Photonic beam-combiner for visible interferometry with SCExAO/FIRST: laboratory characterization and design optimization
Manon Lallement Elsa Huby Sylvestre Lacour Guillermo Martin Kevin Barjot Guy Perrin Daniel Rouan Vincent Lapeyrere Sebastien Vievard Olivier Guyon Julien Lozi Vincent Deo Takayuki Kotani Cecil Pham Cedric Cassagnettes Adrien Billat Nick Cvetojevic Franck Marchis
 
Exploring the magnetic and thermal evolution of a coronal jet
    [
    September 9, 2024
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§ ABSTRACT 
Integrated optics are used to achieve astronomical interferometry inside robust and compact materials, improving the instrument's stability and sensitivity. In order to perform differential phase measurements at the Hα line (656.3 nm) with the 600-800 nm spectro-interferometer FIRST, a photonic integrated circuit (PIC) is being developed in collaboration with TEEM Photonics. This PIC performs the interferometric combination of the beams coming from sub-apertures selected in the telescope pupil, thus implementing the pupil remapping technique to restore the diffraction limit of the telescope. In this work, we report on the latest developments carried out within the FIRST project to produce a high performance visible PIC. The PICs are manufactured by TEEM Photonics, using their technology based on K_+:Na_+ ion exchange in glass. The first part of the study consists in the experimental characterization of the fundamental properties of the waveguides, in order to build an accurate model, which is the basis for the design of more complex functions. In the second part, theoretical designs and their optimization for three types of combiner architectures are presented: symmetric directional coupler, asymmetric directional couplers and ABCD cells including achromatic phase shifters.







*Manon Lallement,  manon.lallement@obspm.fr

2   




§ INTRODUCTION

 



Interferometric techniques in astronomy consist in recovering the source spatial intensity distribution with an angular resolution as fine as λ/2B, with B the baseline length. Implemented on a monolithic telescope, sparse aperture masking (SAM) is a technique providing a spatial resolution down to λ/2D with λ/D the telescope diffraction limit. This technique consists in applying a non-redundant mask with holes at the pupil plane defining several sub-apertures. At the telescope focal plane, the image no longer corresponds to the point spread function of the telescope's pupil but to the superimposition of fringe patterns created by each pair of interfering sub-aperture beams. The non-redundant configuration of the mask means that each baseline, i.e each pair of sub-apertures separated by a baseline vector B⃗, produces a unique fringe pattern. The information carried by each baseline can be retrieved independently. This is well illustrated in the Fourier domain, in which each baseline information, i.e phase and contrast of the associated fringes, is carried by a single peak isolated from the others. All this ensures that there is no blurring effect between the fringes in the presence of residual atmospheric turbulence or phase perturbations due to the optical bench, and that one can recover information at the diffraction limit of the telescope, and even below. The SAM technique limitations are (1) the reduced collecting area and thus the sensitivity limit and (2) the speckle noise which remains at the scale of one sub-aperture, limiting the contrast of the fringes.

To address these limitations, the pupil remapping technique<cit.> theoretically gives access to the whole pupil: instead of using a sparse aperture mask, a micro-lens array samples the whole pupil in several sub-apertures and injects their light in single-mode fibers spatially filtering the wavefront, thus removing the speckle noise. Sub-aperture pairs are recombined non redundantly or pairwise so that the information carried by each baseline can be retrieved independently. In addition, the interferograms can be spectrally dispersed. The spatial intensity distribution of the source is thus recovered with an angular resolution of λ/2D with an increased accuracy compared with SAM and with a better (u,v) plane and spectral coverage.



FIRST, which stands for Fibered Imager FoR a Single Telescope, was built at the Observatoire de Paris to validate the concept of pupil remapping coupled with spectroscopy. From 2010 to 2013, the instrument was used on the 3m-Shane Telescope at the Lick Observatory<cit.>. In 2013, FIRST was installed at the 8.2 m Subaru Telescope <cit.> on the Subaru Coronagraphic Extreme Adaptive Optics platform (SCExAO)<cit.>, enhancing the ultimate angular resolution of the instrument, with λ/D = 16.5 mas at 656.3 nm.

SCExAO delivers a Strehl ratio of 50% to 60% in the visible at 750 nm. FIRST is thus leveraging the wavefront stability provided by SCExAO, making long exposure up to 1 second possible, without loosing the fringe contrast.

Two versions of the FIRST instrument are currently on the SCExAO's bench: FIRST version 1 (FIRSTv1) and FIRST version 2 (FIRSTv2) depending on the interferometric combination method. Table <ref> shows their respective features. Close binary stars were detected and spectrally characterized with FIRSTv1 using closure phase measurements <cit.>. Currently, our efforts are focusing on pushing the detection limits of the instrument, with the ultimate goal of detecting exoplanetary systems. For development purposes, a replica of FIRSTv2 has been built in the laboratory of Observatoire de Paris<cit.>, where the PIC prototypes are characterized prior to further validation on the sky. 

















Young gas giant exoplanets are particularly interesting for FIRSTv2. Studies based on the populations of exoplanets detected by radial velocities<cit.> show that the distribution of gas giant is supposed to be maximal for systems with a separation of 1-3 au, which corresponds to an angular separation of 7-28 mas at 140 pc, the distance to the Taurus Nuclear Cloud. This region cannot be probed by classical imager on 8m-telescopes, but the angular resolution offered by the interferometric technique places it within reach of FIRST.

Moreover, in the visible, a dynamic range of 10^6 to 10^9 is required to differentiate between the light from an exoplanet and its host star<cit.>. This is currently out of reach for interferometric techniques like FIRST. However, at the protoplanetary state, gas giants are less than 4 Myr old and are still accreting matter from their surrounding disk, inducing a strong emission in the hydrogen line at 656.3 nm (Hα)<cit.>. As a consequence, the contrast between the planet and the star at this particular line is lowered down to 10^2 - 10^3, making them easier to detect in the visible. Three protoplanet detections have been confirmed using Hα imaging. This is the case for protoplanets PDS70b and PDS70c detected using the MUSE integral field spectrograph<cit.> as well as AB Aurigae b, detected at various wavelengths by several instruments and in particular, at the Hα line with VAMPIRES installed on SCExAO<cit.>. 

With a spectro-interferometer, Hα differential phase measurement is the equivalent of Hα imaging. It consists in comparing the complex visibility phase at the Hα line (where the protoplanet is brighter) to the complex visibility phase in the continuum (where it is too faint to be detected). This technique has recently been implemented using the high precision phase measurements with the GRAVITY instrument to detect the broad line region around a quasar<cit.>. The FIRST instrument can perform this measurement at high angular resolution in the visible, targeting the Hα line. For that purpose, its spectral resolution, sensitivity and dynamic range are currently being enhanced as specified in Table <ref>.



In the FIRST instrument, the telescope's pupil is divided into sub-apertures thanks to a micro-lens array which couples the light from the sub-apertures into Polarization Maintaining (PM) Single-Mode Fibers (SMF). 

Optical delay lines are used to compensate for the fiber length difference and reach the null Optical Path Difference (OPD). In FIRSTv1, SMFs are used to remap the input pupil sub-apertures into a non-redundant configuration in the output pupil<cit.>. Thanks to this multiplexing, each pair of sub-apertures produces a fringe pattern with a unique spatial frequency. The interferometric combination of the beams is achieved in free-space following a Young slit-like experiment. FIRSTv2 is currently under development on a testbed at the Observatoire de Paris. In this upgraded version of the instrument, SMF inject the sub-apertures light into a Photonic Integrated Circuit (PIC) where the beams are recombined pairwise. 

The output signal is dispersed with a spectrograph with R ∼ 400 in FIRSTv1 and R ∼ 3600 in FIRSTv2. 

Fig. <ref> presents FIRSTv2 optical system and the sample of data acquired in the laboratory: 5 sub-apertures are recombined accounting for 10 baselines. In the configuration presented in Fig. <ref>, each baseline is encoded in four horizontal spectra: the interferometric combinations are performed in the PIC by 2x2 directional couplers, i.e there are two outputs per baseline. The light from both PIC outputs is vertically split by a Wollaston prism to avoid fringe blurring induced by the birefringence in the setup, and horizontally dispersed on the camera thanks to a Volume Holographic Grating (VPH). It covers the  600-800 nm spectral band with a resolution of R∼ 3600 at 670 nm. To further sample the fringes, a segmented mirror is used to temporally modulate the phase of the input beams, by applying piston commands to the segments corresponding to each sub-aperture. 


 




The PIC device is the heart of the upgraded version of the instrument FIRSTv2, that will enhance the stability, accuracy and sensitivity compared to FIRSTv1. The design and characterization of a high performance visible PIC is a critical step in these developments, and we report in this paper on the latest developments carried out in collaboration with TEEM Photonics to optimize the building blocks required to produce a complete device. Specifications for the PIC component suited for FIRSTv2 are listed in Section <ref>, and the manufacturing process developed by TEEM Photonics based on K_+:Na_+ ion exchange technology (ioNext) is described. In Section <ref>, the characterization of TEEM Photonics' standard straight waveguides in terms of single-mode spectral range and mode field profile are presented. These measurements are critical to define a model of the waveguide 3D diffused index profile thanks to the BeamPROP modeling software, which is the working basis for the design of more complex functions. In Section <ref>, we present the theoretical models and design optimization of three types of combiners investigated so far: 1) directional couplers are the simplest design and were characterized in laboratory, showing a weak chromatic behavior, 2) asymmetric directional couplers can be tuned to make the coupler even less chromatic, 3) ABCD cells are more complex and comprise splitters, phase shifters and combiners, but are more convenient as they provide additional measurements of the interference state, thus avoiding the need for temporal modulation of the phase. The latest PIC prototypes and test chips including single combiner functions for characterization and validation are also presented. In Section  <ref>, we conclude and present the next steps.






§ FIRSTV2 VISIBLE PHOTONICS INTEGRATED CIRCUIT FOR HIGH PERFORMANCE BEAM COMBINATION

 



 The FIRSTv2's high throughput visible beam combiner is being developed and characterized<cit.> to enhance the stability and accuracy of the measurements of interferometric observables. Compared to bulk optics beam combination, photonic integrated circuit combination is more robust, less sensitive to thermal variations, mechanical constraints and alignment errors. This is critical to meet FIRSTv2 dynamic range specification presented in Table. <ref> as phase measurement accuracy directly limits the achievable dynamic range. In FIRSTv1, the free-space interferometric combination leads to spatially sampled fringes<cit.>. In FIRSTv2, the fringes are temporally modulated by applying piston commands to a segmented mirror located between the pupil and the micro-lens array. The signal of interest is thus condensed into a few pixels, instead of a few hundreds, reducing the read-out noise and enhancing the phase measurement SNR. Our efforts currently focus on performing the combination of 5 input beams, but for future FIRST upgrades, PIC devices can relatively easily be scaled to a higher number of sub-aperture pairs by densifying the design or by duplicating the PIC devices.
 





 §.§ Architecture of the FIRSTv2 photonic beam combiner

 
FIRSTv2 interferometric combination scheme consists in combining the light of 5 sub-apertures pair by pair, as presented in Fig. <ref>. This design is called a 5-telescopes combiner, or 5T-combiner, by keeping the same name convention as PICs developped for long baseline interferometers combining the light of different telescopes.

Each of the 5 inputs are split in four thanks to two cascaded Y junctions and are recombined with the other four thanks to combiners. As presented in  Fig. <ref>, combiners can be Y junctions (10 outputs in total, one per baseline), directional couplers (20 outputs) or ABCD cells (40 outputs). Fig. <ref> presents an overview of these combiners. In order to properly sample the fringes around the null OPD when using directional couplers and Y junctions, a phase modulation of the input beams is required. This phase modulation is achieved by adding piston to the corresponding segments of the segmented mirror following a 20 steps sequence running at 20 Hz. 
 The ABCD cell provides four measurements of the interference state between the input beams, with four distinct phase differences, allowing the fringe reconstruction with a single image. The use of ABCD cells thus increase the observing time efficiency, however they are more complex to design, as they require splitters, directional couplers and achromatic phase shifters.


 





 §.§ Specifications for the FIRSTv2 5T-combiner

 
As explained in Sec. <ref>, the FIRSTv2 science case is the detection of accreting protoplanets by differential phase measurement at the Hα line. It consists in comparing the complex visibility phase at the Hα  line (at 656.3 nm, where the protoplanet is brighter) to the complex visibility phase in the continuum (from 630 to 650 nm and from 660 to 780 nm, where it is not detected). Therefore, we defined PIC specifications depending on the wavelength, as shown in Table. <ref>, with stronger constraints around the Hα line, where the signal of interest is expected. Indeed, the photon transmission in this spectral channel is critical, while the continuum signal is evaluated on ∼ 700 channels when using the whole spectral range. The specifications concern several aspects:

    
  * overall transmission: FIRSTv2 PIC overall transmission must be greater than 50% in the 630 to 780 nm spectral band and greater than 75% around the Hα line.
    
  * insertion or coupling loss: it specifies the amount of energy lost because of the mismatch between the fundamental modes of the PM-630HP fiber, used for the injection, and the PIC waveguide.
    
  * Polarization Extinction Ratio (PER): it characterizes the capability of the PIC building blocks to keep light linearly polarized and propagating along a principal axis. In that case, the PER should be close to 100%. After injecting linearly polarized light along the principal axis, the Polarization Extinction Ratio (PER) corresponds to the ratio between the output power of the light linearly polarized along the principal axis and the output power of the light orthogonally polarized.
    
  * cross-talk: it defines the amount of unwanted light that gets transferred from one waveguide to another. This specification mainly applies to waveguides crossing each other and does not apply to splitters nor combiners.








Specification of FIRSTv2 5T-combiner building blocks, i.e splitters and combiners, are presented in Table. <ref>.  The internal losses are specified in order to keep the overall transmission above the specified level. Transfer rate specifications for splitters and combiners are meant to evenly distribute the light in order to maximize fringe contrast over the spectral band. Each of the 5 inputs are split in four thanks to two cascaded Y junctions. The Y junction transfer rate should be 50± 5% meaning that each of the Y junction outputs contains 50± 5% of the input flux. These 5 inputs are recombined with one another thanks to either the Y junction, directional coupler or ABCD cell combiners. For an ABCD cell, the 25±5% specification means that each of the four ABCD cell outputs contains close to 25% of the input flux.






 §.§ TEEM Photonics K_+:Na_+ ion exchange technology (ioNext)

 
The FIRSTv2 PIC is developed in collaboration with TEEM Photonicshttps://www.teemphotonics.com/, a company based in Grenoble, France. TEEM Photonics ioNext technology consists in K_+:Na_+ ion exchange in a glass substrate. A lithographic mask is used to control the regions where the ion exchange is performed, thus creating gradient-index waveguides featuring a precisely controlled mode field profile and effective index<cit.>. At a final stage, a glue layer and a counter-blade are deposited on top of the glass substrate. They allow for: 1) a larger front surface where V-grooves are bonded ; 2) for a protection of the waveguide which would otherwise be directly on the surface, and thus sensitive to scratches and dust ; and 3) for the symmetrization of the waveguide mode in order to better match the modes of the inserting and collecting optical fibers. 

The SM-630 fiber insertion loss is specified by TEEM Photonics to be 0.13 dB over the single-mode spectral band for a 2 μm wide waveguide. Straight waveguide propagation loss is about 0.25 dB/cm at 780 nm. 




 §.§ The first 5T-beam combiners prototypes



Prior to the present work, two 5T PIC prototypes fabricated with TEEM Photonics ioNext technology were characterized on the FIRSTv2 testbed at the Observatoire de Paris <cit.>. These PICs perform the interferometric recombination of 5 sub-apertures beams with two different types of combiners: one is based on Y junction combiners and is called 5TY, while the other one is based on symmetric directional couplers or X couplers, further detailed in Sec. <ref>, and is called 5TX.

As illustrated in Fig. <ref>, Y junction consists in two single-mode waveguides which merge into one output single-mode waveguide. If the fields propagating in input waveguides are in phase (resp. in phase opposition), they are coupled into the fundamental mode (resp. in radiative modes) of the output waveguide, meaning that half of the interferometric signal is lost in radiative modes in a Y junction. For that reason, Y junctions were not further investigated and are not part of the present study.



The 5T prototypes were characterized in terms of throughput and cross-talk, as defined in Sec. <ref>. Light was injected in one of the five input waveguides and the leakage in the other waveguides was measured. Cross-talk has a mean value of about 1% in both PIC prototype but can reach, for some inputs, 10% for the 5TX PIC and 20% for the 5TY PIC. The throughput was estimated to about 15% (5TY) to 30% (5TX)<cit.>. This low throughput is mainly due to non-optimized combiners and bend curvature radii. Further developments are thus needed to reach the specifications presented in Sec. <ref>, motivating the work presented in the following sections. 

The methodology adopted to develop the PICs is indeed an iterative process. A numerical model of the waveguides manufactured with the ioNext technology is required to feed the BeamPROP modeling software. Combiners are designed and optimized based on the numerical simulations of their performance. Once manufactured, these combiners are characterized in the laboratory to refine the numerical model and tune the design parameters for the next manufacturing run.




§ FUNDAMENTAL PARAMETER ESTIMATION FOR WAVEGUIDE MODELING



In order to model the waveguides built with the ioNext technology and design FIRST PIC's combiners, fundamental parameters estimation of standard, i.e straight, waveguides is needed. In particular, the fabrication process, i.e K+ NA+ ion exchange in glass, creates a diffused index profile in a glass substrate which needs to be recovered. This section presents the laboratory measurements of single-mode spectral range and mode field diameter of ioNext standard straight waveguides which are used to defined the 3D diffused index profile of the waveguide on the BeamPROP software.



 §.§ Single-mode spectral range laboratory measurement


 Figure <ref> presents single-mode spectral range measurement performed by TEEM Photonics engineers. The throughput P=10·log(P_out/P_in) is measured for two straight waveguides test samples G1 and G2. The white light source is fibered and the measurement is performed with a fibered spectrometer. The input power P_in is thus estimated by connecting directly the source fiber to the spectrometer fiber. The output power P_out is measured by inserting the photonic integrated circuit straight waveguide between these two fibers. 

For wavelengths greater than 820 nm, the input light is not guided by the waveguide. When the wavelength decreases and as soon as the input light gets guided through the fundamental mode (m=0) of the waveguide, P_out increases explaining the peak at the cutoff wavelength around 820 nm. In the single-mode range, the coupling efficiency between the fiber and the waveguide fundamental modes decreases with the wavelength. 
An increasing amount of the input light is coupled into the waveguide second mode (m=1) and immediately lost because of the low confinement of this second mode. As soon as the wavelength gets small enough, the energy coupled in the second mode becomes confined and is transmitted through the waveguide: a second peak in the output power appears at this second cutoff wavelength of about 530 nm. The measured single-mode spectral range extends between these two cut-off wavelengths, approximately between 530 to 820 nm for 2 μm wide waveguides.

 


It can be noted that the throughput value should be lower than 0 dB because this passive straight waveguide should only insert losses. This could be explained by the measurement noise or by losses induced by the fiber-to-fiber alignment, performed inside a standard connector, while the fiber-to-waveguide alignment is precisely tweaked, potentially leading to an underestimation of the input power measurement. Also, no information is given regarding the input polarization and as the waveguides are known to be birefringent, a different single-mode spectral range is to be expected between p- and s-polarized light.



 §.§ Mode field diameter laboratory measurement


TEEM Photonics waveguide mode field diameter (MFD) measurements were performed for p- and s-polarized light injection. In this particular experiment, p-polarized (resp. s-polarized) electric field lies in a plane parallel (resp. orthogonal) to the plane of the PIC. The Thorlabs PM-630-HP reference fiber and TEEM Photonics waveguide outputs are imaged with a x40 objective on a Thorlabs CDD. The MFD are calibrated for p- and s-polarized light injection thanks to a reference polarization maintaining fiber with a known 4.5 ± 0.5 μm MFD. The MFD measurements for a horizontal and vertical cross-section of TEEM Photonics waveguide mode are presented in Fig. <ref> and Table. <ref>.





MFD measurements reveal the mode asymmetry induced by the fabrication process: the ion exchange is performed at the glass substrate surface meaning that the waveguides are not buried into the glass, inducing a form birefringence in spite of the glue layer and the counter-blade being used for symmetrization. Currently, the precision of the mode field diameter measurement is not sufficient to define a polarization-dependent model of the ioNext technology that could be taken into account in the BeamPROP simulation to refine the design. As a consequence, a non-polarized mode field diameter is defined by taking the mean values of polarized mode field dimensions. Effective indexes and throughput in both polarizations will be further investigated thanks to the characterization of new PICs, see Sec. <ref>.





















 §.§ Modeling of TEEM Photonics waveguide diffused index profile with the BeamPROP software



Based on the single-mode spectral range and non-polarized mode field diameter measurements, a 3D diffused index profile of TEEM Photonics waveguide is derived on the BeamPROP modeling softwarehttps://www.synopsys.com/photonic-solutions/rsoft-photonic-device-tools/passive-device-beamprop.html. The 3D diffused index profile parameters are presented in Fig. <ref>. 



The glue layer is about 20 μm thick in the y>0 region and is considered as semi-infinite. Its refractive index is n_c=1.49 at 635 nm. The width of the lithographic mask used for the diffusion process is w. The diffusion length in the horizontal (resp. vertical) direction is h_x (resp. h_y). The maximum refractive index difference between the glass substrate and the waveguide core produced by the diffusion process is Δ n.
The waveguide diffused index profile cross-section n(x,y) in the glass substrate (y<0) is defined by: 

    n(x,y<0)=n_o+[Δ n· g(x)· f(y)]^γ,

with n_0=1.52 at 635 nm the glass substrate (y<0) refractive index, and where the functions g(x) and f(x) describe the region where the diffusion takes place: 

    g(x)=1/2{erf[(w/2-x)/h_x)]+erf[(w/2+x)/h_x]  },


    f(y)=exp(-y^2/h_y^2).

For the sake of simplicity in this preliminary model: 1) the diffusion process is assumed to be equivalent in both x and y directions, i.e. h_x=h_y=h ; 2) the relation between the ion concentration and the index n(x,y) is supposed to be linear, i.e γ=1 in Eq. <ref>. In practice, the diffusion length h and maximum refractive index difference Δ n parameters are optimized thanks to laboratory measurements as reported in Sec. <ref> and Sec. <ref>.





§ DESIGN AND OPTIMIZATION OF THREE TYPES OF COMBINERS




Different types of combiners are considered in this section: symmetric and asymmetric directional couplers, and ABCD cells (composed of Y junction, directional couplers and achromatic phase shifters). Laboratory characterization of symmetric directional couplers and design of asymmetric directional couplers and ABCD cells are presented. Directional couplers, also called X directional couplers according to their shape, recombine the light from two input waveguides and lead to two outputs, corresponding to the coupling of the two waveguides with a phase difference of 0 and π. ABCD cells involve a more complex design, including splitters, phase shifters and directional couplers. They provide four outputs which correspond to the coupling of the two input waveguides with a phase difference of 0, π/2, π and 3π/2, see Fig.<ref>. 
 



 §.§ Symmetric directional couplers






  §.§.§ Theoretical model


The directional coupler standard geometry is presented in Fig. <ref>: two waveguides come very close to one another, inducing a coupling between the two by evanescent waves in the so-called interaction zone. For simulation and characterization purposes, the waveguide called "Through" (resp. "Cross") is the waveguide in which the light is injected (resp. not injected). The interaction zone is defined by its length L and the gap distance g between the waveguides. The directional coupler is called symmetric if both waveguides in the interaction zone have the same geometrical properties. 


The normalized output powers P_through and P_cross of a lossless directional coupler<cit.> are given by: 

    P_through = 1-P_cross = 1-(κ^2/Δ^2)· sin^2(Δ· L)


with κ the mode coupling coefficient and Δ defined as:  


    Δ=√((β_through-β_cross/2)^2+κ^2)

with β_through and β_cross the propagation constants of the directional coupler waveguides. In Equation <ref>, F=κ^2/Δ^2 corresponds to the maximum power that can be transferred from one waveguide to the other. For a symmetric directional coupler, as both waveguides in the interaction zone have the same geometrical properties, propagation constants are equal (β_through = β_cross). Therefore F is equal to 1, meaning that all the power in one waveguide can be transferred into the other one, if the interaction zone length is properly chosen. The shortest length of the interaction zone for which the power is entirely transferred from one waveguide to the other is called L_0:100. Equation <ref> shows that the transferred power P_cross is periodic with a period proportional to 2L_0:100, with L_0:100=π/2Δ.
 
The directional coupler transfer rate A:B (with A+B=1) is defined as the proportion of the output flux in each output : 


    P_through/P_through+P_cross : P_cross/P_through+P_cross


The specification for the PIC directional coupler is to transmit 50 ± 10% of the light in each output over the 600 to 800 nm spectral band, i.e to have an achromatic transfer rate of 50:50. In order to get a 50:50 transfer rate, the shortest interaction zone length L_50:50 is L_0:100/2. Other solutions L_50:50,k for the interaction length are derived from Eq. <ref>:


    L_50:50,k=1/Δ(π/4+kπ/2) = L_0:100(1/2+k)   with   k∈ℕ


As the interaction zone length L_0:100 depends on the wavelength, the transfer rate of a symmetric coupler is necessarily chromatic<cit.>. However, a directional coupler can be designed with asymmetric waveguides, leading to β_through≠β_cross. The asymmetry can be tuned in order to compensate for the transfer rate chromaticity, as explained in Sect. <ref>. 




  §.§.§ Laboratory characterization


Three symmetric directional couplers were manufactured by TEEM Photonics using the ioNext technology presented in Sec. <ref>. 

The symmetric directional couplers all have a gap g = 2 μm and three different interaction zone lengths: L=50 μm, 100 μm or 150 μm. Measurements are performed using a P2-830A fiber connected to a white SLED source to inject non-polarized light into the PIC's symmetric directional couplers. A x4 objective is used so that the light from the PIC output feeds an OceanOptics spectrometer. A linear polarizer, located between the objective and the spectrometer, selects the p- or s-polarized light. In this particular case, p-polarized (resp. s-polarized) electric field lies in a plane parallel (resp. orthogonal) to the plane of the PIC.

Transfer rate measurements, as defined by Eq. <ref>, are presented in Fig. <ref>, for both p- and s-polarized light. The objective is twofold: 1) find the interaction zone length L_50:50 to get a 50:50 transfer rate and 2) investigate the wavelength dependence of the transfer rate over the 600 to 750 nm spectral band.


The results presented in Fig. <ref> confirm that the coupling coefficient depends on polarization, i.e waveguides are birefringent. Birefringence is not an issue for our application as long as there is no cross-talk between p- and s-polarized light (i.e no polarization cross-talk) that would reduce the fringe contrast. In the present experiment, the input light is not polarized, such that polarization cross-talk cannot be assessed, but it will be the focus of a future study. Despite the polarization dependency, our experimental measurements show that a compromise can be found for the interaction zone length L_50:50 which is situated between 20 and 40 μm for both polarizations. 
Furthermore, it is noticeable that these symmetric directional couplers are not highly chromatic, in particular for the P polarization. Indeed, the relative standard deviations are smaller than 5.5% over the 600-700 nm range, as shown in Table. <ref>.



























 §.§ Asymmetric directional couplers







  §.§.§ Asymmetric geometries


This section presents two asymmetric directional couplers geometries defined as uniformly asymmetric (UA) and non-symmetric (NS)<cit.> as shown in Fig. <ref>.

  

A uniformly asymmetric directional coupler is composed of one waveguide wider than the other by an amount dw in the interaction zone. For non-symmetric directional coupler, one waveguide is wider than the other one in the interaction zone and it is additionally tapered along the interaction zone. This means that the width is continuously varying from one starting value w to a final value w+dw. 





  §.§.§ Optimization based on BeamPROP simulations


The parameter space (dw,L) is probed to define the optimal set of parameters for each geometry. The optimal solutions found for uniformly asymmetric and non-symmetric directional couplers are detailed in Fig. <ref>. For both types of asymmetric directional couplers, the spectral transfer rates are computed and plotted in Fig. <ref>. For each coupler geometry, the transfer rate is evaluated when injecting in waveguide 1 (“Through” is “waveguide 1” in this case) or waveguide 2 (“Through” is “waveguide 2” in this case). Our simulation results show that the behavior of the coupler weakly depends on the input waveguide. 

A study has also been carried out for tolerancing purposes, by computing the spectral transfer rate for various differential widths and interaction zone lengths around the nominal parameters doublet (dw_n,L_n).


The simulation results are presented in Fig. <ref>, showing that the differential width accuracy is critical, while a change of a few tenths of microns of the interaction zone length has no significant impact. 







 






 §.§ ABCD cell combiners






Y junctions and directional couplers are the simplest combiners that can be designed. However, they provide only one or two measurement points per interfering pairs, which is not sufficient to determine the fringe phase and amplitude. In this case, an additional temporal phase modulation is required, to reach a minimum of four fringe samples. In the FIRST instrument, a segmented mirror located before the PIC is used to vary the relative piston between the beams, pair by pair. For a large number of baselines, the data acquisition procedure thus becomes long and complex, and sensitive to phase perturbations occurring from one frame to the other.
A way to increase the data  acquisition efficiency is to use another type of combiner called an ABCD cell as presented in the following section.



  §.§.§ Theoretical model





The ABCD beam combination is an interferometric scheme<cit.> giving a four point interferometric fringe sampling at its output.  Each of the four outputs is the combination of the two inputs with a different phase as represented in Fig. <ref>.
ABCD cell input 1 and 2, i.e interfering beams, are split in two and are recombined by pairs thanks to two directional couplers. The phases of the four outputs are specified as follows: 


    A: φ_12^A
        B: φ_12^B=φ_12^A+π
        C: φ_12^C=φ_12^A+π/2
        D: φ_12^D=φ_12^A+π/2+π=φ_12^A+3π/2


In order to build an achromatic ABCD cell, a π/2 achromatic passive phase shifter is required. A phase shifter is composed of two parallel waveguides: waveguide 1 and 2. Each waveguide of the same total length, is divided into N segments of optimized widths and lengths. The width variations induce variations of the mode effective index, and thus a difference in the optical path length between the two arms. The phase difference for a phase shifter of length L composed of a N=1 segment is given as follows: 


    Δφ=φ_1-φ_2=2π/λ(n_eff,1-n_eff,2)L


with n_eff,1=A_1+B_1λ+C_1λ^2 and n_eff,2=A_2+B_2λ+C_2λ^2 the effective indexes of waveguides 1 and 2. 
This N=1 segment phase shifter is achromatic if: 


    ∀λ,  ∂φ/∂λ=-2π/λ(Δ n/λ-∂Δ n/∂λ)L=0


with Δ n=n_eff,1-n_eff,2 the effective index difference between waveguide 1 and 2. Following P. Labeye's PhD thesis<cit.>, these equations can be generalized for a design composed of N ≥ 1 segments shifting the phase by φ_0=π/2:


    φ=2π/λ∑_i=1^NΔ n_i L_i=φ_0



    ∂φ/∂λ=-2π/λ∑_i=1^N(A_i/λ-C_iλ)L_i=0

with Δ n_i=n_eff,1,i-n_eff,2,i=(A_1,i-A_2,i)+(B_1,i-B_2,i)λ+(C_1,i-C_2,i)λ^2=A_i+B_iλ+C_iλ^2 the effective index difference between waveguide 1 and 2 in segment i. Equation <ref> is true for all wavelengths only if:


    ∑_i=1^NC_iL_i=0   and  ∑_i=1^NA_iL_i=0.


Consequently, one can derive that:


    ∑_i=1^N(B_iL_i)=φ_0/2π


is also a condition for Eq. <ref> to be valid at all wavelengths. Thus, it is possible to find sets of (A_i, B_i, C_i) parameters that will produce an achromatic phase shift over a given bandwidth.




  §.§.§ Design and simulation of an achromatic π/2 passive phase shifter


For a given waveguide width, the coefficients A, B and C are estimated based on simulations performed with the BeamPROP software. Effective indexes are computed for widths between 1.8 and 3 μm with a step of 0.1 μm, as shown in Fig. <ref>. A second order polynomial fit is used to estimate the A, B and C coefficients for each waveguide width value. Two- and three-segments solutions for achromatic φ_0=π/2 phase shifters are investigated. The lithographic fabrication error on waveguide width ranges from 0.2 to 0.5 μ m and is considered homogeneous throughout the wafer, the glass substrate on which several PICs are manufactured. Therefore, the width error ϵ_w is similar for the two waveguides of the phase shifter. Fig. <ref> shows that n_eff,w1+ϵ_w-n_eff,w2+ϵ_w = n_eff,w1-n_eff,w2 meaning that the phase shift does not change with the fabrication error if this error is homogeneous throughout the wafer. 



The system of equations to be solved is given by Eq. <ref> and Eq. <ref>, which can be rewritten with a matrix formalism: 

    [ L ]=[ M ]^-1·[ ϕ]


with:


L=[ L_1; L_2;     ], M=[ A_1 A_2; B_1 B_2;     ] and [ ϕ]=[      0; φ_0/2π;        ] for the two-segment solution, and


L=[ L_1; L_2; L_3;     ], M=[ A_1 A_2 A_3; B_1 B_2 B_3; C_1 C_2 C_3;     ] and [ ϕ]=[      0; φ_0/2π;      0;        ] for the three-segment solution.





For the N=2 and N=3 segments phase shifters, four and six waveguide widths values have to be optimized respectively. This is done by scanning all parameter grids, and  computing segments lengths for all the possible width combinations. 
When constraining the waveguide width between 1.8 and 2.3 μm, i.e. the single-mode range for wavelength from 600 to 820 nm, optimal parameter sets lead to achromatic φ_0=π/2 phase shifters about a dozen millimeters long. However they can be made shorter if multi-modal waveguides are use, i.e. width larger than 2.3 μm. Indeed, with larger width, the effective index increases and the light gets slowed down more efficiently. In this case, optimal parameter sets can be found for shorter total lengths: 1.8 mm for the two-segment design, and 2.7 mm for the three-segment design.


Mode coupling between segments is performed with tapers. Each taper present in one waveguide must be included in the other one as well, as they would otherwise induce additional phase difference<cit.>. Simulations are run in the three following cases: without tapers, 10 μm and 100 μm long tapers. Results are reported in Fig. <ref>, highlighting the need for tapers and showing that an achromatic phase shift is achieved over the 600-800 nm bandwidth within an accuracy of 2 deg at best. The three-segment solution offers a better phase shift performance at the cost of a longer total length as described in Table. <ref>.










 §.§ Latest prototypes and test functions




A test wafer has been manufactured in order to characterize and validate the models and the designs presented in this paper, in particular regarding the asymmetric directional couplers and the ABCD cells. The laboratory characterization of the different components that have been included in this wafer will be the subject of a future communication.

The complete wafer layout is presented in Fig. <ref>. It comprises different test PICs, labelled from 1 to 8 in the figure. There are test PICs intended for characterizing the individual building blocks, as well as two complete 5T-combiners: 

    
  * PIC 1 is composed of straight and curved waveguides to assess some fundamental properties of the waveguides. The width of the straight waveguides range from 1.8 to 3 μm with a step of 0.2 μm in order to characterize the spectral range where the single-mode behavior is achieved. Various S bends with curvature radii ranging from 10 to 40 mm will help determine the minimum acceptable curvature radius based on loss measurements. This PIC also contains straight waveguides with 10 crossings at various angles of 5, 10, 20, 30, and 45 degrees to evaluate excess losses due to waveguide crossing. PIC 8 contains additional straight waveguides with 10 or 20 crossings at angles ranging from 3 to 15 degrees.
    
  * PIC 2 contains asymmetric directional couplers as presented in Sec. <ref>, with various sets of parameters for the interaction zone geometry, i.e. the differential width dw between the coupling waveguides and the interaction zone length L. Characterization will mainly consists in spectral and polarized transfer rate measurements. Because some fundamental values of TEEM Photonics technology are currently not well known, especially concerning birefringence, this PIC also contains Mach-Zehnder interferometers for spectral effective indexes measurement. Effective indexes in both polarizations will allow to refine the BeamPROP model and include both polarizations in simulations.
    
  * PIC 4 contains 8 ABCD cells with various phase shifters. 
    
  * PIC 6 contains Mach-Zehnder interferometers with phase shifters presented in Sec. <ref>. 
    
  * PIC 7 contains Y junctions with different junction zone geometries.
    
  * PIC 3 is a complete 5T interferometric combination PIC based on ABCD cells, while PIC 5 is a 5T PIC based on uniformly asymmetric directional couplers. Their design has been presented in Fig. <ref>. They are meant to be installed on the FIRST instrument at the Subaru telescope, provided that their performance in terms of throughput and polarization is high enough.

















§ CONCLUSION








Photonics integrated circuits constitute promising devices to perform beam manipulation requiring functions such as beam splitters, phase shifters or combiners. They thus offer stable and robust solutions for interferometric instruments, for long baseline interferometers, as well as pupil remapper instruments. While the technology has been well developed at the telecom wavelength range in the infrared, high performance visible PICs are still difficult to achieve. Within the SCExAO/FIRST project, a visible spectro-interferometer performing pupil remapping at the Subaru telescope, we are working with TEEM Photonics to produce a 5T beam combiner with high enough performance in terms of throughput and chromaticity, that would be suited for on-sky observations.

The development process is iterative and started with a first step to characterize the fundamental properties of the waveguides manufactured with the ioNext technology operated by TEEM Photonics. Based on the effective refractive index profile model of TEEM Photonics' standard waveguide validated by laboratory measurements, we have designed and optimized symmetric and asymmetric directional couplers, as well as achromatic phase shifters intended for ABCD cells, working in the 600-800 nm wavelength range. A wafer containing various test PIC designs has recently been manufactured. The next step will be to characterize these PICs in the laboratory, to further evaluate losses due to propagation, bends or crossings, and to identify polarization dependent behaviors. These results will be the subject of a future paper.




§ ACKNOWLEDGEMENTS

A part of this work was previously published as a SPIE proceeding<cit.> for the SPIE Astronomical Telescopes and Instrumentation 2022 event in Montréal, Québec, Canada. This project is supported by the French National Research Agency (ANR-21-CE31-0005) and the doctoral school Astronomy & Astrophysics of Ile de France (ED 127). Authors also acknowledge  the funding by ASHRA (Action Spécifique Haute Resolution Angulaire) from INSU-CNRS and thank TEEM Photonics for their support and trust.

   
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Manon Lallement (She/Her) is a PhD student in instrumentation for astronomy at the Observatoire de Paris. Her PhD is supervised by Elsa Huby and Sylvestre Lacour and is funded by the doctoral school Astronomy & Astrophysics of Ile de France (ED 127). She received her BS and MS degrees in photonics, theoretical and applied optics from the French Institut d'Optique Graduate School in 2019 and 2021, respectively.
Her current research interests include visible photonics for astronomical interferometry, 3D printed micro-lenses array for injection into single-mode fibers and interferometric data analysis. She is a member of SPIE. 
 https://spie.org/profile/Manon.Lallement-4374172



Biographies of the other authors are not available.




