B are 0-0 peaks of central (C_3 when n=5-6; C_3, C_4 when n=7-8) and distal (C_1, C_2) carbons, respectively. As n increases, the intensity ratio of peak A to peak B increases due to the increase of more central carbons. This can be explained by the locality of core excitations and the alkane structure (without delocalized π structure), which make the atom-specific spectra dominantly determined by the nearest neighboring units. In a long alkane molecule, C_1 directly links to a –CH_2 group, C_2 directly bonds to a -CH_3 and –CH_2 groups on each side, and all other non-equivalent carbons all bond with two –CH_2 groups, which are classified as central carbons. It is worth noting that different central carbons contribute closely similar spectra, as verified by the coincidence of C_3 and C_4 spectra in heptane and octane. Therefore, in an even longer linear alkane, peak A will be extremely high compared to peak B. The total spectra can be considered as contributed from three types of carbons: C_1, C_2, and central carbons. This explains the stable features in longer linear alkanes, with their relative peak intensities adjusted by the ratios of different types of carbons and their relative BE differences.



 §.§ Evolution of distal carbons C_1 and C_2


Figure <ref> clearly demonstrates the evolution of vibrationally-resolved XPS profiles for the two distal carbon atoms C_1 (orange) and C_2 (lightblue). The concept of distal carbons is evident for long alkanes with n=5–8, while all carbons are considered as distal carbons for short alkanes with n=1–4, providing a comprehensive and smooth picture. Methane shows three features i-iii, attributed to the C^*–H stretching vibration. Since ethane, an additional C^*–H bending vibration is involved, contributing to peaks vi (pure 0-2 bending mode excitation) and v (mixed 0-1 excitations with both modes). All other atom-specific spectral profiles show great similarity to the ethane spectrum, with relative peak intensities of the five peaks preserved almost the same. Figure <ref> highlights the evolution of peaks i-iii. These results indicate that the spectrum of ethane can be considered as a “spectral seed”, as different shifted spectra of ethane approximately generate the total spectra of a longer linear alkane. Additionally, a switch of BE order is found for C_1 and C_2. Carbon C_2 starts since n=3, exhibiting a higher binding energy of ca. 0.1 eV than C_1. This energy order is quickly reversed since n=4, and at n=8, the binding energy of C_2 is ca. 0.1 eV lower than C_1.



 §.§ Structural changes induced by core ionizations


Structurally, the linear alkanes are generated by gradually adding methylene groups (–CH_2–) to methane. Each molecule is constructed by linking C-C and C-H single bonds, and studying the structural change induced by the C1s core hole is crucial. In alkanes, the typical C–C bond length hovers around 1.529 Å, with a tiny difference between distal (1.528 Å) and central carbons (1.529–1.530 Å), as shown in Fig. <ref>. The structural changes near the excited carbon are summarized in Table <ref>, revealing discernible patterns. Specifically, the bond lengths C^*-C_r and C^*-C_l in the FCH state exhibit minimal changes (not exceeding 0.02 Å) compared to the ground state structure, indicating marginal influence due to the presence of a C1s core hole. Here, C_l (C_r) is the left (right) carbon atom bonded to C^*  (see relative atomic positions in Fig. <ref>). The angle ∠C_r-C^*-C_l, typically close to 114^∘, experiences an increment of ca. 1^∘ from the ground- to the final-state geometry. This angle at C^* experiences slightly larger bending for central carbons compared to distal carbons. All C^*–H bonds exhibit virtually identical characteristics in both the ground and the final states, with minor shortening of 0.05–0.07 Å.



§ SUMMARY AND CONCLUSIONS


To summarize, we simulated high-resolution, vibrationally-resolved C1s X-ray photoelectron spectra (XPS) for a series of linear alkanes, ranging from methane to octane (C_nH_2n+2, with n=1–8), by combining full core-hole density functional theory with Franck-Condon calculations that account for Duschinsky rotation. Our theoretical spectra align excellently with all available experimental results, and we predicted the spectra of heptane, showing reasonable spectral evolution in this series. This approach has facilitated reliable peak assignments and enabled an investigation into the spectral evolution with increasing chain length, as well as the underlying vibronic coupling rules governing this molecular family.

The spectrum of ethane serves as a “spectral seed”, with the vibronic profile of each longer alkane exhibiting similar characteristics, albeit with shifts in peak positions and slight intensity adjustments. The carbon atoms in these longer alkanes are categorized as central or distal (C_1 and C_2), with central carbons contributing nearly identically to the lowest-energy feature A, while distal carbons contribute to the second-lowest feature B. Detailed assignments elucidate the distinct spectra observed in short alkanes (n=1–4) and their stabilization in longer alkanes (n=5–8). Through comprehensive assignments, we tracked the evolution of specific spectral features across the molecular series. We investigated core ionization-induced structural changes and identified Franck-Condon-active vibrational modes, highlighting two key active modes present in each atom-specific spectrum: the C^*–H symmetric stretching (ca. 3400–3500 cm^-1) and the C^*–H bending (ca. 1400–1500 cm^-1) modes. 

This work extends our FCH-DR approach to flexible molecular systems, enhancing our understanding of vibronic coupling phenomena. The good performance of our protocol in these flexible systems should benefit from the minimal structural changes induced by core ionization. It also proves that the harmonic oscillator approximation works effectively for alkane systems. The insights gained from this analysis of simple alkane systems will be valuable for studying more complex hydrocarbons and other flexible molecules. Ultimately, this study contributes to our ongoing research goal of constructing a comprehensive, high-resolution theoretical XPS library.




§ ACKNOWLEDGMENTS


We thank Prof. Yi Luo for helpful discussions. Financial support from the National Natural Science Foundation of China (Grant No. 12274229)  is greatly acknowledged. M.W. thanks to Fund for Fostering Talented Doctoral Students of Nanjing University of Science and Technology. 


































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 Paper Title: Hybrid electroluminescence device for on-demand single photon generation at room temperature.




fancy





Hybrid electroluminescence device for on-demand single photon generation at room temperature
    Sunder Ali Khowaja Senior Member, IEEE, Parus Khuwaja, Kapal Dev Senior Member IEEE, Hussam Al Hamadi, and Engin Zeydan Senior Member IEEE 

Sunder Ali Khowaja is with School of Computing, Faculty of Computing, Digital and Data, Technological University Dublin, and CONNECT Centre, Ireland. Email: sunderali.khowaja@tudublin.ie

Parus Khowaja is with Institute of Business Administration, University of Sindh, Jamshoro. (e-mail:Parus.khuwaja@usindh.edu.pk).
Kapal Dev is associated with CONNECT Centre and Department of Computer Science and  Munster Technological University, Bishopstown, Cork, T12 P928, Ireland,e-mail: (kapal.dev@ieee.org)
Hussam Al Hamadi with College of Engineering and IT 
University of Dubai, e-mail: (Halhammadi@ud.ac.ae)
Engin Zeydan with Centre Tecnològic de Telecomunicacions de Catalunya (CTTC), Barcelona, Spain, 08860, e-mail: (ezeydan@cttc.es)


    
August 1, 2024

================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================


















A. Rodek, K. Oreszczuk, M. Potemski, P. Kossacki


Institute of Experimental Physics, Faculty of Physics, University of Warsaw, Pasteura 5, Warsaw, 02-093, Poland


Email Address:

aleksander.rodek@fuw.edu.pl, kacper.oreszczuk@fuw.edu.pl,
 marek.potemski@lncmi.cnrs.fr, piotr.kossacki@fuw.edu.pl


M. Hajdel, A. Kafar, M. Aktas, Ł. Marona, C. Skierbiszewski

Institute of High Pressure Physics, Polish Academy of Sciences, Sokołowska 29/37, Warsaw, 01-142, Poland

Email Address:

hajdel@unipress.waw.pl, ak@unipress.waw.pl, lucja.marona@unipress.waw.pl, czeslaw@unipress.waw.pl 

M. Potemski

Laboratoire National des Champs Magnétiques
Intenses, CNRS-UGA-UPS-INSA-EMFL, 25 Av. des Martyrs Grenoble, 38042, France

CEZAMAT-CENTERA2 Labs, Warsaw University of Technology, Koszykowa 75, 00-662 Warsaw, Poland











§ ABSTRACT 

Recent research focused on single photon emitters (SPEs) hosted by layered semiconductors, particularly hexagonal boron nitride (hBN), has revealed a promising alternative to quantum dots (QDs) for generating single, indistinguishable photons. hBN-based SPEs offer lower material costs, room temperature emission, and easy integration into potential optoelectronic devices due to the layered structure of the host crystal. This work presents compact hybrid electroluminescence devices, in which GaN laser diodes (LDs) are used for bottom-to-top excitation of hBN nanoflakes deposited on the laser facets. Our approach circumvents the issue of electroluminescence generation from hBN and provides access to the SPE's signal without optical driving by an external laser. Using laser diodes upgraded with Bragg reflectors a room-temperature generation of single photons from hBN is confirmed by an 80%-dip in their g^(2) second-order correlation. The on-demand emission of single photons at room temperature is demonstrated by driving the laser diodes in pulsed operation, with confidence supported by a measured g^(2)(0) value of 0.37.
 






§ INTRODUCTION

Solid-state single photon emitters (SPEs) are nowadays one of the most promising platforms for providing commercially viable devices that could be utilized in emerging technologies, which take advantage of the quantum properties of light, e.g. quantum information storage, communications, and sensing <cit.>.  Significant progress has been made in the last decade, particularly for the system of quantum dots (QDs) based on III-V materials, which resulted in the fabrication of nanostructures able to deterministically emit indistinguishable single photons with high efficiency<cit.>. The QD-based single photon sources are already commercially available, underscoring their practicality, but their production is not trivial.  Future progress in implementing quantum technologies on a larger scale requires solutions that are, among others, production-time-effective, reliable, and reproducible. All these issues are particularly obstructive for the platform that ultimately depends on the manual selection of the most optimal stochastically-produced single-photon-emitting sites, or frequently even the entire photonic structures<cit.>.  In consequence, there is currently a strong drive to identify alternatives to the QD-based single photon sources, which has already led to many breakthrough works concerning single-photon emission from layered semiconductor materials: transition metal dichalcogenides (TMDs) and hexagonal boron nitride (hBN)<cit.>. 
 Indeed, various processing methods have been recently developed, allowing for the deterministic manufacture of efficient single-photon-emitting sites in TMDs and hBN<cit.>. Particularly, the latter material offers an important opportunity to observe quantum emitters under ambient conditions<cit.>. Simultaneously, full integration of layered materials into various optoelectronic devices, e.g. light emitting diodes requires efficient electrical driving of their luminescence signal. However, the essential step of controlled doping of TMDs and hBN remains still a largely unresolved challenge<cit.>. As such the up-to-date demonstrations of electroluminescence devices based on layered materials almost universally depend on time- and work-extensive manufacturing processes like the design and lithography of electrical contacts or the manual stacking of flakes with micrometer precision<cit.>. In that context, an alternative approach has been recently demonstrated that utilized hybrid devices, where monolayers of TMDs were directly deposited on top of GaN-based blue-emitting LEDs <cit.>. This bottom-to-top excitation scheme has proved able to generate not only the standard excitonic spectra of TMD luminescence but also single-photon-emission without the necessity of driving by an external laser, albeit exclusively at cryogenic temperatures. Such electrically-driven single photon emission, which could ideally operate at room temperature, would thus greatly reduce the size and operational complexity of any potential device adding to its scalability on an industrial level. 
Here we present a prototype device consisting of a GaN/InGaN laser diode for the bottom-to-top excitation of commercially available hBN nanocrystals deposited directly on laser facets. We demonstrate the generation of single-photon-emission in ambient conditions both under continuous-wave (CW) excitation and on-demand pulsed driving schemes. Importantly, the operation of our single photon source has been made viable with the amended design of InGaN/GaN laser diodes, which have been equipped with specially developed bandpass filters (Bragg reflectors). 




§ RESULTS



 §.§ SPE hybrid device

The essential components of our device include: the InGaN/GaN-based laser diode (LD) <cit.> serving as the excitation source, a band-stop filter, and commercially available nanocrystals of hBN. Such hBN crystal powder has been used in different works in recent years as an easily obtainable source of single photon emitters, which can survive at room temperature <cit.>.
Narrow emission lines observed in its photoluminescence signal may span the entire visible to near-infrared spectral range<cit.>. The visualization of the device concept is shown in the Fig. <ref>



There are two fundamental requirements concerning the excitation source for it to be suitable for the on-demand optical driving of the SPEs in hBN. Firstly, the optical power density has to be rather large compared to TMDs, e.g. WSe_2. Previous reports show that typically 1-2 mW/μ m^2 of optical power have to be delivered to produce the single photon emission at room temperature <cit.>. Secondly, it should be able to generate excitation impulses with a temporal width lower than the typical lifetime of the single photon emitter. GaN/InGaN-based LD meets all of the requirements given above with additional advantages related to this device family such as high reliability and compact size. The characteristics of the LD and Bragg filters used in the hybrid electroluminescence device are shown in Fig. <ref>.



In continuous wave operation, LD delivers 100mW of optical power at 115mA of driving current at room temperature (see supplementary data for details). Given the size of the lasing area below 5 μ m^2 the device far exceeds the intensity required for exciting the SPEs in hBN (∼ 1-2 mW per μm^2). In particular, this high margin allows us to reduce the heating effects, for example by lowering the laser duty cycle in a pulsed operation mode. For such a case, the necessary average power of ∼2mW would correspond to a duty cycle of single percents. The laser wavelength is 422.8nm, which is suitable to efficiently excite hBN SPEs.
Except for the laser emission signal presented in Fig. <ref>(a), the nitride LD also emits photons in a spontaneous electroluminescence process, with their wavelengths spanning from the visible spectrum up to the near IR. Above the lasing threshold this spontaneous broad-band emission is orders of magnitude weaker than the intensity of the laser radiation. It is, however, strong enough to influence the correlation measurements of single photons. Therefore it is crucial to provide efficient spectral filtering of this addition. For this reason, we developed a laser facet mirror coating that acts as a band-stop filter between 600 and 800 nm. This wavelength range corresponds to the long-wavelength limit of the SPEs obtainable from hBN. It has been chosen to minimize the noise-to-signal ratio related to either the spontaneous emission of the diode, which is additionally filtered by the mirror coating, or the broad emission of the hBN <cit.>. Both of these contributions display pronounced extinction with the increasing wavelength. 

The band-stop filter mirror was deposited on top of the front facet of the LD. It consisted of a stack (5 or 6 repetitions) of a low refractive index MgF_2 (with 129.4 nm thickness) and a high refractive index Ta_2O_5 (thickness 91.6 nm) layers. In Fig. <ref>(c) the simulation of the reflectance spectrum of the mirror stack repeated 5 times is presented together with the experimental data for 2, 3, 5, and 6 repetitions. The data displays a saturating behavior of the maximal obtainable value of reflection with the increasing number of layers. A total of 5 repetitions of mirror coatings were used in the final device and showed sufficiently high reflection in the red spectrum regime (up to 92%) and reasonably beneficial low reflection for the lasing wavelength (5%). In Fig. <ref>(b) we present a comparison of the long wavelength spectra of two LDs originating from the same epitaxial process with and without the band-stop filter. For the coated LD the signal at 740 nm drops to 7.2% of that of the LD with no coatings.


Importantly, we also observe a near 90% reflectance value in the spectral range of 650-700 nm even up to a relatively high collection angle of 40 degrees in both polarizations(see supplementary). The back side of the LD facet was additionally coated with a standard highly reflective dielectric mirror. 

After the deposition of the filters, the coated LDs were mounted in standard TO-56 laser mounts ready for the process of hBN placement. 




 §.§ Room-temperature single-photon emission

The straightforward method of obtaining well-isolated single-photon emitting spots is to distribute hBN powder on a polydimethylsiloxane (PDMS) film, which is then stamped onto a substrate surface, very often on Si/SiO_2 wafers, and on laser facets in our case. After removing the PDMS film, a certain amount of the hBN powder grains remain on the substrate surface, with their density depending on the initial distribution of hBN on the PDMS film. This process can be further tuned by additional stamping and is sufficient to isolate the SPE sites effectively. Such a simple approach has been applied in the fabrication of our device, while more elaborate techniques, such as large-scale deterministic positioning of hBN nanocrystals<cit.> by capillary assembly, may be advantageous <cit.> to explore in the future. 




 
In Fig. <ref> we show SEM images of the laser facet before (a) and after (b) the hBN powder deposition. The lasing area is marked in blue, while the hBN nanocrystals are marked in red, displaying their presence on top of the excitation region. In the CW photoluminescence measurements, the laser diode was powered by rectangular electric pulses with the period T=5 ms and filling factor of 10-25% to minimize heating effects at room temperature. The driving voltage was chosen so that the average excitation power density was in the order of hundreds of microwatts per μ m^2(P_avg=100-700 μW/μ m^2). The PL signal was collected through an apochromatic, high NA=0.65 microscope objective and measured with a standard Czerny-Turner spectrometer equipped with a CCD camera. Spatial filtering of the signal in the horizontal axis was performed by the entrance slit of the spectrometer, while the vertical axis filtration was done either by a multitrack feature of the CCD camera or, especially for the case of autocorrelation measurements, an additional slit placed in the intermediate imaging plane of a collimating setup added in the optical path. In Fig. <ref> (c) we present examples of the obtained PL spectra of hBN narrow emission lines measured at room temperature. Each spectra corresponds to a different deposition process of the hBN powder onto the laser diode showcasing the possibility of stochastic tuning the emission energy between 600-800 nm, by consecutive stamping.  Typical linewidth values reside between 1.5 and 5 nm. To characterize the single-photon emission of the hBN narrow spectral lines we performed g^(2) autocorrelation function measurements in a standard Hanbury Brown and Twiss configuration.




 In Fig. <ref> we present the autocorrelation measurement (b) of an exemplary narrow emission line (a) obtained from the device. The g^(2) function was fitted with a standard formula that takes into account a long timescale bunching effect often present in this material<cit.>.
 
    g^(2)(τ)=1-A_1 exp(-|τ|/t_1)+A_2 exp(-|τ|/t_2)

Here A_1 — is the antibunching depth and A_2 — bunching amplitude. t_1 corresponds to the emitter lifetime and t_2 is the timescale of the bunching effect. 




For the case of the presented emitter, its lifetime is two orders of magnitude faster than the detected bunching timescale and nicely correlates with the literature-reported values of single nanoseconds<cit.>. After renormalization of g^(2)_r=g^(2)/1+A_2 we obtain the final value determining the single-photon character of the emission g^(2)_r(0)=0.19±0.05, which satisfies the general criteria for a single photon source (g^(2)(0) < 0.5). Having thus confirmed the room temperature single-photon emission from the device we now turn to the pulsed excitation schemes for on-demand generation of such emission.




 §.§ On-demand generation of single photons

To generate nanosecond electric impulses with sufficiently high currents we, once again, take advantage of the recent progress in the technology of nitride-based electronics <cit.> and use a home-made power supply in which we incorporate the commercially-available EPC8002 GaN field-effect transistor with a specialized driver LMG1020. The presented laser diode is capable of pulsed operation where the temporal width of the impulses does not exceed the typical lifetime of studied emitters. This property, crucial for the on-demand generation of single photons, was verified with autocorrelation measurements of the laser emission. In Fig. <ref>(a) we show the g^(2) function of the LD under pulsed driving with a period T=50 ns. The resultant autocorrelation peaks display a Full-Width-Half-Maximum of 1.8 ns and decay slope of t_dec=0.56 ns. The operating speed of the LD is limited by its capacitance, which can be estimated to be C≈10 pF. Here, we assumed an RC circuit and considered the impedance of the utilized coaxial cables of R=50 Ohm.



In Fig. <ref>(b) we show an example of the autocorrelation function of an hBN emitter's photoluminescence excited by the LD under pulsed driving. Exponential peaks fitted to the data yield a lifetime of 11 ns and g^(2)(0)=0.37±0.06. Given that this value satisfies the criterion for single-photon emission we thus confirm the generation of a single photon per excitation pulse from the LD device.



§ CONCLUSIONS

With the ever-increasing demand for single-photon-generating devices driven by the rapid development of photon-based quantum technologies, layered semiconductors have emerged as promising materials and have begun to take center stage in nanophotonics research. In this work, we overcome one of the most limiting factors for producing hBN-based optical devices by manufacturing structures that utilize a novel bottom-to-top excitation scheme. We presented that the produced hybrid device, even at this early stage of research, when compared to the historical progress in QD-based technologies, already provides on-demand generation of single photons at room temperature and without the necessity for external laser excitation. The achieved values of the zero-delay autocorrelation function indicate, however, the requirement for further optimization to match the industry standards. This may be partly due to the selected material of hBN nanocrystals. Its manual deposition on the laser facet provides luminescence from different SPEs in the 600-800 nm range and, while being time/cost-efficient, it does not employ any further optimization of the emitter's position or annealing treatments<cit.>. In the future, certain methods for deterministic generation of efficient single-photon emitting sites could be integrated to improve both the efficiency and scalability of the production of these devices<cit.>. This is particularly promising while considering the recently reported achievement of visible Hong-Ou-Mandel effects in the two-photon interference of hBN SPE's signal, which is the next big step towards the generation of indistinguishable photons<cit.>. 




Supporting Information Supporting Information is available from the Wiley Online Library or from the author.






FundingThis work was supported by the National Science Centre, Poland under projects: 2020/39/B/ST3/03251, 2022/45/B/ST7/03964, 2020/39/B/ST7/02945, and Foundation for Polish Science under TEAM-TECH POIR.04.04.00-00-210C/16-00 and TEAM POIR.04.04.00-00-1A18/16-01 (ATOMOPTO) projects.





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 Paper Title: Femtosecond switching of strong light-matter interactions in microcavities with two-dimensional semiconductors.










	
	

    
    Contributed equally
 	Dipartimento di Fisica, Politecnico di Milano, Piazza Leonardo Da Vinci 32, 20133 Milano, Italy
  
  	
    Contributed equally
    Dipartimento di Fisica, Politecnico di Milano, Piazza Leonardo Da Vinci 32, 20133 Milano, Italy
    NanoPhotonics Centre, Cavendish Laboratory, Department of Physics, JJ Thompson Ave, University of Cambridge, Cambridge, UK
    
	
	Dipartimento di Fisica, Politecnico di Milano, Piazza Leonardo Da Vinci 32, 20133 Milano, Italy

    
	Department of Physics, University of Exeter, Stocker Road, EX4 4PY, Exeter, UK

   	
    Dipartimento di Fisica, Politecnico di Milano, Piazza Leonardo Da Vinci 32, 20133 Milano, Italy
    Department of Mechanical Engineering, Columbia University, New York, NY 10027, USA

    
    Dipartimento di Fisica, Politecnico di Milano, Piazza Leonardo Da Vinci 32, 20133 Milano, Italy
    
    
    Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Università di Parma, Parco Area delle Scienze 7/A, 43124 Parma, Italy
    
    
    Department of Physics and Astronomy, University of Sheffield, Hounsfield Road, S3 7RH, Sheffield, UK
    
    
    Department of Physics and Astronomy, University of Sheffield, Hounsfield Road, S3 7RH, Sheffield, UK

  	
	Department of Physics and Astronomy, University of Sheffield, Hounsfield Road, S3 7RH, Sheffield, UK
 
	
	Advanced Materials Laboratory, National Institute for Materials Science, 1-1 Namiki, Tsukuba, 305-0044, Japan
 	
	Advanced Materials Laboratory, National Institute for Materials Science, 1-1 Namiki, Tsukuba, 305-0044, Japan
 
    
	Department of Physics and Astronomy, University of Sheffield, Hounsfield Road, S3 7RH, Sheffield, UK
 
    
	Department of Physics, University of Exeter, Stocker Road, EX4 4PY, Exeter, UK
 
    
    
	Dipartimento di Fisica, Politecnico di Milano, Piazza Leonardo Da Vinci 32, 20133 Milano, Italy
 
	
     a.tartakovskii@sheffield.ac.uk
	Department of Physics and Astronomy, University of Sheffield, Hounsfield Road, S3 7RH, Sheffield, UK
 
 	
    giulio.cerullo@polimi.it
	Dipartimento di Fisica, Politecnico di Milano, Piazza Leonardo Da Vinci 32, 20133 Milano, Italy
    CNR-IFN, Piazza Leonardo da Vinci 32, Milano, 20133, Italy
	
	Femtosecond switching of strong light-matter interactions in microcavities with two-dimensional semiconductors
    Giulio Cerullo
    August 5, 2024
==============================================================================================================





Ultrafast all-optical logic devices based on nonlinear light-matter interactions hold the promise to overcome the speed limitations of conventional electronic devices. Strong coupling of excitons and photons inside an optical resonator enhances such interactions and generates new polariton states which give access to unique nonlinear phenomena, such as Bose-Einstein condensation, used for all-optical ultrafast polariton transistors. However, the pulse energies required to pump such devices range from tens to hundreds of pJ, making them not competitive with electronic transistors. Here we introduce a new paradigm for all-optical switching based on the ultrafast transition from the strong to the weak coupling regime in microcavities embedding atomically thin transition metal dichalcogenides. Employing single and double stacks of hBN-encapsulated MoS_2 homobilayers with high optical nonlinearities and fast exciton relaxation times, we observe a collapse of the 55-meV polariton gap and its revival in less than one picosecond, lowering the threshold for optical switching below 4 pJ per pulse, while retaining ultrahigh switching frequencies. 
As an additional degree of freedom, the switching can be triggered pumping either the intra- or the interlayer excitons of the bilayers at different wavelengths, speeding up the polariton dynamics, owing to unique interspecies excitonic interactions. 
Our approach will enable the development of compact ultrafast all-optical logical circuits and neural networks, showcasing a new platform for polaritonic information processing based on manipulating the light-matter coupling.






§ INTRODUCTION


All-optical switches based on nonlinear optical materials have been extensively investigated to overcome the speed limitations of electronic circuits, thanks to their potential to work at much higher frequencies owing to the inherently fast light-matter interactions underlying their operation <cit.>.
Demonstrations of ultrafast all-optical logic gates have been achieved in a plethora of solid state platforms, exploiting optical nonlinearities (χ^(2) and χ^(3)) <cit.> and saturable absorption <cit.>.
Notable examples employed microring resonators <cit.>, plasmonic nanostructures <cit.>, photonic crystals <cit.>, metasurfaces <cit.>, 2D materials <cit.> or even single molecules <cit.>. 
Such devices showcased switching times down to tens of femtoseconds, but usually at the expense of the switching energy or the on/off contrast <cit.>. More recently, an optimal combination of femtosecond switching times and femtojoule operating energies has been obtained in an all-optical nonlinear device based on a lithium niobate waveguide <cit.>, but with millimeter-scale lengths hindering the production of densely on-chip integrated circuits. 
Therefore, combining integration capabilities, full scalability and high performances in all-optical switches is still an open challenge. 


Excitons in semiconducting materials embedded in optical resonators can be used for all-optical switching, as they show a highly nonlinear response when they are in strong coupling (SC) with resonant photons confined in the structure. In such a regime, the rate of coherent energy transfer between the energy-degenerate excitons and photons is higher than the loss rate, and new hybrid light-matter quasiparticles arise, called polaritons <cit.>. The energy splitting of the polariton states (Rabi splitting) is a direct measure of their coupling strength, which is proportional to the quality factor of the resonator and to the exciton absorption cross section.

Harnessing polaritonic nonlinear interactions is of key importance for a broad range of phenomena and applications, such as lasing <cit.>, optical parametric amplification <cit.>, Bose-Einstein condensation (BEC) <cit.>, or for quantum effects (polariton blockade) <cit.>.
A combination of blueshift of the polariton states and gain in photoluminescence intensity occurring above the threshold of BEC has been used as operational principle of all-optical polariton logic devices and neural networks <cit.>. 
Ultrafast optical switching (with switching times ≤ 1 ps) relying on BEC has been demonstrated even at room temperature <cit.>, with the drawback of using high pulse energies to reach the condensation threshold (≈ 30-300 pJ per pulse at least). 

Alternative strategies for ultrafast all-optical switching in the SC regime rely on the modulation of the light-matter coupling strength acting on the exciton absorption, which produces a spectral shift of the polariton states, acting as a gate for the light transmitted/reflected by the device.
Varying strongly the coupling strength would eventually lead to a complete transition from strong to weak coupling regime or viceversa <cit.>. SC can be switched off in strongly coupled optical microcavities comprising GaAs quantum wells through optical saturation of excitons <cit.> or via electrically-tuned charge build-up <cit.>. Alternatively, it can be switched on by optically induced absorption, i.e. for inter sub-band transitions <cit.>. 
However, achieving complete on/off switching cycle below 1 ps acting on the coupling strength in these material platforms is not possible due the long excitons and excited carriers lifetime <cit.>. 

Atomically thin transition metal dichalcogenides (TMDs) are promising nonlinear optical materials <cit.>, where excitons remain stable up to room temperature due to large binding energies and oscillator strength <cit.>. 
Owing to these properties, TMDs can easily enter the SC regime, when integrated in optical resonators <cit.>. Recent studies of TMD polaritons aim to maximise nonlinear interactions going beyond the use of 1s neutral excitons, exploring higher Rydberg excitonic states, charged excitonic complexes, moiré or dipolar excitons <cit.>. Nonlinearities in MoS_2 bilayers have been shown to be particularly pronounced, due to a combination of Coulomb dipole-dipole interactions, phase space filling and interspecies interactions between hybridized intra- and interlayer excitons <cit.>. Moreover, the fast dynamics of TMD excitons <cit.> makes these materials very promising for ultrafast logic gates. While strong exciton nonlinear interactions can quench the Rabi splitting in TMD-based microcavities <cit.>, all-optical ultrafast SC switching in atomically thin TMD systems has not been demonstrated.

Here we introduce a new strategy to perform all-optical switching in microcavities made of hBN-encapsulated MoS_2 bilayers, exploiting optical saturation of excitons and their fast relaxation times to produce an ultrafast collapse and revival of the SC regime. We employ femtosecond transient reflectivity (TR) spectroscopy to demonstrate sub-picosecond switching of the SC regime at pulse energies lower than 4 pJ, in ultra-compact devices comprising an atomically thin active material. We show the full tuneability of this process through different degrees of freedom, such as pumping frequency, pulse power and bilayer configuration. 
The SC switching leads to a strong modulation of the polariton peaks splitting, reducing the initial energy separation from 42 meV to less than their linewidth, making them indistinguishable from a single peak. The Rabi splitting modulation is further enhanced by placing a stack of two bilayers separated by hBN in the cavity, going from 55 meV to a complete collapse, resulting in an effective extinction ratio of about 7.5 dB working in reflection.
We further demonstrate an on/off SC switching frequency as high as 250 GHz, which can be extended up to 1 THz.





§ RESULTS




 §.§ Static and dynamic optical behaviour of MoS_2 bilayers


The TMD structures used in our work are made of monolayers (MLs) and bilayers (BLs) of MoS_2 encapsulated in hBN, placed on distributed Bragg reflectors (DBRs), for the subsequent fabrication of optical microcavities. All the spectroscopy experiments in this work are performed at T = 8K. Figure <ref>a shows the Reflectance Contrast (RC) spectra of ML and BL MoS_2 outside the cavity. The absorption of the intralayer A exciton in the BL (X_A-BL) is higher than in the ML, due to presence of the additional layer. In the BL a new excitonic resonance appears at ≈ 2 eV, which is attributed to dipolar hybridized interlayer excitons (hIX) with a high oscillator strength, resulting from the coherent tunnelling of holes between the valence bands of the two layers (Fig. <ref>a inset) <cit.>.

We study the ultrafast response of MoS_2 excitons by ultrafast TR micro-spectroscopy. We deliver two collinear ≈ 100-fs pulses, a narrow-band pump and a broad-band probe, focused on the sample using a microscope objective (Figure <ref>b). We then vary the delay time τ between them and monitor the changes in the broadband reflectivity spectrum of the probe (see Methods for experimental details). Figure <ref>c shows the differential reflectivity (ΔR/R) map measured as a function of the probe photon energy and pump-probe delay for a BL MoS_2, tuning the energy of the pump pulses at ≈ 1.94 eV, in resonance with the X_A-BL. 

Generally, the shape of the TR spectra in TMD MLs is a result of multiple effects, such as optical saturation (photo-bleaching), line broadening and spectral shift of the exciton peaks, leading to positive and negative TR signals around the exciton energies <cit.>. 
To show more clearly the temporal evolution of the exciton features in our system, we perform an analysis of the transient ΔR/R response based on the Transfer Matrix Method (TMM) to extract the time-dependent RC spectra of the material (see Methods for details) <cit.>. Figure <ref>d shows the MoS_2 BL dynamic RC spectrum at a delay of 0.3 ps (purple curve), compared to the one before the pump pulse (-0.3 ps, orange curve). We fit the dynamic RC with three Lorentzians (solid lines in Fig. <ref>d) to extract the time-varying intensity and energy shift of each excitonic mode. After excitation, the X_A-BL peak is quenched and slightly blue-shifted, as a result of phase-space filling and Coulomb interactions occurring at high exciton densities <cit.>. The small shoulder at 1.91 eV appearing at positive delays is instead related to a photo-induced absorption of the trion (X^*_A-BL) <cit.>. 
Since the excitation pulses are in resonance with X_A-BL (shaded yellow area in Fig. <ref>d), at lower energies compared to hIX, we would expect negligible optical saturation of the latter if the two excitonic species were totally uncoupled. On the contrary, we observe a photo-bleaching of the hIX absorption, although less intense than in the X_A-BL case. This demonstrates their hybridization with intralayer excitons due to the coherent hole tunnelling between the valence bands of the two layers and to the fermionic interactions between holes of X_A-BL and hIX sharing the same valence band (see inset of Fig. <ref>a) <cit.>.

Tracking the RC peak intensity as a function of the delay time, we can extract the ultrafast dynamics of the excitonic species (Fig. <ref>e). For both X_A-BL and hIX, the exciton population rises instantaneously (within the ≈ 100 fs temporal resolution of our setup), then decays exponentially, with about 50% of the initial population already relaxed within 2 ps (a comparison with  MoS_2 ML exciton dynamics is reported in Supplementary Note S1). 
The simultaneous build-up of hIX population is again an indication of strong hybridization and fermionic interactions with intralayer excitons.
The fast exciton decay time, resulting from both radiative and non-radiative relaxation processes of bright excitons <cit.>, the latter being significant in TMD bilayers, is also similar between X_A-BL and hIX. 






 §.§ Femtosecond switching of the strong coupling regime


We exploit the highly nonlinear exciton interactions in MoS_2 BL to drastically modify the light-matter coupling strength in microcavities on ultrafast time-scales. 
The microcavity samples are fabricated by covering the hBN-encapsulated MoS_2 heterostructures placed on DBRs with a transparent polymeric spacer (polymethylmethacrylate, PMMA) and a top silver (Ag) mirror, as illustrated in Fig. <ref>a. 
We perform k-space (Fourier) spectroscopy to image the angular dispersion of the monolithic cavity embedding the MoS_2 BL (Fig. <ref>b). Two distinct anticrossings appear when the cavity mode is in resonance with X_A-BL and hIX energies, a clear signature of the SC regime, resulting in upper, middle and lower polariton branches (UPB, MPB, LPB). Fitting the dispersion with a three coupled oscillators model, we extract Rabi splittings of Ω_A_BL= 42 meV and Ω_hIX= 23 meV for X_A-BL and hIX respectively. The Rabi splitting for strongly coupled intralayer excitons in a ML cavity is reduced to Ω_A_ML= 28 meV, due to the lower oscillator strength (see Supplementary Note S2 for the coupled oscillators model fit of the ML cavity).

We use ultrafast TR spectroscopy to excite the MoS_2 BL-based microcavities with narrowband ultrashort pulses tuned at the energy of X_A-BL. To better visualize the dynamic behaviour of polariton spectrum, we plot directly the reflectance (1-R) spectra measured on the cavity as a function of delay time and probe photon energy (Fig. <ref>c,d), while we include the TR data of the same measurement in Supplementary Note S3. Considering that in the spectral region of interest the reflectance of the cavity without the TMD is close to 1, plotting 1-R as a function of time is equivalent to showing the dynamic RC.
We focus our analysis on incidence angles close to normal, on the anticrossing between the cavity mode and the X_A-BL, resulting in the MPB and LPB. 
In a stark contrast to the out-of-cavity experiments, we do not observe a strong quenching of the absorption, but rather measure huge shifts of the polariton states. At negative delays, MPB and LPB are clearly separated, located at 1.953 eV and 1.911 eV respectively. When the pump and probe pulses are synchronous, the two polariton peaks collapse symmetrically in one broad central peak at ≈ 1.94 eV (purple line in Fig. <ref>c). Already after 2 ps, the two polariton branches start to reappear, while after 100 ps they have almost completely recovered. The 1-R map as a function of the time delay (Fig. <ref>d) shows more clearly the complete collapse and revival of polaritons, which can be only explained as a reversible transition from the strong to the weak coupling regime. We note that this trend follows well the dynamic absorption of the excitonic species measured outside the cavity (Fig. <ref>e), being a direct consequence of density-dependent optical saturation of excitons.   

We performed a quantitative analysis of the ultrafast behaviour of MoS_2 BL polaritons by fitting the experimental 1-R peaks with Gaussian functions. The results are shown in Fig. <ref>e, where the extracted peak energies and linewidths are plotted against the time delay up to 1.5 ps. Within few hundreds of femtoseconds from the zero-delay, the LPB shows a blueshift of about 27 meV, while the MPB redshifts by about 14 meV, merging in a single peak at about 250 fs. Such huge shifts cannot be explained just taking into account the bare exciton energy variations, which are in the order of only 5 meV (Fig. <ref>d). When the energy separation of the polariton states is lower than the linewidth of the cavity mode or the exciton, the anticrossing is not visible anymore and the system falls into the weak coupling regime (red star in Fig. <ref>e). 
Already after ≈ 500 fs, the SC is recovered. 
We note that in the weak coupling, the cavity mode is strongly broadened by the background absorption of the excitons, already broad due to excitation-induced dephasing <cit.>. Such broadening also affects the polariton peaks after the collapse, as shown in the color bars of Fig. <ref>e, which become more discernible only after 2 ps. 
We consider that the SC switching has been reached when the Rabi splitting is equal to the unperturbed exciton linewidth (the FWHM of X_A-BL is ≈ 20 meV in static conditions).
On the other hand, considering as switching threshold the splitting being equal to the broader polariton linewidths would be a less stringent criterion. In the latter case the pump energy to induce a peak splitting collapse would be even lower, also resulting in a faster SC recovery.


We observed a similar SC collapse also in microcavities embedding a ML of MoS_2, but in that case the longer exciton lifetimes led to a much slower SC recovery, while the smaller Rabi splitting worsened the on/off contrast, i.e. the signal intensity ratio between the 1-R spectra of the cavity in the unperturbed SC and weak coupling conditions respectively (see Supplementary Note S4).




The pump pulse energy plays a major role in the SC switching dynamics, as shown in Fig. <ref>a where MoS_2 BL cavity spectra taken at a delay of 250 fs for different excitation pulse energies demonstrate the gradual quenching of the Rabi splitting. The SC collapse in TMD cavities is a direct consequence of exciton nonlinear interactions, which scale proportionally to their density <cit.>. 
We demonstrate this effect by carrying out theoretical simulations of the cavity 1-R spectra using the TMM (Fig. <ref>b), employing the MoS_2 BL optical constants calculated from the exciton nonlinear absorption as a function of the density (see Supplementary Note S5). The match between experiments and simulations proves that the main cause behind the observed femtosecond switching of the SC regime is the optical saturation of MoS_2 BL excitons at high excitation densities, which recovers very rapidly due to the fast radiative and non-radiative exciton relaxation mechanisms in this system. 

Figure <ref>c reports the MPB-LPB energy difference against the time delay, normalized with respect to its value before excitation. In this figure, the blue dots are related to the experiment reported in Fig. <ref>, performed at 3.75 pJ (pump fluence: 212 μJ cm^-2), and show the recovery of SC occurring on two different time-scales, a fast one within 1 ps and a slow one which is concluded after ≈ 100 ps. We ascribe those two recovery steps to the population decay dynamics of the bare excitons (Fig. <ref>e). The dashed horizontal line represents the threshold for the weak coupling regime, when the polariton splitting is smaller than the exciton linewidth. Increasing or decreasing by few picojoules the excitation energy, we can extend the temporal window of weak coupling regime (7.5 pJ, red dots in Fig. <ref>c) or suppress completely the transition (2.5 pJ, yellow dots in Fig. <ref>c). 

We provide a theoretical explanation on the energy dependent dynamic SC switching upon direct excitation of the intralayer excitons, considering several possible contributions to the dynamics. We note that the ≈ 1 ps timescale for the fast recovery coincides with the bare X_A-BL exciton fast decay. 
However, this short timescale cannot account for the later slower recovery (∼100 ps) of the Rabi splitting, indicating that a significant exciton population remains in the sample and nonlinear phase space filling impacts the spectrum. 
One plausible explanation is the existence of long-lived dark excitonic states at lower energies <cit.>. In this case photoexcited excitons can be transferred to such states which interact with light weakly, forming a reservoir that contributes to the nonlinear phase space filling. In a bilayer MoS_2, low-energy states are represented by spin-forbidden states due to spin-orbit coupling, or momentum-forbidden states <cit.> due to indirect bandgap 
<cit.>. These states possess a very long lifetime and can explain the TR dynamics. The corresponding model is summarized in the Methods, while we provide more details about the simulated polariton dynamics in Supplementary Note S6.

Leveraging on the ultrafast recovery times of SC in our samples, we demonstrate the possibility to modulate light-matter interactions at very high frequencies, illuminating the cavity with two subsequent pump pulses at 1.94 eV delayed by only ≈ 4 ps. To produce such pulse pair, we used a birefringent YVO_4 crystal with optical axis rotated by 45^∘ with respect to the polarization of the incoming pump pulse, followed by a linear polarizer (see Fig. <ref>d and Methods for more details). The first pulse energy was tuned to be slightly lower than the experiment in Fig. <ref> in order to get a faster SC recovery, while the second pulse energy was adjusted to take into account the residual exciton population after the first pulse. The resulting transient 1-R map shows two reversible on/off cycles (Fig. <ref>e), proving a very fast switching frequency of ≈ 250 GHz. We note that this value was limited by technical constraints (the fixed delay between the pulse pair determined by the thickness of the available YVO_4 crystal), while the theoretical limit is given by the recovery time of the SC.





 §.§ Ultrafast SC switching by interspecies interactions


We exploit the interspecies exciton interactions specific of MoS_2 BL to generate optical saturation of X_A-BL acting on the hIX, exciting selectively the latter and probing the quenching of the Rabi splitting on X_A-BL (Fig. <ref>a). This process relies on nonlinear fermionic interactions (i.e. involving a single charge carrier constituting the exciton) between the two excitonic species: the X_A-BL valence band is shared with hIX, therefore exciting the latter causes optical quenching of the former, due to Pauli blocking of holes for X_A-BL <cit.>. Figure <ref>b shows the transient 1-R map of the MPB-LPB, pumping the hIX of the BL: the femtosecond switching of SC regime occurs very clearly also in this case. Comparing this result with the previous case of resonant X_A-BL pumping (Fig. <ref>c), the fast SC recovery is even more distinct, with the two polariton peaks being clearly visible and well separated already after 1 ps, as shown in Figure <ref>c.

The dynamics of the nonlinear response when pumping in resonance with the hIX is also consistent with the developed model based on the nonlinear saturation and phase space filling from the long-lived states (see Methods). 
In this case, we considered similar lifetimes for hIX compared to X_A-BL, but we assumed the rate for transferring the pumped hIX to the long-lived reservoir states contributing to the phase space filling  
to be smaller than in the X_A-BL case. 
This may be understood as the result of the hIX’s wavefunction spreading in the out-of-plane direction. This implies that hIX is less 2D than X_A-BL, leading to a weaker scattering effect with disorder and to a smaller transfer rate. Furthermore, thermalization through exciton-phonon scattering is another important mechanism converting the bright states into momentum-dark states which may also gives smaller transfer rate upon hIX pumping. 
These scattering effects yield a faster recovery of SC, in agreement with our experimental observation (see Supplementary Note S6). 

Another effect leading to faster recovery in the hIX pumping scheme is the mitigation of the Pauli blockade. In contrast to X_A-BL pumping, the pumped exciton only shares holes with the probed exciton but not the electrons <cit.>, see Fig. <ref>a, leading to a weaker saturation effect. This can also result in a faster recovery of SC if similar conditions as in X_A-BL case are used (except the pump photon energy). Combining the effects of a smaller bright-to-dark exciton transfer rate and weaker Pauli blockade, the influence of the reservoir long-lived states is less significant exciting the hIX. Therefore, a faster recovery time of SC is easier to achieve in this case.
Such fast recovery would allow to further increase the switching frequency, up to ≈ 1 THz. We underline that to achieve such hIX-induced Rabi quenching we use an excitation energy of 4.37 pJ, only moderately higher compared to the resonant excitation case. Higher pulse energies will increase the recovery time and the weak coupling time window.





 §.§ SC switching in a double BL microcavity


Finally, we fabricated a device comprising two vertically stacked bilayer MoS_2 separated by an hBN spacer of  40 nm. We placed this structure in a microcavity made of the same DBR and silver mirror used in the single BL cavity, with a PMMA spacer between the TMD stack and the top mirror (Fig. <ref>a). Similarly to microcavities with multiple quantum wells <cit.>, the SC is enhanced in this sample, as the Rabi splitting is increased to 55 meV due to the additional BL unit (see Supplementary Note S7 for the coupled oscillators model fit of the strongly coupled cavity dispersion).
Figure <ref>b shows a comparison between the static 1-R spectra of a single BL (blue line) and a double BL (red line) cavity, taken at small angles. It clearly appears that the MPB-LPB peaks are more separated in the double BL cavity compared to the single BL sample, being also redshifted, as an effect of the more negative detuning of the former. 
The negative detuning also leads to a decrease of the LPB linewidth in the double BL cavity, being more cavity-like.

By exciting the double BL cavity with pump pulses resonant with X_A-BL, we observed again an ultrafast collapse of the MPB-LPB polariton peaks into a weakly coupled cavity mode, followed by their later recovery (Fig. <ref>c). We note that the pulse energy used for this experiment (8.25 pJ) was not adjusted to obtain a sub-ps SC recovery, but just to demonstrate the SC switching. 
Figure <ref>d shows the full dynamics of the SC collapse and recovery in the double BL cavity, where the energy separation between MPB and LPB (blue and red lines respectively) drops from ≈ 55 meV to zero immediately after the pump pulse.
Considering the unperturbed SC condition as the on state of the optical switch and the weak coupling as the off state, we calculated the spectral power extinction ratio (ER) from the 1-R spectra before and after the pump pulse, taken at -350 fs and 350 fs respectively, where ER(dB)=10log((1-R_on)/(1-R_off)). 
The ER is increased significantly in the double BL cavity compared to the single BL device, over a broad energy range, as shown in Fig. <ref>e. For both the devices, the maximum ER in absolute value is reached around the energy of the weakly coupled cavity mode, between the LPB and MPB, which transmits (reflects) more during the off (on) state. While for the single BL the ER absolute value reaches 3.2 dB at 1.932 eV, it is enhanced up to 7.5 dB at 1.915 eV in the double BL.
It is worth to mention that the ER is also high in the spectral regions of the LPB and MPB, where it shows opposite sign, meaning that the optical switch can be used in direct or reverse mode just by changing the operational wavelength.
We note that working at high frequencies the effective extinction ratio for a second switching event decreases due to the residual exciton population after the first pulse excitation (e.g. by about four times in the double pump pulse experiment shown in Fig. <ref>). We foresee that this drawback can be mitigated by reducing the long exciton decay component, for example suppressing the exciton scattering to dark states and using a different optical resonator with higher quality factor and smaller mode volume, to induce a strong Purcell effect. 




§ DISCUSSION


In summary, we exploit the transient behaviour of MoS_2 exciton-polaritons to demonstrate ultrafast optical switches, whose operational principle is based on the instantaneous transition from the strong to the weak coupling regime due to optical saturation of excitons, which can recover on the sub-picosecond timescale. The MoS_2 BL system uniquely combines nanometric thickness, large nonlinearities and large Rabi splitting with short lifetimes, with the latter enabling observation of the exceptionally fast recovery. 
Furthermore, we show that by increasing slightly the pump pulse energies above the threshold for the SC collapse, the weak coupling time window can be significantly extended and deterministically controlled, being sensitive to energy variations of hundreds of femtojoules and below, crucially important for sensing and low light applications <cit.>. 
This system, also, offers additional degrees of freedom. It can operate at different excitation energies, for example in resonance with either intra- or interlayer excitons, leveraging on the strong interspecies interactions between these excitonic species, unique to the MoS_2 BLs. We foresee this property to be particularly useful for multiplexed logic operations <cit.>. 
Owing to the fast recovery of SC, we were able to perform subsequent switching events delayed by 4 ps, demonstrating an operational frequency of ≈ 250 GHz. Considering the sub-ps SC switching time, this frequency can be pushed up to 1 THz, surpassing even the fastest electronic transistors demonstrated so far <cit.>. 
We also demonstrate that using a microcavity with two stacked MoS_2 BLs can boost the Rabi splitting and greatly enhance the on/off extinction ratio, reaching a maximum of 7.5 dB in a single switching event. An improvement of the optical resonator quality factor and a shorter exciton lifetime will lead to even greater on/off contrast for high frequency switching. 

Our work highlights atomically thin TMDs as a flexible system with rich physics in which sub-ps all-optical switching can be achieved and finely controlled.   
The insights provided can be pivotal for the development of TMD-based high speed all-optical circuits. Moreover, the developed ultrafast nonlinear switching unit can improve the performance of optical neural networks <cit.> acting as an all-optical nonlinear activation function. 
We believe that further progress in boosting the nonlinear interactions of TMD excitons, for example employing localized excitons in moiré heterostructures, could reduce even more the energy required for the ultrafast switching processes, reaching eventually the quantum regime of operation down to a few photons, eventually leading to single-photon transistor-type devices <cit.>. 
The ultrafast switching operation can be extended in principle up to room temperature using our devices, owing to the large binding energies of TMD excitons. 
Considering also the nanometric thickness of each hBN/BL/hBN stack, a microcavity could be filled by many TMD units, greatly increasing the Rabi splitting and subsequently the spectral shifts when used as ultrafast switches, which will lead to enhanced on/off extinction ratio.
Moreover, the integration of electrical contacts in the microcavity structures <cit.> would enable the fabrication of electro-optical interfaces by tuning the electrostatic doping and electric field, which can provide giant shifts of the hIX energy in MoS_2 BLs <cit.>, also enhancing their nonlinear interactions. 
Finally, highly nonlinear TMD systems combined with novel light-confining nanostructures supporting photonic resonances of high quality factor, such as quasi-bound states in the continuum <cit.>, will allow the development of ultra-compact nanodevices, easy to be integrated in planar photonic circuits architectures.

Developing ultrafast all-optical switches based on the transition from the strong to the weak coupling regime would be crucial also to unveil more exotic physical phenomena. 
The crossing point between the strong and weak coupling regimes, called exceptional point, enfolds exotic physics arising from the non-Hermitian Hamiltonian describing such condition <cit.>. 
Both the Hamiltonian eigenvalues and their corresponding eigenstates collapse in exceptional points, which for systems of coupled exciton and photon states translates in their energy and losses becoming identical, making the system very sensitive to external perturbations.
Changing non-adiabatically the parameters of the coupled system in order to encircle the exceptional point can produce topological effects and chiral mode conversions, where the direction of the encirclement dictates the final state of the system <cit.>. Owing to the ultrafast collapse and recovery of SC, our system could be used to demonstrate for the first time this effect in planar microcavities.





§ METHODS




 §.§ Sample Fabrication

The hBN/MoS_2/hBN heterostructures were assembled using a polydimethylsiloxane (PDMS) polymer stamp method. The PMMA spacer for the monolithic cavity was deposited using a spin-coating technique, while a silver mirror of 45 nm thickness was thermally evaporated on top of it.



 §.§ Optical Measurements

For the transient reflectivity measurements 100-fs pulses from an amplified Ti:Sapphire laser at 2 kHz repetition rate are used. The laser output is split in two beams. A portion of the laser output is utilized to drive a non-collinear optical parametric amplifier (NOPA), which allows tuning the pump wavelength. The rest is used for the generation of the broadband white light probe pulse by focusing the beam on a sapphire plate. The delay between pump and probe pulses is controlled by a mechanical delay line. The pulses are combined collinearly and focused on the sample using a 50x objective, resulting in a spot size of ≈ 1.5 μm. The sample is kept in a helium cryostat at 8 K. The differential reflectivity (Δ R/R) spectra are recorded at various time delays τ to track the changes induced by the pump. Specifically, the reflectivity spectrum of the probe with the pump on, R_PumpOn, is compared at each delay with a reference spectrum obtained when the pump is off, R_PumpOff. These are used to calculate Δ R/R = R_PumpOn - R_PumpOff/R_PumpOff, shown in the TR maps. The dynamic probe maps are extracted from a combination of the measured RC spectrum with the pump off and the Δ R/R maps, as R_PumpOn=R_PumpOff(1+Δ R/R). For the double pump pulses experiments, we use a thick YVO_4 birefringent crystal with optical axis rotated at 45^∘ with respect to the vertical pump polarization, which produces a replica of the pulse with horizontal polarization delayed by ≈ 4 ps. The rotation of a subsequent polarizer is changed to finely adjust the energy of each pulse in order to ensure the SC recovery after each excitation pulse.



 §.§ Transfer Matrix Method Analysis


In order to extract the spectral and temporal evolution of the excitonic optical properties from the Δ R/R maps, we follow a procedure recently reported in refs <cit.>. The transient reflectivity R(ω,τ) of MoS2 BL at each delay time is determined from the equilibrium reflectivity R(ω), which is reconstructed from the TMM fit of the static RC spectrum, and the transient reflectivity Δ R/R(ω,τ), following this relation:



R(ω,τ)=R(ω)(Δ R/R(ω,τ) + 1)



Then, the dynamic RC is obtained by applying the formula: 
RC(ω,τ) = 1-(R(ω,τ)/R_sub),
where the substrate reflectivity R_sub is simulated with the TMM. See Supplementary Note S5 for more details on the TMM simulations.





 §.§ Theoretical Model for the Polariton Dynamics


To gain further insight into the dynamics in the system, we develop a mean-field model that captures the main trends in our experiment over different excitation regimes. The Hamiltonian corresponds to the coupled cavity-photon system, where the X_A-BL mode (being the probed A exciton of a homobilayer) hybridizes with the cavity mode. The Hamiltonian reads


    H=[          E_c+iκ 1/2g(n_X)Ω_A_BL; 1/2g(n_X)Ω_A_BL       E_A_BL+iγ ],


where E_c and κ are the energy and linewidth of the cavity photon, and E_A_BL and γ are the energy and linewidth of the probed exciton. In the Hamiltonian above Ω_A_BL is the Rabi splitting at weak pumping, and g(n_X)=e^-α n_X is the dimensionless nonlinear coupling, with α being the nonlinear phase space filling (saturation) coefficient <cit.>. The magnitude of the Rabi splitting is dependent on the total number of excitons in the system, n_X. In general n_X(t)=n_p(t)+n_R(t) is time-dependent, and includes excitons (electron-hole pairs) from different states. Specifically, we separate the two fractions corresponding to n_p and n_R being the population of the pumped exciton and the long-lived excitons in the reservoir. Crucially, both contribute to the nonlinear saturation effect. The dynamics of excitonic fractions can be described by rate equations defining the transfer and population redistribution, which read

    dn_p/dt=   -γ_p n_p-r n_p+Θ(t),
    dn_R/dt=   -γ_R n_R+r n_p,

where γ_p is the pumped exciton decay rate, γ_R is the decay rate of the long-lived exciton in the reservoir, and r is the rate constant for transferring the pumped excitons into the reservoir. The function Θ(t) depends on the pump laser profile in time. For example, Θ(t) can be a Heaviside step function to model the pump as an on-off switching field. In this model, we assume γ_R≈γ_p/100 for the long-lived states corresponding to 100 ps decay time, and consider the decay timescale being similar to that of spin-forbidden dark states <cit.>. In fact, the decay time may be different, but this does not change our later conclusion in a qualitative way. Furthermore, when considering a spin-conserving process, we let the transfer rate be comparable to the pumped exciton decay rate <cit.>, r≈γ_p, such that this allows the pumped exciton transfer into the reservoir. With this, we find a good qualitative agreement between the theoretical spectrum and the experimental measurement (see Supplementary Note S6). Particularly, the theory demonstrated the excitation pulse energy dependence of the recovery time of SC. 



§ ACKNOWLEDGEMENTS


AG and GC acknowledge support by the European Union Marie Sklodowska-Curie Actions project ENOSIS H2020-MSCA-IF-2020-101029644. CT acknowledges the European Union’s Horizon Europe research and innovation programme under the Marie Skłodowska-Curie PIONEER HORIZON-MSCA-2021-PF-GF grant agreement No 101066108. AG, CT, SDC and GC acknowledge funding from the European Horizon EIC Pathfinder Open programme under grant agreement no. 101130384 (QUONDENSATE). This work reflects only authors’ view and the European Commission is not responsible for any use that may be made of the information it contains.
AIT and SR acknowledge financial support of the EPSRC grants EP/V006975/1, EP/V026496/1 and EP/S030751/1. OK and KWS acknowledge the support from UK EPSRC grant EP/X017222/1.


The authors declare no competing interests.



§ AUTHOR CONTRIBUTIONS


AG, CL, CC and CT carried out the optical spectroscopy experiments with contribution from GDB and GS. CL and SR fabricated the 2D samples. KW and TT synthesized the high quality hBN. PC, RJ and DGL fabricated the microcavities. AG, CL, CC, GDB and GS analyzed the data with contribution from SDC, AIT and GC. AG performed the transfer matrix simulations. KWS and OK developed the theory on dynamic optical saturation in microcavities. AG, CL and GC wrote the manuscript with contribution from all other co-authors. DGL, OK, SDC, AIT and GC managed various aspects of the project. SDC, AIT and GC supervised the project.



§ DATA AVAILABILITY


The data that support the findings of this study are available from the corresponding authors upon reasonable request.



§ CODE AVAILABILITY


The computer codes and algorithms used to process the data included in this study are available from the corresponding authors upon reasonable request.



§ INCLUSION AND ETHICS STATEMENT


All collaborators of this study that have fulfilled the criteria for authorship required by Nature Portfolio journals have been included as authors, as their participation was essential for the design and implementation of the study. Roles and responsibilities were agreed among collaborators ahead of the research. This work includes findings that are locally relevant, which have been determined in collaboration with local partners. This research was not severely restricted or prohibited in the setting of the researchers, and does not result in stigmatization, incrimination, discrimination or personal risk to participants. Local and regional research relevant to our study was taken into account in citations.



§ ADDITIONAL INFORMATION


Supplementary Information is available for this paper.





 Paper Title: Floquet Engineering of Polaritonic Amplification in Dispersive Photonic Time Crystals.



APS/123-QED





These authors contributed equally to this work
 
These authors contributed equally to this work




 Elmore Family School of Electrical and Computer Engineering 
 Birck Nanotechnology Center, Purdue University, West Lafayette, IN 47907, USA





 

 
 













       

§ ABSTRACT 
In this study, we investigate the dynamics of dispersive photonic time crystals (PTCs) and their potential applications for controlling light-matter interaction. By employing the Lorentz-Drude model, we analyze theoretically and via numerical simulation the effects of periodic modulation of dispersion parameters, revealing the emergence of hybrid bandgaps from interaction of polaritonic branches with unique characteristics. Our study demonstrates that dispersive PTCs offer novel excitation channels and amplification possibilities, that require lower modulation frequencies compared to non-dispersive systems thus alleviating experimental challenges for the realization of PTCs in the optical regime. These findings pave the way for advancements in polaritonic lasing and resonant Raman scattering. 










               
Floquet Engineering of Polaritonic Amplification in Dispersive Photonic Time Crystals
    Vladimir M. Shalaev
    August 5, 2024
=====================================================================================







§ INTRODUCTION

The study of time-varying photonic media has a history spanning several decades <cit.>. Nevertheless, recent years have witnessed a resurgence of interest in this field revealing numerous new physical phenomena <cit.> such as the creation of squeezed and entangled photon states <cit.>, spatiotemporal beam steering <cit.>, and the breaking of Lorentz reciprocity <cit.>. One of the most notable outcomes of engineering temporal dynamics is the concept of photonic time crystals (PTCs)<cit.>, where the permittivity of the host material undergoes a periodic modulation with time, analogous to spatial photonic crystals which are periodically patterned in real space <cit.>. Thanks to the non-Hermitian properties of PTCs these systems exhibit unique light-matter interactions, leading to intriguing phenomena such as generation and amplification of radiation fields from free electrons and dipoles <cit.>.

Abrupt and strong changes in material's optical properties over time result in reflections and refractions at temporal interfaces <cit.>, akin to spatial boundaries. However, unlike spatial boundaries, temporal ones require dynamical modulation of optical properties, creating an open system and allowing time-reflected and refracted waves to have different energies than the incident beam. PTCs exploit this to create and amplify electromagnetic radiation in their momentum bandgap region, where time-reflected and refracted waves constructively interfere <cit.>. This amplification resembles parametric amplification <cit.> but differs in two distinct features.  Firstly, the presence of strong modulation of material parameters, such as strongly modulating permittivity around the epsilon near zero point<cit.>, breaks the applicability limits of the conventional perturbative nonlinear treatment. Secondly, it is assumed that the entire medium undergoes simultaneous modulation while the probe beam propagates inside the medium, relaxing the typical phase-matching conditions <cit.>. 


Amplification within the bandgap of PTCs fundamentally arises from the transfer of energy from the modulation to the probe. This process necessitates interactions among various frequency components within the medium, leading to the creation of new components at the probe frequency. In a non-dispersive medium, where only a single frequency mode is supported for each momentum, effective amplification requires that the modulation frequency be twice that of the probe beam. Thus, accessing the momentum bandgap in non-dispersive media demands modulation at twice the probe beam frequency, which poses significant challenges for the realization of PTCs at optical or near-infrared (NIR) frequencies. This is why time reflection and exponential growth in the momentum bandgaps have so far only been observed experimentally in the microwave domain <cit.>. However, recent advances in optical materials that can be strongly modulated on an femtosecond time scale <cit.> create a potential for experimental realization of PTCs also in the optical domain <cit.>. These materials are typically strongly dispersive and lossy. The material dispersion enforces electric field to evolve continuously in time, and potentially makes it more challenging to observe unique PTC behavior<cit.>.  Additionally, the dispersive nature of materials couples energy from the light to mechanical motion in the material forming hybrid states of light and matter known as polaritons, which are new quasiparticles of the coupled system with well-defined energy and momentum. Building on the nondispersive foundations of recent PTC studies <cit.>, we explore the inclusion of material dispersion in our analysis which opens up the possibility of investigating the interaction between matter and radiation field in the presence of optical modulation. We further discuss and examine more realistic experimental scenarios exploiting the full potential of such materials. 


Recent studies for this exciting prospect have begun by investigating plane wave propagation in a dispersive time-varying media <cit.>, a two-level model with periodic modulation of resonance frequency <cit.> and widening of momentum bandgap near material or structural resonance <cit.>. A broader investigation of polaritonic interactions and its effect on PTC amplification is still needed. In this paper we focus on analyzing the overall PTC dispersion, also referred to as optical band-diagram, in the presence of Lorentz-Drude type material dispersion. Specifically, we explore the modulation of the coupling between different polaritonic branches. We observe the emergence of hybrid bandgaps which stems from interaction between upper and lower polaritonic branches. These hybrid bandgaps have experimentally favorable conditions. 

Our main findings reveal that when the parameters of dispersive materials are modulated, additional bandgaps with lower modulation speeds emerge in the system. These bandgaps are accessible from the modes of the unmodulated medium and enable excitation channels for coherently interfering time-reflected and refracted waves, establishing exponentially growing fields. This relaxes the fast modulation requirement of nondispersive materials and suggests more accessible experimental pathways for observing such effects with much slower modulation frequencies. 

As amplification in PTCs stems from non-perturbative nonlinear interactions, the amplification of waves within the polaritonic PTCs is intricately connected to the nonlinear interaction among polaritons <cit.>. Thus, by examining polaritonic PTCs we may gain valuable insights into the physics of these strong, non-perturbative interactions among polaritons that pave the way forward for enhanced resonant Raman scattering and polaritonic lasing <cit.>.



§ POLARITONIC BANDS WITH MODULATION


When the optical properties of a medium are periodically modulated with a frequency Ω, its properties remain time-translation invariant under a temporal shift of T = 1/Ω, i.e. Ψ(t+T)=Ψ(t), where Ψ is a physical quantity such as electromagnetic field amplitude or polarization in the medium and t is the time. This temporal periodicity causes the ω-k dispersion of the supported modes to also become periodic in frequency with period Ω. Therefore, the dispersion of the system is completely characterized by the frequency range of 0 to Ω, known as the 1st Brillouin zone.  Owing to this periodicity, the modes of the medium can be described using the Floquet theory, by expanding the relevant quantities in the form Ψ(t)=e^iω_kt∑_n=-∞^∞Ψ_ne^inΩ t
,with ω_k+2π n=ω_k, where n is an integer. 

In the case of a stationary, nondispersive optical medium, the dispersion relation is given by the expression ω=kc/n, where c is the speed of light in a vacuum, and n is the refractive index. This relation defines a single line for positive ω and k shown in Figure <ref> by black dashed lines. The effect of periodic modulation of the dielectric function on the dispersion relation for such nondispersive materials can be visualized as dispersion lines of the unmodulated medium being translated along the frequency axis by integer multiples of the modulation frequency Ω . In this case the dispersion for negative frequencies also becomes relevant as they get shifted to the positive frequency domain and mix with other dispersion lines. The translation of the dispersion lines causes the branches in different Brillouin zones to cross each other leading to complex frequency eigenvalues, ω_k, with degenerate real parts. This contrasts with the unmodulated, lossless medium, which has only real eigenfrequencies. Following the analogy of the frequency band gaps in spatial photonic crystals, these regions with imaginary eigenfrequencies have been called momentum bandgaps with the characteristic signature of exponentially growing photon fields <cit.>. The emergence of the momentum bandgaps and complex eigenvalues are illustrated for a nondispersive medium in Figure <ref>. The modes, as they are linear, cross each other at frequency Ω/2 resulting in a bandgap at that frequency. 




In the case of a dispersive medium, the energy no longer remains purely in the fields. The electromagnetic waves hybridize with material excitations and form polaritons. The response of the materials that possess bound charges can be described classically with Lorentz type frequency-dependent dielectric function.

    ε(ω)=ϵ_∞+ω_p^2ω_0^2-ω^2-iγω

where, ω_0, γ, and ε_∞ are respectively the central frequency, Drude damping and high-frequency permittivity of the material and ω_p^2=4π e^2N/m^* is the plasma frequency of the material defined by free carrier concentration N and effective mass m^*. This form of dielectric function follows from the harmonically oscillating polarization density of the material excited by electromagnetic waves of the same frequency.

    d^2P/dt^2+γdP/dt+ω_0^2P=Ne^2/m^*E

Polaritons in such media are formed when electromagnetic waves couple to the mechanical oscillations of these bound charges through induced polarization created due to their harmonic motion. The dispersion relation for such material follows from Maxwell’s equations and is given by k^2=ω^2ε(ω)/c^2. The solution of this quadratic equation leads to two dispersion branches above and below characteristic frequency ω_0/√(ε_∞), which are conventionally called upper and lower polaritonic branches. These two branches arise from the coupling of phononic and photonic dispersion lines. Due to the coupling, the branches bend outward and form avoided crossing with an energy difference Δ E known as the Rabi splitting. An example of such dispersion curves is shown in Figure <ref>.a calculated for normalized oscillator parameters ω_p=1.6Ω, ω_0=0.6Ω, and ε_∞=1 . 


We now incorporate temporal changes in material properties using dispersion parameters. Although our model includes dispersive effects the response of dispersion parameters to external stimuli is still assumed to be instantaneous. Experimentally the most accessible modulation is through optical pumping of semiconductors and changing conduction band electron density or effective mass. These two parameters directly affect the plasma frequency, therefore in our model, we assume to be modulating the plasma frequency.

    ω_p^2(t)=ω_p0^2[1+Δ^2cos(Ω t)]


Due to this periodicity, the fields inside the medium can be written as Floquet expansion and the following eigenvalue equation is obtained from Maxwell's equations.

    ω_kΨ_p=∑_n=-∞^∞(M_p-n-nΩ I_4×4δ_pn)Ψ_n

where Ψ_p^k=[E_xp,H_yp,J_xp,P_xp]^T, with superscript T meaning transpose operation and J_xp=∂_t P_xp, and matrices M̅_m are given by


    M_m=
      [[           0 -ckδ_m0/ϵ_∞    4π iδ_m0           0;     -ckδ_m0           0           0           0;       -if_m           0        iγ_m     iω_om^2;           0           0      -iδ_m0           0 ]]

The summation in Eq.<ref> runs over all possible Floquet modes, however in numerical implementation, we consider only finite number modes and truncate this infinite system of equations into generalized eigenvalue problem:

    ω_kΨ^(k)=A̅Ψ^(k)

with Ψ = [Ψ_1^(k),Ψ_2^(k),...,Ψ^(k)] and A̅ is a matrix formed by elements from Eq.<ref>. 
Here for an isotropic medium, without losing the generality we assumed the waves propagate along the z-axis. The equations for waves with orthogonal polarization Ψ=[E_y,H_x,J_y,P_y]^T have very similar form, hence we will focus only on one of them for our discussions. The details of the calculation can be found in the Supplementary Information(SI).

Solving the eigenvalue Eq.<ref> results in the band diagram for a dispersive photonic time crystal along with the corresponding eigenfunctions. Introducing modulation to a dispersive medium, like in the nondispersive case, causes the dispersion branches to be translated with the modulation frequency. The overlapping polaritonic branches interact and give rise to bandgaps or avoided crossing based on the type of dispersion line as shown in Figure <ref>.b. The dashed lines show the translated dispersion lines of the stationary medium and the blue lines show the dispersion line of the modulated medium. The dispersion parameters and eigenfrequency of the Floquet modes are expressed relative to the modulation frequency Ω.

The presence of polaritonic bands and their modulation introduce novel phenomena into the physics of time-varying medium. One of the most prominent effects for this case is the interaction between lower and upper polariton branches. The crossing of lower and upper branches belonging to positive and negative frequency quadrants causes the real part of the frequency eigenvalues to split and give rise to a frequency bandgap highlighted in Fig <ref>.c which is otherwise not observed in nondispersive case. 

The interaction of polaritonic branches causes the emergence of bandgaps as well. First, a bandgap with similar characteristics to non-dispersive PTC bandgaps arises from the interaction between two lower branches, highlighted in Fig <ref>.d as same branch bandgap. These bandgaps occur in the middle of the unit cell at frequency Ω/2, with a flat dispersion line. A distinguishing factor for these bandgaps from the non-dispersive case is the additional excitation channels made available by the upper branch with matching momentum. 

Furthermore, the crossing of upper and lower polaritonic branches leads to the formation of a novel bandgap which we term as hybrid bandgaps with distinct features. First, hybrid bandgaps does not occur on a flat line resulting in nonzero group velocity. Therefore, the energy is amplified directionally. Secondly, the modulation frequency required to create this bandgap is not necessarily twice the frequency of the unmodulated branch, but it is the energy difference between the upper and lower branches. Therefore, the requirement of having the modulation speed twice the probe beam's frequency does not apply. 

The hybrid bandgap's size and amplification heavily rely on the coupling strength between the upper and lower branches, and is maximized when the group velocities of the branches match to allow a more profound overlap. This alignment occurs when the hybrid bandgap coincides with the momentum region where Rabi splitting is present between the unmodulated branches. We accentuate this effect using dispersion parameters of a well-known phononic optical material hexagonal boron nitride (h-BN)(see Table <ref>)<cit.>. The band diagram for this case is given in Fig <ref>.a.




For a phononic medium, the minimum separation of the lower and upper branches is equal to ω_p/√(ε_∞) which is proportional to the difference between the transverse phonon (TO) and longitudinal phonon (LO) frequencies such that Δω=√(ω_LO^2-ω_TO^2) [36]. The modulation frequency has been chosen to be half of the Rabi splitting (Δω) between the upper and lower branches. This allows maximal overlap coupling between lower and upper branches where their group velocities match, as shown in Fig <ref>.b, creating profound bandgaps. We investigate the characteristics of wave propagation in these bandgaps and the possibility of obtaining amplification with slower modulations in the subsequent section.



§ EXCITATION OF MODES


For a given medium, the eigen solutions and their dispersions discussed in the previous section characterize the natural modes supported in that medium. When this medium is excited by external sources, the induced field can be represented analytically as time-dependent superpositions of these eigenmodes. Such analytical formulations give great flexibility to study the dynamics of the modes in the system. In particular, it allows us to observe how individual modes are excited, and how they evolve and interact among themselves. To demonstrate this, we discuss the dynamics of a dispersive, time-varying medium under dipole and plane wave excitations. 

We first illustrate the system dynamics using a dipole excitation. This study builds on previous discussions regarding a non-dispersive medium <cit.>. Specifically, we examine the excitation of the system whose band diagram is depicted in Figure <ref>.b. The system is, excited by a time-harmonic dipole source with an oscillation frequency ω_d = 0.7Ω, chosen to fall outside of any bandgaps. Fig <ref>.a shows the excitation spectrum as density plots superimposed by the band diagram (green dashed lines). 


The excitation frequency is represented by the black line. Initially, the light couples most strongly to the branches of the dispersion diagram closest to the excitation frequency. However, the entire band is also excited because the radiated light from a point source has all the momentum components. Additionally, higher-order bands are excited through modulation-induced interaction. Similar to the non-dispersive case, waves excited at the momentum bandgap grow exponentially and appear with the strongest amplitude.

Exponential growth can also be observed from the time-momentum graph in Figure <ref>2b. Initially, the momentum values corresponding to the supported modes of the system at the source frequency are excited. As time progresses, the modes with the correct momentum range to couple into the bandgaps grow exponentially, eventually surpassing fields at any other wavenumber.

Excitation via dipole sources displays the range of modes. However, experimental excitation is usually confined to propagating or evanescent waves inside the medium, which are bound within the supported momentum range. To investigate excitation with a wave we employ FDTD simulations with a probe beam propagating within a modulated medium. The details of the calculation are provided in the SI.

Before modulation, the probe beam can have momentum and frequency residing only on one of the dispersion branches of the non-modulated medium. Once the modulation starts, the beam’s momentum remains constant while its frequency couples to the Floquet modes. When the probe beam momentum is within the bandgap, the beam couples into the momentum gap mode and starts growing. For a nondispersive medium,
the bandgap always occurs at k=Ω/(2n_ avgc) <cit.>. In such a medium, for a wave to have this momentum, its frequency should always be Ω/2 Hence, exciting the bandgap requires very fast and strong modulation of the optical properties, making it challenging to possibly observe photonic time crystals in the optical regime. However, the situation is dramatically different for dispersive materials and these strict conditions can be relaxed as there are multiple features, including new types of band gaps and avoided crossings in the band diagram that lead to different excitation regimes.

 First, we investigate the same-branch bandgaps. We again use the dispersion parameters of the medium in Figure <ref>.c. In this case the bandgap occurs at 2.5 k=Ω/c. Additionally, contrary to the non-dispersive case there are two frequencies where propagation with this momentum is allowed for the non-modulated medium, on the upper and lower branches. 
We excite this medium from the upper branch with ω= 3Ω as highlighted in Figure <ref>.a. 

Once the modulation starts some of the energy remains at the original branch and continues to propagate without a change in the amplitude. This is a result of the original branch being almost identically present in the dispersion of the modulated medium and the bandgap does not occur on this branch. 
As shown in figure <ref>.b some portion of the energy of the incident beam couples into the bandgap created by the lower branches occurring at Ω/2 and at every integer multiple of Ω/2. The light in this branch shows the same characteristics as the bandgap mode in a nondispersive medium. As the modulation starts, the beam coupled to the bandgap is equally separated into forward and backward propagating modes and grows exponentially. The bandgap occurs on a flat band, therefore, the group velocity of the beam is zero and the wave remains localized in space until modulation ends. It is observed that the new modes are excited in both quadrants equally, which means the forward and backward propagating beams have equal amplitudes. The video of the temporal evolution of the beam for this excitation is given in Supplementary Video 1. Higher order frequencies are also visible at 3Ω/2 as a result of harmonic generation with the modulation. Although this case demonstrates transferring energy between polaritons of the medium, the bandgap mode still occurs at half the frequency of the modulation.

This is not the scenario for the hybrid bandgap. We investigate the band diagram shown in Figure <ref>.a. The hybrid bandgap occurs around k=7Ω/c where the upper and lower branches cross each other. To couple into the bandgap mode, we excite from the upper branch at ω = 4.23Ω and k =6.7Ω/c as shown in Figure <ref>.a. After modulation is introduced, the bandgap occurs on the branch where the beam was already propagating before the modulation, therefore all the energy directly couples to the bandgap mode resulting in amplification. Notably, the bandgap in this case is not a flat band as in the same-branch case but is instead slanted. Therefore, the resulting amplified beam propagates with non-zero group velocity. This is a distinct difference between the dispersionless cases investigated previously. Figure <ref>.b shows the ω-k relationship of the beam inside the medium during the modulation where one can observe the growth in the field amplitude and the excitation of the lower branches Ω away in frequency from the probe beam. Supplementary Video 2 shows the temporal evolution of this excitation.



One notable observation of this structure is the requirement of modulation speed is more than 8 times lower than that of a non-dispersive medium. The interaction between the polaritonic branches allows for PTC bandgap-like amplification to happen at significantly slower modulation. In the example shown a modulation at 13 THz gives rise to directional amplification of a beam at 55 THz. This feature is a clear path forward for observing PTC bandgaps in the infrared (IR) regime and possibly in the optics regime with appropriate selection of material and modulation properties.



§ CONCLUSION


In this work, we have investigated the effects of periodic temporal modulation on dispersive mediums. The dispersion lines of the material start translating and mixing via modulation. The resulting band diagrams exhibit bandgaps that emanate from the interaction of polaritonic branches. Due to dispersive materials being able to support several modes at a given momentum, there are more possibilities for interactions. This allows for the occurrence of new bandgaps and new channels for excitation. 

One significant feature observed in the polaritonic PTCs is the hybrid bandgap that occurs through the interaction of a lower and an upper dispersion branch. The region where the two branches have the closest energy and exhibit avoided crossing is where they are most strongly coupled. When driven by their energy difference in this regime, it is possible to create a strong interaction between polaritons and the modulation, that allows for energy to be transferred from modulation to the probe beam. This is achieved with much lower modulation frequencies than what is necessary for non-dispersive PTCs. Allowing for experimental observations in higher energy regimes of the electromagnetic spectrum.

When modulated with a frequency proportional to the energy separation of the LO and TO phonons, phononic materials can be used for polaritonic amplification. Although creating strong optically induced changes in these materials are challenging, it has been shown that it is possible to drive nonlinear response in phononic materials through Born effective charge and dielectric screening of the electric field by electrons. This nonlinearity has been used to observe parametric amplification of optical phonons.<cit.> 

Apart from employing the phononic material dispersion one other possible realization of polaritonic amplification is through constructing the desired dispersion lines through structured materials. Bulk and surface plasmons of thin films hybridize near the epsilon near zero region of the thin films to give rise to Berreman and ENZ modes, which can be coupled through relatively low frequency modulation to give rise to amplified polaritons.
The interaction of polaritonic modes in dispersive photonic time crystals (PTCs) has the potential for amplification across a broad electromagnetic spectrum. This enhancement can impact the scope and applications of polaritonic lasing, resonant Raman scattering, and polariton nonlinearities.
 

Authors acknowledge support by the U.S. Department of Energy under Award DE-SC0017717 (Control of Light), and the Air Force Office of Scientific
Research under Award FA9550-21-1-0299






 Paper Title: Advanced pure tilt actuator for testing tilt-to-length coupling in space-based gravitational wave detection.







MOE Key Laboratory of Fundamental Physical Quantities Measurement & Hubei Key Laboratory of Gravitation and Quantum Physics, PGMF and School of Physics, Huazhong University of Science and Technology, Wuhan 430074, China


*yanhao2022@hust.edu.cn 



 
Tilt-to-length (TTL) coupling, caused by the jitter of test masses or satellites, is a significant noise source in space-based gravitational wave detection. Calibrating and suppressing TTL coupling noise at the sub-nanometer level is essential. One main challenge in current ground-based TTL coupling testing is the residual translational movement of the tilt actuator. This paper introduces the development of an advanced pure tilt actuator (APTA) specifically designed for testing TTL coupling. The APTA provides precise tilt motion and is monitored by a four-beam interferometer, which measures the displacement of attached array pyramids. We present a detailed theoretical model and experimental setup. Experimental results demonstrate that this optical test bed, equipped with the APTA, can achieve subnanometer-level TTL coupling calibration. In addition, a typical heterodyne interferometer was tested using the APTA test bed. Comparative testing demonstrated that the imaging system is capable of effectively suppressing TTL coupling errors. The TTL coupling coefficients were reduced from over ± 30  μm/rad to within ± 5  μm/rad across a range of ± 200  μrad, meeting the preliminary requirements for the TianQin mission. This APTA test platform has the potential to be widely utilized for ground-based TTL coupling inspection.

©2024 Optica Publishing Group under the terms of the https://www.osapublishing.org/submit/research-articles/authors/open-access-publishing-agreementOptica Publishing Group Open Access Publishing Agreement






§ INTRODUCTION

In space gravitational wave detection missions such as TianQin<cit.>, the Laser Interferometer Space Antenna (LISA) <cit.>, and Taiji <cit.>, the demand for displacement detection sensitivity is extremely high. Measurement optical path changes caused by tilt jitter, known as TTL coupling, have become a significant potential noise source, severely affecting the accuracy of displacement measurements along sensitive axes. TTL coupling occurs in both test mass interferometers and long-arm interferometers. The TianQin mission, a space-based gravitational wave detection project in geocentric orbit, constructs a large-scale laser interferometer with an arm length of approximately 1.7 × 10^5 km . It aims to measure inter-satellite displacement changes with a sensitivity limit of 1 pm/Hz^1/2 at 6 mHz [1]. This requirement imposes stringent demands on the suppression of various noise sources, particularly TTL coupling noise. To achieve this goal, the TTL coupling must be minimized to less than 10 μ m/rad for tilts of 10 nrad/Hz^1/2 at 6mHz.

Reaching this goal directly is almost impossible, but we can achieve it through several steps. First, we should minimize misalignment error as much as possible during design and installation. However, it is not feasible to perfectly align all components in the system using this method alone, so we must meet the required benchmarks as closely as possible at this phase. Next, we need to calibrate the misalignment. The beam alignment mechanism (BAM), composed of a pair of parallel glass windows, can adjust the transversal position of the beam <cit.>. The fast steering mirror can correct the beam's angular direction. These two devices are integrated into the system for compensatory adjustments to minimize alignment errors after installation. Finally, misalignments can occur during launch, transfer to orbit, and over time. The remaining TTL noise needs to be subtracted in ground-based post-processing by measuring angular jitter <cit.>, ensuring continuous compliance with gravitational wave detection requirements. By following the three steps, the TTL coupling can be suppressed to the level below our required threshold. However, before achieving this, a clear understanding of the TTL coupling formation mechanism is essential, along with the design and construction of ground-based experimental testing setups.

TTL coupling can arise from various mechanisms, broadly categorized into geometric and non-geometric types <cit.>. In the test mass (TM) interferometer, geometric TTL coupling occurs when the tilt of the TM increases the optical path length from the origin of the measuring light to the detector surface, a phenomenon known as the light lever effect. Misalignment between the TM's center of mass and the reflection point of the measurement laser can also result in a piston effect. Additionally, changes in the angle of the incident measurement light on a plane transmission mirror can alter the optical path along the sensitive axis. Non-geometric TTL coupling involves factors such as the sizes and positions of the beam waists of the two interfering beams, which can affect the readout of the interference signal. Wavefront errors, including higher-order modes and wavefront distortions, modify the wavefront curvature of the interfering beams, potentially impacting the signal readout and inducing TTL coupling errors.  To date, research in this area has been limited. Beyond the interference light itself, the characteristics and geometry of the optoelectronic detector—a critical component in detecting interference signals—directly influence signal readout. Factors such as the detector’s area, slit dimensions, and electrical non-uniformity still need to be accurately characterized.

Ground-based experimental testing on TTL coupling is crucial for the development of space laser interferometers. However, a challenge in experimental testing setups is how to generate pure rotation without introducing parasitic displacement along the sensitive axis, as such displacement would affect the ultimate precise calibration of TTL coupling. Various methods have been proposed in numerous studies, such as using the characteristic of identical optical parameters of two interference beams in a zero-difference interferometer, monitoring and deducting the parasitic displacement of the actuator with a large single element photo diode (SEPD), successfully calibrating the TTL coupling of the device, and verifying that a double lens imaging system can suppress TTL coupling <cit.>. To realistically simulate the LISA optical platform, the LISA team constructed an optical testing platform using a silicate bonding technique, successfully deducing the parasitic displacement of the actuator with technologies such as phase locking between multiple beams <cit.>. This device can simultaneously simulate TM interferometers and interstellar interferometers <cit.>. However, this device has a complex optical path structure, and since most optical components are glued on the substrate, it is not easily modifiable or adjustable to test different settings, such as different distances or beam characteristics. To make the TTL coupling calibration device more flexible, the LISA team developed the advanced tilt actuator (ATA), taking advantage of the characteristic that a conical prism is insensitive to changes in the angle of incident light, effectively suppressing the parasitic displacement produced by the actuator <cit.>. The ATA was designed and built as an independent experimental platform; hence it can be used not only for a specific TTL coupling experiment but also for other experiments requiring three-degree-of-freedom (DOF) motion of the ATA.

Based on our research on experimental testing setups for TTL coupling, we noticed that the pyramid produces a lateral displacement in the incoming light during rotation, thus affecting the final readout of the interference signal and thus influencing the calibration of the TTL coupling. To address this issue, we designed a self-aligning four-beam interferometer <cit.>. The feature of this device is that the conical prism does not cause lateral displacement of the incoming light when tilted. The device is constructed using a UV adhesive <cit.>, providing compactness and significantly improved displacement measurement sensitivity. This makes it potentially applicable for high-precision TTL coupling experiments in the future.

The paper is organized as follows. Section 2 provides an overview of the mechanisms of TTL coupling formation, which helps us to gain a clearer understanding of TTL coupling. Section 3 details the working principles, optical design, and error analysis of the ATPA, and how this device is applied to ground-based experimental calibration of TTL coupling. Section 4 presents the ground-based experimental testing setup for TTL coupling, demonstrates the calibration of system TTL coupling, and shows the suppression of TTL coupling using a confocal imaging system. Section 5 contains the discussion.



§ TTL COUPLING AND IMAGING SYSTEM



 §.§ Mechanism of TTL coupling

TTL coupling in laser interferometers is a complex effect. Geometric TTL coupling originates from the situation whereby one of the beams, usually the one carrying the main measurement signal, tilts and deviates from the initial interferometer light path, leading to a propagation distance greater than that of the completely aligned reference beam. The optical path difference between beams results in a phase difference, for example, the power fluctuation of the photodiode in a zero-difference interferometer, or the phase shift of the heterodyne signal in a heterodyne interferometer. 


Taking the TM interferometer as an example, when the TM tilts, geometric TTL coupling errors can occur due to the presence of a lever arm between the TM and the detector, a phenomenon known as the light lever effect. As shown in Fig. 1, assuming the laser is a plane wave, the expression for this effect can be calculated through simple geometric relationships as:

    OPD_lever ≈d_lever /2α^2.

Furthermore, an offset exists between the reflection point of the measuring light on the TM surface and the TM rotation point (center of mass), leading to an extra optical path, known as the piston effect. Its expression can be written as:

    OPD_piston ≈- d_lat α+ d_long α^2.

where d_lever is the lever arm length, d_lat is the lateral offset of the reflection point, d_long is the longitudinal offset, and α is the beam tilt angle. All the above results are retained to the second order of the tilt angle α.

However, in a real heterodyne interferometer, lasers have their own wavefront curvatures and power distributions. For instance, the TM interferometer uses Gaussian light while the interstellar interferometer uses both Gaussian light and flat-top light. If the two interference lasers are not perfectly matched and aligned on the detector surface (for example, due to laser axis offset, wavefront tilt, and distortion), they would interfere with the interference light field distribution on the detector surface and subsequently impact the final sensitive-axis optical path signal readout. This type of effect is known as a non-geometric TTL coupling. The sensitive axis optical path signal LPS readout can be written as the phase of the interference light field at the active area of the detector, integrated over that part,

    LPS=1/k(∬ E_m E_r^* d S_PD).

Where E_m represents the electric field of the measurement light, and E_r refers to the electric field of the reference light, any factors affecting this integral will cause TTL coupling, such as the previously mentioned mismatch of light parameters, properties of the photodetector <cit.>, and even different definitions of the sensitive axis optical path signal can result in various TTL couplings, as mentioned in literature <cit.>. In interstellar interferometers, the interfering beams are Gaussian light and flattop light, and the sensitive axis optical path signal needs to replace the electric field expression in the integral equation. The TTL coupling generation mechanism is roughly the same in both kinds of interferometers. However, due to the intercept and magnification of the far-end light by the interstellar telescope, the beam tilt angle is amplified, thereby affecting the level of TTL coupling.





 §.§ Imaging system

Imaging systems are an effective method for suppressing the TTL coupling <cit.>, as shown in Fig. 2. A strict optical design is employed to place the TM reflection surface in the entrance pupil of the imaging system and the quadrant photodiode (QPD) in the exit pupil. The Fermat principle is utilized to suppress the change in the geometric optical path of the measurement beam during tilting, as well as to restrain the offset of the light spot on the surface of the photodetector. However, imaging systems can alter the optical parameters of the interfering beams and the tilt angle of the measurement beam, potentially increasing non-geometric TTL coupling. Consequently, imaging systems can suppress geometric effects while leaving residual non-geometric effects. To reduce the system's TTL coupling below the required level, modifying the design of the imaging system is the only method to deliberately introduce optical path changes to offset non-geometric TTL coupling.

In this paper, we design a set of classic confocal imaging systems using existing lab lenses. Through fine-tuning the installation positions of the imaging system and QPD, we attempt to combine and test their suppression effects on TTL coupling within the system. The imaging system is installed on a three-axis displacement stage, allowing for precise multiaxis adjustments of the lens position. The detector installation is based on thermally stable single-blade aluminum flexors with ultrafine precision screws, achieving micrometer-level positional accuracy and milliradian angular accuracy.



§ APTA: ADVANCED PURE TILT ACTUATOR



 §.§ Measurement principle


In TTL coupling calibration, it is necessary to subtract the residual longitudinal displacement of the tilting actuator at the measurement light reflection point. This requires monitoring the three degrees of freedom (longitudinal displacement, yaw and pitch) at the reflection point to effectively cancel out the actuator's residual noise. As I understand it, two non-collinear lines define a plane, and similarly, three non-collinear points determine a plane. During plane movement, we can infer the three degrees of freedom at any point on the plane from the displacement information at these three points. By leveraging this property, we can monitor and subtract the residual longitudinal displacement at the measurement light reflection point while also reading the tilt angle of the measurement light. To achieve these objectives, two critical conditions must be met. Firstly, the precise measurement of the longitudinal displacement information at three points on the plane; Secendly, Accurate positioning of these three points.


Once these conditions are satisfied, we can theoretically decouple the three degrees of freedom at the measurement light reflection point. As shown in Figure 4, assume three points on the same plane in space have coordinates P_1(x_1, y_1, z_1), P_2(x_2, y_2, z_2)  and P_3(x_3, y_3, z_4), where the Z-axis is the sensitive direction. Given that the plane's rotation angle is very small (of the order of a hundred microradians), we assume the coordinates remain unchanged, treating x_1,2,3 and y_1,2,3 as constants during the tilt, while the longitudinal displacements z_1,2,3 change over time. The normal vector of the plane containing the mirror can be expressed through the cross-product of vectors between the three points:

    n⃗=P_1 P_2×P_2 P_3=(n_x, n_y, n_z).

Substituting the coordinate information, the elements of the normal vector can be expressed as:

    n_x=z_1 ·(y_3-y_2)+z_2 ·(y_1-y_3)+z_3 ·(y_2-y_1), 
        n_y=z_1 ·(x_2-x_3)+z_2 ·(x_3-x_1)+z_3 ·(x_1-x_2), 
        n_z=x_1 ·(y_2-y_3)+z_2 ·(y_3-y_1)+z_3 ·(y_1-y_2).

where n_z is a constant. Any point P(x,y,z) on the plane containing P_1, P_2 and P_3 can be expressed as:

    z=-n_x ·(x-x_1)+n_y ·(y-y_1)/n_z+z_1.


As can be seen from the equation above, knowing the longitudinal displacement and position information of three points on the plane enables accurate measurement of the longitudinal displacement z at any other point on the plane. By utilizing this property, the longitudinal displacement of the measurement light reflection point can be measured and the actuator's residual noise can be subtracted, thus calibrating the TTL coupling error in the interferometer more accurately.



 §.§ Optical design

The APTA (Advanced Pure Tilt Actuator) consists of two parts: a compact four-beam interferometer and a conical rotary actuator, as shown in Figure 5. The compact four-beam interferometer is composed of a collimator, a lateral displacement prism, a beam-splitting prism, and a punctured plane mirror, with all optical elements adhered to an ultra-low expansion microcrystalline glass substrate using UV glue <cit.>, ensuring a structure with robust thermal stability.


The optical path design is shown in Figure 6. The laser emitted from the collimator is split into two layers, upper and lower, with four beams in total, by the lateral displacement prism. The two layers of beams are completely parallel. Each beam is then split again via the beam-splitting prism, with three of the beams exiting after being transmitted through the punctured reflecting mirror. The position of the hole in the mirror is precisely measured to ensure that there is no truncation of the light spot. The remaining transmitted laser beam reflects off the mirror and returns along the original optical path, interfering with another beam reflected from the beam-splitting prism. Since the reflecting mirror does not move, this interference signal serves as a reference to remove noise from the upstream fiber link and environmental noise. The conical rotary actuator consists of a piezoelectric ceramic mirror flipper and three hollow conicals. Piezoelectric ceramic acts as an actuator by providing three degrees of freedom (pitch rotation, yaw rotation, and longitudinal translation) controlled by voltage. The hollow conicals serve as target mirrors, reflecting the three transmitted beams back to the punctured reflecting mirror. Thanks to the property of conical prisms, which keeps the incident and outgoing beams parallel, adjusting the attitude of the perforated reflector will ensure each beam reflected by the cone exits the same way it entered. Then these interfere with their corresponding reference light, forming three independent measurement signals.


APTA's unique structure provides several advantages. First, when the target cone rotates, the contrast of the interference signals hardly changes, maintaining a high level. This is because the measurement beam always returns along the same path, thus ensuring consistently well-aligned interfering light <cit.>. Second, the optical path within the hollow cones is a constant value, solely dependent on the height of the cones and not the angle of incidence. This makes the cone insensitive to the angle of incidence - TTL coupling can thus be considered insensitive to an interferometer using a cone as the target mirror. Third, its versatile use. As an independently designed and built experimental platform, the overall structure of APTA is compact and easy to transport. Therefore, it can be used not only for specific TTL coupling experiments but also for other experiments requiring multi-degree of freedom motion. Fourth, APTA has three independent measurement interferometers that can detect the three degrees of freedom motion information of the target mirror. Moreover, multi-beam measurement has a larger linear range and lower non-linearity [20], allowing for future comparison and calibration of the DWS signal in the test platform.



 §.§ Error analysis

APTA will be used for TTL coupling calibration, so it is necessary to analyze its own errors to avoid introducing them into the calibration results. The first concern is the dihedral angle error <cit.>. Due to the machining process, it is impossible to ensure that the three faces of the corner cube prism are perfectly perpendicular to each other, which affects the property of the constant optical path within the corner cube, thereby introducing calibration errors into the APTA. In this context, we need to account for the dihedral angle errors in the internal optical path calculations of the corner cube. Let's consider the dihedral angle errors as δ_12 and δ_13, where δ_12 is the dihedral angle error between the first and second reflective surfaces, and δ_13 is the dihedral angle error between the first and third reflective surfaces.


At this point, we only consider the change in the geometric optical path of the measurement light within the corner cube. Assuming the measurement light is a ray, we first assume that the vertex of the corner cube is positioned at the origin "(0,0,0)^T"  of the coordinate system. The installation position of the corner cube is as shown in Figure 7(a). The normal vector of the exit plane is n_0=(0,0,1)^T, and this plane is 1 meter away from the origin. The exit point of the light ray is P_0(x_0, 0,0), with the propagation direction l⃗=(0,0,-1)^T. Using ray-tracing methods for analysis and calculation, we can derive the analytical solution for the direction of the exit ray after reflection by the corner cube with errors (accurate to the second order of the dihedral angle error),

    l^'≈([ 0; 0; 1 ])+1/2([   -2 δ_12-2 √(2)δ_13-4 δ_12δ_13+√(2)δ_12^2; -2 √(2)δ_12+2 √(2)δ_12δ_13-δ_12^2+2 δ_13^2;          -3 δ_12^2-2 √(2)δ_12δ_13-2 δ_13^2 ])+([ -δ_12+√(2)δ_13; -δ_12+√(2)δ_13;              0 ])([ η; ϕ; 0 ]).

Here, η and ϕ represent the pitch and yaw angles of the corner cube prism, respectively. The first term in the equation can be understood as the direction of the reflected ray after passing through an ideal corner cube reflector, which is completely parallel and opposite to the direction of the incident ray. The second term represents the deviation in the direction of the exit ray when the corner cube reflector with dihedral angle errors is in a static state. The third term represents the coupling deviation in the direction of the exit ray due to the rotation of the corner cube reflector with dihedral angle errors. During the rotation of the corner cube reflector, the introduced geometric optical path error is:

    OPD_error≈   ( x_0 δ_12 + √(2) x_0 δ_13 + √(2)δ_12δ_13 + ( 32 + x_0√(2)) δ_12^2 + δ_13^2 )_static term
        + [ x_0 δ_12 - √(2) x_0 δ_13] η_first-order term+ {( 1 - x_0/√(2)) δ_12^2 - 2 δ_13^2 }η_first-order term
       + [ ( √(2) - x_0/2) δ_12^2 - ( 2 + √(2) x_0 ) δ_12δ_13 + x_0 δ_13^2 ] ϕ_first-order term
        + [ -x_0/2δ_12 - x_0/√(2)δ_13] η^2_second-order term+ [ -√(2) x_0 δ_12] ηϕ_second-order term

The first term in the equation (static term) represents the additional optical path introduced when the corner cube reflector is static, accounting for the exit point position and its dihedral angle error. The second and third terms represent the linear coupling of the pitch angle η. However, the third term involves second-order polynomials of the dihedral angle error, indicating that if the dihedral angle deviation is small, its contribution to the optical path length change is negligible. The fourth term denotes the change in optical path length corresponding to the second-order polynomial of the pitch angle η. Similarly to the third term, its impact is negligible for small rotation angles and dihedral angle errors. The fifth term shows the linear coupling of the yaw angle ϕ. Given that the second-order polynomial of the dihedral angle deviation is very small, the influence of the fifth term on the optical path length change is not significant. Similarly, the contribution from the sixth term may be negligible.


In practice, the dihedral angle error can be controlled in 3 arcseconds (1.45 μrad), the deflection of the corner cube reflector is approximately -300 ∼300 μrad, and the accuracy of displacement measurement is within the picometer range, so the influence of the third-order term can be ignored. To verify the correctness of the analytical solution, we numerically solved the ray-tracing process using Matlab, and compared the results of the two methods. Assuming the exit point is s⃗(0.005  m, 0,0) and the deflection of the corner cube reflector is -300 ∼ 300  μrad, and there is only one dihedral angle error δ_12=0, δ_13=0∼40 μrad, only the coupling term (excluding the static term) is calculated.When the corner cube undergoes pitch deflection η, the additional optical path introduced by the coupling of different dihedral angle errors is shown in Figure 8(a), where the solid line represents the numerical calculation results and the dashed line represents the analytical calculation results. It can be seen that the maximum TTL coupling coefficient is ∼0.28 μm/rad@δ_13=0∼40 μrad, and the analytical results are in good agreement with the numerical results.

Next, we consider the incident angle error of the measurement light on the corner cube prism. Ideally, the three measurement laser beams should be parallel to each other in pairs and perpendicular to the plane of the corner cube; otherwise, the measurement interferometer will not be able to accurately measure the longitudinal displacement of the corner cube vertex. In the experiment, we first use a flat mirror to reflect the three measurement beams back along their original paths, while the other end of the mirror is adjusted using an independent interferometer to reflect its measurement beam back along its original path. During the experiment, the power of the light at the return port of the circulator can be used to determine whether the measurement beam is returning along its original path. When the return power of all measurement beams is maximized, the mirror is replaced with a corner cube combined with a mirror, which forms the corner cube rotation actuator. Then, by adjusting its alignment to restore the return power of all measurement beams, it can be assumed that the three measurement beams in the APTA are aligned: parallel to each other in pairs and perpendicularly incident onto the plane containing the corner cube vertex. Despite efforts to align the beam angles, some residual misalignment may still exist. As shown in Figure 7(b), assuming the incidence angle of the beam is i=δ, the optical path S_pyramid of the laser inside the corner cube is calculated using the ray tracing method as follows,


    S_pyramid =2 n H/√(1-sin i^2/n^2).

where n is the refractive index of the corner cube material. For a hollow corner cube n, and H is the height of the corner cube (distance from the vertex to the incident plane). Since a hollow corner cube is used in this experiment, when the laser is incident non-orthogonally (i ≠ 0^∘) on the surface of the corner cube, the optical path error approximated to the second order is,

    OPD_error ≈ 2 H·(1+i^2/2)-2 H=H i^2.


In this experiment, the height of the corner cube H is 0.01 m. From the above equation, the optical path difference under various incidence angles can be analyzed and compared with the numerical calculation results, as shown in Figure 8(b). The simulation results indicate that the optical path error increases with the incidence angle; therefore, the incidence angle should be kept as small as possible. Based on the current experimental testing level, the incidence angle should be controlled below 100 μrad.

Finally, considering the installation errors related to the position of the corner cube, as derived in Section 3.1, it is necessary to combine the position and displacement information of the corner cube vertex to obtain the displacement information at the measurement light reflection point in the TTL coupling experiment. Therefore, the position of the pymarid vertex is crucial for the final calibration of the TTL coupling. In this experiment, a perforated metal base is used to install the hollow corner cube prism, which is then fixed by metal-set screws. This means that the installation accuracy of the corner cube vertex position depends on the machining accuracy. Ideally, the coordinates of the vertices of three corner cubes in the XY plane are P_1(-7  mm, 7  mm), P_2(7  mm, 7  mm) and P_3(7  mm,-7  mm). At present, the machining accuracy of the metal base is ±0.05 mm. Since the machining errors are random, in order to realistically reflect the actual situation, random errors within the range of ±0.05 mm  are assigned to the coordinates of the vertices of the three corner cubes in the XY plane, as shown in Table 1.





Assuming only pitch rotation around the X-axis occurs, substituting the above errors into formulas (6, 7), the errors in the three degrees of freedom at the origin O are given by:

    Δϕ_error ≈ 5.831 × 10^-3·η
       Δη_error ≈-4.157 × 10^-3·η
       Δ z_error ≈ 1.317 × 10^-3·η

where ϕ is the yaw angle, η is the pitch angle, and Δϕ_error, Δη_error and Δ z_error are the yaw angle error, pitch angle error, and longitudinal displacement error at the origin, respectively. From the above equations, it can be seen that even if the plane of the corner cube only undergoes one degree of freedom motion, errors will still be generated in all three degrees of freedom. This means that the lateral displacement of the vertex position will lead to cross-coupling between each readout signal of the APTA. The longitudinal displacement error Δ z_arb at any point on the vertex plane of the pymarid is expressed as:

    Δ z_arb≈ (-x_arb·Δϕ_error) + (-y_arb·Δη_error) + Δ z_error.

where x_arb and y_arb are the lateral position offsets of any point on the corner cube vertex plane with respect to the initial origin in the X and Y axes. Calculating the longitudinal displacement errors at three different positions, with coordinate information x = 0 mm,  y = 0, 2, 4, 6 mm, the results are shown in Figure 9. It can be seen that the longitudinal displacement increases with the increase in the offset, and exhibits a significant first-order linear relationship, which cannot be ignored.

To suppress the above errors, intentional introduction of errors can be used to offset the known errors. The specific method is to adjust the position of the measurement light-reflection point during the decoupling process to offset the existing linear errors. The feasibility of this method has been preliminarily verified in previous studies <cit.>.



§ EXPERIMENTAL SETUP AND RESULTS



 §.§ Experimental setup

A ground-based experimental setup was established to test the TTL coupling in the quality inspection interferometer, where APTA and the imaging system are used respectively for calibration and suppression of the TTL coupling. As shown in Figure 10, the entire setup consists of two parts: the TM interferometer and APTA. The heterodyne interferometer is used to simulate the quality inspection interferometer, while the APTA is used to simulate the tilt of the TM attitude and to compensate for the errors of longitudinal displacement during motion. The following sections will explain each part in detail.


In this experiment, a low-noise narrow-linewidth NPRO laser (Coherent, Mephisto) with a central wavelength of 1064 nm was used as the light source. A high resolution dual channel waveform generator (Keysight, 33600A series) was utilized to generate two sinusoidal signals with frequencies f_1 = 40 MHz and f_2 = 40.01 MHz, respectively, achieving a heterodyne frequency of 10 kHz for the heterodyne interferometer. The orthogonal interference signals were received by a variable-gain InGaAs detector (Thorlabs, PDA20CS2) and a quadrant photodetector (Kyphotonics). Furthermore, a data acquisition (DAQ) system with low quantization loss (NI-PXIe 4464) was used for phase acquisition and demodulation. The sampling rate was set to 1 MHz, with a phase readout rate of 50 samples/s.

A single-frequency laser is transmitted to the optical workstation through a single-mode polarization-maintaining fiber. In the optical workstation, the laser undergoes frequency shifting via a pair of acousto-optic modulators. The two laser beams, which have a frequency difference of 10 kHz and nearly equal optical path length and power, are used as the measurement beam f_1 and the reference beam f_1, respectively, and are injected into the space interferometry portion. To transmit the two heterodyne beams into the APTA and the heterodyne interferometer, each beam is split using a 50:50 beam splitter, resulting in a total of four laser beams coupled into the fiber collimators.

In the TM interferometer, the measurement beam and the reference beam pass through nonpolarizing beam splitters, respectively, where the two reflected beams interfere. The interference beam is detected by a large-area single-quadrant photodetector through a mirror, forming a reference signal used to subtract fiber noise and environmental noise at the front end of the TM interferometer. The measurement beam, after passing through another nonpolarizing beam splitter and reflecting off the APTA mirror, generates beam tilt and reflects again through the. At this point, the tilted measurement beam and the reference beam are combined in NPBS, forming two separate interference beams. One of these interference beams is detected by a quadrant photodetector without passing through the imaging system, using differential wavefront sensing to obtain the tilt angle between the two beams and measure the TTL coupling magnitude within the TM. The other interference beam, passing through a confocal dual-lens imaging system, is detected by another quadrant photodetector. This setup is used to verify the imaging system's ability to suppress TTL coupling. The parameters for the imaging system and the TM interferometer are listed in Table 2.


In high-precision experiments, interferometers are typically built on ultra-low expansion glass substrates and placed in a vacuum environment to suppress environmental disturbances such as temperature fluctuations and air currents. However, in this experiment, the test platform was constructed on a stainless steel honeycomb breadboard and placed in ambient air. The entire system was enclosed with an acrylic cover for passive temperature control and acoustic isolation. To further suppress noise that is not related to the tilt modulation signal and to more accurately calibrate the TTL coupling in the interferometer, we employed coherent filtering techniques <cit.>. This filtering technique requires applying a modulation signal to the tilt actuator and collecting multiple cycles of the interferometer signal. By performing a fast Fourier transform on the interferometer signal, we can consider any signal frequencies in the frequency domain that do not equal the modulation frequency or its harmonics as noise. By only computing the inverse transform corresponding to the modulation frequency and its harmonics, we retain the displacement signal related to the beam tilt and reconstruct the TTL coupling curve. To avoid frequency leakage, we applied a 0.5 Hz sinusoidal modulation signal to the tilt actuator, with the post-downsampled sampling rate of the phase meter set to 50 Hz, an integer multiple of the modulation frequency. Clock synchronization was performed between the modulation signal source and the phase meter, and the sampling time was approximately 10 minutes.


 §.§ Test results

After completing this setup, we compared the interferometer signals on the reference QPD and the measurement QPD, which respectively represent the original TTL coupling in the TM interferometer and the suppressed TTL coupling after the imaging system. By comparing these two signals, we can assess the effectiveness of the imaging system in suppressing TTL coupling. In the experiment, the longitudinal displacement signal LPS was calculated using the arithmetic mean phase of the four quadrants of the QPD,

    LPS^AP = ϕ_A + ϕ_B + ϕ_C + ϕ_D/4k.

where ϕ_A represents the phase measured by quadrant A of the QPD, and similarly for the other quadrants. k = 2π/λ is the wavenumber, with the laser wavelength being 1064 nm. The beam tilt angle information was obtained through the DWS signal.

The calibration results of TTL coupling in the TM interferometer before and after suppression by the imaging system are shown in Figure 11. After careful alignment and match of the reflection points <cit.>, the imaging system was aligned and the QPD measurement was adjusted to minimize TTL coupling. Without the imaging system, the longitudinal displacement readout of the system could change up to 3 nm within a tilt angle range of ± 200 μrad. After incorporating the imaging system, the longitudinal displacement readout variation was reduced to below 0.6 nm. It is evident that the TTL coupling curve exhibits a significant quadratic effect with respect to angle, although higher-order terms still remain. To calculate the variation of the angle-to-twist ratio coefficient, we fitted the TTL coupling curve with a fourth-order polynomial and plotted the first derivative against the angle. It can be seen that within a beam tilt range of ± 100 μrad, the TTL coupling coefficient, after suppression by the imaging system, decreased from ± 15 μm/rad to below ± 5 μm/rad.




§ CONCLUSION

TTL coupling is ubiquitous and any misalignment can lead to its occurrence. It is one of the main noise sources in space-based gravitational wave detection missions such as TianQin, LISA, and Taiji. Therefore, studying the formation mechanism of TTL coupling and the methods to suppress it are crucial for the development of ultra-high-precision space laser interferometers. One challenge in the experiments involving TTL coupling is that the longitudinal displacement error of the tilt actuator directly couples into the TTL calibration and interferes with the final results. To address this, we propose a novel tilt brake (APTA) to eliminate the longitudinal displacement error of the actuator, providing the interferometer system under test with a pure tilt drive. The APTA comprises two parts: a compact four-beam interferometer and a retroreflector tilt actuator. The APTA reflects the measurement beam in the interferometer under test and measures the three degrees of freedom information at the reflection point, thus enabling accurate TTL coupling calibration. The APTA features a compact and portable structure, making it applicable for TTL coupling experiments for future interferometers or any experiment requiring three degrees of freedom motion.

Additionally, we employ a classical confocal dual-lens imaging system to suppress TTL coupling. Experimental results show that after aligning the lenses and fine-tuning the detector position, the imaging system significantly suppresses TTL coupling. However, since off-the-shelf lenses were used and the actual layout of the optical platform in a space interferometer was not considered, future optimizations of the imaging system design based on the actual optical platform configuration are necessary.

During TTL coupling calibration, APTA still experiences high-order noise effects. We believe that misalignments during assembly and manufacturing errors in components such as the retroreflector are likely causes. Despite efforts to minimize errors during assembly, residual errors still interfere with the final calibration results. This study analyzes three error mechanisms within the APTA and employs data processing to minimize their impact on calibration. More research is required to investigate the error mechanisms within the APTA and optimize its structure and assembly process.

TTL coupling experiments show that without the imaging system and within a tilt range of ± 200 μrad, the measured TTL coupling reaches up to 3 nm, with a coupling coefficient of ± 15 μm/rad. Using the imaging system, the TTL coupling can be reduced to below 0.6 nm throughout the tilt range, with the coupling coefficient decreasing to within ± 5 μm/rad.

Furthermore, within a small tilt range of ± 100 μrad, the slope can be reduced to below 3 μm/rad, meeting the preliminary requirements of the TianQin mission.


Funding
National Natural Science Foundation of China (12105375).National Key Research and Development Program of China (2022YFC2203901).

Acknowledgments
We thank members of Prof. xxx for useful discussions. The authors acknowledge the experimental facility support from the PGMF (National Precision Gravity Measurement Facility).

Disclosures
The authors declare no conflict of interest.

Data Availability Statement
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request. 

Supplemental document
See Supplement 1 for supporting content. 







 Paper Title: Silicon Double-Disk Optomechanical Resonators from Wafer-Scale Double-Layered Silicon-on-Insulator.



SOISilicon-on-Insulator

WGMWhispering Gallery Mode
EBPGElectron-Beam Pattern Generator
EBLElectron-Beam Lithography
RIEReactive-Ion Etching
IPAisopropyl alcohol
HFhydroflouric acid
VPEVapor Phase Etching
FSRFree Spectral Range
SNRSignal-to-Noise Ratio
PECProximity Effect Correction
SEMScanning Electron Microscope
FEMFinite Element Method
Sisilicon
SiO_2silica
PDphotodector
PSDPower Spectrum Density
FWHMFull Width at Half Maximum
DRIEDeep Reactive Ion Etching
BOXburied oxide
PCPolarization Controller
HWHMHalf Width at Half Maximum





1School of Mathematics and Physics, The University of Queensland, St Lucia, Queensland 4072, Australia.

2ARC Centre of Excellence for Engineered Quantum Systems, St Lucia, Queensland 4072, Australia.

3Centre for Quantum Dynamics, Griffith University, Nathan, Queensland 4222, Australia.

4ARC Centre of Excellence in Quantum Biotechnology, St. Lucia, Queensland 4072, Australia

*w.bowen@uq.edu.au 



 
WGM optomechanical resonators are a promising technology for the simultaneous control and measurement of optical and mechanical degrees of freedom at the nanoscale. They offer potential for use across a wide range of applications such as sensors and quantum transducers. Double-disk WGM resonators, which host strongly interacting mechanical and optical modes co-localized around their circumference, are particularly attractive due to their high optomechanical coupling. Large-scale integrated fabrication of silicon double-disk WGM resonators has not previously been demonstrated. In this work we present a process for the fabrication of double-layer silicon-on-insulator wafers, which we then use to fabricate functional optomechanical double silicon disk resonators with on-chip optical coupling. The integrated devices present an experimentally observed optical quality factors of the order of 10^5 and a single-photon optomechanical coupling of approximately 15 kHz.





§ INTRODUCTION

	
Optomechanical WGM resonators have applications in many fields including precision sensors <cit.>, biosensors <cit.>, memory <cit.>, communications <cit.> and for quantum information devices <cit.>. Such resonators support optical modes confined to their periphery by the refractive index contrast at their outer boundary. They also support mechanical degrees of freedom that strongly overlap spatially with the confined optical field <cit.>, leading to significant optomechanical coupling. WGM optomechanical resonators have been experimentally demonstrated in structures such as microspheres <cit.>, microtoroids <cit.>, (single) microdisks <cit.>, and double-disks <cit.>.

A double-disk WGM resonator is made by fabricating two coaxial disks separated by a narrow air gap. Evanescent coupling between the two disks results in hybridization of the WGMs, forming modes that have a high field intensity in the gap<cit.>. This is illustrated in Fig. <ref>(a). Modulation of the air gap by out-of-plane mechanical motion results in large optomechanical coupling; particularly for the symmetrical out-of-plane oscillation (as shown in Fig. <ref>(b)). This is compounded by the tendency for out-of-plane deformations to be much larger than in-plane deformations (of equal energy), due to bending typically requiring less energy than stretching or compressing a material <cit.>. As such, double-disk systems achieve some of the largest optomechanical coupling rates in WGM optomechanical devices demonstrated to date <cit.>.

The optomechanical coupling can also be increased by improving the confinement of the optical mode and ensuring strong co-localization of the optical and mechanical modes<cit.>.
Both objectives can be achieved by moving to smaller resonators fabricated from materials with higher refractive indices. Double-disk optomechanical resonators have previously been fabricated from materials such as silica (SiO_2, n≈ 1.44) <cit.>,  silicon nitride (Si_3N_4, n≈ 1.98) <cit.>, and lithium niobate (LiNO_3, n≈ 2.21)    <cit.>. There is much interest in developing Si optomechanical devices for many applications <cit.>. Silicon's larger refractive index (n=3.48) permits tighter confinement of optical modes and hence higher coupling rates and further miniaturization of devices. This potential is further bolstered by silicon's compatibility with electronics fabrication processes and operation with light in the telecommunications C-band, leading to opportunities for integration with a range of technologies and systems. In particular, Si optomechanical resonators present ready compatibility with silicon photonic, semiconducting, and superconducting circuits. Further, recent studies have demonstrated control of defect-induced single photon emissions in silicon substrates <cit.> offering the potential for integration with quantum information systems utilizing such photon sources. 

Previous reports have demonstrated chip-scale bonding to produce high-purity Si double-layers with the necessary oxide spacing layer (∼ 100 nm) for production of free-standing Si double-disks <cit.> or oxide-clad ring resonators <cit.>. However, due to the complicated nature of their production, demonstration of large-scale Si double-disk WGM resonator fabrication and experimental investigations of their optomechanical performance remained elusive.

In this work we present a truly wafer-scale approach for double layer SOI substrate fabrication, with subsequent chip-scale fabrication and characterization of free-standing double-silicon-disk WGM optomechanical resonators. We achieve this by bonding two entire 4-inch SOI wafers prior to device fabrication. This allows for both chip- and wafer-scale double-layer SOI photonics, which are compatible with the aforementioned technologies. Further, we present FEM modeling and comparison to experimental results of the fabricated devices. We observe optical and mechanical resonances of the disks under vacuum, and characterize the single-photon optomechanical coupling rate (g_0) through measurements of the optomechanical spring effect.




§ DEVICE DESIGN AND FABRICATION


As described above, double-disk resonators consist of two coaxial Si microdisks separated vertically by a SiO_2 sandwich layer of thickness z_g, acting as a spacer. This SiO_2 layer is selectively etched around the periphery, resulting in an air gap between the disks. When the gap is sufficiently small, evanescent coupling results in the emergence of a hybridized optical mode that exists in both disks and the gap simultaneously; this is depicted in Fig. <ref>(a). As expected, our optical FEM simulation (COMSOL Multiphysics) shows that the fundamental (transverse) WGMs are confined to the outer ∼ 2 m of the disks, which is commensurate with the region of greatest deformation caused by mechanical modes. The fundamental symmetric flapping mode is shown in Fig. <ref>(b).

To characterize the impact of the motion on the optical WGM, we first consider the effective refractive index (n_eff), which can be determined from the weighted integral method<cit.>:

    n_eff = (∫_V E(r⃗) n^2(r⃗)  dV/∫_V E(r⃗)  dV)^1/2.

Here, E and n are the electric field amplitude and material refractive index, respectively, at position r⃗ within the simulation volume V. The effective index of the WGMs with varying disk separation is presented in Fig. <ref>(c). When there is no air-gap (i.e. z_g = 0) the WGM behaves as if contained within a single 400-nm disk, with the field well-confined inside the Si; thus, n_eff approaches n_Si. Conversely, when the gap is very large (e.g. z_g ≥ 300 nm) the mode resembles two independent WGMs with negligible shared field (see Fig. <ref>(c) insets). The effective refractive index is lower than that of the 0-nm gap because a larger fraction of the optical field extends evanescently beyond the Si.

The angular resonance frequencies of an optical WGM (ω_m) with a radius of r are approximately given by ω_m = mc/r n_eff, where m is the mode number associated with the number of wavelengths within the cavity and c is the speed of light in vacuum. The dependence of ω_m on n_eff, which is modulated by mechanical motion, ensures that changes in the disk separation will result in a shift in optical resonance frequency. For small deformations, the dependence can be summarised by the optomechanical coupling  (we suppress the subscript for the mth mode),

    G_OM = δω_m/δ z_g.

A high G_OM value is advantageous for applications such as sensing and transduction <cit.>. Our FEM model is presented in Fig. <ref>(d). Here, it can be seen that the optomechanical coupling diminishes with increasing separation. The maximum coupling rate occurs as z_g approaches zero; however, this regime is experimentally impracticable due to large van der Waal's and Casimir forces that can promote collapse of the disks, effectively destroying the mechanical modes of interest. For these reasons, and to ensure fabrication consistency, a separation of 60 nm was chosen. With a disk separation of 60 nm we predict an optomechanical coupling of G_OM/2π≈150 GHz/nm, which is among the highest reported in optomechanical cavities <cit.>.



A second key figure of merit is the single-photon optomechanical coupling rate, which 
is given by:


    g_0 = x_zpfG_OM,


where x_zpf = √(ħ/2m_effΩ) is the zero point motion of the oscillation with resonance frequency of Ω/2π, ħ is the reduced Planck's constant, and

    m_eff =∫_Vρ(r⃗) |X⃗(r⃗)|^2/| X⃗_max|^2  dV
 
is the effective mass of the mechanical oscillation. Here ρ is the material's density, X⃗ is the displacement, and X⃗_max is the maximum displacement amplitude of the excitation. The single-photon coupling rate is equal to the frequency shift per zero-point displacement of the oscillator, and also determines the optical force per photon <cit.>.

Using the data extracted from both optical and mechanical FEM simulations (see S.1 for the mechanical FEM), a 120 m radius Si double–disk resonator with an air gap of 60 nm and undercut of 3.35 m yields a single photon coupling rate of g_0/2π = 116 kHz. 



 §.§ Further design considerations


The design of our double-disk devices broadly follows the concepts of those presented in <cit.>. However, unlike previous reports—which demonstrated only chip-scale double-layer SOI—we present a wafer-scale fabrication process utilizing two whole wafers of commercially available SOI (see Sec. <ref> & S.2.1). Furthermore, unlike many previous works that have coupled light directly from tapered optical fibers into the WGM cavity <cit.>, our work utilizes coupling via integrated on-chip waveguides, enabling superior flexibility and scalability.

Our wafer-scale double-layer SOI approach allows for rapid production of many devices, allowing for a range of devices of varying radii and waveguide-to-disk coupling distance to be fabricated and explored on the same chip. The radii of our devices range from 15 m to 120 m. A minimum radius of 15 m is used to ensure radiative optical losses in the devices are negligible <cit.>. This range of radii presents free spectral ranges (FSRs) in the range of 1–7 nm, allowing for expedition of experimental probing.

The coupling distance ranges from 100–400 nm. The on-chip waveguides are 500 nm wide, which ensures single mode operation around 1550 nm wavelength. The waveguides' two right-angled bends form an S-shaped configuration, minimizing the potential for stray light (e.g., light reflected off the substrate) to couple into the output. The waveguides are tapered down to 100 nm over the course of 50 m, allowing for mode-matched coupling to and from tapered optical fiber tips in a similar manner to that presented in <cit.>. Thin tethers along the length of the waveguides minimize buckling due to stress in the Si films. These tethers are narrower than the waveguide to suppress their scattering cross-section.





§ FABRICATION







 §.§ Double-layer silicon-on-insulator wafer fabrication


Wafer-scale fabrication of double-disk devices requires double-layer SOI wafers. However, such wafers are not commercially available and so were produced in-house in a similar—though scaled-up—manner to that which is presented in <cit.>. Here SOI wafers (4 inch) consisting of a 215-nm upper layer of crystalline Si, a 3-m buried oxide (SiO_2) layer, and a 750-m Si carrier layer were purchased from WaferPro. The wafers were thermally oxidized in a dry oxidation furnace (HiTech benchtop oxidation furnace) at 900^∘C for two hours, forming an SiO_2 layer at the surface of the wafers. This oxide layer forms one half of the spacer layer in the final double-disk devices, as depicted in Fig. <ref>(b & c).  Thin-film reflectometry thickness analysis (SCI FilmTek 2000m) revealed an oxide thickness of 30 nm and underlying silicon layer thickness reduced to 200 nm. This increase in overall thickness is consistent with prior work <cit.>, and occurs because of silica's lower density than Si. The oxidized SOI wafers were then inspected for cleanliness with darkfield microscopy; pristine surfaces are a requirement for successful wafer bonding. After determining the surfaces were free of contaminants, the wafers were surface activated by exposure to oxygen plasma (200 sccm, 50 W, Oxford Instruments PlasmaPro80). The bonding process was then performed using an EVG520 IS Hot Embosser. The process consists of two pistons being pushed together. Mounted onto the top piston is a piece of quartz and the bottom piston is flat steel. When the process starts the temperature is slowly ramped up to 600^∘C and a force of 3.1 kN is applied. The high temperature reduces the buildup of stress between the wafers, and increases the bonding strength <cit.>. Bond strength was then assessed by a razor blade test, as described by <cit.>.

Following the bonding, one side of the stack is exposed to HF VPE in order to remove any SiO_2 grown on the back of the wafer during the thermal oxidation process (as the underside of the wafers were also exposed in the oxidation furnace).
Afterwards, DRIE (Plasma-Therm Versaline DRIE) is used to remove the 750-m silicon handle layer on one side of the bonded stack. Here a two-step Bosch process is used, consisting of C_4F_8 gas, which forms a passivating layer, followed by an isotropic Si etch with SF_6. This process was repeated and the progress of the etch was monitored periodically by removing the wafer stack and visually inspecting. Photographs of the etch progression throughout the DRIE process can be found in the Supplementary Material (Sec. S.2, Fig. S3), showing non-homogeneous removal of material. As a result, a final liquid HF etch, used to remove the 3-m BOX layer that is now partially exposed, leaves approximately 60% of the wafer area usable for fabrication of devices. Finally, the double-layer SOI substrate was diced into chips of 20 mm× 20 mm for the subsequent fabrication of optomechanical silicon double-disks.



 §.§ Device fabrication


The double-disks are fabricated using standard EBL and RIE. A schematic of the fabrication process is depicted in Fig. S4. 


Electron-resist AR-P 6200.13 (CSAR, AllResist) was spincoated  onto the double-layer SOI chips (1500 RPM), followed by 180^∘C bake producing a ∼580-nm coating (confirmed with thin-film reflectometry). As the total thickness of material that needs to be etched is relatively thick (2×200 nm Si and 60 nm SiO_2), and the selectivity of the RIE is quite low (approaching 1:1) a thicker resist than typical SOI fabrication was found to be desirable to protect the top device layer. To compensate for the thicker resist and heterogeneous stack, electron scattering simulations and resultant PEC were performed with GenISys Tracer and GenISys Beamer software respectively. The chips were then patterned with a Raith EBPG-5150 electron beam pattern generator (100 kV accelerating voltage, beam current 1–100 nA) for a total dosage of approximately 300 C/cm^2, followed by developing  with AR600-546 developer, O-xylene and IPA for 90, 5, and 15 seconds respectively.

After patterning, the chips were then etched with RIE in a three-stage process to remove the Si, SiO_2 and Si device layers sequentially. Here, the Si etches consist of SF_6/CHF_3 20/35 sccm (82 W RF power) for 450 s, and SiO_2 etch of CHF_3/Ar 25/45 sccm (200 W) for 120 s.
Afterwards, any residual resist was removed by soaking the chips in Remover-PG at 60^∘C until the resist was visibly removed upon inspection with an optical microscope. The final step in the fabrication process is release of the design with HF VPE (at 35^∘C).



The devices were inspected with optical microscopy, due to concerns surrounding degradation of their optical Q-factors when subjected to scanning electron microscopy (SEM). Any devices imaged with SEM were not used for subsequent measurements. SEM pictures of a fabricated double-disk device and waveguide are included in Fig. <ref>(b–d). The pictures are taken at an oblique angle of 45^∘. The high contrast between the edges of the waveguides and disks indicate the existence of an air gap.




§ DEVICE CHARACTERIZATION






 §.§ Optical properties of Si double-disk resonators


Optical transmission spectra were utilized to confirm the functionality of the waveguides and access the optical interaction with the double-disk resonators. For this, a fiber-coupled laser (EXFO T100S-HP model CU) was connected to a single sided taper, which was aligned with the waveguide using a 6-axis positioning stage. Another single sided taper was similarly aligned on the other end of the S-bend waveguide and connects to a PD (New focus 1811-FC-AC, Newport Inc.). The PD is connected to an oscilloscope (Tektronix DPO 3034 Digital Phosphor Oscilloscope), a computer (through a National Instruments Data Acquisition (DAQ) board) and a spectrum analyser (Liquid instruments, Moku:Pro), as depicted in Supplementary Fig. S6. The emission wavelength of the laser was swept whilst monitoring the transmitted intensity, as is presented in Fig. <ref>. Here, a high density of sharp dips are observed from the spectrum. These correspond to a FSR of ∼ 2.8 nm.
The FSR  (Δλ) can be approximated as 

    Δλ≈λ^2/2π r n_eff,

which for an r = 45-m,  z_g = 60-nm double-disk  results in an FSR of 2.88 nm, in good agreement with experiment.

The fundamental WGMs of the double-disks are accompanied by higher order resonances, as observed from the additional families of sharp dips with similar FSRs. The broader features of the transmission spectrum are attributed to the polarization-dependent coupling efficiencies between the fiber tapers and on-chip waveguides <cit.>. These can be compensated for by manipulating the PC. However, it is evident from Fig. <ref> (inset) that these polarization effects are negligible over the  line-width of individual resonance dips, so we will neglect them in the following characterization.


 

Characterization of the optomechanical interaction was performed using a slow wavelength sweep with the devices held in vacuum (see section <ref> for details). An exemplar transmission trace is shown in Fig. <ref>. The asymmetrical lineshape is unexpected, as most WGMs exhibit symmetrical Lorentzian profiles <cit.>. This asymmetry is independent of the PC, optical power, and sweep rate, which suggests that it is not caused by thermal broadening <cit.>. Rather, the asymmetry appears to be due to interference between the high-Q optical mode of the resonator and low-Q optical resonances supported by the waveguide, i.e., a Fano resonance. The Fano-type lineshape is given by <cit.>:


    T(ω) = T_0+(T_∞-T_0)(q-ϵ)^2/1+ϵ^2,


where T_0 and T_∞ are the transmissions at resonance and far off-resonance respectively, q is the Fano asymmetry parameter, and ϵ = 2Δ/κ is the detuning Δ = ω_0-ω_L normalised by the cavity (amplitude) loss rate, κ/2. Here, ω_0 is the cavity resonance frequency and ω_L is the laser frequency.
This lineshape was found to be in excellent agreement with the measured data (Fig. <ref>(b)). The extracted energy decay rate of the cavity is κ = 2π× 1.71 GHz, which is in good agreement to the estimate based upon the the FWHM of the transmission dip. This yields an optical Q-factor of (Q = ω_0/κ) ∼ 2×10^5,
which is comparable to reports of other double-disk photonic resonators (which range from 2.5 × 10^4<cit.> to ∼10^6<cit.> for Si and SiN devices respectively). The fitted Fano parameter is q=+0.1744.



 §.§ Optomechanical properties of Si double-disk resonators


Initial investigations of the mechanical resonances of the disks were performed by detuning the laser around the HWHM point (sideband detuning) of one optical mode and monitoring the PSD of the transmitted light with a spectrum analyzer. The mechanical displacement is transduced into intensity fluctuations of the detected signal. The resulting power spectrum shows, as expected, that squeeze-film damping greatly degrades the mechanical Q-factor under ambient pressures, resulting in mechanical Q-factors of less than 10 (see S.3.1, & Fig. S7). This is consistent with prior works <cit.>. Despite this, we also see the existence of a mechanical resonance at 24 MHz, as predicted by our FEM simulation for double-disks with an undercut of ∼3.35m (Fig. S1).

To overcome the effects of squeeze-film damping and study the intrinsic optomechanical properties of the disks, further measurements were performed under vacuum (∼ 10^-6 mbar). The vacuum apparatus is depicted in Fig. <ref>. Here, a free-space laser (New Focus 6300 High-Power Velocity Tunable Diode Laser) is coupled into a single-mode fiber. A 99:1 beamsplitter is used to monitor the input power during the experiment prior to insertion into the chamber, and the output fiber is connected to a  PD (New Focus 1811-FC-AC, Newport Inc.). An oscilloscope (Tektronix, DPO3034) is used for data acquisition.

Single-sided tapered fibers are inserted into the vacuum chamber using feedthroughs. The fiber tips are glued on aluminum supports and mounted on positioning stages (SmarAct Linear Stages). Two windows (top and side view) allow for monitoring of the coupling of the fiber tips to the waveguide. 
Fig. <ref> shows the vacuum chamber experimental set-up with a mounted chip. The chamber is pumped down with a roughing-pump-backed turbomolecular pump.

 

Fig. <ref>(a) depicts the PSD of the transmitted signal when sideband-detuned (with detuning Δ = -0.744 GHz). Here, what was a very broad and nondescript feature in the spectrum at ambient pressure is revealed to be four distinct mechanical modes. Fitting Lorentzian lineshapes to each peak shows that the mechanical Q-factors of the four modes are on the order of ∼10^3, which is comparable to previous reports on silica double-disks in vacuum conditions <cit.>; this is, however, the first measurement of silicon double-disk resonators under vacuum. The complete fit model and parameters are summarised in S.3.2 and Table S.1.



Determining the optomechanical coupling rate is less straightforward, but can be achieved by exploiting the optical spring effect, wherein a detuned laser drive causes a shift of the mechanical resonance frequency <cit.>. The expected shift of angular frequency δΩ_i is given by


    δΩ_i/Ω_i =  [ 1 - C_i(0)/Q_i·2ϵ/(1+ϵ^2)^2]^1/2-1,


where Q_i is the zero-detuning mechanical Q-factor[We use the optomechanically shifted Q-factor in the calculation of C, which introduces an error on the order of 0.5%. This is far less than the uncertainty in the fit, and therefore negligible. See S.3.2 for details.], and κ is the optical linewidth (as discussed in section <ref>). We have introduced the (resonant) optomechanical cooperativity, C_i(0), which is related to the single-photon optomechanical coupling rate by


    g_0 = √(C_i(0) κΓ_i/4n(0)),


where n(Δ) is the number of intracavity photons when the cavity is pumped at detuning Δ. Thus, we can recover g_0 by fitting multiple mechanical spectra recorded at different detunings.

Detuning-dependent power spectra are shown in Fig. <ref>(b). The corresponding optical spring effect fittings (Eq. <ref>) are shown in Fig. <ref>(c), with numerical values quoted in Table <ref>. Two of these modes (1 and 4) achieve cooperativities greater than unity, which is an important threshold for onset of instability in optomechanical amplification <cit.>.

To convert C_i(0) to g_0 we require the resonant intracavity photon number (n(0) ≃ 5 × 10^4), which can be calculated from the approximate waveguide photon number and the coupling strength into the cavity (see S.3.3 & S.3.4 for details). We find that the optomechanical coupling rates lie between g_0/2π = 7.78 kHz and g_0/2π = 15.47 kHz for the modes presented in this study (see Table <ref>). These are consistent with reports for hybridized silica WGM resonators such as those presented in <cit.>. Note that we were unable to observe optical resonances in our low-radius devices (likely due to fabrication imperfections), which were expected to have the largest optomechanical coupling rates. Our devices can therefore be further optimized to improve these g_0 values beyond those obtained in silica. The maximum experimentally determined value of g_0 is ≈ 1/8 of the expected valued based on the FEM (Sec. (<ref>)), which is reasonably congruent. The disparity likely resulting from fabrication inconsistencies or imperfections.
 





§ CONCLUSION

We have demonstrated the first wafer-scale fabrication of double-layer silicon-on-insulator wafers. These custom wafers were then utilized in the fabrication of freestanding Si double-disk WGM optomechanical resonators with associated on-chip silicon waveguides, enabling integration with an enormous variety of existing Si-based technologies. 

Further, we presented the first observation of the optomechanical spring-effect in silicon double-disks. Four mechanical modes were identified that show a strong change in mechanical resonance frequency as a function of detuning. The optomechanical cooperativity for each mode is extracted from fitting to the experimental data. Using this, we determine the single-photon coupling rates in these devices.

Future devices could be improved by increasing the undercut between the disks and reducing the disks' diameters. Increasing the undercut makes the mechanical modes less sensitive to fabrication imperfections, and also tends to improve the mechanical quality factor. Reducing the disk radius is also predicted to lead to higher optomechanical coupling rates.

Further research with these devices could involve the integration of these devices with a wide range of applications including single-photon defect emitters, opto-electronics or microwave resonators which could prove useful for the fields of quantum computing and quantum state transfer. 











Funding
This work was financially supported by the Australian Research Council (ARC) Centres of excellence for Engineered Quantum systems (EQUS, Grant No. CE170100009) and Quantum Biotechnology (QUBIC, Grant No. CE230100021). This project was in part funded under the Next Generation Technologies Fund, which is now administered through the Advanced Strategic Capabilities Accelerator.


Acknowledgments
The authors acknowledge the facilities, and the scientific and technical assistance, of the Australian Microscopy & Microanalysis Research Facility at the Centre for Microscopy and Microanalysis, The University of Queensland. This work was performed in part at the Queensland node of the Australian National Fabrication Facility, a company established under the National Collaborative Research Infrastructure Strategy to provide nano- and microfabrication facilities for Australia's researchers. This work was financially supported by the Australian Research Council (ARC) Centres of excellence for Engineered Quantum systems (EQUS, Grant No. CE170100009) and Quantum Biotechnology (QUBIC, Grant No. CE230100021). This project was in part funded under the Next Generation Technologies Fund, which is now administered through the Advanced Strategic Capabilities Accelerator.

Disclosures
The authors declare no conflicts of interest.

Supplemental document
See Supplemental Material for supporting content, including further FEM modeling, greater detail about the double-layer SOI wafer production process and the device fabrication, results on optomechanics at ambient conditions, and investigations into the cavity photon number and photon coupling rates. 










 Paper Title: Low-depth, compact and error-tolerant photonic matrix-vector multiplication beyond the unitary group.





















Russian Quantum Center, Bolshoy Boulevard 30, building 1, Moscow, 121205, Russia
Faculty of Physics, M. V. Lomonosov Moscow State University, Leninskie Gory 1, Moscow, 119991, Russia

saygin@physics.msu.ru

Sber Quantum Technology Center, Kutuzovski prospect 32, Moscow, 121170 Russia
Faculty of Physics, M. V. Lomonosov Moscow State University, Leninskie Gory 1, Moscow, 119991, Russia

Laboratory of Quantum Engineering of Light, South Ural State University (SUSU), Prospekt Lenina 76, Russia, Chelyabinsk, 454080 Russia 



Sber Quantum Technology Center, Kutuzovski prospect 32, Moscow, 121170 Russia

Russian Quantum Center, Bolshoy Boulevard 30, building 1, Moscow, 121205, Russia
Faculty of Physics, M. V. Lomonosov Moscow State University, Leninskie Gory 1, Moscow, 119991, Russia


§ ABSTRACT 

Large-scale programmable photonic circuits are opening up new possibilities for information processing providing fast and energy-efficient means for matrix-vector multiplication. Here, we introduce a novel architecture of photonic circuits capable of implementing non-unitary transfer matrices, usually required by photonic neural networks, iterative equation solvers or quantum samplers. Our architecture exploits compact low-depth beam-splitter meshes rather than bulky fully connected mixing blocks used in previous designs, making it more compatible with planar integrated photonics technology.  We have shown that photonic circuits designed with our architecture have lower depth than their standard counterparts and are extremely tolerant to hardware errors. 






                              
Low-depth, compact and error-tolerant photonic matrix-vector multiplication beyond the unitary group
    Stanislav S. Straupe
    Received ...; accepted ...
====================================================================================================










§ INTRODUCTION



Programmable photonic circuits are indispensable ingredients of classical <cit.> and quantum information processing devices <cit.>. For these devices to achieve the ultimate efficiencies promised by photonics, namely, accelerated speed and energy-efficiency of computation, it is necessary to use programmable interferometers capable of implementing large-scale and high-quality linear transformations. However, fabrication of programmable interferometers with the required scale and quality is challenging due to hardware errors <cit.>. Therefore, developing interferometer architectures that are less susceptible to errors, while  optimally exploiting the resources of a specific photonic platform is very important and timely <cit.>.



The fundamental mathematical operation of matrix-vector multiplication (MVM) can be efficiently performed by programmable photonics. In particular, this operation is essential for data transformation, weight adjustments, and learning processes in photonic neural networks <cit.>. In recent years, the development of novel architectures has been primarily focused on programmable interferometers that implement multiplications by unitary transfer matrices. In particular, the planar meshes of Mach-Zehnder interferometers (MZIs) proposed by Reck <cit.> and Clements <cit.> can implement arbitrary unitary transfer matrices. These architectures are well suited for fabrication by the mature integrated photonics technology enabling massive production of sophisticated optical circuits <cit.>. Sadly, errors in the constituent directional couplers introduce errors in the implemented multimode transformations <cit.>. The effect of errors can be alleviated by adding extra elements into optical circuits that correct the errors, however, this makes the circuits larger and harder to program <cit.>. Several error-tolerant architectures for programmable unitaries that do not require extra phase-shift elements have been proposed <cit.> and demonstrated in experiments <cit.>.

A class of linear transformations broader than unitaries is usually required for photonic information processing, in particular, in classical photonic neural networks <cit.>, iterative equation solvers <cit.> or quantum graph problem solvers <cit.>. Fortunately,  one can use programmable unitary circuits to implement these generic linear transformations. Common approaches embed the transformation in a larger unitary leading to an overhead in the circuit depth that is typically two times larger than the number of modes in the desired transformation. It turns out that this depth scaling is suboptimal <cit.>.

Recently, Tang et al. <cit.> have shown that arbitrary transfer matrices can be implemented by programmable circuits with twice lower depths than in the aforementioned approaches. To achieve that, the authors considered a low-depth version of the architecture <cit.>, known to be universal and exhibiting high tolerance to hardware errors. This architecture relies on static fully-connected mixing blocks requiring interaction between all participating modes at once, which is costly to accomplish, especially by planar integrated photonics technologies.

Here we propose an architecture of programmable interferometers that are simultaneously low-depth and error-tolerant. The architecture relies on partially mixing static blocks conveniently realized by a minor modification of the beam-splitter (BS) mesh, which we have studied previously as means to implement programmable unitaries <cit.>. The proposed programmable circuits are more compatible with planar manufacturing technologies, since they do not require multimode mixing contrary to <cit.>. As a result, the overall footprint, that includes both static and programmable elements, is more compact than that of the known counterparts.

The paper is organized as follows. In Sec. <ref> we review the methods of constructing programmable multimode optical circuits relevant for understanding our results that follow. In Sec. <ref> we introduce our architecture of programmable circuits and study its performance. We conclude in Sec. <ref>.








§ PROGRAMMABLE MULTIMODE OPTICS


The  transformation of a linear N-mode optical device can be described by its complex-valued N×N transfer matrix U linking the input, 𝐚=(a_1,…,a_N)^T, and output, 𝐛=(b_1,…,b_N)^T  amplitude vectors:

    
    𝐛=U𝐚.


Multimode interferometers performing linear transformations have the form of sequence of layers, each consisting of tunable phase-shifts and static multimode blocks, as illustrated in Fig. <ref>a. The transfer matrices of such interferometers can be written as

    
    U=Φ^(L+1)(φ^(L+1))∏_l=1^LV^(l)Φ^(l)(φ^(l)),


where Φ^(l)(φ)=diag(e^iφ_1,…,e^iφ_N-1,1) is the diagonal matrix of a tunable  layer with index l (l=1,L+1) with φ_j being tunable phase-shifts, V^(l) – the transfer matrix of a static layer with index l. In the following, we consider interferometers with programmable transfer matrices, which is achieved by properly setting the tunable phase-shifts.



    


The transformation (<ref>) with N-1 phase-shifts per layer corresponds to the circuit with densest possible packing of these elements. In this work, we are interested in low-depth architectures of programmable circuits and, thus, from the very beginning consider interferometers of this kind. However, it should be noted that not all programmable circuits can be reduced to the form. For example, the unitary architectures <cit.>  can be compactified to form (<ref>) using the method proposed in <cit.>, whereas the  architecture proposed in <cit.> cannot. 




 §.§ Unitary transformations





In real programmable photonic devices tunable phase-shifts are relatively large and lossy. This hinders the scaling of the programmable circuits and motivates the development of low-depth architectures. Therefore, an important parameter that should be kept in mind at designing a programmable circuit is its depth D, which we define as the number of phase-shift layers, i.e. D=L+1. For universal unitary architectures the minimal depth of an N-mode circuit is D=N+1, which corresponds to the total of N^2-1 phase-shifts — the number of real parameters necessary to describe an arbitrary N×N unitary matrix up to a global phase.

In (<ref>), the choice of particular static transfer matrices V^(l) specified by a chosen architecture, has a strong effect on the circuit functionality. For example, according to our previous work <cit.>, error-tolerance of programmable unitaries is achieved by choosing static multimode mixing linear optics for which V^(l) are dense matrices. However, the experimental realizations of such transfer matrices requires organization of interference between all participating modes in each layer. As a result, the area required to allocate the circuit  scales unfavorably with the size of the target transformations N, thereby limiting the size of achievable circuits. To illustrate this issue, Fig. <ref>b shows an example of a multimode mixing block implemented by coupled waveguide arrays  <cit.>. The transformation performed by the coupled waveguides is governed by the coupling taking place between neighbouring waveguides, which necessitates increasing the interaction length l_int as N grows in order to achieve a fully mixing matrix V^(l). Moreover, the bends required to couple the waveguides should have longer length l_bend with increasing N (see Fig. <ref>b). Another way to implement fully mixing matrices V^(l) by integrated photonics is using the multimode interferometers (MMI) <cit.>, shown in Fig. <ref>c. Similarly to the coupled waveguides arrays, the MMIs' length l_mmi scales with the size N of the programmable circuits. More sophisticated methods of designing mixing blocks in planar integrated photonics do not aid in precluding the scaling of the static blocks either <cit.>.

As an alternative, compact circuits can be realized by another programmable unitary architecture that uses meshes of beam-splitters (BSs), which has been proposed in our previous work <cit.>. The corresponding static block used in this architecture is depicted in Fig. <ref>d. The block consists of two BS layers, so that the static block transfer matrices read:

    
    V^(l)_BS=∏_k∈Ω_2T_BS,k^(l,2)∏_j∈Ω_1T_BS,j^(l,1),  (l=1,L),


where 


    T^(l,m)_BS,j=(
    		[          1          ⋯          ⋯          ⋯          ⋯          0;          ⋮          ⋱                                           ⋮;          ⋮             ρ_j^(l,m) iτ_j^(l,m)                     ⋮;          ⋮            iτ_j^(l,m)  ρ_j^(l,m)                     ⋮;          ⋮                                           ⋱          ⋮;          0                     ⋯          ⋯                     1 ])


designate the transfer matrix of a single BS acting on modes j and j+1 in the sub-layer with index m (m=1,2). Ω_1 and Ω_2 denote the ordered sequence of BSs in the first and second sub-layer, respectively. The BS matrix (<ref>) has all diagonal elements equal to one except those labeled by reflectivity ρ_j^(l,m), and all off-diagonal elements equal to zero except those labeled by transmissivity τ_j^(l,m) (τ_j^(l,m)2+ρ_j^(l,m)2=1). Obviously, matrices (<ref>) are radically different from the ones of the multimode mixing blocks, e.g. implemented by coupled waveguide arrays, in that they provide much lesser connectivity.

At the same time the BS-based architecture with static blocks depicted Fig. <ref>c is advantageous compared to the architecture based on multimode mixing blocks, such as those shown in Fig. <ref>b. This is due to the fact that the length of the static block l_BS is independent on the scale of the multimode transformation N. In addition,  the BS-based architecture is better suited for fabrication by intergated photonics technology, wherein it enables the creation of deeper programmable circuits as demonstrated recently in <cit.>. Despite having a lower degree of connectivity and experimental complexity than the architectures based on fully mixing static blocks, the BS-based architecture still offers ultimate error-tolerance <cit.>. 




 §.§ Transformations beyond the unitary group


Numerous applications require programmable interferometer architectures that can implement arbitrary transfer matrices beyond the unitary group. Hereinafter, we will designate such non-unitary matrices by A, in order to distinguish them.

It is known that programmable unitaries can be utilized to implement non-unitary matrices. The architectures capable of performing this task can be divided into two categories as illustrated in Fig. <ref>. Firstly, using a singular value decomposition (SVD) method, an N×N matrix A can be represented as a product of three N×N matrices:

    
    A=WΣU,


each of which can be implemented by optics, as shown in Fig. <ref>a. Here, U and W are N×N unitary matrices that can be implemented by any programmable unitary interferometer. The realization of the non-unitary diagonal matrix Σ=diag(σ_1,…,σ_N), comprised of singular values σ_j, requires individual modulation of field amplitudes. In Fig. <ref>a this is attained by N parallel Mach-Zehnder modulators (MZIs) programmed by phase-shifts located in their arms. It should be noted that such amplitude modulation is limited to singular values  |σ_j|≤1. However, it does not diminish the applicability of the SVD-based interferometers, since in photonic information processing devices, in particular, in photonic neural networks, scaled target transfer matrices A (e.g., divided by the maximum singular value to meet the requirements) have the same utility as the original ones. The circuit depth of the SVD-based interferometers scales as:

    
    D=2N+3.




    


The second approach is based on considering a target N×N non-unitary matrix A as being a submatrix a larger M×M unitary one (M>N). This embedding becomes obvious when we use the fact that any A with the singular values obeying |σ_j|≤1 can be always represented as a part of a 2N×2N unitary matrix U of the form:

    
    U=([             A √(𝕀_N×N-AA^†); √(𝕀_N×N-A^†A)          -A^† ]),


where 𝕀_N×N is an N×N identity matrix, e.g. <cit.>. In this approach, a straightforward method to implement arbitrary non-unitary matrices A (with singular values |σ_j|≤1) is using a universal  programmable 2N×2N interferometer, as shown in Fig. <ref>b. The circuit depth of the optical scheme scales as D=2N+2.

The circuit depth can be made substantially shallower than (<ref>). As has been demonstrated in the recent paper by Tang et al. <cit.>, arbitrary non-unitary N×N matrices can be embedded into 2N×2N programmable unitary circuits with depth scaling as D=N+2 — a two-fold improvement over the previously known results for complex-valued matrices. The architecture by Tang et al. <cit.> uses the unitary architecture depicted in Fig. <ref>a with L=N+1 multimode mixing blocks. In this work, we propose an even more compact and practical architecture. 



    





§ LOW-DEPTH ERROR-TOLERANT ARCHITECTURE


The proposed architecture is illustrated in Fig. <ref>. Here, the constituent static blocks V^(l) take the form of BS meshes acting on 2N modes, which are more compact than the previously used multimode mixing blocks <cit.>. This blocks resemble the planar meshes used in the unitary programmable architecture <cit.> shown in Fig. <ref>d. However,  there is an important modification making the circuit non-planar. Namely, there is an additional BS coupling two distant modes, thereby the whole mesh has the circular topology shown in Fig. <ref>b, with the transfer matrices having the following form: 

    
    V^(l)_circ=∏_k∈Ω_circT_BS,k^(l,2)∏_j∈Ω_1T_BS,j^(l,1)=T^(l,2)_BS,NV^(l)_BS,(l=1,L),


where Ω_circ is a sequence of BSs acting on pairs of neighbouring modes in a circular manner (see Fig. <ref>b). The sequence Ω_circ is simply the sequence Ω_2 from (<ref>) with an added BS described by the following transfer matrix:


    T^(l,2)_BS,N=(
    		[  ρ_N^(l,2)          0          ⋯          0 iτ_N^(l,2);          0          1                     0          0;          ⋮                     ⋱                     ⋮;          0          0                     1          0; iτ_N^(l,2)          0          ⋯          0  ρ_N^(l,2) ]).



As will be shown below, the seemingly minor modification of the planar BS mesh has a strong effect on the capability of the programmable circuits, especially on their tolerance to hardware errors. We notice that the modification of the planar mesh does not present an issue from the hardware perspective, since the waveguide crossings, necessary to implement the circular mesh on a planar chip, is the standard element accessible in mature integrated photonic technologies <cit.>. Moreover, modern  facilities offer fabrication of multilayer photonic integrated circuits, making the proposed architecture even easier to implement <cit.>. 

To study the capabilities of the architecture in implementing non-unitary transfer matrices, we consider N×N submatrices A of the transfer matrix of the 2N-mode interferometer corresponding to the field amplitudes, a_n and b_n, occupying the inputs and outputs alternately, as illustrated in Fig. <ref>a. To quantify the fidelity of the programmable circuits, we use the normalized square error (NSE):

    
    NSE(A^(0),A)=1/N∑_i,j=1^N|A^(0)_ij-A_ij|^2,


which compares the target  matrix A^(0) consisting of elements A_ij^(0) and the actual transfer matrix A consisting of elements A_ij realized by the interferometer, where N is the size of the matrices. Provided that the matrices A^(0) and A are equal, (<ref>) gets its minimal value of NSE=0.

Until now, we did not specify the required parameters of the programmable circuits, namely, the depth D and the values of BS's transmissivities τ_j^(l,m), corresponding to high-fidelity implementation of non-unitary transfer matrices. To find the minimal depth D=L+1 necessary to implement target matrices of given size N, we performed numerical simulations at various values of D. In parallel with this, we analyze the dependence of (<ref>) on τ_j^(l,m)'s that allowed us to investigate the error-tolerance capabilities of the circuit at specified N and D. First, we study the effect of coherent errors where all BSs are identical τ_j^(l,m)=τ, but τ can take various values.




 §.§ Non-unitary transfer matrices



We investigate the capabilities of our architecure to implement randomly generated non-unitary matrices. We used SVD (<ref>) to generate a target matrix A^(0), wherein the unitary complex-valued matrices U and V were drawn from the Haar random distribution using the method based on the QR-decomposition of random matrices from the Ginibre ensemble <cit.>. The diagonal matrix Σ was filled with independently generated values from a uniform distribution in the [0,1] range. 

There is no known analytical solution for the values of phase shifts that minimize the fidelity measure (<ref>), except for some specific architectures <cit.>, thereby necessitating the use of a numerical optimization algorithm (see Appendix <ref> for details). Given a target matrix A^(0), the algorithm was searching towards a global minimum of NSE over the space of phase-shifts φ^(l) (l=1,D). To decrease the chance of falling into local minima, we used multiple runs of optimization (∼100) with random initial values of the
phases. Each numerical experiment involved optimization over a series of target matrices A^(0).



    


Using the optimization method, we studied the effect of different values of τ on attained NSE values. In addition to the circuits with the circular BS mesh, we also conducted simulations for the circuits based on the  traditional planar BS meshes (see Fig. <ref>c).  Fig. <ref> shows the obtained dependencies of NSE on τ for the proposed circular programmable BS meshes (Fig. <ref>a,b) and the planar programmable BS meshes (Fig. <ref>c,d). For the circuits built with the circular BS meshes, two dependencies are plotted  that correspond to 

    
  * a value of depth D_ET at which the circuit exhibits a high-fidelity plateau with NSE≈0, where a particular value of τ does not matter, and
    
  * a value of depth D_ET-1, to demonstrate the threshold-like appearance of the NSE≈0 plateau.

Specifically, for N=5 and N=10 we arrive at D_ET=7 and D_ET=12, respectively. Therefore, this suggests that D_ET=N+2, which is the same as shown in <cit.> for the architecture based on the multimode mixing blocks, implemented as coupled waveguides (Fig. <ref>b) or MMIs (Fig. <ref>c). 

Obviously, the appearance of a wide plateau of τ's with NSE≈0 suggests that the programmable circuits are tolerant to hardware errors that comes in the form of deviations in the constituent static BS elements. Also notice that the NSE≈0 plateau is so wide that the corresponding static errors in BS elements, that should fall well in this plateau, are far beyond what is likely to be encountered in practice. 

We used the obtained depth values D_ET and D_ET-1 to plot the corresponding dependencies for the planar BS meshes in Fig. <ref>c,d. One can see that, in strike contrast to the circular BS meshes, the  planar BS meshes do not exhibit the high-fidelity plateau, necessary for tolerance to hardware errors. 

In addition to the coherent errors, assuming correlated deviations of the BS parameters,  we studied the influence of incoherent BS errors. For this, we generated τ_j^(l,m) independently from the gaussian random distribution for various  distribution widths. The high-fidelity plateau, similar to the case of coherent errors, is observed, thereby proving the tolerance to hardware errors of our architecture.






 §.§ Special cases




    


The analysis of the capabilities of the proposed architecture on randomly sampled non-unitary matrices by no means prove its operation for arbitrary transfer matrices that can be encountered in real uses. Therefore, we have also considered a special case of transfer matrices that fall out of the analysis conducted above. 


In particular, we analyse the implementation of generic unitary transfer matrices, which is also a case very unlikely to be caught by sampling non-unitary matrices of the larger size.
We generated the target matrix A^(0) to be a unitary matrix sampled from

the Haar random distribution using the method based on the QR-decomposition of random matrices from the Ginibre ensemble. As before, we used the fidelity measure (<ref>).  Fig. <ref>a and Fig. <ref>b demonstrate the obtained results for unitary transfer matrices of N=5 and N=10, respectively. As can be seen from the figures, the circular interferometer does not work in error-tolerant way, as it did for non-unitary target matrices. In addition to generic unitary matrices, we also studied the implementation of permutation matrices which are unitary, but present a more narrower class. The obtained results turned out to be similar to the ones illustrated by Fig. <ref>.






 §.§ Relaxed requirements for target matrices





    

To achieve high-fidelity matrix-vector multiplication by programmable photonic circuits in general and our circular architecture, in particular, we suggest the following trick. In many applications, matrix-vector multiplication is allowed to be performed up to a global complex value s. In other words, multiplication by the matrix sA^(0) can be as useful as  multiplication by the specified one A^(0). For example, this is usually the case in opto-electronic devices <cit.>, wherein the vector result of the photonic multiplication is immediately measured and converted to electronic format for further processing. The scaling operation present a simple routine, which can be easily performed by electronics.  

We used the modified fidelity measure that takes into account the freedom of scaling factor:

    NSE_s(A^(0),A)=1/|s|^2NSE(sA^(0),A),

which we optimize in the space of phase-shifts {φ^(l)} and the scaling factor s (see Appendix <ref> for details). It should be noted that, because lower  values of |s| are associated with higher losses introduced to the transformation, higher |s| are usually preferable. Our optimization algorithm used to minimize (<ref>) opts for higher values of |s|.

We challenge our architecture again in the task of implementing unitary transfer matrices. The results obtained in optimization of (<ref>) are shown in Fig. <ref>. Comparing these results with those shown in Fig. <ref>, we see the significant effect of using the relaxed fidelity measure (<ref>) on the quality of implemented unitary matrices. The high-fidelity plateau with NSE_s≈0 is observed, which suggest the tolerance to hardware errors.





§ CONCLUSION


In this work we proposed a novel architecture of programmable circuits capable of implementing transfer matrices beyond the unitary group. Our architecture has the advantage of having a lower depth than the previously known counterparts and possess tolerance to hardware errors. In addition, the programmable circuits designed with the architecture is compatible with planar integrated photonics. All these properties, simultaneously offered by our architecture, facilitate the fabrication of large-scale programmable circuits performing photonic matrix-vector multiplication by mature integrated photonics technologies. Taking into account the typical scale of matrix-vector multiplications  required by modern computing, in particular, deep neural networks, the advantages of our 
circuits are decisive for practical implementation in real-world computing systems. This makes us believe that the results of this work are of interest for both fundamental and applied fields.









§ OPTIMIZATION ALGORITHM FOR NSE


The optimization algorithm used to find the phase-shift values corresponding to a minimum of the NSE (<ref>) works as follows. 

Consider one phase-shift layer with index l, so that the interferometer transfer matrix  A = CΦ^(l)B, where B and C are the matrices describing the parts of the interferometer acting  before and after the phase-shift layer, respectively (see Fig. <ref>a).

Then, the NSE  may be rewritten as

    NNSE(A^(0),A)=
    Tr(A^(0)†-B^†Φ̅^(l)C^†)(A^(0)-CΦ^(l)B)=∑_i,j=1^N|A^(0)_ij|^2+
    
        +Tr(Φ̅^(l)C^†CΦ^(l)BB^†)-2[Tr(Φ̅^(l)C^†A^(0)B^†)].

Next, it is possible to determine the NSE minimum with respect to a single phase-shift ϕ^(l)_k, 0<k<N in layer l, using the condition:

    
    ∂NSE(A^(0),A)/∂ϕ^(l)_k=0,


which yields

    
    [e^-iϕ^(l)_k(∑_j=1
    j ≠ k^N(C^†C)_kje^iϕ^(l)_j(BB^†)_jk
        -(C^†A^(0)B^†)_kk)]=0,


assuming ϕ^(l)_N=0 and is not changeable. From (<ref>) follows the analytical solution corresponding to a minima of the NSE with respect to the phase-shift parameter:

    ϕ^(l)∗_k=((C^†A^(0)B^†)_kk-∑_j=1
    j ≠ k^N(C^†C)_kje^iϕ^(l)_j(BB^†)_jk).



Our algorithm iteratively chooses a programmable phase-shift layer and a phase-shift within it, to which the analytical solution (<ref>) is applied. Namely, one iteration updates phase-shifts ϕ^(l)_k one-by-one using (<ref>) by sequentially selecting  layer index l and phase-shifts index k. We run this process until one of the following conditions is met: 1) NSE<10^-10, 2) the relative difference between successive iterations (NSE_t+1-NSE_t)/NSE_t<10^-8, or 3) the maximum number of iterations 10^6 has been reached. To find optimal phase-shift values corresponding to a global minimum of (<ref>), the optimization procedure was conducted 100 times, each time initialized by random starting values.







§ OPTIMIZATION ALGORITHM FOR NSE_S




The optimization algorithm used to find the phase-shift values and scaling factor s simultaneously corresponding to the highest possible quality and lowest level of introduced losses was as follows.


For a target matrix A^(0) and actual transfer matrix A the scaling factor s minimizing NSE_s can be found analytically. For this,  NSE_s is written explicitly

    NNSE_s(A^(0),A) =∑_i,j=1^N|A^(0)_ij|^2-
    
        -2[e^-iarg(s)Tr[A^†A^(0)]]/|s| + ∑_i,j=1^N|A_ij|^2/|s|^2,

which readily yields the optimal value 

    s = ∑_i,j=1^N|A_ij|^2/Tr[A^†A^(0)].



To optimize the NSE_s, we modify the procedure for NSE by adding updating the s value after each iteration of the algorithm. If NSE_s<10^-10 was reached, then the next optimization procedure starting from a new random point had a lower bound on |s|. This bound was gradually increased as the iterations proceeded. This was done in order to determine the scaling factor with the maximum possible amplitude, which is preferable to minimize losses.









 Paper Title: Giant Second-Order Susceptibility in Monolayer WSe2 via Strain Engineering.












dangylei@cityu.edu.hk



srol@hku.hk












§ ABSTRACT 
  

Monolayer WSe_2 (ML WSe_2) exhibits high second harmonic generation (SHG) efficiency under single 1-photon (1-p) or 2-photon (2-p) resonance conditions due to enhanced second-order susceptibility compared with the off-resonance state <cit.>.   
Here, we propose a novel strain engineering approach to dramatically  boost the in-plane second-order susceptibility (χ_yyy) by tuning the biaxial strain to shift two K-valley excitons (the A-exciton and a high-lying exciton (HX)) into double resonance. 
We first identify the A-exciton and HX from the 2D Mott-Wannier model and  compare the enhanced χ_yyy at single 1-p and 2-p resonance states. 
The strongest enhancement arises from the 2-p resonance state of HX. 
By applying a small uniform biaxial strain (0.16%), we observe an excitonic double resonance state (E_HX = 2E_A, E_HX and E_A are the exciton absorption energies). Strain-tuning can yield up to a 50X enhancement in χ_yyy compared to the single 2-p resonance state of HX, resulting in a further three orders of magnitude enhancement in SHG efficiency. 
Further exploration of  strain states near 0.16% reveals  double resonance also occurs at other wavevectors near the K valley, leading to a further increase in χ_yyy, confirming that strain engineering is an effective methodology for enhancing χ_yyy. 
Our findings suggest new avenues for strain engineering the optical properties of 2D materials  for novel nonlinear optoelectronics applications. 







Giant Second-Order Susceptibility in Monolayer WSe_2 
 via Strain Engineering
    David J. Srolovitz^1,
    
=============================================================================






















Second harmonic generation (SHG) is a process where incident photons interact with a nonlinear, noncentrosymmetric crystal to generate new photons with twice the initial frequency <cit.>. 
SHG finds significant applications in monolayer materials due to their atomic thickness induced phase matching between the incident and second harmonic waves for efficient SHG output, as well as exciton states that can enhance SHG intensity via excitonic resonance <cit.>. 
An important strategy to enhance the second-order susceptibility (χ^(2)) is through one-photon (1-p) or two-photon (2-p) resonance <cit.>; the SHG intensity is proportional to the square of the second-order susceptibility <cit.>. 
Achieving these two single resonance states entails the alignment of virtual states with the eigenenergy levels of the material  during the SHG process. 
In monolayer materials, this can be achieved by matching the energy of one (1-p) or two (2-p) incident photons with the exciton energy. 
Lin et al.  <cit.> reported a narrow and intense SHG peak from a 716 nm continuous-wave (CW) laser resonantly driving the A-exciton in ML WSe_2. 
The observation of a strong SHG signal from a low pump irradiance (CW rather than femtosecond laser) provides strong evidence of the 1-p A-exciton resonance inducing a large second-order susceptibility <cit.>. 
Wang et al.  <cit.> reported a 2-p resonance state in ML WSe_2 by targeting 2-photon excitation energy at the A-exciton, leading to an up to  3 orders of magnitude enhancement in SHG efficiency (compared with the off-resonance state).





To further enhance the second-order susceptibility, we propose to use a combination of 1-p and 2-p processes at a single  wavevector; i.e. "double resonance"  (Fig <ref>b). 


Double resonance is a promising  approach to inducing a giant enhancement in the second-order susceptibility compared to single 1-p or 2-p resonance state <cit.>. 
However, to our knowledge, double resonance has never been observed in real experiments due to its restrictive band configuration criteria: three bands must form two continually nested states (in  reciprocal space); this is an extension of the two-band nesting concept  in first-order linear absorption processes <cit.>.

A recent experiment identified a novel bright exciton with an in-plane dipole and negative effective mass, termed the "high-lying exciton" (HX), at the K valley <cit.>. Remarkably, the energy of the HX is close to twice that of the A-exciton in ML WSe_2, rendering it an ideal candidate for achieving double resonance combined with the A-exciton. 


In this study,  we investigate the potential of the high-lying exciton for boosting  in-plane second-order susceptibility (χ_yyy) through double resonance with A-exciton in ML WSe_2. 
We explore the application of biaxial strain (η) engineering to tune the band energies around the K valley and two exciton states in order to  enhance χ_yyy. 
First, we identify the exciton binding energy and the absorption peaks (E_A, E_HX)  within a 2D Mott-Wannier model and first-principle calculations. 
We then compare the single 1-p and 2-p resonance state enhanced χ_yyy and find that the strongest on-resonance χ_yyy comes from the 2-p resonance state at HX in  unstrained ML WSe_2. 
By applying a biaxial tensile strain  (η = 0.16%) to satisfy the excitonic double resonance condition (E_HX = 2E_A), we observe a ∼50-fold enhancement in χ_yyy compared to the single 2-p resonance at HX, leading to a further three orders of magnitude enhancement in the SHG efficiency 


By exploring different strain states to achieve double resonance at other wavevectors near the K valley (0.16% ≤η ≤ 0.67%), the double resonance effect also enhances χ_yyy compared to the single 1-p or 2-p resonance state. 
Our findings demonstrate the potential of strain engineering in 2D materials for optimizing optical properties. 
Materials with strongly enhanced second-order susceptibility hold considerable potential in different nonlinear optoelectronic applications. 



We obtain the optimized crystal structures and self-consistent wavefunctions using density functional theory (DFT) within the projected augmented wave (PAW) formalism implemented in VASP <cit.>. 
We compare the Perdew-Burke-Ernzerhof (PBE) and Heyd–Scuseria–Ernzerhof (HSE) hybrid functionals for describing electron exchange-correlation, with previously reported self-consistent GW results  <cit.>. 
Our main results are based on the PBE scheme; the PBE and HSE band shapes exhibit negligible difference near the K valley (see Fig. S1), while the choice of PBE effectively reduces the computational demands compared with HSE and GW-based calculations. 
Neither PBE nor HSE band structures compare well with GW results using only a single-step scissor correction at the band gap. 
Therefore, we apply different corrections for different conduction band valleys to ensure that the corrected bands align well with the previously reported GW results <cit.> (scissor correction details and comparisons among exchange-correlation schemes are provided in Sections I and II of the SM).

We determine the second-order susceptibility tensor (χ^(2)) within the independent-particle approximation (IPA) while with adequate correction, which well describes our target excitons (A-exciton, HX) for discussion. 
In addition to the PAW-PBE, we also describe electron and hole wavefunctions using the maximally localized Wannier function (MLWF) basis <cit.>. 
Comparisons and computation details may be found in Section IV of the SM. 
Wannier interpolation is utilized to efficiently compute  energy and momentum matrix elements on an arbitrary k grid in reciprocal space. 
It is important to note that achieving double resonance  leads to numerical difficulties (divergences) in evaluating the second-order susceptibility  using regular Monkhorst-Pack sampling. 
Hence, we employ a  method in which the target Brillouin zone is divided into triangles across which  the energy and momentum matrix elements are linear within each triangle <cit.> to achieve faster convergence and improved accuracy  of our second-order susceptibility results (see Section V, Fig. S5, S6 of the SM).





Our first step is to identify the A-exciton and HX based on the scissor-corrected band structure and the Mott-Wannier model. 
We use the experimentally-defined lattice constant (3.282 Å) in our calculation model for direct comparison with previously reported experimental results. 
Fig. <ref>a illustrates the atomic structure and band structure of ML WSe_2, exhibiting a direct band gap at the K valley of 2.18 eV. We then determine the exciton binding energy using the Mott-Wannier model,   regarding the electron-hole pair as a  "pseudo-hydrogen atom" <cit.>. 
The 3D Mott-Wannier exciton model must be modified to account for  the  non-local and wavevector-dependent screening  in 2D materials. 
Hence, we employ the 2D Mott-Wannier model <cit.> to calculate the binding energy as E^B=(8 μ)/(1+√(1+32 παμ / 3))^2, where μ represents the reduced effective mass of the electron-hole pair forming the exciton (μ^-1=m_e^-1+m_h^-1). 
μ  can be  fit from the band structure as ħ^2(d^2 E(k) / d k^2)^-1, where ħ is the reduced Planck constant and E(k) is the energy band dispersion in  momentum space. 
α denotes the internal polarizability, which is linearly fit to the wavevector-dependent dielectric constant. 

In ML WSe_2, the A-exciton and HX can be depicted as an electron and hole in three bands forming two pairs in Fig. <ref>a (A: VB, CB^1; HX: VB, CB^2). 
Based on the  PBE functional results, the A  and HX  exciton binding energies are 0.48 eV and 0.58 eV in ML WSe_2, respectively, in good agreement with reported BSE results (A: 0.45 eV, HX: 0.6 eV). Therefore, the absorption energy of A (E_A) and HX (E_HX) in the unstrained ML WSe2 is at 1.70 eV and 3.35 eV, respectively, by subtracting each binding energy from the GW gap at the K valley. By calculating the oscillator strength (|<ψ_v|p̂| ψ_c>|^2/(E_c-E_v), v∈{VB}  c∈{CB^1, CB^2}) at the K valley for the two excitons in Fig. S9 (c), we found that the oscillator strength ratio also correspond with previous BSE results (i.e., the intensity ratio HX: A ≈ 1:18), but at much lower computational cost <cit.> (Section III of the SM). In summary, our PBE-based description of two excitons, with appropriate corrections, enables a good representation of the A-exciton and HX for subsequent strain engineering.


























Based on the identification of two excitons, we explore how single 1-p and 2-p resonance from the A-exciton and HX enhances χ_yyy. 
Fig. S7 shows the χ_yyy spectrum when the incident photon energy ħω  ranges from 0.5-4 eV, covering the energies from half the A-exciton energy to the HX. 
The four peaks on the spectrum (from low to high energy) correspond to the 2-p A, 2-p HX, 1-p A, and 1-p HX on-resonance state-induced χ_yyy enhancement. 
Due to the relatively small value of the 1-p HX peak and  its uV emission energy, we focus  on the first three resonance states. 
Fig. <ref>(a)-(c) separately shows three single resonance effects near the resonance incident energy, and we label the highest on-resonance χ_yyy peak as χ^∗ to observe the maximum enhancement fold (we isolated the 1-p and 2-p states in the χ_yyy calculation - see Section V, SM). 

Fig. <ref>(a) shows as an example describing the 2-p A resonance. 
We clearly observe that χ_yyy  is 30-40X that of the off-resonance state susceptibility at ħω = 0.85 eV  (i.e., χ_2pA^∗),   
in agreement with the previous report  3 orders of magnitude enhancement of the SHG efficiency from 2-p A resonance <cit.>. 
Among the three enhancement states, the strongest  comes from 2-p HX resonance; this is likely because the intermediate virtual state energy is nearly aligned with the A-exciton energy, leading to a state that is  closest to double resonance amongst the three. Hence, we use the 2-p HX resonance result to benchmark the double resonance enhancement effect in the next step.





We then proposed to tune the energies of the A-exciton and HX into excitonic double resonance;  i.e., to ensure that  E_HX = 2E_A where E_HX and E_A are the absorbance energies for these two excitons. Without external tuning, E_HX < 2E_A in ML WSe_2. We tune these energy levels via strain engineering (via easy to implement biaxial  strain)  to shift the bands relative to the Fermi level and the excitonic levels relative to each other <cit.>. Fig. <ref>(a) shows how E_A and E_HX respond to strain (η, -1% ≤ η ≤ 1%) in ML WSe_2. They both redshift with increasing η, consistent with  experimental reports <cit.>. To interpret the variation trend, we examine the effect of strain η on the VB, CB^1 and CB^2 band energies (E^o_v, E^o_c1, E^o_c2) and the exciton binding energies  (E^B_A, E^B_HX) at the K valley  (Fig. S9). The relative shift rate  between the CB^1 and VB is almost four times larger than that between CB^2 and VB due to different bonding nature in different bands (Fig. S10). And the A and HX exciton binding energy E^B  decreases with increasing η. This may be attributed to more spatially separated electron-hole pairs, with weaker Coulomb interactions, with increasing η.







Fig. <ref>(a)  illustrates the relationship between E_A(η) and E_HX(η) as well as the excitonic double resonance condition (E_HX = 2E_A).
This condition is satisfied at η = 0.16%, where E_A = 1.67 eV and E_HX = 3.34 eV. 
 By calculating the relative enhancement  χ_yyy/χ^∗_2pHX under the double resonance condition, we see (Fig. <ref>b)  that this strain leads to an up to ∼50X enhancement in χ_yyy compared with the single 2-p HX resonance result in unstrained ML WSe_2. 
This ∼50X enhancement in χ_yyy can produce a 3-order of magnitude increase in the in-plane SHG intensity based on the single 1-p or 2-p resonance result via easy-to-implement biaxial strain engineering, making ML WSe_2 considerably potent for many nonlinear optoelectronic applications. 













The 50-fold enhancement in χ_yyy (based on  single 2-p HX resonance) is an important tuning outcome. 
More enhancement  is possible through tuning based on the excitonic energy over the entire exciton band, as other double resonance states are possible in addition to the K valley. 
Hence we first examine other strain state-induced χ_yyy enhancement - see Fig. <ref>(a). 
The strongest enhancement  appears near ħω = 1.67 eV; i.e., the  K-valley double resonance energy. 
Besides the strongest enhancement at η = 0.16%, other strains lead to an enhancement of χ_yyy relative to the single on resonance χ_yyy (i.e.,  χ_2pHX^∗, χ_1pA^∗, and χ_2pA^∗) in the unstrained monolayer. 
Fig. <ref>(a) demonstrates that  strain tuning is an effective approach for enhancing the second-order susceptibility  compared to  previously reported single 1-p and 2-p resonance methodologies.  

To explain the strain tuning effects outside of the K valley, we explore the exciton energy (from our band diagrams) and the binding energy. 
Fig. S13 shows contours of the energies for the A-exciton  (Δ E_A = E^o_c1-E^o_v-E^B_A) and HX (Δ E_HX= E^o_c2-E^o_v-E^B_HX)  in k_x-k_y near the K valley in ML WSe_2. 
We quantify the A-exciton energy Δ E_A(k, η) by  fitting the 2D E-k diagram to a circle as Δ E_A(k,η)=Δ E_A(K, η)+(ħ^2/2m_c1(η)-ħ^2/2m_v(η))(k_x^2+k_y^2), where m_c1 and m_v are the CB^1 and VB effective masses in the K valley (Fig. S11, S12). 
Δ E_HX exhibits a much smaller variation ratio (with larger anisotropy) than Δ E_A (Fig. <ref>b). Hence, we approximate Δ E_HX(k, η) = Δ E_HX(K, η), where Δ E_HX(K, η) is the HX band energy difference in the K valley.





To achieve double resonance,  Δ E_HX(k, η) = 2Δ E_A(k, η) must have a solution η(k) in √(k_x^2 + k_y^2)≤ 5 × 10^-2 Å^-1 (where both excitonic envelope functions are greater than 0). 
This occurs for 0.16%≤η≤ 0.67% (dashed lines in Fig. <ref>a). At η = 0.16%, only the K-valley wavevector satisfies the double resonance condition, leading to the strongest enhancement in χ_yyy. 
As η increases from 0.16%, the double resonance condition is met at other wavevectors, forming a circle of wavevectors satisfying double resonance. 
At η = 0.67%, the double resonance wavevector circle reaches the edges of the A-exciton range. 
We illustrate this by plotting the energy of |Δ E_HX-2Δ E_A|^-1 and  the Δ E_A contour versus k on the same figure, in Fig. <ref>b (for η = 0.32%). 
We observe a circle of overlapping states around the K valley; |Δ E_HX-2Δ E_A|^-1 is a maximum (red) and overlaps the  Δ E_A = 1.67 eV contour. 
This  proves the existence of double resonance near the K valley, corresponding with the χ_yyy result in Fig. <ref>a and Fig. S8 for the  2X enhancement of  χ_yyy peak intensity  (compared with χ_2pHX^∗  of the unstrained monolayer) for an incident photon energy of 1.67 eV. 












Our first-principle calculations  indicate that  a small strain can produce a very large enhancement of the second-order susceptibility and  the corresponding SHG output, reminiscent of previous experiments. 
We find that different experimental measurements of second-order susceptibility in ML WSe_2 yield very different results (up to two-orders of magnitude) <cit.>.
We speculate that this difference may arise from the strain-induced double resonance effect when ML WSe_2 is placed on a substrate for SHG measurements. 
Rahman et al. <cit.> estimated that a 0.35% strain was presented in a wrinkled monolayer of CuInP_2S_6 (CIPS) and showed that this led to an 160-fold increase in the SHG intensity (which was sensitive to incident photon energy). 
Their experimental observation of a giant SHG enhancement  may indeed be the result of the excitonic double resonance mechanism suggested here.
This provides further impetus for further experiments as a function of  strain (and possibly strain gradients) to engineer the second-order susceptibility.







In conclusion, we explored the potential of exploiting high-lying exciton double resonance (with the A-exciton) to induce a giant enhancement (∼50-fold) in second-order susceptibility (χ_yyy) of ML WSe_2.
We further demonstrated that  biaxial strain is an effective approach to tune the band energies and  excitonic states into double resonance. 
We identify two excitonic absorbance peaks (E_A, E_HX) and their binding energies (E^B) within the 2D Mott-Wannier model. 
Our results exhibit good agreement with those from previous  Bethe-Salpeter equation BSE results (with much lower computational demands) and experiments. 
Comparing the 1-p and 2-p resonance state enhanced χ_yyy in the unstrained ML WSe_2 shows that the strongest enhancement arises from the 2-p resonance state of HX.
Subsequently, we strain-tune the excitonic absorption energy relationship to achieve  double resonance  (E_HX = 2E_A).
A tensile biaxial strain of 0.16% leads to a maximum (50X enhancement) in χ_yyy compared with the on-resonance 2-p HX result of the unstrained monolayer. 
Further exploration of the effect of strain reveals the existence of other double resonance states near the K valley that also enhance χ_yyy.
Overall, our results suggest new possibilities for strain-tuning 2D material properties to greatly enhance optoelectronic properties for novel potential future optoelectronic applications.


Acknowledgment - The authors acknowledge the financial support by the Research Grants Council of Hong Kong (YX, ZP, and DL- GRF Grant No. 15304519, DJS and ZG - GRF Grant No. 11211019), and support from the National Natural Science Foundation of China (HW - Grants No. 12172347 and No. 12232016).  





 Paper Title: Effective Magnetic Switching of THz Signal in Planar Structured Spintronic Emitters.



APS/123-QED





 eugenk@ipmras.ru










^1Institute for Physics of Microstructures RAS, Nizhny Novgorod,  Russia
^2Lobachevsky State University of Nizhny Novgorod, Nizhny Novgorod, Russia
^3 MIREA – Russian Technological University, Moscow, Russia


             

§ ABSTRACT 
We demonstrate an efficient swithcable spintronic THz emission from a Co(2nm)/Pt(2nm) bilayer split into a grating of micro- or nanostripes in applied external magnetic field of the order of 10mT.
The THz signal is effectively emitted if the samples are magnetized perpendicular to the stripes direction and decreases upon a 90-degree rotation of a sample. The maximal 27-fold ratio of amplitude change is achieved for the sample with a 2 μ m width of stripes. For the samples with easy magnetization axis directed perpendicular to stripes the THz signal is linearly polarized in the direction of the stripes and abruptly switches during the sample re-magnetization process. If the easy axis is along the stripes the THz source is almost switched off in a zero magnetic field.










                              
Effective Magnetic Switching of THz Signal in Planar Structured Spintronic Emitters
    Elena D. Mishina^3
    August 5, 2024
===================================================================================





Figure/



§  INTRODUCTION

In the last decade, promising sources of THz radiation based on nanostructures consisting of nano-thick ferromagnetic layers (FM) having boundaries with nano-thick layers of a non-magnetic (usually heavy) metal (NM) have been actively studied and promoted <cit.>. The minimal effective structure consists of two layers: FM/NM. Numerous studies have shown that such structures can act as broadband THz emitters, comparable in efficiency to semiconductor analogues, ZnTe, semiconductor-based antennas (see for review <cit.>). Besides they have the sufficient advantage because THz emission can be manipulated by magnetic field in this case <cit.>.

When such a system is irradiated with a femtosecond high-intensity optical pulse, a bunch of hot electrons is generated. In a magnetic material, these hot electrons are spin-polarized, which results in a short pulse of “pure” spin current flowing from the FM to the NM. The spin current induces an electric (charge) current in the NM via the inverse spin Hall effect. Such a non-stationary electric current emits a short and broadband electromagnetic wave pulse corresponding to the terahertz frequency range <cit.>. Using this technology, THz fields of 1.5 MV/cm have already been achieved <cit.>.The highest efficiency of optical-to-THz conversion was obtained for double <cit.> and triple <cit.> layer structures. 

Structures in which in-plane anisotropy is introduced make it possible to completely control the polarization of the emitted THz wave by changing magnitude of the magnetic field <cit.>. Controlling amplitude by changing magnitude of the magnetic field while polarization remains constant has also been demonstrated in a spin valve type structure consisting of two layers of iron separated by a 4 nm thick layer of platinum with one of the iron layers being attached to antiferromagnetic IrMn layer <cit.>. In those structures, both Fe layers inject spin current into Pt. Changing magnetization from parallel to antiparallel results in antiparallel or parallel charge current from opposite sides of the structure, thus providing minimal (zero) or maximal THz amplitude. The IrMn layer also emits THz waves when a spin current is injected into it from the neighboring iron layer, but an order of magnitude less efficiently than Pt <cit.>. As a result, in <cit.> the ratio of THz signal intensities in the “on” and “off” modes of the device is approximately 15. In <cit.>, modulation of THz intensity by up to 87% is achieved in On-Chip Wave-guided SiNx/Fe/Pt/SiNx structure. THz signal modulation is a primary area of research in THz communication, focusing on key parameters such as materials and structures, modulation depth, modulation speed, and others <cit.>. Electrical modulation, while having shown good results, requires further improvement to achieve deeper and faster modulation. Optical modulation, on the other hand, already demonstrates excellent characteristics in these areas, but there is still room for improvement, making it a promising direction for further scientific research and development <cit.>.

In this study, we propose a structure that provides a novel mechanism for controlling the amplitude of a THz signal from a spintronics emitter using an external magnetic field. We used a standard cobalt-platinum bilayer and etched a grating in it with a sub-wavelength period throughout the entire depth of the bilayer. This array acts as a THz antenna, where a magnetic field applied perpendicularly or parallel to the stripes allows the amplitude of the THz wave to be modulated with a high modulation depth. The efficiency of amplitude modulation has been studied depending on the width and period of the stripes, and a theoretical justification for the observed dependencies has been given.


The paper is organized as follows. In Section <ref> we describe preparation of the samples and the experimental measurement technique. Section <ref> compiles the experimental data for THz signal from all samples. In Section <ref>, the results of measurements are discussed and appropriate theory is verified showing reasonable fitting parameters.



§  METHODS AND SAMPLES




The spintronic terahertz emitters are Co(2)/Pt(2) bilayer films (hereafter, thickness is in nm, order of layers is from substrate) grown on quartz substrates by magnetron sputtering. An external magnetic field of approximately 1 kOe was applied during growth in order to induce a uniaxial anisotropy in the sample plane. Details of growth technology may be found elsewhere <cit.>. The 1cm × 1cm-size gratings with microns to hundreds of microns period (4, 50, 100, 300, 500, 1000 μ m) and 1/2 filling factor are then made from these films by optical lithography (Figure <ref>(a)). We also make a sample with 8 μ m period and 3/4 filling factor (6 μ m stripe width). 
Samples of periodic spintronic emitters with anisotropy both along the easy axis (E.A.) relative to the stripes and across the stripes were manufactured.






Sample planar structure characterization was performed by optical microscopy. Magnetic properties were checked by longitudinal magnetooptical Kerr rotation (MOKE) investigation with the use of He-Ne laser with the wavelength of 630 nm.
THz emission measurements were conducted by using a standard THz time-domain spectroscopy method. A Ti:Sapphire laser with an amplifier system generated pulses with a central wavelength of 800 nm, repetition rate of 3 kHz, and pulse duration of 35 fs. Samples of spintronic THz emitters were placed in a constant external magnetic field of up to 2 kOe, created by an electromagnet.
A magnetic field was applied along the Y-axis of the laboratory coordinate system (Figure <ref>(c)). This study considered two geometries for magnetic field application: magnetizing samples along an easy axis H ∥E.A. and along a hard axis H ∥H.A.. These geometries are achieved by a 90-degree rotation of a sample in the laboratory coordinate system. This is shown schematically in Figure <ref>(b,c) for structures in which the easy axis is aligned along the direction of the stripes. These pictures also illustrate the mechanism of THz generation due to ISHE in both configurations (arrows indicate the directions of the spin current Js injected from the ferromagnetic layer into the layer of non-magnetic Pt and the charge current Jc generated as a result of ISHE).

The generated THz signal was focused onto a nonlinear optical crystal, ZnTe, using a system of parabolic mirrors. To detect the THz signal, a standard electro-optical sampling technique was employed with a setup that included the ZnTe detector, a quarter-wave plate, a Wollaston prism, and a balanced photodetector. The polarization of the pump and probe beams was aligned parallel to the X-axis of the laboratory coordinate system (Figure <ref>(c)) and the ZnTe axis (-110). The laser spot size on the samples was 3 mm at the FWHM level, corresponding to a pump energy density of 0.47 mJ/cm^2. A wire-grid polarizer installed in the THz radiation path, along with the THz polarization-sensitive ZnTe, enabled the analysis of the THz radiation polarization. All measurements presented in this work were performed for the Ex component of the THz field. A more detailed description of the THz detection setup can be found in Ref. <cit.>.



§  EXPERIMENTAL RESULTS

All samples are verified via the MOKE measurements along and perpendicular to the easy axis. Typical MOKE curves are shown in Figure <ref> for the initial film and a grating with 2 μ m wire width.

For micron-sized stripes, the easy axis (E.A.) is along the magnetic field applied during the sample growth regardless of the stripe direction. The magnetization reverses abruptly in 20-40 Oe if the magnetic field is applied along the easy axis for all the samples. For the magnetic field applied along the hard axis (H.A.) the saturation is reached at approximately 100 Oe.

We perform time dependence of THz field amplitude for the setup when the grating is irradiated by an optical pump from front (sample) and back (substrate) side. The main quantitative difference is the magnitude of signal due to different absorption coefficients for optical and THz signal in a quartz substrate. 
Note that the results are qualitatively the same both for front and back side. 
Typical result for irradiation from front side is shown in Figure <ref>(d) for 1000 μ m and 500 μ m grating period with 1/2 filling factor and close geometry. 








Figure <ref> shows the hysteresis loop for the THz signal when an external magnetic field is applied either along or perpendicular to the easy axis for different grating periods. The hysteresis loop is constructed as follows. The THz signal is measured at a delay position of 0 ps, which corresponds to the peak maximum (as shown in Figure <ref>(d)). Then, at this position, a magnetic field is applied, and the corresponding THz amplitude is recorded for each value of the magnetic field. Typically, the THz signal polarized in the direction perpendicular to the magnetic field applied to the sample is recorded. We designate this usual signal polarization as X-polarization. In the left column of Figure <ref>, the hysteresis loops are shown for samples with the E.A. perpendicular to the stripes at grating periods of (a) 1000 μ m, (b) 50 μ m, (c) 8 μ m. In the right column of Figure <ref>, the hysteresis loops are shown for samples with the E.A. parallel to the stripes at grating periods of (d) 300 μ m, (e) 100 μ m, (f) 8 μ m. The insets at the top of Figure  <ref> show the orientation of the easy axis (E.A.) and the hard axis (H.A.) relative to the direction of the grating. The black curves correspond to magnetization along the easy axis (E.A.), while the red curves correspond to magnetization along the hard axis (H.A.).


One can straightforwardly see that the signal amplitude in a saturating magnetic field depends on whether the sample is magnetized along or perpendicular to the stripes while the loop shape depends on the direction of easy axis with respect to grating orientation. The decrease in THz signal amplitude after the samples are magnetized along the hard axis and then the magnetic field is turned off is due to the change of the magnetic state: the samples are split into magnetic domains in which the magnetization turns along local easy axis direction. Non-zero remanent magnetization is due to a non-uniform polycrystalline structure of the sample.

We aslo measure the other linear polarization (which we denote as Y-polarization) of THz signal (see Supplementary Materials) where the polycrystalline structure of the sample is even more pronounced. The main feature of these measurements is that for narrow stripes the Y-polarized signal is very small when the magnetic field is applied perpendicular to the stripes. This matches the fact that these samples poorly emit THz signal when magnetized along the stripes. Thus almost one linear polarization of the THz signal is emitted well from narrow stripes; the amplitude of this signal can be effectively controlled by the direction of applied magnetic field with respect to the stripes orientation.

To conclude this section, Figure <ref> shows the THz signal from the sample with 2 μ m-wide stripes and a period of 4 μ m, with the easy axis aligned along the stripes. In Figure <ref>(a), the time-domain THz signal is shown for different magnetic field orientations: +0.4 kOe and -0.4 kOe along the hard axis (H.A.) and +0.4 kOe and -0.4 kOe along the easy axis (E.A.). For the THz signal measurements along the easy axis (E.A.), the signal was amplified 10 times for better visibility in the plot. The blue line at 0 picoseconds corresponds to the peak maximum, as shown in Figure <ref>(b). In Figure <ref>(b), the THz peak amplitude is plotted as a function of the applied magnetic field. It can be seen that there is a drastic 27-fold difference in the signal magnitude upon a 90-degree rotation of the sample.





§  DISCUSSION

The results for the THz signal ratio versus the stripe width extracted from the experimental data for different stripe width from 150 μ m down to 2 μ for both front and back side measurements are shown in Figure <ref> (in logarithmic scale).

One can see that the ratio grows as the stripe width becomes smaller, but quite slowly. Similar dependencies are observed for front side and back side measurements.


In order to provide a fit of the experimental data we develop the following theoretical model. For the sample magnetized perpendicularly to the stripes, they are supposed to be infinite. Hence the electric current flows with no charge accumulation. We suppose that the power injected into a unit square of the emitter from magnetic system p_i is constant. The electric field emitted per one stripe is then proportional to the electric current through it: I_∥∼√(2 p_i l^2 h/ρ), where ρ is resistivity, l, h are length and thickness of the stripe, respectively. For sample magnetization along the stripes a permanent current would split charges with the velocity of light and then stop. Since the current source provided by ISHE is alternating we ought to calculate the high-frequency conductance. It consists of active resistance determined by film conductivity and reactance mainly determined by electrical capacitance of the system. We may roughly estimate it by supposing that the charge is conducted at very thin wires with the diameter equal to the layer thickness h (Figure <ref>).

We then calculate the energy power accumulated at such an edge in the electric field of all other edges with alternating charge sign and take into account that the electric field of the charge that lies at the metal side is screened by a factor of ξ:

    p_c = (1-ξ)I_⊥^2/2 l ω Log(L/w),

where ω is the frequency of electric current I_⊥, L and w are the characteristic grating width and the stripe width. Taking into account power absorption due to the wire resistance it is then easy to obtain an estimation of the electric current that emits THz signal:

    I_⊥∼√(2 p_i l^2 h/ρ + 1-ξ/ωh/w Log(L/w)).

It is then straightforward to calculate the THz signal ratio as:

    Ratio = I_∥/I_⊥ = √(1 + h/L1-ξ/ρωL/w Log(L/w)).


The results of experimental data fit with the use of (<ref>) are shown in Figure <ref> by solid (dashed) line for front (back) side configuration. Note that we use data for relatively big (> 10 μ m) stripe width for the fit. The characteristic grid coherence length L is determined from the fit and is equal to 264 μ m for front side measurements and 184 μ m for back side ones. These values are of the same order; the difference may be attributed to sensitivity of this parameter to small deviations of experimental values caused by noise. One can note that although the real grid size is of the order of 1cm we obtain L of the order of THz signal wavelength. This is governed by the fact that equation (<ref>) is obtained in the quasistatic approximation which works well for distances less than the wavelength. The other fitting parameter is the stripe effective resistivity ρ. Supposing that the screening parameter ξ << 1 we evaluate it from our fit as ρ = 8 · 10^-8Ω· m for measurements from both sides. Considering the structure as a parallel connection of Pt  and Co conductors with the same cross section and resistivity equal to 10.6 · 10^-8Ω· m and  6.2 · 10^-8Ω· m respectively <cit.> we arrive at a very close resistivity equal to 8.4 · 10^-8Ω· m. Thus our rough model fits the experimental data quite well.

It is worth noting that the model (and the experiment) is almost not sensitive to the distance between stripes unless this distance is of the order of value of the stripe width w. This is explained by a logarithmic dependence of the energy stored in the accumulated charge on this distance which leads to a small constant correction under the root in (<ref>). Indeed, the point put for 6 μ m stripe width in Figure <ref> is measured at the 2 μ m distance between stripes. Although it is not used for theoretical fit it corresponds to that fit quite well. On the other hand, the ratio value equal to 27 obtained for the 2 μ m stripe width is slightly greater than that predicted by Equation <ref> (approximately 15). This may be attributed to the fact that we do not take into account some other effects such as the circuit inductance in our model. This makes the THz signal ratio even greater.





§ CONCLUSION

In summary, we show that the planar grating that consists of stripes made of typical ferromagnet / heavy metal spintronic terahertz emitter may act as a THz source which intensity can be effectively controlled by application of an external magnetic field: the source emits THz signal well if magnetized perpendicular to the stripes direction and much worse if magnetized along the stripes. The ratio of THz signals is approximately 27 (almost two times greater than that for a structure with two ferromagnetic layers divided by a relatively thick 4nm Pt interlayer <cit.>) for the stripe width 2 μ m which is easily achieved by optical lithography. Although the overall THz signal amplitude obtained from our samples is less than from solid films by a filling factor our method may give a greater signal than that suggested in literature <cit.> because of a much smaller total thickness of layers that we use.
Our samples emit the THz signal linearly polarized along the stripes; thus we straightforwardly intergate a linear polarization THz antenna into the spintronic emitters. 



The THz signal from the samples with crystalline anisotropy perpendicular to the stripes direction almost abruptly switches from one THz field polarization direction to an opposite one upon a remagnetization process which gives two controllable THz signal states. On the other hand, for the samples with crystalline anisotropy parallel to the stripe direction the THz field goes from a maximum in a saturating field to a very small value in zero field thus allowing to controllably achieve three states of THz signal. This opens up a possibility to create either two- or three-state controllable sources for possible use in THz logic devices <cit.>.


This work was supported by Center of Excellence “Center of Photonics” funded by The Ministry of Science and Higher Education of the Russian Federation, contract 075-15-2022-316. THz measurement experiments were supported by the RSF (Grant No. 23-19-00849).






*





 Paper Title: Multilevel Fast Multipole Algorithm for Electromagnetic Scattering by Large Metasurfaces using a Static Mode Representation.







Department of Electrical Engineering and Information Technology, Università degli Studi di Napoli Federico II, via Claudio 21,  Napoli, 80125, Italy
 

Department of Electrical Engineering and Information Technology, Università degli Studi di Napoli Federico II, via Claudio 21,  Napoli, 80125, Italy


Università degli Studi di Cassino e del Lazio Meridionale, Via G. Di Biasio n. 43, 03043 Cassino (FR), Italy



Università degli Studi di Cassino e del Lazio Meridionale, Via G. Di Biasio n. 43, 03043 Cassino (FR), Italy
 

[]carlo.forestiere@unina.it
Department of Electrical Engineering and Information Technology, Università degli Studi di Napoli Federico II, via Claudio 21,  Napoli, 80125, Italy


§ ABSTRACT 
Metasurfaces, comprising large arrays of interacting subwavelength scatterers, pose significant challenges for general-purpose computational methods due to their large electric dimensions and multiscale nature. This paper introduces an efficient boundary element method specifically tailored for metasurfaces, leveraging the Poggio-Miller-Chang-Harrington-Wu-Tsai (PMCHWT) formulation. Our method combines the Multilevel Fast Multipole Algorithm (MLFMA) with a representation of surface current density unknowns using static modes, a set of entire domain basis functions dependent only on object shape. The reduction in the number of unknowns by the static mode expansion (SME), combined with the 𝒪(N log N) complexity of MLFMA matrix-vector products, significantly reduces CPU time and memory requirements for solving scattering problems from metasurfaces compared to classical MLFMA with RWG basis functions. The MLFMA-SME method offers substantial benefits for the analysis and optimization of metalenses and metasurfaces.


Multilevel Fast Multipole Algorithm 
 for Electromagnetic Scattering by Large Metasurfaces 
 using a Static Mode Representation
    Carlo Forestiere
    August 5, 2024
===============================================================================================================================






§ INTRODUCTION


Metasurfaces are collections of interacting subwavelength scatterers, typically operating near resonance. By carefully designing the geometry and arrangement of these scatterers, metasurfaces can engineer effective constitutive relations.  They are also versatile in shaping and manipulating electromagnetic waves for various applications, including imaging and analog computing <cit.>. Metalenses are one common class of metasurfaces, in which the unit cells are designed to impart a specific, position-dependent phase shift to transmitted light, resulting in a global focusing effect. Accurate simulation of metasurfaces and metalenses, with efficient use of time and memory, is crucial for understanding their fundamental properties and optimizing their performance. 

Metasurfaces design often relies on the “unit cell” approximation <cit.>. This approach constructs a library of the electromagnetic response of meta-atoms, such as disks, pillars, or nanofins, as a function of a design parameter, like the disk radius, pillar height, or the nanofin orientation, assuming the structure to be locally periodic.  Unfortunately, the “unit-cell” approach has several limitations: the most critical limitation is the assumption that the interaction between adjacent meta-atoms is similar to that in periodic arrays, which necessitates a slowly changing structure. This constraint limits the design space, resulting in narrow-bandwidth operation, low efficiencies and many types of non-ideality.

General-purpose electromagnetic simulation tools, based on either integral or differential formulation, face significant challenges. Differential formulations are easy to implement and result in sparse matrices but require truncation of the computational domain through the introduction of convenient boundary conditions.   Among them, the  Finite-Difference Time-Domain (FDTD) is widely used to model metasurfaces. However, the finite resolution leads to inaccuracies in wave speed, resulting in phase accumulation errors as the size of the scattering region increases <cit.>. Hence, even though the complexity of FDTD simulations nominally increases linearly with the size of the array, maintaining a fixed error requires increasing the resolution. This makes the approach prohibitive when applied using traditional computing platforms. 

Integral formulations are appealing because they define the unknowns only within the objects' volumes or on their boundaries if the objects are spatially piecewise homogeneous, and naturally satisfy radiation conditions at infinity. However, they usually involve dense matrices, which are computationally demanding to invert, even with acceleration techniques like the fast multipole method <cit.>.

A brute-force strategy for scaling full-wave simulations involves adapting general-purpose computational electromagnetic algorithms to massive parallelization for CPU-based <cit.> or GPU-based computing platforms <cit.>. However, it is more efficient to tailor existing methods to the specific characteristics of the problem of electromagnetic scattering from metasurfaces, which generally consist of repeated particle shapes arranged in aperiodic positions and arbitrary orientations, typically subwavelength.

 In the present work, we propose a fast boundary integral method specifically tailored for large arrays of subwavelength scatterers, leveraging the Poggio-Miller-Chang-Harrington-Wu-Tsai (PMCHWT) formulation <cit.>. We combine the Multilevel Fast Multiple Method with the boundary integral equation method employing an expansion of the unknowns of the Poggio-Miller-Chang-Harrington-Wu-Tsai (PMCHWT) equation in terms of a particular class of entire-domain basis functions, defined on the individual meta-atom, and denoted as static modes. As demonstrated in Ref. <cit.>, static modes drastically reduce the total number of unknowns in scattering problems involving arrays of penetrable particles compared to discretization based on sub-domain basis functions, such as RWG.


This work is organized as follows: In Section <ref>, we recall the PMCHWT formulation and its discretization. In Section <ref>, we briefly review the static mode expansion <cit.>, and in Section <ref> we combine the the fast multipole method with this expansion. In Section <ref>, we first validate the proposed method by comparing, for arrays of spheres, the results with those obtained using the multiparticle Mie theory <cit.>. We then assess the computational burden in terms of CPU time and memory requirements by comparing the static mode expansion to a more traditional approach using RWG basis functions. Finally, we present results for three large particle arrays: a golden angle spiral array, a two-layer Moiré superlattice, and a 100 λ× 100 λ canonical metalens.



§ PMCHWT SURFACE INTEGRAL EQUATION

 
	Let us consider a system composed of  particles illuminated by a time-harmonic electromagnetic
	field Re{E⃗_inc e^jω t}. Each particle is a linear, homogeneous, isotropic material that occupies the three-dimensional domain Ω_, with = 1 ,...,. The material has permittivity
	ε^+(ω), permeability μ^+(ω) and it is surrounded by a background
	medium with permittivity ε^-(ω) and permeability
	μ^-(ω). The equivalent electric j⃗_e^
	and magnetic j⃗_m^ surface current densities, defined on ∂Ω_, are
	solutions of the Poggio-Miller-Chang-Harrington-Wu-Tsai (PMCHWT) <cit.> surface integral problem
 
 
 
 
    [ 𝒵^11    ⋯  𝒵^1;    ⋮    ⋱    ⋮; 𝒵^ 1    ⋯   𝒵^ ][ j⃗^1;    ⋮;  j⃗^ ] = [ v⃗^1;    ⋮;  v⃗^ ] ,

	
with

    𝒵^ = [   ζ^- 𝒯_-^ + ζ^+ 𝒯_+^δ_           𝒦_-^ + 𝒦_+^δ_;       -(𝒦_-^ + 𝒦_+^δ_^) 1/ζ^-𝒯_-^ + 1/ζ^+𝒯_+^δ_ ],

where δ_ is the Kronecker delta, and

    j⃗^ = [ j⃗_e^; j⃗_m^ ], v⃗^ = [ e⃗_0^; h⃗_0^ ],

	
    e⃗_0^ = - n_×n_×E⃗_inc |_∂Ω_, h⃗_0^ = - n_×n_×H⃗_inc |_∂Ω_.


The operators 𝒯_±^ and 𝒦_±^ are, respectively, the EFIE and MFIE integral operators, defined as

    𝒯_±^{u⃗} (r⃗) =
       j k^±n_×n_×∫_∂Ω_ g^± (r⃗-r⃗') u⃗(r⃗') ' + 
       1/jk^±n_×n_×∫_∂Ω_∇ g^± (r⃗ - r⃗') ∇'_S ·u⃗(r⃗') ',


    𝒦_±^{u⃗}(r⃗) = n_×n_×∫_∂Ω_u⃗(r⃗') ×∇' g^±(r⃗ - r⃗') ',

where ∇_S' · denotes the surface divergence and g^± is the homogeneous space Green's function of the region Ω_±, i.e.

    g^± (r⃗ - r⃗') = e^-jk^±r⃗ - r⃗'/4πr⃗ - r⃗',
 
with k^± = ω√(μ^±ε^±) and ζ^± = √(μ^± /ε^±).

Apart from a multiplicative factor, the operator 𝒯_±^{u⃗} (r⃗) takes a surface electric current density u⃗ defined on the ν-th particle and outputs the tangential component of the electric field it generates on the μ-th particle. Similarly, the operator 𝒦_±^{u⃗} (r⃗) takes a surface magnetic current density u⃗ defined on the ν-th particle and outputs the tangential component of the electric field it generates on the μ-th particle.



 §.§ Galerkin equations

Aiming at the solution of the PMCWHT eq. (<ref>), we represent the equivalent electric and magnetic surface currents on the particle Ω_
as a combination of a set of basis function ℬ_φ^ = {φ^_n(r⃗)}_n = 1^N, so:

    j⃗_e^(r⃗) = ∑_n = 1^Nα^,e_n φ^_n (r⃗) 
    j⃗_m^(r⃗) = ∑_n = 1^Nα^,m_n φ_n^ (r⃗).

We find the finite-dimensional approximation of the PMCHWT by substituting eq. (<ref>) in eq. (<ref>) and by projecting along the same set of basis functions, according to a Galerkin projection scheme:

    [ Z^11    ⋯  Z^1;    ⋮    ⋱    ⋮; Z^ 1    ⋯   Z^ ][ J⃗^1;    ⋮;  J⃗^ ] = [ V⃗^1;    ⋮;  V⃗^ ] ,

where

    Z^ = [     ζ^- T_-^ + ζ^+ T_+^δ_             K_-^ + K_+^δ_;          -(K_-^ + K_+^δ_) 1/ζ^- T_-^ + 1/ζ^+ T_+^δ_ ],



    [T_±^]_mn = ⟨φ_m^ | 𝒯_±^ | φ_n^⟩,  	[K_±^]_mn = ⟨φ_m^ | 𝒦_±^ | φ_n^⟩,



    J⃗^= [ J⃗_e^; J⃗_m^ ], V⃗^ = [ E⃗_0^; H⃗_0^ ],


    J⃗^_e = [α_1^,e, … ,α_N^,e]^T, J⃗_m^ = [α_1^,m, … ,α_N^,m]^T,



    [E⃗_0^]_n = ⟨φ_n^ | e⃗_0^⟩,  [H⃗_0^]_n = ⟨φ^_n | h⃗_0^⟩,


    ⟨u⃗ | v⃗⟩ = ∫_∂Ωu⃗^*(r⃗) ·v⃗(r⃗) ,

If the particles are identical, each block Z^ is a square matrix of dimension 2N. Therefore, the finite-dimensional system (<ref>) has 2N degrees of freedom, making the resolution of problems with a large number of particles prohibitive when using sub-domain basis function as the RWG <cit.> (where N is the number of edges on the surface mesh of the particle), even when acceleration techniques like fast multipole algorithms are implemented.
In the following, we denoted as φ the sub-domain basis function, such as the Rao-Wilton-Glisson <cit.>.

Let us introduce another set of basis functions ℬ_ψ^ = {ψ^_n(r⃗)}_n=1^N. These functions are related to the basis ℬ_φ^ through the linear transformation

    ψ_i^(r⃗) = ∑_j=1^N m_ijφ^_j(r⃗),

where M = [m_ij] is the transformation matrix. To find the representations of the operators T_±^ and K_±^ in the new basis ℬ_ψ, we can transform them from the basis ℬ_φ representation using

    {A_±^}_ψ = M {A_±^}_φ M^T,

where {A_±^}_φ denotes the representation of the operators A_±^ = T_±^, K_±^ in the basis ℬ_φ.




§ STATIC SURFACE CURRENT MODES



Guided by the fact that the particles constituting the building blocks of a metasurface, i.e., the meta-atoms, are typically smaller than the wavelength, we seek a set of entire-domain basis functions defined on the surface of an individual meta-atom that can efficiently describe the electric and magnetic surface current densities in this regime. Mathematically, we aim for basis functions that diagonalize some of the operators involved in the PMCHWT formulation in the static limit. This set of entire-domain basis functions, defined on the surface of an individual meta-atom and termed static surface current density modes, has recently been introduced and proven effective in representing surface current density fields  <cit.>. These modes consist of two distinct subsets: longitudinal modes, characterized by a zero surface curl, and transverse modes, with zero surface divergence. These subsets are constructed by introducing two separate auxiliary eigenvalue problems. These two eigenvalue problems are associated with two operators, 𝒯_0^⊥ and 𝒯_0^∥, which are obtained from the static-limit of the first and second operators in Eq. (<ref>), respectively.


Let Ω be one of the meta-atom {Ω_}_ = 1^ in the array, whose boundary ∂Ω is “sufficiently regular" <cit.>; n is the normal to ∂Ω pointing outward. The longitudinal static surface current modes are nontrivial solutions of the eigenvalue problem:

    𝒯_0^{j⃗^_k }( r⃗)  =  γ_k^ j⃗^_k     on ∂Ω,

where

    𝒯_0^{w⃗}( r⃗) = n×n×∇∮_∂Ω g_0 ( r⃗  - r⃗') ∇' _S  · w( r⃗' ) ',

∇_S· denotes the surface divergence, and g_0 is the homogeneous space static Green's function

    g_0 ( r⃗ - r⃗' ) = 1/4 π 1/r⃗ - r⃗' .

Eq. (<ref>) can be numerically solved by using as expansion and test functions the star basis functions <cit.>.
Longitudinal static modes constitute a basis for the square-integrable functions defined on ∂Ω and having zero surface curl, and diagonalize the first operator in Eq. (<ref>) in the small-particle limit.


The transverse static surface current modes are nontrivial solutions of the eigenvalue problem:

    𝒯_0^⊥{j⃗^⊥_k }( r⃗)  = γ_k^⊥j⃗^⊥_k     on ∂Ω,

with 

    𝒯_0^⊥{w⃗}( r⃗) =  -n×n×∮_∂Ω g_0 ( r⃗ - r⃗' ) w⃗( r⃗' ) '.
 

This equation can be numerically solved by using as expansion and test function the loop functions <cit.>.
Trasverse static modes constitute a basis for the square integrable functions defined on ∂Ω having zero surface divergence, and diagonalize the second operator in Eq. (<ref>) in the small-particle limit.
Several characteristics make the use of static modes appealing: i) The retarded Green's function, serving as the kernel for integral operators in surface integral formulations, can be decomposed into the static Green's function, which possesses an integrable singularity, and a distinct term that is a regular function. The integral operators that include the static Green's function are diagonalized by the static modes, thus regularizing the overall problem. ii) Employing the expansion of static modes, along with suitable rescaling and rearrangement of the unknowns, renders the surface integral formulation resistant to the low-frequency breakdown issue. iii) Static modes are solely dependent on the object's shape, allowing for their universal application across diverse operating frequencies and material properties of the object. This enables a consistent, frequency-independent representation of any scattering scenario involving objects of a particular shape, surpassing other basis sets, such as characteristic modes <cit.>, which vary with frequency and material properties.

As demonstrated in Ref. <cit.>, the transverse static modes {j⃗_n^⊥}_n=1^N^⊥, associated with the first N^⊥ eigenvalues γ_n^⊥ (sorted in descending order), and the longitudinal static modes {j⃗_m^}_m = 1^N^,
associated with the first N^ eigenvalues γ_m^ (sorted in ascending order), can be used as basis function to represent the equivalent electric and magnetic surface currents on Ω_, namely ℬ_ψ^ = {ψ^_i}_i=1^N^⊥ + N^ =  {j⃗_1^⊥,...,j⃗_N^⊥^⊥,j⃗_1^,...,j⃗_N^^}. The transformation (<ref>) can be used to express the (<ref>) in the static modes basis starting from the RWG representation. For each couple (,) of particles we have

    T̂^ = [ T̂^_⊥⊥  T̂^_⊥;  T̂^_⊥   T̂^_;        ], K̂^ = [ K̂^_⊥⊥  K̂^_⊥;  K̂^_⊥   K̂^_;        ],


    T̂^_α, β = M^α T^ M^βT, K̂^_α, β = M^α K^ M^βT,


    T̂^ = MT^ M^T, K̂^ = M K^ M^T,

where  α, β = ⊥, and

    M = [ M^⊥;  M^ ].




As the static surface modes are a very effective description of the surface current density field on objects whose dimensions are small compared to the wavelength, the dimension of the operators T̂ and K̂ is typically smaller than that of T and K, the transformation (<ref>) effectively acts as a compression. Consequently, the system (<ref>) now has 2(N^⊥ + N^) degrees of freedom, which is typically drastically smaller than 2n_e derived from the RWG basis, where n_e is the number of edges in the single-particle mesh. In the following, we denoted as ψ the basis function in the static modes basis.
 


§ FAST MULTIPOLE METHOD



We apply the Multilevel Fast Multipole Algorithm (MLFMA) <cit.> to solve the PMCHWT system (<ref>) using a basis of entire-domain static modes to represent electric and magnetic surface currents. MLFMA accelerates the matrix-vector product by enabling groups of basis functions to interact collectively rather than through pairwise interactions between every possible pair of basis functions.

Initially, basis functions are grouped.

Following this, the impedance matrix Z is decomposed as the sum of two matrices: the near-field and the far-field matrices. The near-field matrix is associated with interactions among basis functions within adjacent groups, while the far-field matrix is associated with interactions between non-adjacent basis functions. 

Traditionally, grouping basis functions involves a hierarchical tree structure, such as a Quadtree or Octree <cit.>. The bounding box, i.e., the smallest box containing the entire object, is first identified. This box is then recursively subdivided until, at the finest level, the size of the box is a fraction of the wavelength, typically λ/4. 


At the finest level, standard Quadtree/Octree approaches might divide a meta-atom into different groups, making the grouping of entire-domain basis functions ambiguous. Leveraging the fact that we deal with meta-atoms whose dimensions are smaller than the wavelength, we propose grouping basis functions by considering each particle as a finest-level group, as shown in Fig. <ref>(a). At this level, all static modes ψ^_i are physically associated with the center of the particle Ω_.  For coarser levels, we hierarchically group the basis functions using a standard Quadtree structure (Fig. <ref>b), which is particularly suitable for structures where the lateral dimensions significantly exceed the vertical ones, as in the case of metasurfaces and metalenses. 

Following the fast multipole prescriptions, we decompose the impedance matrix Z as the sum of its block-diagonal part (near part) and the off-diagonal part (far part). For identical particles, the self-particle blocks Z^ are identical, i.e. Z^ = Z_s for = 1,...,. Therefore, Z_s can be evaluated and stored once, leading to significant savings in memory requirements and CPU time. To address the singularities in Green's function when r⃗  = r⃗', the numerical integration of shape function times Green's function or its gradient are evaluated using the techniques introduced in Refs. <cit.> and <cit.>.

According to (<ref>), the far-field part of Z is 


    Z^ = [   ζ^- T_-^       K_-^;      -K_-^ 1/ζ^- T_-^ ] ≠,

with , = 1,...,. Since only k^- appears when ≠, in the following we simplify the notations by omitting the "-" symbol, thus k = k^-.

To evaluate the operators T^ and K^ when Ω_ is far from Ω_ (i.e. ≠), we rewrite the equations (<ref>) and (<ref>) as

    T_mn^= -jk ∫_∂Ω_ψ_m^(r⃗) ·∫_∂Ω_G(r⃗ , r⃗')ψ_n^(r⃗'⃗)  '  ,


    K_mn^=  ∫_∂Ω_ψ_m^(r⃗) ·∫_∂Ω_ψ_n^(r⃗'⃗)  ×∇ ' g(r⃗ - r⃗')   '  ,

where G is the dyadic Green's function

    G(r⃗,r⃗')  = ( I - 1/k^2∇∇') g(r⃗- r⃗'),

and I is the unit dyad. Then, using the addition theorem <cit.>, Green's function can be expressed as

    e^-jkr⃗ - r⃗'/4πr⃗-r⃗' = 
    ∫_𝕊^2 e^-jkk·(r⃗ - r⃗_) T_L(k, r⃗_)  e^jkk·(r⃗' - r⃗_)dk,

with r⃗∈Ω_, r⃗' ∈Ω_, r⃗_ = r⃗_ - r⃗_, where r⃗_ and r⃗_ are, respectively, the centers of the particles Ω_ and Ω_, 𝕊^2 = {k∈ℝ^3 : k_2 = 1} is the unit sphere,  T_L is the translation function <cit.> 

    T_L( k, r⃗_) =  
    -jk/(4π)^2∑_ℓ = 0^L (-j)^ℓ (2ℓ +1) h_ℓ^(2) (kr⃗_) P_ℓ (k·r_),

where h_ℓ^(2) is the second kind spherical Hankel function of order ℓ, P_ℓ is the Legendre Polynomial of order ℓ. 

Using the expansion (<ref>) in Eqs. (<ref>) and (<ref>), we obtain


    T^_mn =  -jk∫_𝕊^2ψ̂^_m(k)· T_L(k, r⃗_)(I - kk) ·ψ̂^*_n(k)  dk,



    K^_mn = -jk∫_𝕊^2k×ψ̂^_m(k)· T_L(k, r⃗_)  ψ̂^*_n(k) dk,

where and ψ̂^_m is the k-space representation of the basis function

    ψ̂^_m(k) = ∫_∂Ω_ψ_m^(r⃗) e^-jkk· (r⃗ - r⃗_).


It is well known that truncating the series (<ref>) at a finite order L requires careful handling due to numerical instabilities caused by the nature of spherical Hankel functions <cit.>. Ideally, one would choose a high order to minimize error, but these functions are highly oscillatory at higher order, meaning that increasing L does not necessarily enhance accuracy <cit.>. Furthermore, these functions diverge at small arguments, that is, when the electrical distance between two centers is too small, leading to a low-frequency breakdown <cit.>. However, in this field of application, such instabilities are not observed at optical frequencies, because the particles are not too close together.



 



§ RESULTS AND DISCUSSION




The execution of the numerical algorithm can be subdivided into five stages:

	
  i) Assembly of the self-interaction impedance matrix Z_s of an individual meta-atom and the right-hand side vector V⃗ using RWG basis functions. Since the particles are identical, Z_s only needs to be evaluated and stored once. The computational complexity and memory requirement for assembling and storing Z_s, in terms of the number of particles, is (2n_e)^2𝒪(1).
	
  ii) Generation of the static modes of an isolated object. Specifically, the longitudinal static modes are obtained by solving the eigenvalue problem (<ref>) using star functions as both expansion and testing functions; the transverse static modes are obtained by solving the eigenvalue problem (<ref>) using loop functions as both expansion and testing functions. In Fig. <ref>, we present the calculated first ten longitudinal and ten transverse static modes for a nanodisk.

	
  iii) Evaluation of the MLFMA operators, consisting of the translation function T_L, the radiation and receive operators ψ̂ and ψ̂^†. The time complexity and memory requirements for assembling T_L are on the order of 𝒪(log). Since the operators ψ̂ and ψ̂^† are evaluated for each particle, their time and memory complexities are on the order of 𝒪(). These complexities can be reduced to 𝒪(1) if the particles have the same spatial orientation (i.e., there is no rotation between particles).
	
  iv) Compression stage, i.e. transitioning from the representation of the matrices Z_s, ψ̂ and ψ̂^† in terms of the RWG basis to the representation in terms of the static modes basis. This stage can be integrated into steps i) and iii), as there is no need to directly store the RWG representation of these operators.
	
  v) Solution of the PMCHWT system using an iterative method accelerated by the Multilevel Fast Multipole Algorithm (MLFMA), which reduces the computational complexity of the matrix-vector product from 𝒪(^2) to 𝒪(log). This stage is the most time-consuming, as the entire system is solved with a complexity of 𝒪(I log), where I is the number of iterations required. To accelerate the convergence of the Generalized Minimal Residual Method (GMRES) <cit.>, we employ a block-diagonal preconditioner, where each block is the single-particle self-interaction impedance matrix. We choose a relative tolerance of 10^-4. It effectively reduces I to 𝒪(^α), with α <1.

In the following simulations, the truncation number L of the translation function (<ref>), is initially set to 10 at the finest level. This value is subsequently doubled at each successive level <cit.>. The numerical evaluation of the integrals over the unit sphere is performed using K = 2L^2 point <cit.>.
For interpolation, we employ a 4× 4 Lagrange interpolation scheme.
The FORTRAN code used to implement this algorithm operates on a single CPU (Intel Xeon-Gold 6140 M 2.3 GHz).



 §.§ Validation

We validate the MLFMA-SME method by investigating the scattering from a particle array composed of 100 identical gold nano-spheres, each of radius R = 100 nm, and arranged according to a golden angle spiral pattern <cit.>. The i-th particle is centered at (x_i,y_i), according to 

    x_i = r_i cos(i  α)
    
    y_i = r_i  sin(i  α),

where i  = 1,...,; r_i = c√(i), c is a constant scaling factor, α = 2π/ϕ^2 is the golden angle, where ϕ = (1+√(5))/2 is the golden ratio. We choose the scaling factor as c = √(3)R.

The gold permittivity is obtained by interpolating experimental data <cit.>. The array is illuminated by a linearly polarized plane, propagating along the axis orthogonal to the array's plane, linearly polarized along the x-axis, with varying wavelength λ in the interval [300,700]nm. Each particle's surface is discretized by a triangular mesh having 500 nodes, 996 triangles, and 1494 edges. The scattering cross section, C_sca, is plotted in Fig. <ref> a as function of λ. The solution, computed by the MLFMA-SME, utilizing N^ = N^⊥ = 10 modes per particle, corresponding to a total number of 4000 unknowns, is compared against the multiparticle Mie theory <cit.>, where the field expansions are truncated at the order M = 15. The  relative error on the scattering cross section is defined as 

    ϵ{C_sca} = |C_sca - C̅_sca|/C̅_sca,

where C̅_sca  is the reference solution obtained by the multiparticle Mie theory. 
The achieved error is lower than 0.02 all over the investigated spectral range.

For the same array, in Fig. <ref>b we show the normalized scattered far field for a fixed wavelength λ = 600 nm. The reference Mie solution, obtained for M = 30, is included for comparison.  An excellent agreement is found.


 §.§ Computational costs


We conducted a complexity study on metasurfaces consisting of a varying number up to 5· 10^4 of identical gold nano-spheres, each of radius R = 100 nm. The particles are arranged according to a golden-angle spiral pattern.  This aperiodic pattern is particularly convenient for our study as it is deterministic, and allows us to increase the dimension of the array one particle at a time without appreciably changing the array's density. The array is excited by a linearly polarized plane wave of wavelength λ = 600 nm, propagating along the axis orthogonal to the array's plane. Each sphere has been discretized by a surface mesh, shown in the inset of Fig. <ref> (a), with 100 nodes, 196 triangles, 294 edges, which results in an average edge length approximately equal to λ/15.
In Fig <ref>, we present a comparison between the CPU time and memory requirements using the MLFMA with RWG basis function and static mode expansion with N^ = N^⊥ = 10.  Simulation results show that the use of MLFMA, in both cases RWG and SME, reduces the computational complexity of the matrix-vector product from 𝒪(^2) to 𝒪(log), as expected. The number of degrees of freedom, and the computational time of each case study reported in Fig. <ref>  are listed in Tab. <ref>.
In particular, with the use of the SME, this simulation is approximately 3 ÷ 5 times faster. 



However, the speed-up depends on various factors, such as the number of points used for interpolation/anterpolation between different levels of the quadtree, or the number of points used to sample the transfer function T_L on the unit sphere. Above all, mesh density on each particle plays a very important role. Indeed, the advantages introduced by the MLFMA-SME become much more apparent with denser meshes, necessary for discretizing irregular objects, such as nanofins and nanorods typically used in metalens. The reason why the speed-up is less than the number of unknowns' compression ratio is attributable to the k-space representation of the MLFMA operators, which means that the computational complexity is not only linked to the number of unknowns but also to the electrical size of the array.



Another crucial advantage introduced by SME in the MLFMA concerns memory requirements. Although the translation function T_L is the same in both RWG and SME, as it doesn't depend on the basis function, the matrices representing the radiation and receive operators are κ times smaller, and the self impedance matrix Z_s is κ^2 times smaller, where κ = n_e/(N^∥ + N^⊥) is the total number of unknowns' compression ratio. Additionally, after m iterations, computing the GMRES solution requires storing all the basis vectors of the Krylov subspace 𝒦_m, leading to a memory requirement of 𝒪(mN), where N is the total number of unknowns. As N increases, the storage cost may become prohibitive, even for a small number of iterations m. As previously mentioned, SME can drastically reduce N, allowing for the simulation of larger structures that were impracticable before.


 



 §.§ Electrically large particle arrays



The use of the MLFMA-SME finds its natural application in the numerical solution of the electromagnetic scattering problem by large metasurfaces, where an array, whose dimension can be much larger than the incident wavelength, is made of  objects with identical shape but with different orientation and size. We show three exemplificative case studies of the application of this method to metasurfaces and metalenses: a golden angle spiral of gold bricks, and a Moirè lattice of gold disks and a canonical metalens of TiO_2 nano-fins.

In all three cases, we model the shape of the meta-atoms as a superellipsoid, whose boundary has the implicit equation 

    [ (x/a_x)^2/m + (y/a_y)^2/m]^m/n + (z/a_z)^2/n - 1 = 0,

We used the public domain code developed by Per-Olof Persson and Gilbert Strang <cit.> to generate the surface mesh. 

We first consider a finite-size golden angle spiral array having linear dimension of 115 λ. The array is made of 40 · 10^3 gold bricks of dimension 200 × 200 × 100 nm. We model this shape as a superellipsoid, using Eq. (<ref>) with m = n = 0.25, a_x = a/2, a_y = a_z = a. Each particle is described by a surface mesh having 436 nodes, 868 triangles, and 1302 edges. The i-th particle is rotated with respect to x by an angle θ_i = 2π (i-1)/. The array is excited by a plane wave, linearly polarized in the plane of the array, along the 𝐱̂-axis, and propagating in the direction orthogonal to the array's plane with wavelength λ = 600 nm. The array size is about 70 μm. In Fig. <ref>a, we show the electric field magnitude on the particles' surface. The solution is computed by using the MLFMA-SME as detailed in this work using the static mode expansion with N^ = N^⊥ = 10, i.e. 40 unknowns per particle. The total computational time is about 20.4 hours on a single CPU. 








Next, we now consider an example of 2.5D metastructures, which are stacked layers of interacting metasurface layers. They provide sufficient degrees of freedom to implement efficient multifunctional devices. Specifically, we consider a two-layer Moiré superlattice array of 20160 gold nanodisks of radius R = 100 nm. We model this shape as a superellipsoid, whose boundary has the implicit equation (<ref>), with m = 1, n = 0.25, a_x = a_y = R, a_z = R/2.  Each particle is described by a surface mesh having 588 nodes, 1172 triangles, and 1758 edges. The centers of the particles are arranged at the vertices of two layers, obtained by periodically replicating a hexagon with a side length of 400 nm. The second layer is located at a height h = 300 nm above the first and is rotated by an angle θ = 5° with respect to it. The array is excited by a plane wave, linearly polarized in the plane of the array, along the x-axis, and propagating in the direction orthogonal to the array's plane with wavelength λ = 600 nm. The array size is about 46 × 53  μm, which is roughly (76× 87)λ. In Fig. <ref>b we show the electric field magnitude on the particles' surface, while in Fig. <ref> we show the longitudinal and transverse modes. The solution is computed using the MLFMA-SME with N^ = N^⊥ = 10. The total computational time is less than 6 hours on a single CPU.

Eventually, we consider an example of a canonical metalens <cit.>, composed of 10^4 TiO_2 nanofins of dimension 410× 85× 600 nm. We model this shape as a superellipsoid, whose boundary has the implicit equation (<ref>), with m = n = 0.25. The total computational time is about 15 hours on a single CPU. Each particle is described by a surface mesh having 198 nodes, 392 triangles, and 588 edges.  To achieve the desired phase profile, the i-th particle, centered at (x_i,y_i), is rotated with respect to x
by an angle <cit.>

    θ_i = π/λ(f - √(f^2 + x_i^2 + y_i^2)),
 where f is the focal length. The metalens has a desired numerical aperture (NA) of 0.8, resulting in a focal length f = 37.5λ. The array is excited by a plane wave, circularly polarized in the plane of the array and propagating in the direction orthogonal to the array's plane with wavelength λ = 660 nm. The array size is 66 × 66  μm, which is  (100× 100)λ. In Fig. <ref>c we show the electric field magnitude on the particles' surface. The solution is computed using the MLFMA-SME with N^ = N^⊥ = 10. In Fig. <ref>d we show the normalized intensity E⃗(x,z)^2 on the x-y cross-sectional plane. The achieved focal length is consistent with the prescribed value. 




§ CONCLUSION

In this work, we proposed a fast integral equation solver suitable for large particle arrays, even those hundreds of wavelengths in diameter, comprising numerous arbitrarily shaped objects with varying orientations and sizes. These scenarios are commonly encountered in the electromagnetic modeling of metasurfaces and metalenses. Our approach combines the Multilevel Fast Multipole Algorithm (MLFMA) with the PMCHWT formulation, utilizing a particular class of entire domain basis function defined on the isolated particle, termed Static Mode <cit.>.

To validate the accuracy of the proposed method, we analyze the scattering from finite aperiodic arrays of nano-spheres and compare the results with those obtained using the multiparticle Mie theory <cit.>. We report the CPU-time and memory requirements as functions of the number of particles, comparing the static mode expansion to a more traditional approach using RWG basis functions. Our findings indicate that the MLFMA-SME is 3 to 5 times faster and requires about 5 times less memory than the traditional MLFMA employing RWG basis functions. Furthermore, this speed and memory advantages significantly improve as the density of the surface mesh increases.

Additionally, we demonstrate the capabilities of the MLFMA-SME method on different arrays: a golden-angle spiral array with a diameter of 115 λ, consisting of 40,000 gold bricks; a two-layer Moiré superlattice made up of 20,160 gold nanodisks, with overall dimensions of 76λ× 87λ; a canonical metalens composed of 10,000 TiO_2 nanofins, covering an area of 100λ× 100λ and having focal length f=37.5λ.

Our results show that SME drastically reduces the total number of unknowns, leading to significant savings in CPU time and memory requirements for the numerical solution of scattering problems from arrays of identical particles compared to the classical MLFMA implementation with sub-domain basis functions. This advancement enables the simulation of large structures that were previously impracticable.



This work was supported by the Italian Ministry of University and Research under the PRIN-2022, Grant Number 2022Y53F3X “Inverse Design of High-Performance Large-Scale Metalenses".


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 Paper Title: Limitations on bandwidth-integrated passive cloaking.






Department of Electrical and Computer Engineering, Princeton University, Princeton, New Jersey 08544, USA


Department of Chemistry, Princeton University, Princeton, New Jersey 08544, USA


Department of Electrical and Computer Engineering, Princeton University, Princeton, New Jersey 08544, USA


Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA


Department of Engineering Physics, Polytechnique Montréal, Montréal, Québec H3T 1J4, Canada


Department of Electrical and Computer Engineering, Princeton University, Princeton, New Jersey 08544, USA

§ ABSTRACT 
We present a general framework for the computation of structure-agnostic bounds on the performance of passive cloaks over a nonzero bandwidth.
We apply this framework in 2D to the canonical scenario of cloaking a circular object.
We find that perfect cloaking using a finite-sized isotropic cloak is impossible over any bandwidth, with the bounds scaling linearly with the bandwidth before saturating due to the finite size of the cloak and the presence of material loss.
The bounds also exhibit linear scaling with material loss in the cloak and linear scaling with the inverse of the radial thickness of the design region before saturation due to finite-size effects or the presence of material loss.
The formulation could readily find applications in the development of cloaking devices, setting expectations and benchmarks for optimal performance.


Limitations on bandwidth-integrated passive cloaking
    Alejandro W. Rodriguez
    August 5, 2024
====================================================





Introduction.— The ability to engineer optical cloaking of objects to make them undetectable by observers has wide-ranging implications for radar <cit.>, obscurance <cit.>, and sensing <cit.> applications.
Several techniques have been proposed to achieve (near-)invisibility at a single operating optical wavelength, including transformation optics <cit.>, scattering-cancellation via plasmonic <cit.> or mantle cloaking <cit.>, waveguide cloaking <cit.>, transmission-line networks <cit.>, or the use of anomalous localized resonances <cit.>, to name a few.
A fundamental and practically relevant question is whether one can achieve (near-)invisibility over a wide spectral window, e.g., the entire visible spectrum.
As pointed out by Pendry et al. using an argument based on the phase and group velocities of light in the presence of dispersion <cit.>, and more recently by others using more formal arguments for the inevitable distortion of a pulse wave by materials with local response <cit.>, perfect cloaking of an isolated object in vacuum over a nonzero bandwidth is impossible due to causality; the finite speed of light and the presence of dispersion limits fully effective cloaking to a single frequency.
However, the Pendry et al. and Miller proofs say little about imperfect cloaking (a nonzero but small scattering cross section).
For instance, engineering a cloak to attain near-perfect invisibility at a single frequency does not preclude “good” cloaking over some nonzero bandwidth around that frequency.
Hashemi et al. <cit.> presented an alternative proof that agrees with the results of Pendry et al. and Miller for the impossibility of perfect cloaking over a nonzero bandwidth for physical, causal materials, but additionally derived scaling relations with implications for design, e.g., the allowed bandwidth over which one may expect practical cloaking scales inversely with the diameter of the cloaked object.

In this article, we adapt recently developed constrained optimization approaches <cit.> to study the following question
related to bounds on cloaking performance: Given a specified cloak material along with an operating
frequency and bandwidth, what is the best possible performance of a passive cloaking device that can be formed out of that
material within a prescribed region surrounding a cloaked object?
In addition to capturing scaling behaviors, the presented formalism allows for quantitative assessments of expected performance of cloaks. Typical realizations of cloaks require anisotropic metamaterials with extreme and exotic properties, such as permittivities and permeabilities less than one and even equal to zero in some regions of the cloak <cit.>.
This raises the natural question of what is the best achievable performance using the simplest materials possible.
While the numerical bounds lack the transparency and intuition of analytic expressions, they can still provide insight into scaling properties and general trends, and even lead to fairly tight limits (within an order of magnitude of structures discovered via topology optimization).

Formulation.—
In line with the majority of previous works on cloaking systems <cit.>, and as a proof of concept, we study the common scenario of a cloaked object illuminated by an incident plane wave emitted by a far-away source. We consider extinction, a nonnegative quantity for passive systems <cit.>, as the fundamental figure of merit, for the following reasons. First, it is better suited for describing the practical goals of cloaks as opposed to scattered or absorbed power alone: 
minimizing extinction minimizes the scattered field over both near- (absorbed power) and far-field  (scattered power) domains <cit.>, implying that an object becomes truly undetectable regardless of the position of an observer as the figure of merit approaches zero.
Second, analyticity in the upper-half of the complex frequency plane means that one can exploit the residue theorem to evaluate the spectral average over a nonzero bandwidth via a single scattering calculation <cit.>, which greatly simplifies computations.

Throughout, we use the language of scattering theory laid out in previous descriptions of electromagnetic bounds <cit.>, wherein the 𝕋 operator describing bound polarization currents in the medium is defined by the relation

    𝕀 = 𝕋(𝕍^-1 - 𝔾_) = (𝕍^-1 - 𝔾_)𝕋.
 
The operator 𝔾_ represents the background Green's function, which for
vacuum satisfies [ ∇×∇× -
  ω_0^2/c^2]𝔾_(𝐫,
𝐫'; ω_0) = ω_0^2/c^2δ^(3)(𝐫 - 𝐫'). (Note that in keeping with prior work, this definition includes an additional factor of ω_0^2/c^2 compared to the standard convention <cit.>).
The 𝕍 operator is the scattering potential (susceptibility) relative to this background medium (whatever additional material response was not included in the definition of 𝔾_), and |𝐄_i> and |𝐉_i> are defined as the incident electric fields and electric currents in the background, respectively. 
When acting on incident fields, the 𝕋 operator produces the generated current |𝐉_g⟩ = -ik_0/Z𝕋|𝐄_i⟩,
with Z = √(μ_0/ϵ_0) denoting the impedance of free space.

Employing the 𝕋 operator relation to a scattering problem involving a prescribed incident field <cit.>, the extinguished power may be written as,

    P_ext(ω_0) = 
        1/2<𝐄_i|𝐉_g> = 
        1/2Zk_0<𝐄_i|𝕋|𝐄_i>,

where k_0 is left inside Im because it will be continued to the complex plane to map this quantity to a bandwidth-averaged quantity <cit.>. Namely, since real sources emit light over a nonzero bandwidth a key figure of merit is the bandwidth average of extinct power.
Considering a Lorentzian window function L(ω) ≡Δω/π/(ω-ω_0)^2 + Δω^2 centered at ω_0 with a bandwidth Δω, the average extinction power for incident plane waves,

    ⟨ P_ext⟩≡∫_-∞^∞P_ext(ω)L(ω)ω→ P_ext(ω̃),

may be evaluated by closing the contour and picking up the residue at the pole of the Lorentzian in the upper-half complex frequency plane, yielding Eq. (<ref>) with all vectors and operators evaluated at a complex frequency ω̃≡ω_0 + iΔω and complex wave number k̃≡ (ω_0 + iΔω)/c, simplifying the calculation so that one no longer needs to consider power at each individual frequency within the Lorentzian window.
Note also that working at a complex frequency requires the evaluation of χ(ω̃), which is mathematically equivalent to using modified materials at a real frequency ω_0 <cit.>.

To isolate the impact of the cloak on scattering from a fixed object, we let 𝔾_ refer to the vacuum Green's function and shift the known scattering properties of the fixed object into the background of the scattering operator describing the cloak, with 𝔾_ denoting the Green's function of the fixed object in isolation, so that 𝔾_ satisfies [ ∇×∇× - ω_0^2/c^2(𝕍_ + 𝕀)]𝔾_(𝐫,
𝐫';
ω_0) = ω_0^2/c^2𝕀δ^(3)(𝐫 - 𝐫').
In particular, let 𝕋_ = (𝕍_^-1 - 𝔾_)^-1 be the scattering operator of the fixed object (the object to be cloaked) in isolation, and let 𝕋_ = (𝕍_^-1 - 𝔾_)^-1 be the scattering operator of the designable object (the cloak) dressed by the fixed object. After some straightforward algebraic manipulations (see Appendix), we find

    P_ext(ω_0)
        = 1/2Zk_0<𝐄_i|𝔾_^-1𝔾_𝕍_|𝐄_i>
    
        + 1/2Zk_0<𝐄_i|𝔾_^-1𝔾_𝕋_(𝔾_𝕋_ + 𝕀)|𝐄_i>.






The first term is the extinguished power when no cloak is present while only the second term depends on the cloaked object and quantifies the interaction between the cloak and cloaked object.

Our derivation of bounds exploits the optimization procedure based on Lagrange duality laid out in Refs. <cit.>.
The loosest such bound only imposes that the optimal scattering operator satisfies the
conservation of power (optical theorem <cit.>)
over the entire design domain, and not the full scattering equations.
Defining |𝐄_⟩≡ (𝔾_𝕋_ + 𝕀)|𝐄_i⟩ = iZ/k_0𝔾_|𝐉_i>, the total field in the presence of only the fixed object, and |𝐓_⟩≡𝕋_|𝐄_⟩ so that -ik_0/Z|𝐓_⟩ is the induced current in the cloak, we relax the problem such that, instead of optimizing over 𝕋_ with support in the design region, we optimize over |𝐓_⟩ with support in the design region, i.e., optimize over all possible polarization currents within the design region so that the bound considered here automatically takes into account all possible distributions of vacuum and the prescribed cloak material within the design region.
It should be noted, however, that this bound formalism does not yield the optimal material distribution of the cloak <cit.>, a task that calls for more computationally demanding topology optimization methods (NP-hard problems that preclude guarantees of globally optimal solutions) <cit.>.
Similar techniques have recently been used to derive bounds on deterministic scattering and fluctuational electrodynamic phenomena <cit.>.
Concretely, we wish to solve the following problem:


    |T_>min  1/2Z   k_0<𝐄_i|𝔾_^-1𝔾_(𝕍_|𝐄_i>+ℙ_|T_>)
    s.t. ∀ m, 
         ⟨𝐄_|ℙ_m|𝐓_⟩
        -⟨𝐓_|𝕌_m|𝐓_⟩ = 0,


where ℙ_ is a projection operator into the design region Ω_, ℙ_m is a projection operator into a subregion Ω_m⊆Ω_ contained within the design region, and 𝕌_m≡ℙ_(χ_^-1†ℙ_m - 𝔾_^†ℙ_m)ℙ_.
Taking the real and imaginary parts of the complex-valued constraints enforce the conservation of resistive and reactive power, respectively, spatially integrated over a region Ω_m⊆Ω_ <cit.>.
This is a quadratically constrained quadratic program over the field |𝐓_⟩ and we numerically compute bounds on Eq. (<ref>), the primal objective, by evaluating and optimizing the corresponding concave Lagrange dual function <cit.>.





Applications.— We now exploit the framework above to numerically obtain performance bounds for passive cloaks composed of an isotropic electric susceptibility. For simplicity, we focus on 2D settings and consider the scenario of a fixed circular object of radius R_, a common benchmark example for cloaking <cit.>, with the cloak restricted to an annular design region with an inner radius R_ and outer radius R_ [see Fig. <ref>(a) bottom-right schematic] for the case of incident TM (electric field out of plane) plane waves.

First, we consider the situation of illumination of broadband light from the far-field (plane waves). For a fixed χ(ω_0 + iΔω) we find that the bounds scale linearly with the bandwidth before saturating due to the finite size of the design domain and due to the presence of material loss, see Fig. <ref>(a).
As seen in the figure, enforcing the conservation of power within the design domain leads to nontrivial bounds with interesting trends and scaling behavior that are also fairly tight (within an order of magnitude of the objective values of discovered structures).
The structures found via topology optimization perform well within a bandwidth Δω around the “center” frequency ω_0, but at the expense of more scattering in nearby regions of the spectrum, see Fig. <ref>(b).
This is in agreement with the findings in Ref. <cit.>, which showed that any passive cloak made of a linear, nondiamagnetic material respecting causality always increases the scattering and extinction integrated over all frequencies compared to the original uncloaked object.
Thus, near-invisibility in a given frequency window necessarily implies significant scattered/extinct power at other frequencies, and therefore such cloaks may be detected more easily than the original uncloaked objects when illuminated by sufficiently broadband light (short pulses).
As seen in Fig. <ref>(b), the electric field in the lossy metallic object is expelled from and rerouted around the object with the introduction of the cloak. The dielectric object with a smaller loss, on the other hand, 
may try to create zero-field regions or it may allow the electric field to penetrate the object if it is advantageous as is the case for the dielectric cloak example shown in Fig. <ref>(b).

Next, we investigate the performance limits of 2D cloaks of isotropic susceptibility as a function of the material loss, see Fig. <ref>.
The bounds (solid lines) follow trends seen in topology-optimized designs (circular markers) over a broad range of material loss values and both demonstrate, unsurprisingly, improved performance as the material loss in the cloak decreases, exhibiting linear scaling with decreasing material loss before eventually saturating. 
This scaling can be understood as follows.
Writing the total 𝕋 operator in 2× 2 block-form over the two regions, namely,

    𝔾^ = 
        [ 𝔾^_, 𝔾^_,; 𝔾^_, 𝔾^_, ],


    𝕍^-1 =
        [ 𝕍^-1_     0;     0 𝕍^-1_ ],


    𝕋^-1 =
        [ 𝕋_^-1 -𝔾^_,; -𝔾^_, 𝕋_^-1 ],

leads, after the use of the Woodbury formula for a matrix inverse, to a total 𝕋 operator expression of the form

    𝕋
        
        
        
        
           ≡[       𝕐_ 𝕋_𝔾^_,𝕐_; 𝕋_𝔾^_,𝕐_       𝕐_ ].

Here, we introduced 𝕐_ which is the scattering operator of the designable cloak dressed by the presence of the fixed object, as seen from 𝕐_≡ (𝕋_^-1 - 𝔾^_,𝕋_𝔾^_,)^-1 = (𝕍_^-1 - (𝔾^_, + 𝔾^_,𝕋_𝔾^_,))^-1 and recognizing 𝔾_ +𝔾_𝕋_𝔾_ = 𝔾_ as the Green's function of the fixed object in isolation in vacuum.
Likewise for 𝕐_ but with the fixed and design regions switching roles.
Letting χ_ = χ_' + iχ_” be the separation of the susceptibility into its real and imaginary parts, we find

    𝕋_
        = 
        (χ_'ℙ_)[(𝕀 - 𝔾^_,𝕍_')^-1
    
        + ∑_n=0^∞n(𝔾^_,)^n(χ_')^n-1(iχ_”)] 
     + (iχ_”ℙ_)(𝕀 - 𝔾^_,𝕍_')^-1 + O[(χ_”)^2]

and similar for 𝕐_ but with 𝔾_ replaced with 𝔾_.
Using these expansions for 𝕋_ and 𝕐_ in the 2× 2 block-form for the total scattering operator, Eq. (<ref>), leads to the prediction that the extinction at a single frequency, P_ext(ω) = k_0/2Z<𝐄_i|𝕋|𝐄_i>, scales linearly with χ_” towards a constant for a fixed geometric structure as χ_”→ 0.
The numerical Lagrange dual bounds follow the same scaling behavior as the primal objective function for a fixed structure, demonstrating that faster scaling is not possible.
The approach to a positive constant rather than 0 as the material loss of the cloak vanishes is in agreement with the findings of Hashemi et al. <cit.>,
which showed that if the attainable refractive index contrast is bounded from below (as is the case for a cloak of isotropic susceptibility considered here) then there is a bound on the reduction of the scattering cross section for transformation-based invisibility cloaking of an isolated object; in particular, a minimum achievable refractive index contrast greater than 0 necessarily implies a positive cross section.
A Pendry cloak can achieve perfect invisibility at a single frequency with a finite device footprint but it requires a vanishing permittivity and permeability at the inner surface of the cloak <cit.>.
As shown, however, the presented formalism can provide quantitative lower bounds supporting such qualitative observations.

Figure <ref> plots limit values as a function of the radial size of the design region, for a given object radius.
Unsurprisingly, the lower bounds are monotonically nonincreasing as the allowed cloak footprint increases since any polarization current with a given performance can be contained in an enlarged design domain. 
In the single frequency case, Δω→ 0, the bounds scale inversely with the thickness of the design region, R_ - R_, over a notable range of thicknesses.
Hashemi et al. <cit.> argued that for bounded refractive indices, the cloak thickness must scale proportionally to the thickness of the object being cloaked, and the scaling of our numerical bounds demonstrates this behavior.
For positive bandwidths, Δω > 0, the bounds saturate to a positive value, with saturation occurring rather quickly (around R_/R_≈ 2) for bandwidths Δω/ω_0≳ 10^-5.
This has interesting implications for the design of cloaks since it is typically not clear how large a device needs to be to achieve near-optimal or even reasonable performance.
For our chosen parameters, the bounds show that for nonzero bandwidths a cloak design region with a thickness equal to that of the cloaked object essentially saturates the possible performance; larger device footprints will not always lead to significant improvements in performance.

Lastly, the presented inverse designs and bounds were restricted to plane-wave sources coming from a single direction of incidence.
To consider cloaking robust to the direction of incidence, the figure of merit may be modified to encompass multiple plane wave directions (including fully angle-integrated extinction), such that ⟨ P_ext⟩ = 
  ∑_a = 1^N1/2Zk̃<𝐄_i^(a)|𝕋|𝐄_i^(a)>
where each |𝐄_i^(a)⟩ is a plane wave incident from a different angle.
This leads to an optimization problem of the form


    {|T_^(a)>}min  ∑_a=1^N1/2Z   k̃<𝐄_i^(a)|𝔾_^-1𝔾_𝕍_|𝐄_i^(a)>
    
    +
    ∑_a=1^N1/2Z   k̃<𝐄_i^(a)|𝔾_^-1𝔾_ℙ_|T_^(a)>
    s.t. ∀ a, b, m, 
         ⟨𝐄_^(a)|ℙ_m|𝐓_^(b)⟩
        -⟨𝐓_^(a)|𝕌_m|𝐓_^(b)⟩ = 0,


where |𝐄_^(a)⟩≡ (𝔾_𝕋_ + 𝕀)|𝐄_i^(a)⟩ and |𝐓_^(a)⟩≡𝕋_|𝐄_^(a)⟩.
Without the constraints between the a≠ b terms, the problem reduces to a sum of decoupled optimization problems. 
Although each angle is a different scattering problem, 
the polarization currents induced in each scenario are generated by the same structured media and this fact is enforced by the a≠ b “cross constraints”, leading to additional tightening of the bounds <cit.>.
Since the incident field is the same up to a rotation, if the object and design region are invariant under rotations then the bounds are the same for decoupled problems and any tightening should come from the cross-constraints. 
To consider a more complicated scenario involving more than a single incident source, Fig. <ref>
shows computed bounds and inverse design performance values for a rectangular object discovered starting from random initializations for the case where N = 2, comprising horizontally and vertically propagating plane waves. 


















Conclusion.— 
In conclusion, we presented a formalism for computing bounds on passive, broadband cloaking systems that can provide benchmarks and set expectations for future cloaking devices, particularly those developed through the use of topology optimization.
The many variables available for tuning means an exhaustive study of their relationship to the bounds on performance is complex.
We presented a few key examples to demonstrate the capabilities of the formalism and to extract intuition for the obvious parameters one may wish to vary in experiments.
Our numerical results support the claims that broadband cloaking of electrically large objects is practically impossible using passive linear cloaks <cit.> and provide a quantitative assessment of the limits on performance.
We believe this represents an important result for the science of cloaking, suggesting that new concepts and designs, including opening to the field of nonlinear and active metamaterials <cit.>, are necessary for increasing the cloaking bandwidth
and compensating the unwanted scattering required by linearity and causality in the presence of absorption losses <cit.>.
In closing, we note that while our examples focused on 2D and isotropic permittivities, the formalism is valid in 3D settings, and may be easily extended to consider anisotropic permittivities and permeabilities.


Acknowledgments.— 
We acknowledge the support of a Princeton SEAS Innovation Grant and by the Cornell Center for Materials Research (MRSEC) through Award DMR-1719875.
S.M. acknowledges financial support from the Canada First Research Excellence Fund via the Institut de Valorisation des Données (IVADO) collaboration.
The simulations presented in this article were performed on computational
resources managed and supported by Princeton Research
Computing, a consortium of groups including the Princeton Institute for Computational Science and Engineering
(PICSciE) and the Office of Information Technology's
High Performance Computing Center and Visualization Laboratory at Princeton University.







§ OBJECTIVE FUNCTION DERIVATION



In this section, we derive an expression for the total (including the fixed and designable objects) scattering operator which isolates the contribution of the designable object.
A useful operator for this purpose is the 𝕎 operator defined by 

    𝕀 = 𝕎(𝕀-
      𝕍𝔾_) = 
      (𝕀-𝕍𝔾_)𝕎.

From Eqs. (<ref>) and (<ref>), it is clear that 𝕀 = 𝕎𝕍𝕋^-1 and 𝕎 = 𝕀 + 𝕋𝔾_.
The 𝕋 operator takes an incident field and gives ∝ the generated current, |𝐉_g⟩ = -ik_0/Z𝕋|𝐄_i⟩, while the
𝕎 operator takes an initial current and gives the total current, |𝐉_t⟩ = 𝕋𝕍^-1|𝐉_i⟩ = 𝕎|𝐉_i⟩.
We will make use of the nesting property of the 𝕎 operator 

    𝕎_tot = 𝕎_𝕎_,

where 𝕎_tot is the 𝕎 operator of the total system (including all the scatterers), 𝕎_ is the 𝕎 operator of the fixed object only, and 𝕎_ is the 𝕎 operator for the designable body with the fixed object contained in the background, i.e., 𝔾_ replaced by 𝔾_𝕎_ = 𝔾_(𝕀 + 𝕋_𝔾_) = 𝔾_ + 𝔾_𝕋_𝔾_ (which is the total Green's function of the fixed object in the background, 𝔾_) in the defining relation for 𝕎_. 
The nesting property holds since 𝕎_^-1𝕎_^-1 = (𝕀 -𝕍_𝔾_)𝕎_^-1 = (𝕀 - 𝕍_𝔾_𝕎_)𝕎_^-1 = (𝕎_^-1 - 𝕍_𝔾_) = (𝕀 - 𝕍_𝔾_ - 𝕍_𝔾_) = 𝕎_tot^-1.
Using 𝕎_tot𝕍_tot = 𝕋_tot and the nesting relation Eq. (<ref>) yields

    𝕋_tot   = 𝕎_tot𝕍_tot
        = 𝕎_𝕎_𝕍_tot
       =
        𝕎_(𝕎_𝕍_ + 𝕎_𝕍_)
        
       = 𝕎_(𝕎_𝕍_ + 𝕋_)
        
       = 𝕎_(𝕍_ + 𝕋_(𝔾_𝕋_+𝕀)) 
       = 𝕋_ + 𝔾_^-1𝔾_𝕋_𝔾_𝔾_^-1,

where we made use of 𝕎_ = 𝕀 + 𝕋_𝔾_ = 𝕀 + 𝕋_𝔾_𝕎_ = 𝕀 + 𝕋_𝔾_𝕋_𝕍_^-1.
Define |E_>≡(𝔾_𝕋_ + 𝕀)|𝐄_i> = iZ/k_0𝔾_|𝐉_i> and |T_>≡𝕋_|E_>.
Use 𝕋_tot = 𝕎_(𝕍_ + 𝕋_(𝔾_𝕋_ + 𝕀)) to find that the total extinct power by all the scatterers, relative to a vacuum background, is given by

    P_ext   =
        1/2Zk_0<𝐄_i| 𝕋_tot|𝐄_i>
       = 1/2Zk_0<𝐄_i|𝕎_𝕍_|𝐄_i>
         + 1/2Zk_0<𝐄_i|𝕎_𝕋_(𝔾_𝕋_ + 𝕀)|𝐄_i>
       = 1/2Zk_0<𝐄_i|𝕎_𝕍_|𝐄_i>
         + 1/2Zk_0<𝐄_i|𝕎_|T_>

with 𝕋_ subject to the fundamental relation 

    𝕀_   = 
        𝕀_(χ_^-1𝕀 - 𝔾_)𝕀_𝕋_,

where χ_ is the susceptibility used for the object in the design domain (the cloak).
This constraint is similar to that in previous works <cit.>, but where the relevant Green's function is 𝔾_ and not the vacuum Green's function.
Use of 𝕎_ = 𝔾_^-1𝔾_ leads to the primal objective function presented in the main text.



§ TOPOLOGY OPTIMIZATION PROCEDURE



To calculate inverse designs, we made use of the NLopt package <cit.> and followed standard topology optimization algorithms <cit.> based on the method of moving asymptotes <cit.> where the electric susceptibility value of each pixel in the specified design region is considered as an independent design parameter.
Each pixel in the design region is allowed to explore a continuous range of susceptibility values varying linearly between the vacuum value 0 and the prescribed material value χ_clk.
The continuous range [0, 1] ·χ_clk can be interpreted, after normalization by χ_clk, as a filling fraction of the pixel.

We implemented a 2D Maxwell solver which could be used for arbitrary structuring (allow the pixels within any arbitrary 2D design region to vary).
Doing a thousand function evaluations (iterations of optimization over the susceptibility profile for fixed material parameters, design region, bandwidth, etc.) in 2D takes less than 1 hour for a wavelength-scale design region.
The primary challenge in such optimization problems lies in the computational cost of evaluating the objective function and gradient, which is evaluated several hundred if not thousand times during the course of any one optimization.
Let 𝔾 denote the total Green's function and let 𝕄^-1 = 𝔾 where 𝕄 = c^2/ω_0^2∇×∇ - ϵ(𝐫).
After the discretization of the computational grid, then in the design region ϵ_b = 1 + χ_χ̅_b at pixel b. 
The gradient of the objective function P_ext({χ̅_b};ω_0) =
    1/2<𝐄_i|𝐉_g> = 1/2<𝐄_i|𝕍𝔾|𝐉_i> with respect to the degree of freedom χ̅_a is given by

    ∂ P_ext/∂χ̅_a   =
        1/2<𝐄_i|∂(𝕍𝔾)/∂χ̅_a|𝐉_i>
       =
        1/2<𝐄_i|
        (χ_ℙ_a𝔾 + 𝕍∂𝕄^-1/∂χ̅_a)
        |𝐉_i>
       =
        1/2<𝐄_i|
        (χ_ℙ_a𝔾 - 𝕍𝕄^-1∂𝕄/∂χ̅_a𝕄^-1)
        |𝐉_i>
       =
        1/2<𝐄_i|
        (χ_ℙ_a𝔾 
        + χ_𝕍𝔾ℙ_a𝔾)
        |𝐉_i>
       =
        1/2k_0χ_/iZ<𝐄_i|
        (ℙ_a + 𝕍𝔾ℙ_a)
        |𝐄_t>
       ≡1/2k_0χ_/iZ⟨𝐄_i + 𝐄̃| 
        ℙ_a |𝐄_t⟩

where χ̅_a is a topology optimization degree of freedom, normalized to take values in [0, 1], so that the susceptibility at the a-th pixel in the design region is given by χ_clkχ̅_a, ℙ_a is projection onto the pixel indexed by a, |𝐄_t⟩ is the total electric field, and |𝐄̃⟩≡𝔾^†𝕍^†|𝐄_i⟩ which for reciprocal systems becomes |𝐄̃⟩≡ (𝔾𝕍|𝐄_i^*⟩)^*.
Hence, for a given incident field |𝐄_i⟩ one only needs to perform two additional Maxwell solves to calculate |𝐄_t⟩ and |𝐄̃⟩ at each iteration of the optimization to obtain the required gradient Eq. (<ref>) and the objective function
P_ext({χ̅_b};ω_0)
=
1/2<𝐄_i|𝐉_g>
=
1/2Zk_0<𝐄_i|𝕍|𝐄_t>.
Details on the computational complexities encountered in 3D and potential solution methods can be found in Appendix C of Ref. <cit.> and the therein cited references.





 Paper Title: Non-Uniqueness of Metasurfaces for Wave Transformations.



Non-Uniqueness of Metasurfaces for Wave Transformations]Non-Uniqueness of Metasurfaces for Wave Transformations

    koa201@exeter.ac.uk
 School of Physics and Astronomy, University of Exeter, Stocker Road, Exeter, EX4 4QL, UK
 
 School of Physics and Astronomy, University of Exeter, Stocker Road, Exeter, EX4 4QL, UK

 School of Physics and Astronomy, University of Exeter, Stocker Road, Exeter, EX4 4QL, UK

DSTL, Porton Down Salisbury, Wiltshire, SP4 0JQ, UK

 School of Physics and Astronomy, University of Exeter, Stocker Road, Exeter, EX4 4QL, UK

 School of Physics and Astronomy, University of Exeter, Stocker Road, Exeter, EX4 4QL, UK

 School of Physics and Astronomy, University of Exeter, Stocker Road, Exeter, EX4 4QL, UK





§ ABSTRACT 
We show that a large family of tensorial metasurfaces can be found that perform an identical wave transformation, showing that even when the conditions of reciprocity and passivity are imposed, there still remain many solutions to the design problem.  As an example, we explore the case of a metasurface that rotates a single input polarization, showing we can parameterize the set of equivalent reciprocal metasurfaces in terms of a single complex parameter.  Through allowing dissipation and gain within the response, the surface can have many different functionalities in the orthogonal polarization, opening up a new route for the design of multiplexed metasurfaces.


[
    S.A.R. Horsley
    August 5, 2024
==================








§ INTRODUCTION

Metasurfaces are planar media containing an array of polarizable elements <cit.>, carefully chosen to perform a desired transformation of an incoming wave.  Although this concept covers many areas of wave physics <cit.>, here the focus is on metasurfaces for electromagnetic waves.  Within such surfaces, the elements are typically spaced closer than the wavelength, allowing the characterization of the array in terms of an effective electromagnetic boundary condition <cit.>.  Whereas partially or totally transmitting metasurfaces are typically characterized in terms of sheet transition conditions <cit.>, or a polarizability <cit.>,  impenetrable surfaces are usually characterized in terms of a surface impedance <cit.>. Illustrated in Fig. <ref>.  This amounts to specifying the ratio of the electric and magnetic fields at all points on the surface.  There are a variety of approaches to the design of such metasurfaces, controlling e.g. the local reflection phase over the surface <cit.>, or the coupling of surface waves to free–space radiation <cit.>.  In this work, we consider impenetrable metasurfaces described in terms of a tensorial surface impedance and investigate the uniqueness of the surface parameters with respect to a desired transformation of the field.






Although the terminology of a `metasurface' is little more than a decade old, the idea is a continuation of a large body of earlier work on frequency selective surfaces <cit.>, which was revived when similar ideas began to be implemented in optics <cit.>.  Recent work has taken the metasurface idea significantly further forward.  For example, through matching the electric and magnetic polarizability of the surface elements in a transmitting sheet, all reflection can be eliminated, resulting in so–called `Huygens' metasurfaces <cit.>, which sculpt the field while maintaining its direction of propagation. More recently, more exotic metasurfaces have been considered, including non–Hermitian and PT–symmetric elements <cit.>, which contain carefully designed distributions of loss and/or gain, allowing for an unusual control over e.g. the polarization of the output field.  Novel forms of surface excitations have also been discovered at junctions between metasurfaces <cit.>, and most recently non–reciprocal <cit.> and space–time varying surface parameters have been considered <cit.>, enabling both multiplexed functionality and spectral control.

As mentioned above, here we investigate the uniqueness of the tensorial surface impedance, Z_s, when enforcing a single given transformation of the electromagnetic field.  This is relevant to the problem of designing multiplexed—more specifically polarization multiplexed—metasurfaces, where two or more functionalities are imposed in the same design.  Unless the surface properties are e.g. re–configurable <cit.>, it is well known that this is a difficult design problem, and there have been numerous previous approaches <cit.>, some of which are summarized in Ref. <cit.>.

Here we show that, once a single field transformation has been specified at every point on the surface then there remains a great deal of freedom in the parameters determining the surface impedance.  The greatest freedom occurs for non–reciprocal surfaces, but even reciprocal ones always contain a single free complex parameter.  When we constrain the surface to be both passive and reciprocal, we find that through increasing dissipation within the surface, and thereby decreasing the amplitude of the transformed field, we, in a sense, increase the non–uniqueness of the metasurface.  This shows that, through carefully increasing dissipation within the structure, we gain more freedom to find a metasurface that has a specific polarization multiplexed function.





§ THE TENSORIAL IMPEDANCE BOUNDARY CONDITION


We can characterize the reflection of electromagnetic waves from any thin impenetrable metasurface in terms of an impedance boundary condition written in terms of a 2×2 tensor valued surface impedance <cit.>, Z_s.  To understand the origin of this boundary condition, consider Ohm's law within the metasurface relating the electric field parallel to a boundary, 𝐄_∥, to the surface currents, 𝐣_s, via the surface conductivity, σ_s. By relating the surface currents to the magnetic field via 𝐣_s=𝐧̂×𝐇_∥ where 𝐧̂ is the surface normal, we have an equation relating the surface electric and magnetic fields,

    𝐣_s = σ_s·𝐄_∥ = 𝐧̂×𝐇_∥.

In general, the surface conductivity is a 2× 2 tensor, indicating anisotropy, i.e. the vectorial surface current may not be directly proportional to the surface electric field.  For example, a surface composed of an array of thin parallel wires aligned with the 𝐳̂ direction would be characterized by the surface conductivity σ_s=σ_zz 𝐳̂⊗𝐳̂, indicating that current can only flow along the axis of the wires and only flows in response to the 𝐳̂ component of the electric field.  As a matter of convention, interfaces are characterized not in terms of the conductivity but in terms of the surface impedance, which is its inverse, Z_s=σ_s^-1.  This yields the impedance boundary condition, 

    𝐄_∥=Z_s·(𝐧̂×𝐇_∥).

Note that the component of the cycle averaged Poynting vector normal to the metasurface is given by ⟨ S_n⟩=1/2 Re[𝐧̂·(𝐄_∥×𝐇_∥^⋆)]=-1/2 Re[𝐄_∥·(𝐧̂×𝐇_∥^⋆)] and therefore the power locally entering the metasurface ⟨ P_n⟩=-⟨ S_n⟩ is proportional to the Rayleigh quotient of the impedance matrix,

    ⟨ P_n⟩=1/2 Re[(𝐧̂×𝐇_∥)^⋆·Z_s·(𝐧̂×𝐇_∥)].

Given our assumption that the metasurface is impenetrable, a positive value for ⟨ P_n⟩ indicates a dissipative response, whereas a negative value indicates amplification, or gain.  Given that a lossless metasurface will have ⟨ P_n⟩=0 for all possible incident fields, lossless surfaces have an impedance tensor that is `i' times a Hermitian matrix (a so–called anti–Hermitian operator).  In the general case, we can decompose the impedance into Hermitian and anti–Hermitian parts, Z_s=H_1+ i H_2, where H_1,2 are Hermitian matrices.  The dissipated power is thus proportional to the Rayleigh quotient of H_1, implying that passive surfaces (i.e. those without gain) have a positive semi–definite form of H_1.

Another important constraint on the impedance tensor is provided by reciprocity.  Representing the freedom to interchange sources and detectors, this implies the symmetry of the electromagnetic Green function <cit.> G(x_1,x_2)=G^ T(x_1,x_2), which is equivalent to the symmetry of the operators in Maxwell's equations.  Reciprocity thus requires the surface impedance to be a symmetric tensor, Z_s=Z_s^ T.





§ FIELD TRANSFORMATIONS WITH TENSORIAL METASURFACES


An important application of these metasurfaces is in wave transformations, where a given incident wave is reflected into a specified radiation pattern, which in general can include both near and far–fields.  Such transformations include the redirection, focusing, and absorption of an incident wave.  One simple method for finding the impedance distribution that enacts a given transformation is to directly invert Eq. (<ref>).  To do this, we write our complex valued surface fields in terms of their magnitude, e.g. |E|=(𝐄_∥·𝐄_∥^⋆)^1/2 and “direction” 𝐞̂=𝐄_∥/|E|,

    𝐄_∥   = |E|𝐞̂
    𝐇_∥   = |H|𝐡̂

Substituting this decomposition into the earlier expression for the impedance boundary condition (<ref>) we obtain the requirement

    Z_s·(𝐧̂×𝐡̂)=|E|/|H|𝐞̂

which tells us that at each point on the surface, the impedance tensor must rotate and rescale the complex unit vector 𝐧̂×𝐡̂ so that it is parallel to the electric field unit vector, 𝐞̂.  When our desired field transformation has 𝐞̂ everywhere parallel to 𝐧̂×𝐡̂_∥, this can be achieved with an isotropic metasurface, characterized in terms of a scalar impedance, Z_s=|E|/|H|.  This is possible in e.g. a 2D geometry, where one field component always lies out of the plane and the other always in the plane.

Nevertheless, in the general case a tensor valued impedance is required to satisfy (<ref>).  A simple solution is Z_s=(|E|/|H|) 𝐞̂⊗(𝐧̂×𝐡̂)^⋆, where `⊗' is the tensor product defined as e.g. (𝐚⊗𝐛)_ij=(𝐚𝐛^T)_ij=a_ib_j.  However, as also explained in Ref. <cit.>, this form of surface impedance is most often neither passive (Z_s+Z_s^† not positive semi–definite), nor reciprocal (Z_s≠Z_s^ T).  This certainly represents a challenge for any practical realizations of these metasurface parameters, as both a power source and e.g. magnetic elements would have to be embedded within the surface!  Yet there is an additional freedom that we can exploit to bring the design closer to reality.  We can always perform a transformation Z_s→Z_s+𝐯⊗𝐡̂, where 𝐯 is an arbitrary vector, without affecting condition (<ref>).  This leaves us with an impedance tensor of the general form,

    Z_s = |E|/|H|[𝐞̂⊗(𝐧̂×𝐡̂)^⋆ + 𝐯⊗𝐡̂].

This surface impedance performs the desired transformation (<ref>). Generally, it is non–reciprocal and non–passive, with the freedom to choose both complex components of 𝐯, arbitrarily.  The remainder of this paper explores this freedom in the design of metasurfaces.





 §.§ Reciprocal metasurfaces:


To illustrate special cases of Eq. (<ref>), we adopt a complex basis to express the impedance tensor.  Here, we use the complex basis vectors 𝐡̂ and 𝐧̂×𝐡̂^⋆ and the dual basis, 𝐡̂^⋆ and 𝐧̂×𝐡̂, with the electric field direction being given by 𝐞̂=e_1 𝐡̂+e_2 (𝐧̂×𝐡̂)^⋆=e̅_1 𝐡̂^⋆+e̅_2 (𝐧̂×𝐡̂) and the arbitrary vector is written as 𝐯=α 𝐡̂+β (𝐧̂×𝐡̂)^⋆=α̅𝐡̂^⋆+β̅(𝐧×𝐡̂).  In terms of these basis vectors the impedance tensor (<ref>) becomes

    Z_s=|E|/|H|{e_1[ 𝐡̂⊗(𝐧̂×𝐡̂)^⋆+ (𝐧̂×𝐡̂)^⋆⊗𝐡̂]+e_2 (𝐧̂×𝐡̂)^⋆⊗(𝐧̂×𝐡̂)^⋆+α 𝐡̂⊗𝐡̂}

where we have here ensured that the metasurface is reciprocal, Z_s=Z_s^ T, through taking

    β=e_1,

which determines a single component of the arbitrary vector, i.e. 𝐯·(𝐧̂×𝐡̂)=𝐞̂·𝐡̂^⋆.

From Eq. (<ref>), we can see that, irrespective of the field transformation we seek to enact, it is always possible to find a reciprocal metasurface that will perform this transformation.  Indeed, we have an infinite family of such surfaces as we are still free to choose the complex valued parameter α.   Provided we can implement amplification across the metasurface, this shows that there is a non-uniqueness in the surface parameters.  The physical reason for this design freedom is that at every point we are imposing the response to a given surface field.  Of course this leaves the response to the orthogonal surface field unspecified.  The somewhat surprising fact is that while the requirement of reciprocity removes some of this redundancy, it is not enough to completely determine the form of the surface impedance. 





 §.§ Passive metasurfaces:


As described in Sec. <ref>, for a metasurface to be passive it cannot amplify the incident field, and therefore the power entering the surface obeys ⟨ P_n⟩=-⟨ S_n⟩>0.  In terms of the unit vectors (<ref>) and the basis introduced in the previous section, this is equivalent to the condition,

    ⟨ P_n⟩= Re[𝐞̂·(𝐧̂×𝐡̂^⋆)]= Re[e̅_2]≥ 0.

This is a requirement on the field transformation that we must implement at every point on the surface at the design stage: we cannot hope for a passive surface impedance if our design fields require the metasurface to generate energy!  However, even assuming Eq. (<ref>) holds, it may still be that the impedance (<ref>) exhibits gain when probed with other incident fields (<ref>).

As explained in Sec. <ref>, to ensure the surface is passive, the Hermitian part of the impedance tensor must be positive semi–definite, i.e. for all complex unit vectors 𝐮 we must have,

    Re[𝐮^⋆·Z_s·𝐮]   =|E|/|H| Re[α̅|u̅_1|^2+e̅_2|u̅_2|^2+e̅_1u̅_1^⋆u̅_2+β̅u̅_2^⋆u̅_1]
       ≥ 0.

We can ensure the real part of the quadratic form (<ref>) is always positive through imposing both condition (<ref>) and choosing α̅, such that the determinant of the following Hermitian matrix is positive

    det[( Re[α̅]   β̅+e̅_1^⋆/2
    β̅^⋆+e̅_1/2    Re[e̅_2])]>0

which implies the constraint

    Re[α̅]≥1/ Re[e̅_2]|e̅_1^⋆+β̅/2|^2.

Fig. <ref> uses a randomly generated set of surface electric and magnetic fields to show that imposing this constraint indeed results in a passive metasurface.  Note that if we impose zero dissipation in the response to the design field, Re[e̅_2]=0, the inequality (<ref>) is impossible to fulfill.  This shows that reducing the efficiency of the design (i.e. decreasing the reflected power) can help to find a set of metasurface parameters that are, overall passive.   We shall return to this point again in the next section.

Imposing conditions (<ref>) and (<ref>) therefore ensures that our metasurface design is gain–free.  If, on top of this, we also impose the constraint (<ref>), we find a surface that is both passive and reciprocal.  However, there is not enough information in Eqns. (<ref>) and (<ref>) to completely determine both complex parameters α and β.  There thus remains a non–uniqueness in the design of even reciprocal, passive metasurfaces.







 §.§ A special class of reciprocal, passive metasurfaces:


A simple special case of the results given in Secs. <ref> and <ref> is where the unit vector 𝐡̂ defined in Eq. (<ref>) is real valued.  In this case, we no longer have to distinguish the vector basis from its dual, as defined in Sec. <ref>, and barred and un–barred quantities become equal, i.e. α̅=α, β̅=β, e̅_2=e_2, e̅_1=e_1, etc.  Therefore, the three conditions—(<ref>), (<ref>), and (<ref>)–for a passive, reciprocal metasurface become simply

    β   =e_1
     Re[e_2]   ≥0
     Re[α]   ≥1/ Re[e_2]|e_1|^2

which, as discussed above still leaves Im[α] undetermined, indicating an infinite family of passive reciprocal metasurfaces that perform the same transformation.

As an initial example, consider a metasurface that completely absorbs incident waves when the electric field is aligned along the x axis.  For a surface normal 𝐧̂=𝐳̂, the direction of the electric field is 𝐞̂=𝐱̂, and the magnetic field is 𝐡̂=-𝐲̂, with |E|/|H|=η_0 where η_0 is the free space impedance.  In this case, e_1=0, e_2=1 and the reciprocal impedance (<ref>) reduces to

    Z_s=η_0(𝐱̂⊗𝐱̂+α 𝐲̂⊗𝐲̂).

with all of the constraints in Eq. (<ref>) fulfilled so long as Re[α]>0.  Our design formulae therefore reproduces the expected result that a reciprocal, passive metasurface that perfectly absorbs x oriented incident fields, must have the impedance component Z_s,xx equal to the free space impedance, and the orthogonal component, Z_s,zz must have a positive real part.  Note that the imaginary part of α is unconstrained.





§ EXAMPLE: POLARIZATION CONVERSION


We now take a slightly more complex illustration of the family of reciprocal metasurfaces described by Eq. (<ref>), although we retain the assumption of surface uniformity: we consider a polarization converting metasurface. A normally incident plane wave polarized along the x axis impinges on a surface with surface normal 𝐧̂=𝐳̂.  The desired field is reflected with some real amplitude ρ, with a θ change in the polarization angle away from 𝐱̂, as illustrated in Fig. <ref>.  The surface electric field equals,






    𝐄   = [ e^-ik_0z+rcos(θ) e^ik_0z]𝐱̂+ρsin(θ) e^ik_0z 𝐲̂
       z=0→𝐱̂
    +ρ𝐦̂,

where we have introduced the unit vector, 𝐦̂=cos(θ)𝐱̂+sin(θ)𝐲̂.  The previous sections show that the surface impedance enacting this transformation is characterized in terms of the electric and magnetic field unit vectors 𝐞̂ and 𝐡̂, which in this case are given by,

    𝐞̂   =𝐱̂+ρ𝐦̂/√(1+ρ^2+2ρcos(θ))
    𝐡̂   =-𝐲̂+ρ𝐳̂×𝐦̂/√(1+ρ^2-2ρcos(θ)).

In this example, the unit vectors (<ref>) describing the field on the surface are purely real.  Therefore, we can find a passive reciprocal metasurface using the results derived in Sec. <ref>.  The reciprocal surface impedance tensor is given by Eq. (<ref>), with the vector components e_1=𝐞̂·𝐡̂ and e_2=𝐞̂·(𝐧̂×𝐡̂) equal to,

    e_1   =-2ρsin(θ)/√((1+ρ^2)^2-4ρ^2cos^2(θ))
    
        e_2   =1-ρ^2/√((1+ρ^2)^2-4ρ^2cos^2(θ))


In terms of these two components, the constraints for a passive metasurface (<ref>) are fulfilled if both ρ^2≤ 1 (i.e. the reflected wave does not contain a greater energy flux than the incident one) and,

    Re[α]≥1/1-ρ^24ρ^2sin^2(θ)/√((1+ρ^2)^2-4ρ^2cos^2(θ))

with the imaginary part of α free.

Given the uniformity of the surface impedance in this case, we can analytically find the reflection from the surface defined by Eqns. (<ref>), (<ref>), and (<ref>) to verify the above predictions.  The reflection amplitudes, r_xx, r_xy, r_yx, and r_yy are defined such that the first index corresponds to the output polarization, and the second index to the input polarization.  The expressions for these are given in appendix <ref>. The power carried in each polarization channel is given by the square of these amplitudes, e.g. R_xx=|r_xx|^2 tells us the fraction of power reflected in the x channel for incidence in the same channel. Fig. <ref> shows both the results of a full wave simulation, and the reflection computed analytically, verifying both the reciprocity of the surface, R_xy=R_yx and that incident x polarized waves are reflected with the desired polarization. As shown in Fig. <ref>a, when α=0, at designer incidence, the output power is conserved. Meanwhile for the secondary incident polarization, the surface reflects with increased power, even a divergent value when α=2, shown in Fig. <ref>.  For the surface to be passive (as in e.g. <cit.>), our constraint (<ref>) shows that for ρ=1 (i.e. 100% efficiency for x polarized incidence), this is only possible when Re[α]→∞.  Figs. <ref>b and <ref> show this is indeed the case, with the amplification diminishing with increasing α.









To demonstrate the validity of our inequality (<ref>), we work in terms of the eigenvectors of the surface impedance tensor: those directions along which an applied electric field results in a parallel current density.   As discussed in Sec. <ref>, the power dissipated within the surface when the surface current 𝐧̂×𝐇̂_∥ is parallel to one of these eigenvectors 𝐮_i is, from Eq. (<ref>),

    ⟨ P_n,i⟩   =1/2 Re[𝐮_i^⋆·Z_s·𝐮_i]
       =1/2|𝐮_i|^2 Re[z_s,i].

where Z_s𝐮_i = z_s,i𝐮_i. Therefore if Re[z_s,i] < 0 for either i = 1 or i = 2, then the surface cannot be passive. However this is only a necessary condition.  In general, the dissipated power should be positive for a general superposition, c_1𝐮_1+c_2𝐮_2. This power equals:

    ⟨ P_n⟩ = |c_1|^2 ⟨ P_n,1⟩ + |c_2|^2 ⟨ P_n,2⟩ + 1/2 Re[z_s,1 c_1c_2^⋆𝐮_2^*·𝐮_1 + z_s,2 c_1^⋆c_2𝐮_1^*·𝐮_2] ≥ 0 .

When the two eigenvectors are orthogonal, this reduces to ⟨ P_n⟩ = |c_1|^2⟨ P_1⟩ + |c_2|^2⟨ P_2⟩>0, which leads to the requirement that the eigenvalues of Z_s should have a positive real part.  In the current example, the surface impedance matrix is both real and symmetric (provided the free parameter α is real).  Therefore its eigenvectors are orthogonal and we simply require both eigenvalues z_s,1, z_s,2 to have a positive real part.  Fig. <ref> shows the real part of the two eigenvalues of the impedance tensor (<ref>) both as a function of the angle of polarization rotation and the parameter α.   In Fig. <ref> we see that the surface indeed becomes passive once the inequality (<ref>) is satisfied.






§ SUMMARY AND CONCLUSIONS


Impenetrable metasurfaces are often characterized in terms of a scalar valued surface impedance, the value of which encodes both the phase and amplitude of the reflected field.  However, as has been previously recognised, the use of a tensorial surface impedance—practically implemented via an array of anisotropic, polarizable elements—offers a much richer set of possible field transformations.  Here we have explored the problem of designing tensorial metasurfaces for enacting arbitrary field transformations.  We have investigated the uniqueness of the surface parameters for a given field transformation, when the metasurface is constrained to be passive, reciprocal, or both.

Our first finding is that it is possible to find a reciprocal metasurface, whatever the wave transformation we wish to perform.  The tensorial surface impedance is given by Eq. (<ref>) and there are an infinite number of such surfaces, parameterized in terms of a single complex number, α.  In general this family of metasurfaces will exhibit gain and loss for some incident fields, showing that through carefully implementing loss and/or gain within the surface, we may obtain multiplexed functionality.  As an example, we showed in Sec. <ref> that a metasurface rotating x polarized fields by an angle θ could simultaneously show a variety of different functionalities for y–polarized radiation, e.g. infinite amplification, absorption, or rotation.

Nevertheless, despite a large body of fascinating work (e.g. <cit.>) and recent experimental progress <cit.>, metasurfaces with gain are much more difficult to implement than lossy ones.  Our second finding is that, provided the complex parameter α satisfies the inequality (<ref>), we can simultaneously ensure the metasurface is reciprocal and passive (without gain) for an arbitrary incident field.  Surprisingly the surface parameters remain non–unique, and there remain an infinite family of surfaces that perform the same transformation.  One important point is that the right hand side of the inequality (<ref>) diverges when the initial design is made 100% efficient (zero dissipation, all energy reflected).  This actually reduces the design freedom, implying that the free parameter α must be taken to infinity in order that the surface be both lossless and reciprocal. Therefore, when the efficiency of the design is reduced (e.g. reducing the overall amplitude of the reflected field), we find that this increases the range of reciprocal, passive metasurfaces that can enact the transformation.  This was demonstrated in the example of a polarization rotating metasurface, where Fig. <ref>b shows that reducing the efficiency immediately yields an increased range of passive reciprocal surfaces that would otherwise require Re[α]→∞.   We have thus found that a careful choice of loss within the metasurface actually aids the design, allowing for polarization multiplexed functionality, at the price of reducing the efficiency.






§ REFLECTION FROM A TENSOR IMPEDANCE SURFACE

Writing the components of the surface impedance tensor in terms of the Cartesian basis vectors, 𝐱̂and 𝐲̂ as,

    Z_s=(Z_xx   Z_xy
    Z_yx   Z_yy)

and assuming a wave at normal incidence with electric and magnetic fields given by (A_x amplitude incident with x polarization and A_y amplitude incident with y polarization),

    𝐄   =A_x[𝐱̂( e^- ik_0 z+r_xx e^ ik_0 z)+𝐲̂r_yx e^ ik_0 z]+A_y[𝐲̂( e^- ik_0 z+r_yy e^ ik_0 z)+𝐱̂r_xy e^ ik_0 z]
    η_0𝐇   =A_x[𝐲̂(- e^- ik_0 z+r_xx e^ ik_0 z)-𝐱̂r_yx e^ ik_0 z]+A_y[𝐱̂( e^- ik_0 z-r_yy e^ ik_0 z)+𝐲̂r_xy e^ ik_0 z],

the impedance boundary condition defined above in Eq. (<ref>) leads to the following expressions for the four reflections coefficients r_ij,


    r_xx   =(Z_xx-1)(Z_yy+1)-Z_xyZ_yx/(Z_xx+1)(Z_yy+1)-Z_xyZ_yx
    
        r_xy   =2Z_yx/(Z_xx+1)(Z_yy+1)-Z_xyZ_yx
    
        r_yx   =2Z_xy/(Z_xx+1)(Z_yy+1)-Z_xyZ_yx
    
        r_yy   =(Z_xx-1)(Z_yy-1)-Z_xyZ_yx/(Z_xx+1)(Z_yy+1)-Z_xyZ_yx.

These expressions we used to generate Figs. <ref>–<ref> in the main text.














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 Paper Title: Highly-efficient electron ponderomotive acceleration in underdense plasmas.



APS/123-QED





lorenzo.martelli295@gmail.com
LOA, École Polytechnique, ENSTA ParisTech, CNRS, Université Paris-Saclay, 828 Boulevard des Maréchaux, 91762 Palaiseau Cedex, France
THALES AVS - MIS, 2 Rue Marcel Dassault, Vélizy-Villacoublay, 78140, France

LOA, École Polytechnique, ENSTA ParisTech, CNRS, Université Paris-Saclay, 828 Boulevard des Maréchaux, 91762 Palaiseau Cedex, France

LOA, École Polytechnique, ENSTA ParisTech, CNRS, Université Paris-Saclay, 828 Boulevard des Maréchaux, 91762 Palaiseau Cedex, France

LOA, École Polytechnique, ENSTA ParisTech, CNRS, Université Paris-Saclay, 828 Boulevard des Maréchaux, 91762 Palaiseau Cedex, France

LOA, École Polytechnique, ENSTA ParisTech, CNRS, Université Paris-Saclay, 828 Boulevard des Maréchaux, 91762 Palaiseau Cedex, France

LOA, École Polytechnique, ENSTA ParisTech, CNRS, Université Paris-Saclay, 828 Boulevard des Maréchaux, 91762 Palaiseau Cedex, France

LOA, École Polytechnique, ENSTA ParisTech, CNRS, Université Paris-Saclay, 828 Boulevard des Maréchaux, 91762 Palaiseau Cedex, France

LOA, École Polytechnique, ENSTA ParisTech, CNRS, Université Paris-Saclay, 828 Boulevard des Maréchaux, 91762 Palaiseau Cedex, France

LOA, École Polytechnique, ENSTA ParisTech, CNRS, Université Paris-Saclay, 828 Boulevard des Maréchaux, 91762 Palaiseau Cedex, France
Institut Curie, Inserm U 1021-CNRS UMR 3347, University Paris-Saclay, PSL Research University, Centre Universitaire, Orsay, 91471, France

LOA, École Polytechnique, ENSTA ParisTech, CNRS, Université Paris-Saclay, 828 Boulevard des Maréchaux, 91762 Palaiseau Cedex, France






FTMC - Center for Physical Sciences and Technology, Savanoriu Ave. 231, Vilnius, LT-2300, Lithuania


THALES AVS - MIS, 2 Rue Marcel Dassault, Vélizy-Villacoublay, 78140, France

cedric.thaury@ensta-paris.fr
LOA, École Polytechnique, ENSTA ParisTech, CNRS, Université Paris-Saclay, 828 Boulevard des Maréchaux, 91762 Palaiseau Cedex, France

§ ABSTRACT 
Laser-plasma accelerators represent a promising technology for future compact accelerating systems, enabling the acceleration of tens of pC to above 1GeV over just a few centimeters. Nonetheless, these devices currently lack the stability, beam quality and average current of conventional systems. While many efforts have focused on improving acceleration stability and quality, little progress has been made in increasing the beam's average current, which is essential for future laser-plasma-based applications. In this paper, we investigate a laser-plasma acceleration regime aimed at increasing the beam average current with energies up to few-MeVs, efficiently enhancing the beam charge. We present experimental results on configurations that allow reaching charges of 5-30nC and a maximum conversion efficiency of around 14%. Through comprehensive Particle-In-Cell simulations, we interpret the experimental results and present a detailed study on electron dynamics. From our analysis, we show that most electrons are not trapped in a plasma wave; rather, they experience ponderomotive acceleration. Thus, we prove the laser pulse as the main driver of the particles' energy gain process.


Highly-efficient electron ponderomotive acceleration in underdense plasmas
    C. Thaury
    August 5, 2024
==========================================================================





§ INTRODUCTION

Since their proposal <cit.>, laser-plasma accelerators (LPA) have interested the scientific community for their ability to produce accelerating gradients 10^3 - 10^4 times those of conventional systems (i.e., ∼100MV/m). The extremely intense electric field would allow for a scaling-down of future accelerators, representing a cost-effective alternative to state-of-the-art linear accelerators (LINACs) and radio frequency cavities. The production of stable, low-emittance, and highly energetic monochromatic LPA electron beams <cit.>, as well as recent optimization studies <cit.>, have dominated the literature. While these results demonstrate the potential of laser-plasma accelerators, their beam properties are still far from those of conventional systems. For instance, current medical electron LINACs can produce up to 10-10^2 μA, while LPAs deliver only tens-nA <cit.>. Naturally, one direct method of increasing the average current of LPAs comes from increasing the beam's charge. Using high-Z gases such as nitrogen and argon is proven as an effective way of enhancing the charge up to the nC-level <cit.>. Recently, using a 27TW, Ti:Sapphire laser and a pure nitrogen plasma with a density n_e≈3.6×10^19cm^-3, it was possible to accelerate around 15nC to few-MeVs, with divergences exceeding 100s mrad <cit.>. This potentially paves the way for μA-level LPAs, marking an important milestone for laser-plasma-based applications. The acceleration of such beams is believed to be associated with the ionization injection of electrons in multiple plasma periods. Once trapped, the particles are subsequently accelerated by the plasma electric field, producing large energy spectra up to a few-tens of MeV.

In this work, we explore the interaction between a superintense laser pulse and a pure nitrogen plasma with densities in excess of n_e=0.01 n_c, where n_c(cm^-3) = 1.1×10^27/λ_0^2(nm) and here λ_0=800nm. Specifically, through a parametric experimental campaign performed at “Laboratoire d'Optique Appliquée" (LOA), the charge-per-Joule metric serves as a straightforward method in gauging the efficiency of the different configurations of interest. By varying the laser energy, plasma density and gas nozzles, we study the conditions to produce highly divergent (i.e., >100s mrad), few-MeV electron beams with charges of 5-30nC and a maximum laser-to-electron energy conversion efficiency around 14%. Through further investigation of the experimental configurations using Fourier-Bessel Particle-In-Cell (FBPIC) <cit.> simulations, we identify three acceleration mechanisms: Ponderomotive Acceleration (PA), Wakefield Acceleration (WA) and Direct Laser Acceleration (DLA). Specifically, the numerical analysis underlines that, upon interaction with nitrogen, the driver pulse radially expels most electrons through its ponderomotive force. Consequently, our study challenges the wakefield's role in accelerating electrons within the configurations of interest, as commonly attributed.










§ HIGHLY-EFFICIENT ACCELERATION OF CHARGED ELECTRON BEAMS

The experiment was performed using Salle Jaune's 60TW Ti:Sapphire laser system able to produce linearly polarized pulses of λ_0 = 813nm central wavelength and a Full Width at Half Maximum (FWHM) duration of 30fs, the driver beam in Figs. <ref>(a) and (b). The laser pulse is focused using an f/4 Off-Axis Parabola (OAP) on the gas target, as shown in Fig. <ref>(b), leading to a FWHM focal spot of 5± 0.3 μm. We estimate that the maximum laser energy on target is 1.7±0.2J of which ∼ 57% is contained within the central spot, corresponding to the first minima ring. This corresponds to a peak laser intensity I_0 ≈ 1.5×10^20W cm^-2 and a maximum normalized vector potential a_0 ≈ 8.5. We use pure nitrogen and two different cylindrically symmetric supersonic gas nozzles of 2mm and 0.4mm exit diameter. The latter is a fused silica nozzle obtained via hybrid 3D laser machining technique <cit.> produced at the Center for Physical Sciences and Technology in Lithuania. As depicted in Fig. <ref>(b), using a Phasics wavefront sensor <cit.> and a probe laser pulse we are able to perform plasma density measurements.

This figure also illustrates the Beam Profile Monitor (BPM), allowing to perform shot-to-shot charge and beam divergence measurements. Specifically, the beam charge is determined using an absolutely calibrated tritium capsule <cit.>. For each configuration, we calculate the average beam charge over ten consecutive shots, with the statistical error defined as the Root Mean Square Deviation (RMSD). The BPM consists of a motorized Lanex Regular Carestream screen with a diameter of 75mm, positioned on the laser axis, and a 16-bit CCD camera to collect the electron beam within a given solid angle. To measure the energy spectra, instead, we remove the BPM and position a motorized electron spectrometer on the laser axis, as shown in Fig. <ref>(b). Each spectrum is determined as the average over ten shots, with the corresponding RMSD. The spectrometer is composed of a magnetic dipole (B_max = 0.44T) with a 2mm diameter pinhole at the entrance. The selected electrons are subsequently deviated on a Lanex screen calibrated from 1.2MeV to 14.7MeV.


In Fig. <ref>(a) the continuous curves represent the charge-per-Joule (Q_/J,Ω) achieved with the 0.4mm nozzle within a ∼0.25 sr solid angle of the BPM, for three different plasma densities. Here, E_L represents the estimated laser central spot energy. The PIC simulations (dashed lines) reproduce the charges obtained under these experimental conditions. Further details about this set of simulations can be found in the Supplementary Materials (Figs. S1 and S2). In the experimental measurements, at n_e = 0.01 n_c we notice that the charge-per-Joule from 7.5nC/J/sr at E_L=0.24J increases to around 10nC/J/sr for E_L=0.48J. At higher plasma densities, instead, we observe a stronger dependency on the laser energy. Specifically, for n_e=0.02 n_c the charge-per-Joule increases with the laser energy, until reaching E_L>0.24J, where it reaches a maximum of ∼69nC/J/sr and subsequently starts to decrease. Similarly, the curve at n_e=0.05 n_c tends to decrease with the laser energy. The simulations show that this behavior is due to the limited BPM collecting angle: the dimension of the electron beams produced at these laser energies exceeds that of the BPM scintillating screen. Hence, some electrons are not co    llected by the diagnostics, causing a decrease in the charge-per-Joule. 

Furthermore, in Fig. <ref>(a) we also notice that the curves at n_e = 0.02 n_c and n_e = 0.05 n_c superimpose, highlighting the presence of a charge-per-Joule saturation effect. However, the numerical study seems to confirm that the superposition of these curves is somehow magnified by the limited collecting angle. Indeed, even an increase in the plasma density yields more divergent electrons that fall out of the BPM measuring cone. With this nozzle, we estimate a maximum of 82nC/J/sr at E_L = 0.12J (i.e., corresponding to a total charge of 2.5nC), with a conversion efficiency η = 6 %. Here, we define the conversion efficiency as the ratio of the total electron energy to the total laser energy on target. 

In Fig. <ref>(b), instead, we show the results obtained with the 2mm nozzle at n_e = 0.29 n_c and n_e = 0.59 n_c. In this figure, we move the BPM closer to the nozzle, allowing the collecting cone to increase to around 0.5sr. In Fig. <ref>(b) we notice very similar tendencies and values to what we have previously discussed. With this nozzle, we measure a maximum charge-per-Joule of 89nC/J/sr with n_e = 0.59 n_c at E_L = 0.12J (i.e., 5.7nC), corresponding to an energy conversion efficiency η=14.4%. At full laser energy and at n_e = 0.22 n_c, we reach 55nC/J/sr, allowing to measure a maximum charge of ∼ 28 nC and an efficiency η = 9.2%. The same efficiency is also estimated for the same laser energy at n_e=0.59 n_c.

Fig. <ref>(c) and (d) illustrate examples of electron energy spectra measured around the beam center. The curves of Fig. <ref>(c) were obtained with the 2mm nozzle at n_e=0.29 n_c for different laser energies. Thus, we observe that the shape of the energy spectrum and the beam's average energy remain constant across different laser energies ranging from 0.12J to 1J, with an average energy of approximately 5.25±0.12MeV, calculated within the range 1.2MeV and 14.7MeV. We also measured the electron beam energy at different horizontal angles as shown in Fig. <ref>(d). From this figure, it is possible to conclude that the most divergent electrons are also the least energetic. Indeed, the average energy drops from ∼ 6MeV at the central position (i.e., Θ_x = 0) to ∼4.5MeV at the extremes (i.e., Θ_x = ±245mrad), within the range 3-10MeV.

In Fig. <ref>(a-d) we present some single-shot images of the BPM scintillating screen. Specifically, these images refer to the 2 mm nozzle at n_e = 0.29 n_c and n_e = 0.59 n_c for two different laser energies. The white lines represent the transverse and longitudinal beam profiles passing through the maxima of each image. Coherent with our prior discussion, Figs. <ref>(a-d) prove that increasing either the plasma density or the laser energy leads to larger beam sizes. Indeed, from these figures, we estimate FWHM divergences between 200mrad and 440mrad, highlighting the influence of varying plasma density and laser energy on the beam divergence.








§  NUMERICAL ANALYSIS



 §.§ Simulation setup

In this section, we present a numerical study that allows us to interpret the experimental results. We first investigate the charge-per-Joule saturation effect and then discuss the different acceleration mechanisms.

The numerical study was performed employing the 3D code FBPIC, which uses a cylindrical grid with azimuthal decomposition. Concerning the simulation setup, we define a 30 μm density ramp, allowing the laser pulse to focus with limited energy losses at the beginning of a 1500 μm plateau (Fig. S3 in Supplementary Materials). We assume the nitrogen gas is preionized up to N^+3, corresponding to the first three L-shell electrons, for numerical ease. The investigation considers four different plateau plasma densities between n_e=0.02 n_c and n_e=0.18 n_c, corresponding to the full L-shell ionization (i.e., N^+5). The laser considered is a 30 fs-Gaussian beam propagating along z and polarized in the x direction, with energies (E_L) ranging from 0.05J to 1J and a waist w_0=3 μm. Regarding the numerical parameters, we employ a (r,z) mesh with Δ z = λ_0 / 24 and Δ r = 5 Δ z, where λ_0 = 800 nm is the laser wavelength. Finally, three azimuthal modes (m=0-2) are considered, and the macroparticles per cell along r, z and θ are set to 1, 1 and 4 respectively.



 §.§ Efficiency saturation effect


We now intend to explain the charge-per-Joule saturation effect observed experimentally and discussed in the Section <ref>. Thus, in Fig. <ref> we present the charge-per-Joule (Q_/J) within a 4πsr solid angle as a function of the laser energy and plasma density, derived from linearly interpolated numerical results. We only consider electrons with a minimum energy E=2MeV, which can be relevant for a number of low-energy applications such as industrial X-ray tomography. The white dashed lines in Fig. <ref>, instead, refer to charge-per-Joule isolines. From this figure, it is possible to notice that for E_L≳0.12 J and n_e≳0.03 n_c the charge-per-Joule slowly increases from 40nC/J to around 50nC/J with the laser energy. This region is outlined by black dotted lines for visual reference. In accordance with the experimental findings discussed in Section <ref>, we observe that within this region, increasing the plasma density at a fixed laser energy does not result in higher charges-per-Joule.

The numerical analysis underlines that this effect is due to the saturation of the conversion efficiency with respect to the plasma density. More precisely, the laser interacts with the plasma over a characteristic length approximately equal to half the pump depletion length, L_pd∝ n^-1_e <cit.>. Simultaneously, increasing the plasma density results in more electrons being accelerated per unit length. Therefore, despite the reduced interaction length at higher densities, the faster ionization ultimately leads to the same amount of charge being accelerated. This analysis is corroborated by the energy spectra in Fig. <ref>(b), where we consider electrons with energies in the range 2 MeV<E<15 MeV, comprising over 80% of the charge above 2MeV. Indeed, we notice that the curves at E_L=1J for n_e=0.03 n_c and n_e=0.06 n_c overlap, presenting an average energy of 4.7MeV. For these configurations, we estimate the efficiency to be η∼21%. A similar behavior is also observed at E_L=0.12J, where we estimate the average energy to be 5.7MeV and the conversion efficiency is η∼18% for both densities. In other words, regardless of variations in plasma density, the same amount of laser energy ionizes and accelerates the same number of electrons to the same average energy.



 §.§ Electron dynamics and acceleration mechanisms

Before discussing the acceleration mechanisms in detail, we intend to briefly describe the plasma structures that develop during the interaction. Thus, Fig. <ref>(a1) depicts the plasma density (top half) and corresponding radial plasma field (bottom half) obtained with n_e=0.03 n_c and E_L=1J. In this figure, we notice the presence of a long and rapidly-changing channel-like structure, with a characteristic length L≫λ_p, where λ_p=4.7 μm is the plasma wavelength. This massive sheath is formed by nitrogen L-shell electrons and it is filled with K-shell electrons, continuously accumulating and flowing through the channel.

In the bottom half of Fig. <ref>(a1), we highlight that the radial focusing field generated by the L-shell electron sheath confines K-shell electrons close to the laser axis. Consequently, this dense concentration of particles on axis shields the longitudinal wakefield and hinders the formation of ion cavities. Nonetheless, inside of this structure, we can still find rapidly changing density modulations, which can generate a longitudinal wakefield. As we will discuss in the following, these modulations can contribute to the electron acceleration process. Close to the laser intensity peak (i.e., z-ct≈90 μm), instead, the ponderomotive force allows for effective charge separation and the formation of a cavity in front of the channel structure.

In order to understand how electrons gain energy, we calculated the work performed by the laser and plasma electric fields. This task was carried out employing the numerical tool FBPIC-Electric Work Profiler (FBPIC-EWP) <cit.>. This code allows to estimate the work exerted on FBPIC-tracked electrons by both the laser and plasma electric fields, exploiting FBPIC modal decomposition. Hence, if we consider a single electron, in the time interval [0, t] an electric field 𝐄_W,L performs the work

    W_W,L(t) = -e∫^t_0𝐄_W,L·𝐯 dt',

where the subscripts W and L denote the wakefield and laser field contribution respectively, 𝐯 is the electron velocity and e is its charge.

Fig. <ref>(a1) illustrates two examples of K-shell electron trajectories (black curves) on the (z-ct, x) plane experiencing ponderomotive acceleration. L-shell electrons exhibit a similar dynamic as will be discussed in the following and in the Supplementary Materials (Fig. S6). Considering the case denoted by the continuous trajectory in Fig. <ref>(a1), we estimate that the particle crosses N_oc = 12 laser optical cycles before being expelled at the instant t^*=50fs, denoted by a circle in Fig. <ref>(a1). Here, t=0 corresponds to the electron ionization time. Similar to what is observed in other acceleration regimes <cit.>, this sort of dynamic is typical of ponderomotive electrons: they slip through several optical cycles experiencing low energy gains, as shown in Fig. <ref>(a2). Here, we plot the electron kinetic energy and, for simplicity, the work done by the plasma and the laser along the radial direction, defined as W^r_W,L=W^x_W,L + W^y_W,L. We estimate that the laser provides 6.8MeV in radial push. Simultaneously, the electron loses 3.2MeV crossing regions of space where the focusing radial wakefield opposes the ponderomotive push (i.e., at z-ct≈85 μm). Concerning the longitudinal dynamics, instead, the numerical analysis underlines that at t=22fs (i.e., z-ct≈93 μm) the electron crosses the wakefield decelerating region in the front cavity, causing a 1MeV loss, while also the laser performs a negative work around 2.1MeV. Finally, the electron leaves the laser with an energy E=0.5MeV.

The particle denoted with the dashed trajectory in Fig. <ref>(a1) experiences a somewhat similar acceleration process. It slips through the laser field oscillating over N_oc = 21 optical cycles, before leaving the laser field at t^*=83fs, as denoted by the circle on the electron trajectory. From Fig. <ref>(a3) we deduce that in this case, the laser also provides most of the energy, with a radial push of 2.7MeV, while the wakefield exerts a pull of 1.1MeV. Longitudinally, the particle receives 0.7MeV from the plasma field, since it experiences its accelerating phase at t=50fs (i.e., z-ct≈86 μm), while the laser performs a negative work of 0.8MeV. The electron leaves the laser pulse with an energy of E=1.5MeV. Moreover, for t>83fs in Fig. <ref>(a1) we note that this particle remains closer to the laser axis and is radially trapped in the channel-like structure discussed above. Here, it performs radial oscillations until leaving the channel without a significant difference in energy.


Having discussed the ponderomotive acceleration through two examples, we now present a statistically relevant study distinguishing between the number of electrons accelerated by the laser or the wakefield. Moreover, we provide an overview of other acceleration mechanisms we identified, and more details can be found in the Supplementary Materials (Figs. S4 and S5). Therefore, for each tracked electron at the instant t, we now define the laser-to-plasma work ratio as

    χ(t) = |W_L(t)/W_W(t)| - 1,

allowing to compare the laser and plasma field contributions. A positive χ value clearly indicates the laser's central role in driving the electron acceleration process. For coherence with Section <ref>, in Fig. <ref>(b1) we plot the beam angular distribution for both K- and L-shell electrons with E>2MeV at the last iteration. Each electron in the top half of this figure is colored based on its χ value. Thus, we define two criteria for χ and we identify the corresponding electron populations. In the range -1≤χ<0 we find yellow electrons that gain most of their energy from wakefield acceleration.

The red particles, instead, have χ≥0 and are primarily accelerated by the laser. In this population, we find ponderomotive electrons, displaying dynamics similar to the two examples in Fig. <ref>(a1). As discussed below, some of these electrons can also experience direct laser acceleration <cit.>. We estimate that around 70% of the particles exhibit positive values of χ, proving the laser as the primary driver in particle acceleration. In the bottom half of Fig. <ref>(b1), instead, the color represents the energy of each particle. The average energy for E>2MeV is around 9MeV for both laser- and wakefield-accelerated electrons, and we notice a concentration of higher energies closer to the laser axis, which is consistent with the experimental observation. For instance, if we consider a solid angle of 0.25sr (white circle in Fig. <ref>(b1)), the average energy increases to 14MeV. 

Fig. <ref>(b2) depicts the (z-ct, E) phase space of laser- and wakefield-accelerated K-shell electrons at the final iteration of the simulation. This figure highlights the different behaviors of laser-accelerated electrons. As previously mentioned, in this population we find particles undergoing ponderomotive acceleration. We estimate that around 85% of laser-accelerated K-shell electrons gain up to m_ec^2a_p≈8MeV through the ponderomotive push <cit.>. Here, a_p≈16 is the maximum laser normalized vector potential in plasma. Instead, the remaining 15% experiences direct laser acceleration. Fig. <ref> presents an example of electron trajectory undergoing DLA (continuous curve). These particles are trapped in the ion cavity in front of the channel structure via ionization injection. Once trapped, they overlap with the driver pulse and perform oscillations along the laser polarization direction. Subsequently, these oscillations can lead to a gain in longitudinal momentum via the e𝐯×𝐁 term of the laser. The longitudinal wakefield reduces the dephasing between the electrons and the laser, ultimately allowing these particles to reach 10s-100s MeV <cit.>. In the example of Fig. <ref>, the electron reaches a maximum energy of 85MeV, with 79MeV attributed to the laser.

In Fig. <ref>(b2) we can also observe wakefield-accelerated K-shell electrons (yellow dots). Specifically, the numerical analysis highlights the presence of “traditional" wakefield acceleration, where the particles are trapped in phase with the wakefield within the front ion cavity long enough to reach 10s-100sMeV. Additionally, we recognize wakefield accelerated particles displaying a behavior somewhat similar to the dashed trajectory of Fig. <ref>(a1). Once ionized, they initially receive most of their energy from the laser ponderomotive push and they subsequently slip into the channel, where they radially oscillate until escaping the structure with few-MeVs in energy. However, once inside the channel, they undergo acceleration in the longitudinal wakefield phase, induced by the plasma density modulations mentioned earlier. The numerical analysis shows that this plasma field contribution exceeds the laser initial push and, in this sense, they undergo wakefield acceleration. Nonetheless, unlike traditional wakefield acceleration, where particles are trapped in phase with the wakefield within the front ion cavity, these electrons continuously slip through the channel. In Fig. <ref> we provide an example of this sort of non-traditional wakefield acceleration (dashed curve). At the end of the simulation, the particle has gained 4.3MeV in energy, of which ∼3MeV are provided by the wakefield.

Analogously, Fig. <ref>(b3) displays the (z-ct, E) phase space of laser- and plasma-accelerated L-shell electrons at the final iteration of the simulation. We estimate that around 83% of L-shell electrons with E>2MeV undergo PA, while the remaining fraction is mainly accelerated via WA while momentarily crossing the longitudinal wakefield accelerating phase.



§  CONCLUSION AND OUTLOOK

In this paper, we studied a regime allowing the increase of the average current of laser-plasma accelerators with energies at few-MeVs, through the enhancement of the beam charge. With an extensive experimental campaign, we were able to produce charges of 5-30nC, with average energies around 5MeV. Notably, employing a 0.12J laser pulse, we achieve a charge of 5.3nC with a conversion efficiency of 14.4%, one of the highest recorded to date. Improving the laser focal spot quality could lead to efficiencies exceeding 20%, as shown by the numerical study. These results are promising for future LPAs, paving the way for unprecedented average currents. Novel J-class lasers with 100W in average power are emerging <cit.> and they would allow to exceed the μA-level. Considering, for instance, the configuration discussed above (i.e., 5.7nC at 0.12J), we can easily estimate that the maximum achievable average current is ∼5μA, assuming 100W in laser average power. With such capabilities, this electron source emerges as a promising candidate for various applications needing few-MeV electrons, including X-ray tomography <cit.> and irradiation studies.

In configurations similar to those studied here, electron trapping via ionization injection followed by wakefield acceleration is typically considered the dominant acceleration process. However, in this paper, we were able to show that most particles are not injected in plasma cavities, as also discussed in other works present in the literature <cit.>. More specifically, we proved that the electrons primarily gain energy through ponderomotive acceleration, establishing the laser pulse as the main driver in the energy gain mechanism.


This project has received funding from the European Union’s Horizon 2020 research and innovation program under grant agreement n°101020100.


apsrev4-2




