††thanks: Equal contributions††thanks: Equal contributions††thanks: Equal
contributions††thanks: Equal contributions

# Experimental implementation of the optical fractional Fourier transform in
the time-frequency domain

Bartosz Niewelt  Centre for Quantum Optical Technologies, Centre of New
Technologies, University of Warsaw, Banacha 2c, 02-097 Warsaw, Poland  Faculty
of Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland  Marcin
Jastrzębski  Centre for Quantum Optical Technologies, Centre of New
Technologies, University of Warsaw, Banacha 2c, 02-097 Warsaw, Poland  Faculty
of Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland  Stanisław
Kurzyna  Centre for Quantum Optical Technologies, Centre of New Technologies,
University of Warsaw, Banacha 2c, 02-097 Warsaw, Poland  Faculty of Physics,
University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland  Jan Nowosielski
Centre for Quantum Optical Technologies, Centre of New Technologies,
University of Warsaw, Banacha 2c, 02-097 Warsaw, Poland  Faculty of Physics,
University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland  Wojciech Wasilewski
Centre for Quantum Optical Technologies, Centre of New Technologies,
University of Warsaw, Banacha 2c, 02-097 Warsaw, Poland  Faculty of Physics,
University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland  Mateusz Mazelanik
<EMAIL_ADDRESS>Centre for Quantum Optical Technologies, Centre of
New Technologies, University of Warsaw, Banacha 2c, 02-097 Warsaw, Poland
Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland
Michał Parniak<EMAIL_ADDRESS>Centre for Quantum Optical
Technologies, Centre of New Technologies, University of Warsaw, Banacha 2c,
02-097 Warsaw, Poland  Niels Bohr Institute, University of Copenhagen,
Blegdamsvej 17, 2100 Copenhagen

###### Abstract

The fractional Fourier transform (FrFT), a fundamental operation in physics
that corresponds to a rotation of phase space by any angle, is also an
indispensable tool employed in digital signal processing for noise reduction.
Processing of optical signals in their time-frequency degree of freedom
bypasses the digitization step and presents an opportunity to enhance many
protocols in quantum and classical communication, sensing and computing. In
this letter, we present the experimental realization of the fractional Fourier
transform in the time-frequency domain using an atomic quantum-optical memory
system with processing capabilities. Our scheme performs the operation by
imposing programmable interleaved spectral and temporal phases. We have
verified the FrFT by analyses of chroncyclic Wigner functions measured via a
shot-noise limited homodyne detector. Our results hold prospects for achieving
temporal-mode sorting, processing and super-resolved parameter estimation.

### Introduction

Harnessing many degrees of freedom of photons, such as polarization [1],
spatial modes, in particular with orbital angular momentum, or temporal modes
[2] holds continued importance in quantum protocols where researchers seek to
achieve new capabilities or gain enhanced performance and capacity. The time-
frequency domain is exploited to a great effect, but typically only by using
parallel spectral channels, as witnessed by the wavelength-division
multiplexing devices [3, 4, 5, 6]. Nevertheless, from the perspective of
quantum optics, it is clear that more advanced spectro-temporal encoding and
manipulations lead to enhanced performance. For example, one may implement
quantum gates and beamsplitter operations between spectral or temporal modes
[2, 7, 8]. This discrete-mode picture is accompanied by a continuous treatment
of the time-frequency domain [9, 10, 11]. In particular, hybrid approaches
where discrete and continuous spaces are combined are gaining attention in
quantum networking [12]. The continuous domain, implemented via both
processing and detection with resolution exceeding the characteristic coarse-
graining, often allows for accessing the highest available dimensionality of
the quantum system [13, 14, 15]. This is particularly relevant for Einstein-
Podolsky-Rosen type experiments, which provide entanglement as a resource for
high-dimensional quantum key distribution [16, 17]. As far as processing is
concerned, one of the fundamental operations for the continuous variables is
an optical Fourier transform [18] that allows switching between two conjugate
variables. In the spatial domain, these are the position and momentum of a
photon and the Fourier transform can be achieved via a single lens placed a
focal length from the conjugate planes.

The celebrated space-time duality [19, 20] provides the idea of implementing
the same Fourier transform in the time-frequency domain via spectral and
temporal quadratic phases. A combination of those (time lenses and dispersion)
finds applications in temporal imaging [21, 22], photon time-of-flight
spectroscopy [23] and bandwidth-matching of quantum systems [24, 25], with
implementations stretching from ultrafast optics [26] to narrowband atomic
systems [27]. Operations beyond the Fourier transform have been proposed but
scarcely implemented. They range from trivial but useful frequency-translation
operators [28] to complex modulations that could allow largely arbitrary modal
operations [29]. Those more complex operations could allow for super-resolved
sensing of system parameters [30, 31, 32] or enhanced data rate for
communication in the photon-starved regime [33].

Figure 1: (a) Ideational scheme of the evolution of CWF due to temporal lenses
$f_{t}$ and frequency lens $f_{\omega}$. (b) Relevant ${}^{87}\mathrm{Rb}$
energy level configuration. (c) Experimental setup used to perform FrFT, based
on ultracold atoms in a magneto-optical trap. The frequency lens is
implemented using a spatial light modulator (SLM) by imprinting an intensity
pattern on ac-Stark (acS) beam. The acS beam then illuminates atoms, applying
a spatial quadratic phase. The custom shot noise-limited differential
photodiode (DPD) detects the beating between the signal and reference. (d)
Experimental sequence for performing FrFT of two-pulse state. HP and ZP are
respectively Hyperfine pumping and Zeeman pumping.

Here we provide an experimental implementation of the time-frequency domain
optical fractional Fourier transform (FrFT). The FrFT provides a full
generalization of the Fourier transform and provides significant new
capabilities. Notably, in the spatial domain, the FrFT may be implemented by a
different non-focal arrangement of a single lens. With this, the FrFT enables
sorting of orbital angular momentum modes [34]. Optical mode sorting is of
particular interest in the time-frequency domain, as it may provide temporal
and spectral superresolution, enhanced optical communication with noise
rejection, and non-standard coding. The more typical application of the FrFT
in the engineering context is noise filtration. The FrFT is particularly well-
suited to filter out structured noise, for instance with chirp [35], which
goes beyond the typical capabilities of the conventional Fourier transform
[36]. A series of FrFTs and bandpass filters may filter out noise with complex
structures, in particular using adaptive techniques [37, 38]. This is
typically implemented numerically, and the prospect of such noise-filtering
protocols at the level of optical signals could provide significant gains for
noisy communication [39]. The FrFT has also been proposed as a basis of chirp-
based encoding protocols that gain rising interest in the engineering
community [40]. In physics, the FrFT in the phase-space picture arises
naturally as an evolution of a quantum harmonic oscillator.

We implement the FrFT for narrowband photons using a programmable spectral
phase achieved thanks to an atomic optical quantum memory, which combines ac-
Stark modulation of spin waves [41, 42] and gradient-echo storage and
retrieval protocol [43, 44] that inherently allows for spectral manipulation
of stored signal [45, 46]. We verify the performance of the FrFT via an
illustrative example of a rotation of a coherent dual pulse, time-frequency
cat-like state, and further quantify the capabilities of the device by
processing Hermite-Gaussian temporal modes, which are eigenfunctions of the
FrFT [47]. We use homodyne detection and analyze the measured chronocyclic
Wigner functions (CWF) to capture the subtle characteristics of the system.

Figure 2:  CWFs of two-pulse state rotated by angles
$\varphi\in\\{0,\frac{\pi}{3},\frac{\pi}{2},\frac{2\pi}{3}\\}$. The upper row
presents the experimental data. The lower row presents the results obtained
from numerical simulations, taking into account experimental limitations.

### Theory

Let us introduce the fractional Fourier transform by the analogy to the
evolution of a quantum harmonic oscillator with zero energy ground state [48].
With the corresponding Hamiltonian:
$\hat{H}=\nicefrac{{\hat{p}^{2}}}{{2}}+\nicefrac{{\hat{q}^{2}}}{{2}}-\nicefrac{{1}}{{2}}$,
with $\hat{p}=-i\frac{\mathrm{d}}{\mathrm{d}q}$, we define the FrFT as the
evolution operator:

$\widehat{\mathrm{FrFT}}^{(\varphi)}=\exp\left(i\varphi\hat{H}\right),$ (1)

that evolves the initial state $\psi(q)$. The parameter $\varphi$ is then
called the angle of the FrFT, and for $\varphi=\frac{\pi}{2}$ we have the
Fourier transform. Here we deal with the spectro-temporal domain, and we take
$q=\nicefrac{{t}}{{\tau}}$ and $p=\tau\omega$ as unitless quantities
corresponding to the time and frequency of the optical pulses, where $\tau$ is
time unit. Hence we have chronocyclic space in which we can describe the
distributions using CWF, which for the electric field with complex amplitude
$\psi(t/\tau)$ is defined as:

$W(\omega\tau,t/\tau)=\frac{1}{2\pi}\intop\>\mathrm{d}\xi\psi\left(\frac{t}{\tau}+\frac{\xi}{2}\right)\psi^{*}\left(\frac{t}{\tau}-\frac{\xi}{2}\right)e^{i\omega\tau\xi}$
(2)

Figure 3:  (a) Histogram of the fidelity for measured Hermite-Gaussian modes
(measured) projected onto modes (numerical) for $\varphi=\frac{\pi}{4}$. The
colormap represents the phase of the overlap. (b) Histogram of the fidelity
for measured Hermite-Gaussian modes (measured) projected onto modes
(numerical) for $\varphi=\frac{2\pi}{3}$. The colormap represents the phase of
the overlap. (c) CWFs of measured subsequent Hermite-Gaussian modes for
$n\in[0,10]\cap\mathbb{Z}$. The upper row represents $\varphi=0$ and lower
$\varphi=\frac{\pi}{4}$.

The fractional Fourier transform applied to a pulse rotates its CWF by the
angle $\varphi$. Such rotation can be achieved by adding quadratic time and
frequency phases to the pulse [49] in a sequence that represents two temporal
lenses interleaved by a frequency lens with respective focusing powers $d_{t}$
and $d_{\omega}$. Such sequence of transformations applied to two Gaussian
pulses is presented in the figure 1(a). The lenses act on an optical signal as
described below:

$\begin{split}\psi(t/\tau)\xrightarrow{\text{temporal
lens}}\psi(t/\tau)\exp\left[-\frac{id_{t}}{2}\left(\frac{t}{\tau}\right)^{2}\right],\\\
\tilde{\psi}(\omega\tau)\xrightarrow{\text{spectral
lens}}\tilde{\psi}(\omega\tau)\exp\left[-\frac{id_{\omega}}{2}(\omega\tau)^{2}\right],\end{split}$
(3)

where $\tilde{\psi}(\omega\tau)$ is a Fourier transform of the optical signal
$\psi(t/\tau)$. The relation between $d_{t}$ and $d_{\omega}$ for an FrFT can
be found by considering the general rotation matrix in the time-frequency
domain [49] and is given as follows:

$d_{t}(\varphi)=\tan\nicefrac{{\varphi}}{{2}}\hskip
15.0ptd_{\omega}(\varphi)=\sin\varphi$ (4)

where $\varphi$ is the rotation angle. It is important to note that these
equations hold only for angles $\varphi\in(-\pi,\pi)$. Adding each of the
previously described phases to the pulse acts as shearing of the CWF of the
signal as illustrated in Fig. 1(a). Let $T$ and $B$ be the temporal and
spectral widths. Action of the temporal lens leaves the temporal direction
intact but broadens the bandwidth. Therefore, the time-bandwidth area to be
stored in the memory is given as:

$TB^{\prime}=TB(1+|\tan\nicefrac{{\varphi}}{{2}}|)$ (5)

Eigenfunctions of FrFT are the Hermite-Gaussian functions $H_{n}^{G}(t/\tau)$
with eigenvalues being phasors with phase proportional to the angle of
transformation $\varphi$ and order $n$, as described below.

$\widehat{\mathrm{FrFT}}^{(\varphi)}\left[H^{G}_{n}(t/\tau)\right]=H^{G}_{n}(t/\tau)e^{-in\varphi}$
(6)

### Experimental system

Figure 4: (a) The phase of the overlap of subsequent Hermite-Gaussian modes
for $n\in[0,10]\cap\mathbb{Z}$ with fitted functions. Error bars correspond to
$5\sigma$. Shaded regions represent the fitted functions with $\pm\sigma$ of
the fitted parameters. Each line is shifted by $0.75\text{\,}\mathrm{rad}$ to
clarify the plot. (b) Histogram presenting phases of overlaps of Hermite-
Gaussian modes $n\in[0,10]\cap\mathbb{Z}$ and $\varphi=\frac{2\pi}{3}$ with
their numerical equivalents (the same mode, center, and width). The height of
each bar is proportional to number of events i.e. measured phase of overlap is
in a bin range. The bars are color-coded by each mode.

The experiment is based on gradient echo quantum memory (GEM) that is built on
rubidium-87 atoms trapped in a magneto-optical trap (MOT). Atoms form a
cigarette shape $10\text{\,}\mathrm{mm}$ long cloud with an optical density
reaching 85. The ensemble temperature is
$42\text{\,}\mathrm{\SIUnitSymbolMicro K}$. The setup schematic is presented
in figure 1(c). We utilize GEM protocol to map different frequencies of the
signal to specific parts of the atomic cloud according to the formula
$\omega=\beta\times z+\omega_{0}$, where $\beta$ is the value of the magnetic
gradient, $z$ is the position along the $z$-axis and $\omega_{0}$ is the
carrier frequency. We exploit that feature to add an arbitrary phase to the
stored signal by the introduction of spatially varying ac-Stark shifts between
the storage levels during the storage time. The spatial phase imposed by the
beam onto the atomic state is equivalent to the spectral phase of a signal
stored in the atomic cloud. To implement the frequency lens the ac-Stark beam
is shaped to have a Fresnel-lens intensity profile imposing phase according to
(3). The temporal lenses are imposed during the write-in and read-out process
by linearly detuning the coupling laser in time at a matching rate of
$d_{t}/\tau^{2}$. During the read-out we change the sign of detuning rate due
to inverse time caused by different sign of magnetic gradient. The main
experimental limitation is the storage efficiency $\eta$ of the GEM protocol.
Intense coupling laser is necessary for storage but at the same causes
decoherence, thus effectively limiting the efficiency to
$\eta=1-\exp\left(-2\pi\mathrm{OD}/(TB^{\prime})\right)$ [50] where OD is the
optical density of the atomic cloud. Prior to storage, the FrFT protocols
broadens the bandwidth of the signal due to action of temporal lens as
described in Eq. (5). In the experiment we picked $TB=110$ to store first 11
Hermite-Gaussian modes. It follows that $T=\sqrt{110}\tau$ and
$B=\sqrt{110}/\tau$. In our setup the time window is limited by power of
coupling beam that induced a coherence decay at a rate of $\Gamma$ =
$9.1\text{\,}\mathrm{kHz}$, we chose $T\Gamma=0.4$ obtaining $\eta=33\%$,
which sets the $\tau=$4.2\text{\,}\mathrm{\SIUnitSymbolMicro s}$$. The
magnetic field gradient $\beta$ is adjusted to match desired storage bandwidth
$\beta L=B^{\prime}$. The read-out signal is sent to the homodyne detector
connected to RedPitaya STEMlab 125-14 acting as an oscilloscope. For every
measurement, we collect 200 homodyne waveforms of the read-out signal and
average them by subtracting the phase of the LO from each measurement. The
phase of the LO was inferred from additional impulse sent through after the
read-out sequence. Moreover, we found that the GEM protocol itself imposes a
constant quadratic frequency phase, which might be caused by eddy currents
induced in passive elements surrounding the experimental setup. We have
mitigated this issue by applying a constant frequency lens compensating for
this unwanted effect.

### Results

We start the demonstration of the FrFT by preparing a cat-like state input
signal with an envelope:

$\psi^{\text{in}}(t/\tau)=\frac{\exp\left[\frac{-(t-\mu)^{2}}{2(s\tau)^{2}}\right]+\exp\left[\frac{-(t+\mu)^{2}}{2(s\tau)^{2}}\right]}{\sqrt{2\sqrt{\pi}s\tau\left\\{1+\exp\left[-\mu^{2}/(s\tau)^{2}\right]\right\\}}}$
(7)

with $s\tau=$2.4\text{\,}\mathrm{\SIUnitSymbolMicro s}$$ and
$\mu=$7\text{\,}\mathrm{\SIUnitSymbolMicro s}$$. Those parameters were chosen
to efficiently use the whole available bandwidth and temporal window of the
memory. The signal is then sent to the memory performing FrFT with chosen
angles $\varphi\in\\{0,\frac{\pi}{3},\frac{\pi}{2},\frac{2\pi}{3}\\}$ The
measured CWFs of the corresponding readout signals are depicted in Fig. 2 and
compared with matching theoretical predictions. Achieved fidelity between
simulation and experimental data varies between $65\%-88\%$. Next, we
benchmark the FrFT implementation by transforming Hermite-Gaussian input
pulses with an envelope:

$\psi^{\text{in}}_{n}(t/\tau)=\mathrm{H}^{G}_{n}(t/\tau)=\frac{1}{\sqrt{2^{n}n!\tau\sqrt{\pi}}}H_{n}\left(\frac{t}{\tau}\right)\exp\left(\frac{-t^{2}}{2\tau^{2}}\right)$
(8)

The quality of the FrFT is then probed by decomposing the measured readout
signal for different FrFT angles $\varphi$ in the ideal (input)
$\mathrm{H}^{G}_{n}$ basis with $n\in[0,10]\cap\mathbb{Z}$. Obtained complex
decomposition coefficients arranged into transition matrix and presented in
figures 3(a) and 3(b). The coefficients are inferred from complex amplitude
retrieved from homodyne measurement by calculating the overlap:

$\mathcal{F}_{n,m}=\frac{1}{\tau}\int
dt\left[\mathrm{H}^{G}_{n}(t/\tau)\right]^{*}\psi^{\text{out}}_{m}(t/\tau)$
(9)

where $\psi^{\text{out}}_{m}$ is the signal measured at the output of the
memory for set $m$, which is the index of the input Hermite-Gaussian mode and
$n$ is the index of the projection mode. The height of the bars corresponds to
$\left|\mathcal{F}_{n,m}\right|^{2}$, which is equivalent to the fidelity of
measured signal and theoretical predictions. The color of the bars corresponds
to the phase of the calculated overlap
$\phi=\mathrm{arg}\left({\mathcal{F}_{n,m}}\right)$. CWFs calculated from the
measurements of the first 6 Hermite-Gaussian modes are shown in figure 3(c).
After performing FrFT each Hermite-Gaussian mode gains phase $\phi$
proportional to the mode index $n$ as given by equation (6). These phases are
plotted in figure 4(a). We fitted functions $\phi=\phi_{0}+n\frac{d\phi}{dn}$
to each set of the phases $\phi(n)$ of complex eigenvalues and obtained an
effective FrFT angle. Our experiment yielded results presented in table 1. For
each transformation, the set angle is denoted as $\varphi$, the measured
rotation angle is $\nicefrac{{d\phi}}{{dn}}$, while their difference is
represented as $\Delta\varphi$.

set angle $\varphi$ |  measured angle $\nicefrac{{d\phi}}{{dn}}$ |  deviation $\Delta\varphi$ [$\times 10^{2}$]
---|---|---
$0$ |  $0.001\pi$ |  $0.1\pi$
$\pi/6$ |  $0.134\pi$ |  $-3.3\pi$
$\pi/4$ |  $0.218\pi$ |  $-3.2\pi$
$\pi/3$ |  $0.333\pi$ |  $-0.0\pi$
$\pi/2$ |  $0.505\pi$ |  $0.5\pi$
$2\pi/3$ |  $0.677\pi$ |  $1.0\pi$
Table 1: Measured angles of FrFT for 11 consecutive Hermit-Gaussian mode
impulses with their deviations from expected values and fit errors.

In all cases, measured angles differ from desired by less than $0.033\pi$. The
overlap phase of every single measurement with generated Hermite-Gaussian mode
for angle $\varphi=\frac{2\pi}{3}$ is shown on the histogram in figure 4(b).
Obtained phase fluctuations are in order of $0.2\text{\,}\mathrm{rad}$, which
corresponds to a change in the value of the magnetic field by about $0.2\%$.

### Discussion

In this letter, we tackled the problem of the physical implementation of the
fractional Fourier Transform in the spectro-temporal domain. We leveraged the
space-time duality and implemented the FrFT using the quantum-memory time-
frequency processor. Utilizing a CWF we demonstrated a high-fidelity
transformation of two-pulse states. Furthermore, we have characterized the
implementation with the help of Hermite-Gaussian pulses – eigenfunctions of
FrFT – and compared them with theoretical predictions. The results of this
letter introduce many prospects for future research and applications,
especially a method of sorting Hermite-Gaussian temporal modes utilizing FrFT
[34]. Such a mode sorting technique is key for spectral superresolution with
multi parameter quantum estimation [51]. The protocol presented in this letter
opens exciting possibilities for implementation in the antipodal regime of
ultrafast optics, where GEM protocol is not applicable and phases can be
applied directly by dispersion fibers and EOMs. Despite of straightforward
implementation, using ultrafast optics gives rise to new technical challenges
not present in our version of the protocol, but likely to spark interest of
the ultrafast optics community.

##  I Acknowledgments

###### Acknowledgements.

The “Quantum Optical Technologies” (MAB/2018/4) project is carried out within
the International Research Agendas programme of the Foundation for Polish
Science co-financed by the European Union under the European Regional
Development Fund. We thank K. Banaszek for generous support.

## References

  * Han and Li [2005] Y. Han and G. Li, Optics Express 13, 7527 (2005).
  * Brecht  _et al._ [2015] B. Brecht, D. V. Reddy, C. Silberhorn, and M. Raymer, Physical Review X 5, 041017 (2015).
  * Eriksson  _et al._ [2019] T. A. Eriksson, T. Hirano, B. J. Puttnam, G. Rademacher, R. S. Luís, M. Fujiwara, R. Namiki, Y. Awaji, M. Takeoka, N. Wada, and M. Sasaki, Communications Physics 2, 9 (2019).
  * Choi  _et al._ [2014] I. Choi, Y. R. Zhou, J. F. Dynes, Z. Yuan, A. Klar, A. Sharpe, A. Plews, M. Lucamarini, C. Radig, J. Neubert, H. Griesser, M. Eiselt, C. Chunnilall, G. Lepert, A. Sinclair, J.-P. Elbers, A. Lord, and A. Shields, Opt. Express 22, 23121 (2014).
  * Peters  _et al._ [2010] N. A. Peters, P. Toliver, T. E. Chapuran, R. J. Runser, S. R. McNown, C. G. Peterson, D. Rosenberg, N. Dallmann, R. J. Hughes, K. P. McCabe, J. E. Nordholt, and K. T. Tyagi, in  _2010 Conference on Optical Fiber Communication (OFC/NFOEC), collocated National Fiber Optic Engineers Conference_ (OPTICA, San Diego, California United States, 2010) pp. 1–3.
  * Bahrani  _et al._ [2015] S. Bahrani, M. Razavi, and J. A. Salehi, Journal of Lightwave Technology 33, 4687 (2015).
  * Humphreys  _et al._ [2013] P. C. Humphreys, B. J. Metcalf, J. B. Spring, M. Moore, X.-M. Jin, M. Barbieri, W. S. Kolthammer, and I. A. Walmsley, Physical Review Letters 111, 150501 (2013).
  * Campbell  _et al._ [2012] G. Campbell, M. Hosseini, B. M. Sparkes, P. K. Lam, and B. C. Buchler, New Journal of Physics 14, 033022 (2012).
  * Fabre  _et al._ [2022] N. Fabre, A. Keller, and P. Milman, Physical Review A 105, 052429 (2022).
  * Fabre [2023] N. Fabre, Teleportation-based error correction protocol of time-frequency qubits states (2023).
  * Humphreys  _et al._ [2014] P. C. Humphreys, W. S. Kolthammer, J. Nunn, M. Barbieri, A. Datta, and I. A. Walmsley, Physical Review Letters 113, 130502 (2014).
  * Darras  _et al._ [2023] T. Darras, B. E. Asenbeck, G. Guccione, A. Cavaillès, H. Le Jeannic, and J. Laurat, Nature Photonics 17, 165 (2023).
  * Schneeloch  _et al._ [2014] J. Schneeloch, C. J. Broadbent, and J. C. Howell, Physics Letters A 378, 766 (2014).
  * Tasca  _et al._ [2013] D. S. Tasca, Ł. Rudnicki, R. M. Gomes, F. Toscano, and S. P. Walborn, Physical Review Letters 110, 210502 (2013).
  * Xiao  _et al._ [2020] Y. Xiao, Y. Xiang, Q. He, and B. C. Sanders, New Journal of Physics 22, 073063 (2020).
  * Zhong  _et al._ [2015] T. Zhong, H. Zhou, R. D. Horansky, C. Lee, V. B. Verma, A. E. Lita, A. Restelli, J. C. Bienfang, R. P. Mirin, T. Gerrits, S. W. Nam, F. Marsili, M. D. Shaw, Z. Zhang, L. Wang, D. Englund, G. W. Wornell, J. H. Shapiro, and F. N. C. Wong, New Journal of Physics 17, 022002 (2015).
  * Nunn  _et al._ [2013] J. Nunn, L. J. Wright, C. Söller, L. Zhang, I. A. Walmsley, and B. J. Smith, Opt. Express 21, 15959 (2013).
  * Goodman [1996] J. Goodman,  _Introduction to Fourier Optics_, Electrical Engineering Series (McGraw-Hill, 1996).
  * Kolner and Nazarathy [1989] B. H. Kolner and M. Nazarathy, Opt. Lett. 14, 630 (1989).
  * Patera  _et al._ [2018] G. Patera, D. B. Horoshko, and M. I. Kolobov, Physical Review A 98, 053815 (2018).
  * Kuzucu  _et al._ [2009] O. Kuzucu, Y. Okawachi, R. Salem, M. A. Foster, A. C. Turner-Foster, M. Lipson, and A. L. Gaeta, Opt. Express 17, 20605 (2009).
  * Hernandez  _et al._ [2013] V. J. Hernandez, C. V. Bennett, B. D. Moran, A. D. Drobshoff, D. Chang, C. Langrock, M. M. Fejer, and M. Ibsen, Opt. Express 21, 196 (2013).
  * Goda and Jalali [2013] K. Goda and B. Jalali, Nature Photonics 7, 102 (2013).
  * Karpiński  _et al._ [2017] M. Karpiński, M. Jachura, L. J. Wright, and B. J. Smith, Nature Photonics 11, 53 (2017).
  * Foster  _et al._ [2009] M. A. Foster, R. Salem, Y. Okawachi, A. C. Turner-Foster, M. Lipson, and A. L. Gaeta, Nature Photonics 3, 581 (2009).
  * Kauffman  _et al._ [1994] M. T. Kauffman, W. C. Banyai, A. A. Godil, and D. M. Bloom, Applied Physics Letters 64, 270 (1994).
  * Mazelanik  _et al._ [2020] M. Mazelanik, A. Leszczyński, M. Lipka, M. Parniak, and W. Wasilewski, Optica 7, 203 (2020).
  * Kurzyna  _et al._ [2022] S. Kurzyna, M. Jastrzębski, N. Fabre, W. Wasilewski, M. Lipka, and M. Parniak, Opt. Express 30, 39826 (2022).
  * Lu  _et al._ [2019] H.-H. Lu, N. Klco, J. M. Lukens, T. D. Morris, A. Bansal, A. Ekström, G. Hagen, T. Papenbrock, A. M. Weiner, M. J. Savage, and P. Lougovski, Phys. Rev. A 100, 012320 (2019).
  * Mazelanik  _et al._ [2022] M. Mazelanik, A. Leszczyński, and M. Parniak, Nature Communications 13, 691 (2022).
  * Ansari  _et al._ [2021] V. Ansari, B. Brecht, J. Gil-Lopez, J. M. Donohue, J. Řeháček, Z. Hradil, L. L. Sánchez-Soto, and C. Silberhorn, PRX Quantum 2, 010301 (2021).
  * Donohue  _et al._ [2018] J. Donohue, V. Ansari, J. Řeháček, Z. Hradil, B. Stoklasa, M. Paúr, L. Sánchez-Soto, and C. Silberhorn, Physical Review Letters 121, 090501 (2018).
  * Banaszek  _et al._ [2020] K. Banaszek, L. Kunz, M. Jachura, and M. Jarzyna, Journal of Lightwave Technology 38, 2741 (2020).
  * Zhou  _et al._ [2017] Y. Zhou, M. Mirhosseini, D. Fu, J. Zhao, S. M. Hashemi Rafsanjani, A. E. Willner, and R. W. Boyd, Physical Review Letters 119, 263602 (2017).
  * Dorsch  _et al._ [1994] R. G. Dorsch, A. W. Lohmann, Y. Bitran, D. Mendlovic, and H. M. Ozaktas, Applied Optics 33, 7599 (1994).
  * Yasuda  _et al._ [2006] S. Yasuda, Y. Ogasawara, J. Minabe, K. Kawano, M. Furuki, K. Hayashi, K. Haga, and H. Yoshizawa, Opt. Lett. 31, 1639 (2006).
  * Prajna and Mukhopadhyay [2020] K. Prajna and C. K. Mukhopadhyay, Journal of Nondestructive Evaluation 39, 14 (2020).
  * Kumar and Kansal [2017] P. Kumar and S. Kansal, in  _2017 International Conference on Information, Communication, Instrumentation and Control (ICICIC)_, 307 (2017).
  * Raymer and Banaszek [2020] M. G. Raymer and K. Banaszek, Opt. Express 28, 32819 (2020).
  * Ouyang and Zhao [2016] X. Ouyang and J. Zhao, IEEE Transactions on Communications 64, 3946 (2016).
  * Mazelanik  _et al._ [2019] M. Mazelanik, M. Parniak, A. Leszczyński, M. Lipka, and W. Wasilewski, npj Quantum Information 5, 22 (2019).
  * Hosseini  _et al._ [2012] M. Hosseini, B. M. Sparkes, G. T. Campbell, P. K. Lam, and B. C. Buchler, Journal of Physics B: Atomic, Molecular and Optical Physics 45, 124004 (2012).
  * Hosseini  _et al._ [2009] M. Hosseini, B. M. Sparkes, G. Hétet, J. J. Longdell, P. K. Lam, and B. C. Buchler, Nature 461, 241 (2009).
  * Cho  _et al._ [2016] Y.-W. Cho, G. T. Campbell, J. L. Everett, J. Bernu, D. B. Higginbottom, M. T. Cao, J. Geng, N. P. Robins, P. K. Lam, and B. C. Buchler, Optica 3, 100 (2016).
  * Buchler  _et al._ [2010] B. C. Buchler, M. Hosseini, G. Hétet, B. M. Sparkes, and P. K. Lam, Opt. Lett. 35, 1091 (2010).
  * Sparkes  _et al._ [2012] B. M. Sparkes, M. Hosseini, C. Cairns, D. Higginbottom, G. T. Campbell, P. K. Lam, and B. C. Buchler, Phys. Rev. X 2, 021011 (2012).
  * Namias [1980] V. Namias, IMA Journal of Applied Mathematics 25, 241 (1980).
  * Weimann  _et al._ [2016] S. Weimann, A. Perez-Leija, M. Lebugle, R. Keil, M. Tichy, M. Gräfe, R. Heilmann, S. Nolte, H. Moya-Cessa, G. Weihs, D. N. Christodoulides, and A. Szameit, Nature Communications 7, 11027 (2016).
  * Lohmann [1993] A. W. Lohmann, J. Opt. Soc. Am. A 10, 2181 (1993).
  * Sparkes  _et al._ [2013] B. M. Sparkes, J. Bernu, M. Hosseini, J. Geng, Q. Glorieux, P. A. Altin, P. K. Lam, N. P. Robins, and B. C. Buchler, New Journal of Physics 15, 085027 (2013).
  * Tsang  _et al._ [2016] M. Tsang, R. Nair, and X.-M. Lu, Physical Review X 6, 031033 (2016).