# Quantum correlation light-field microscope with extreme depth of field

Yingwen Zhang<EMAIL_ADDRESS>Nexus for Quantum Technologies, University
of Ottawa, K1N 6N5, ON, Ottawa  National Research Council of Canada, 100
Sussex Drive, Ottawa ON Canada, K1A0R6  Duncan England  Duncan.England@nrc-
cnrc.gc.ca National Research Council of Canada, 100 Sussex Drive, Ottawa ON
Canada, K1A0R6  Antony Orth  National Research Council of Canada, 100 Sussex
Drive, Ottawa ON Canada, K1A0R6  Ebrahim Karimi  Nexus for Quantum
Technologies, University of Ottawa, K1N 6N5, ON, Ottawa  National Research
Council of Canada, 100 Sussex Drive, Ottawa ON Canada, K1A0R6  Benjamin
Sussman  Nexus for Quantum Technologies, University of Ottawa, K1N 6N5, ON,
Ottawa  National Research Council of Canada, 100 Sussex Drive, Ottawa ON
Canada, K1A0R6

###### Abstract

Light-field microscopy (LFM) is a 3D microscopy technique whereby volumetric
information of a sample is gained by simultaneously capturing both the
position and momentum (angular) information of light illuminating a scene.
Conventional LFM designs generally require a trade-off between position and
momentum resolution, requiring one to sacrifice resolving power for increased
depth of field (DOF) or vice versa. In this work, we demonstrate a LFM design
that does not require this trade-off by utilizing the inherent correlations
between spatial-temporal entangled photon pairs. Here, one photon from the
pair is used to illuminate a sample from which the position information of the
photon is captured directly by a camera. By virtue of the strong momentum
anti-correlation between the two photons, the momentum information of the
illumination photon can then be inferred by measuring the angle of its
entangled partner on a different camera. By using a combination of ray-tracing
and a Gerchberg-Saxton type algorithm for the light field reconstruction, we
demonstrate that a resolving power of 5 $\mu$m can be maintained with a DOF of
$\sim 500$ $\mu$m, approximately 3 times of the latest LFM designs or $>100$
time that of a conventional microscope. In the extreme, at a resolving power
of 100 $\mu$m, it is possible to achieve near infinite DOF.

##  I Introduction

Utilizing the properties of quantum entangled photons to enhance the
performance of sensing and imaging techniques has been an active area of
research in recent decades. To date, most demonstrated techniques are focused
on noise reduction in the low illumination regime by utilizing the strong
correlations in one of the many degrees of freedom between entangled photon
pairs, such as in time or position/momentum. Examples of this are sub-shot
noise imaging Brida2010 ; Samantaray2017  and quantum ghost imaging
Pittman1995 ; Shapiro2012 ; Padgett2016 . Dispersion immune optical coherence
tomography has also been demonstrated using the two-photon Hong-Ou-Mandel
interference effect Nasr2003 ; Yepiz2022 . With advances in detector
technology, especially with the maturity of single photon sensitive event
cameras Nomerotski2019 ; Defienne2021 , it is now possible to simultaneously
utilize the correlations between multiple degrees of freedom. It has been
demonstrated for further enhanced noise reduction in imaging and sensing by
utilizing spectral-temporal correlations Zhang2020  or spatial-temporal
correlations Defienne2021 ; Zhao2022  in entangled photons. New quantum
imaging techniques utilizing the quantum correlations in multiple degrees of
freedom have also been demonstrated, such as snapshot hyperspectral imaging
Zhang20222 , phase imaging through phase contrast microscopy Hodgson2022  and
Fourier Ptychography Aidukas2019 , Hong-Ou-Mandel-microscopy Ndagano2022 , as
well as light-field/plenoptic imaging Zhang2022 . In this work, we extend the
quantum correlation light-field imaging design Zhang2022  to microscopy and
demonstrate volumetric reconstruction of a microscopic scene at the few micron
resolution with extreme depth of field (DOF).

Light-field or plenoptic imaging Aldelson1992 ; Ng2005 ; raytrix  is a class
of imaging techniques which allows for the reconstruction of the light field
through capturing both the position and momentum (angular) information of
light rays simultaneously. With the reconstructed light field, volumetric
information of an illuminated scene can be obtained in a single measurement,
with no scanning required. This technique has been since adapted to microscopy
Levoy2006 ; Bimber2019 , and has been demonstrated in volumetric non-scanning
imaging of neural activities Wang2022 , microendoscopy Orth2019  and also
shown to have potential applications in optogenetics Schedl2016 . Conventional
light-field microscope (LFM) designs typically make use of a microlens array
(MLA). By placing the MLA one focal length away from an image sensor, each
microlens illuminates a subset of the pixels in the CCD. By knowing which lens
the light ray enters, and onto which pixel it subsequently focuses, one can
obtain both position and momentum information of the light ray simultaneously,
at the expense of sacrificing spatial resolution to gain momentum resolution.
Typical LFM designs have each microlens covering around $10\times 10$ pixels,
thus resulting in a 10 times reduction to the position resolution and
subsequently, resolving power. Increasing the size of each microlens to cover
more pixels will increase the amount of momentum information captured and
subsequently increase the DOF, however at the cost of further reduced position
resolution. Techniques have been developed to improve either the DOF or
resolving power such as through scanning the MLA Lim2009  or the sample stage
Orth2012 . Others include applying 3D deconvolution Broxton2013 , wavefront
shaping Cohen2014 , using a camera array Lin2015 , processing the light-field
information through the Fourier domain Guo2019 , or using the aid of spherical
aberrations in scanning LFM Zhang20223 .

In recent years, a new light-field imaging technique using spatially
correlated thermal or quantum light has been proposed DAngelo2016 ; Pepe2016 ;
DiLena2018 . Here one beam/photon illuminates a scene to capture the position
information, and the momentum information is inferred from the correlated
partner beam/photon that is measured on a separate camera. Using this method,
no trade-off between the position and momentum resolution is needed since each
beam can be captured on separate cameras thus allowing for a much larger DOF
while not sacrificing resolving power. This technique, has since been
demonstrated using weakly correlated thermal light Pepe2017 ; Massaro2022  and
quantum-correlated light Zhang2022 , with a DOF at around 10 mm for a
resolving power in the order of $100$ $\mu$m.

In this work, we present a proof-of-concept demonstration of light-field
microscopy using quantum-correlated photons. We refer to this technique as
quantum-correlation light-field microscopy (QCLFM) throughout the rest of the
manuscript. Unlike the previous work Zhang2022  that only employed ray optics,
we utilize a wave optics approach in order to account for the diffraction
effects that result from the much smaller target features. Our results
demonstrate a resolving power of $5$ $\mu$m (200 lp/mm - line pairs per
millimeter), with a depth of field (DOF) of $\sim 500$ $\mu$m. At a resolving
power of $10$ $\mu$m (100 lp/mm), the DOF can be extended to over $1.5$ mm.
This DOF is $>100$ times that of a conventional microscope using the same
objective lens. Compared to LFM designs, this is an order of magnitude larger
compared to some of the earlier LFM demonstrations Broxton2013 ; Cohen2014
and is $\sim 3$ times larger than some of the most recent LFM techniques
Zhang20223 . In the extreme case, at a resolving power of $100$ $\mu$m, we
observed a nearly infinite DOF, a feat that is practically unfeasible with
current LFM designs based on MLA. Limited by the single photon detection
efficiency, the frame rate of QCLFM is currently in the order of 100 s per
frame, much slower compared to conventional LFM designs that operate at 10-100
Hz. However, with expected advancements in single photon camera technology in
the coming years, we are confident that QCLFM will become a potential
alternative to conventional LFM designs.

##  II Results

###  II.1 Concept

The conceptual setup of the QCLFM system is shown in Fig. 1. Through the
process of type-II spontaneous parametric down-conversion (SPDC), a high
energy photon from a UV pump laser is converted into a pair of lower energy
photons in a nonlinear crystal. The photon pairs have orthogonal polarization
and are entangled in time, position and momentum. It is this strong
correlation property of entanglement that is used to realize QCLFM. The photon
pairs, named signal and idler are first separated into two paths via the use
of a polarizing beamsplitter (PBS). In the path of the signal photon, the
plane of the crystal is first imaged onto the sample placement region and then
onto a time-tagging event camera to record the signal photon’s position
information and arrival time. Through the path of the idler, the Fourier plane
of the crystal is projected onto a different event camera to record the idler
photons’ momentum information and subsequent arrival time. The photon pairs
are then identified through a time-correlation analysis based on their
detection time and the signal photon’s momentum information can then be
inferred from its time-correlated partner.

Figure 1: Conceptual setup of the QCLFM system. Pairs of photons with
orthogonal polarization, correlated in time, position and momentum, are
generated through the process of type II SPDC. The photon pairs, signal and
idler, are separated into two separate paths via the use of a polarizing
beamsplitter (PBS). In the signal path, the plane of the crystal is first
imaged onto the sample placement region and then onto a time-tagging event
camera where position information of the photons are captured. In the idler
path, the Fourier plane of the nonlinear crystal is projected onto a different
event camera to record the partner photons’ momentum information. Time
correlation measurements are then performed on all photons detected on the two
cameras to identify the photon pairs. The momentum information of the signal
photon can then be inferred from its time-correlated partner.

The operation for digital refocusing of a sample placed out of focus by a
distance $z$ can be achieved using two steps. First, using the position and
momentum information of each photon, and knowing the optical elements used
between them, the trajectory of the photons can be reconstructed through a ray
tracing operation. For large features, where diffraction effects are
negligible, this first step, using ray optics, is enough to bring the sample
back into focus Zhang2022 , however, for smaller features, interference and
diffraction effects from wave optics must also be taken into account. In this
case, the image obtained after ray tracing is the diffraction pattern
amplitude of the sample at a distance $z$ when illuminated by a plane wave
propagating in the z-direction (see the Supplementary for details). The second
step is to use a Gerchberg-Saxton type algorithm Saxton1972 ; Fienup1982  to
recover the amplitude of the sample. This refocusing process is illustrated in
Fig.2. Details on the experimental setup and the refocusing procedure can be
found in the Methods section.

Figure 2: Illustration of the digital refocusing process. (a) Using the photon
pairs locations detected on the nonlinear crystal’s image plane and Fourier
plane, a ray-tracing approach is first used to obtain the diffraction pattern
amplitude of the sample as if illuminated by a plane wave propagating in the
z-direction. (b) For a sample of uniform or known phase profile, a Gerchberg-
Saxton type algorithm is used to recover the amplitude profile of the sample.
(c) The recovered amplitude and phase of the sample after 10 loops in the
algorithm. This illustration is that of a 1951 USAF resolution target (group
7) placed at $z=-300$ $\mu$m away from the focus of the microscope objective.

###  II.2 Experimental Results

Figure 3: Refocusing of 1951 USAF resolution target (a) Image of a 1951 USAF
resolution target, obtained after time correlation measurement, placed in the
focus of a 20$\times$ 0.45 NA objective lens. Here, for every photon detected
in the idler beam, there is a signal photon detect in coincidence in the rest
of the image. The region circled in red is the far field imaged by the idler
photons. To not overshadow the rest of the image, the idler beam is only
displayed at 30% intensity. (b) Before and after images of digitally
refocusing groups 6 and 7 of a 1951 USAF resolution target (region in green
square of (a)) placed at different distances from the objective focus with the
smallest resolvable features after refocusing are boxed in blue. The data
acquisition time for the images is 200 s. (c) Digital refocusing of USAF
target groups 2 and 3 placed 1.75 cm away from the focus with a slightly
different lens configuration. This is very close to the Fourier plane as the
target can be clearly seen through ghost imaging in the idler beam (inset).
See the main text for an explanation of ghost imaging. (d) Plot of the
smallest resolution that can be refocused as a function of the distance a
target is placed away from the focus with positive distance being towards the
objective and negative away from the objective. The small peak at 0 $\mu$m is
a consequence of clustering in the raw camera data, see the Methods section
for more details on this. For images of the full dataset of (b) and (c), see
the Supplementary.

For this experiment, only a single time-tagging event camera ($256\times 256$
pixels with 55 $\mu$m pixel pitch) is used. A corner of the camera of around
$100\times 100$ pixels was used for momentum measurement with the rest of the
camera used for position measurement. With the momentum correlation of the
photon pair measured to be approximately 2 pixels wide (13.6$\times 10^{-3}$
$\mu$m-1/17.6$\times 10^{-4}$ rad), effectively around $50\times 50$ pixels
are used for momentum measurement. Though the idler beam is overlapped with
part of the signal beam on the camera, since a coincidence measurement is
performed, the noise in accidental coincidences from this overlap is
negligible. This is based on the fact that the probability of having two
signal photons generated within the coincidence gating time of 10 ns is very
low. See the Method section for more details.

In Fig. 3 we show digitally refocused images of a 1951 USAF resolution target
placed at different distances from the objective focus. We see that a line
spacing of 5 $\mu$m (group 7 element 5) can still be brought back into focus
when the target is placed $\pm 500$ $\mu$m away from the focus of the
objective lens (positive being towards the objective lens and negative being
away from) and for a line spacing of 10 $\mu$m (group 6 element 5) this
distance can be extended to more than 1500 $\mu$m. In the extreme case, we are
able to refocus a target placed at -1.75 cm away from the focus with a line
spacing of $\sim 100$ $\mu$m, as seen in Fig. 3(c) which is very close to the
Fourier plane, i.e. near infinity. This is verified through ghost imaging
Pittman1995 ; Shapiro2012 ; Padgett2016 , a correlation imaging technique
using pairs of photons, in which the idler photon does not interact with the
target being imaged, with only the signal photon interacting with the target.
By time correlating the detection events of the two photons, an image of the
object will appear through the idler photons even though they did not directly
interact with the target. Knowing the idler beam is imaged in the crystal’s
Fourier plane, being able to see the target clearly after time correlation
measurement is an indicator that the target is also placed at/near the Fourier
plane.

Comparing to MLA based LFM demonstrations with a similar resolving power, we
achieved around an order of magnitude improvement in DOF as compared to
earlier LFM demonstration Broxton2013 ; Cohen2014  (group 7 element 3 with
1000 $\mu$m DOF compared 100 $\mu$m DOF), and $\sim$3 times improvement in DOF
compared to one of the latest LFM demonstrations Zhang20223  (group 7 element
3 with 1000 $\mu$m DOF compared to 300 $\mu$m DOF and group 6 element 3 with
2000 $\mu$m DOF compared to 800 $\mu$m DOF), all of which uses more
sophisticated techniques such as deconvolution, wavefront shaping and
spherical aberration assistance for improving position resolution and DOF. In
addition, we see that QCLFM is also insensitive to the direction at which the
target is placed from the objective focus as compared to some conventional LFM
techniques Broxton2013 ; Zhang20223 . On the other hand, a conventional
microscope with an objective numerical aperture of 0.45, as used in this
experiment, would have a depth of field of approximately $5$ $\mu$m (see
Supplementary materials for details).

Next, we test our technique on a more complex 3D scene. Imaging of multiple
overlapping fiber strands, between $5-10$ $\mu$m in diameter, on the edges of
a $\sim 3$ mm thick stack of lens-cleaning tissue can be seen in Fig. 4. A
single measurement was taken on the stack of lens tissue with digital
refocusing performed in post-processing. We can see fiber strands more than 1
mm away from the objective lens focus, and completely not visible at the
objective focus, being brought back into focus. The post-processing results in
a stack of in-focus images obtained at various depths in the sample, some
examples of which are shown in Fig. 4(a). The sum of all of these depth images
results in an “all in focus” image (Fig. 4(b)) in which every strand,
regardless of its position, is bought into focus. Finally, by colour-coding
each image based on its depth, we generate a depth map of the 3D scene shown
in Fig. 4(c).

Some artifacts/halos can seen in the refocused images, this is likely a result
of wavefront distortions from the multiple lenses used in the setup which were
not taken into account during the refocusing procedure. Another likely source
is in the approximations made in the refocusing procedure, such as small angle
approximations and the uniform illumination beam intensity (see the
Supplementary for details).

Figure 4: Refocusing of lens tissue fiber (a) Digitally refocusing onto fiber
strands of lens-cleaning tissue at various distances from the focus of the
objective lens. The image at the lens focus is highlighted in red. The fiber
strands are from the edges of a $\sim 3$ mm thick stack of lens cleaning
tissue. The measurement is taken in with a data acquisition time of 200 s. The
empty circular region on the top right is the part of the camera used for
capturing the idler photons. (b) Image showing all fiber strands brought into
focus through post-processing. (c) Depth map of the fiber strands. A video
showing the refocused images at different depths can be found in the
Supplementary.

##  III Discussion

We have made a proof-of-concept demonstration of a LFM design based on
utilizing the inherent position and momentum correlation of entangled photon
pairs generated through the process of SPDC. Since each degree of freedom can
potentially be measured on separate cameras, one does not need to sacrifice
position resolution for momentum resolution or vice versa as in conventional
LFM designs. This has allowed us to achieve a DOF that is approximately 3
times larger than that of the latest LFM designs, all at an illumination power
of about 3 pW ($15\times 10^{6}$ photons per second at 810 nm). With the high
momentum resolution (effectively $\sim 50\times 50$ pixels at $13.8$ $\mu$m-1
per pixel), it is even possible to digitally refocus objects placed at near
infinity, though with a moderate resolving power. In addition, as compared to
some conventional LFM techniques, QCLFM is also insensitive to whether the
target is placed before or after the objective focus.

The DOF can be further increased if a higher degree of momentum correlation is
achieved with a separate camera used for the momentum measurement on the idler
photon. Momentum correlation can be improved by increasing the pump beam waist
or using a pump with a longer coherence length. Details on this can be found
in the Methods section. Position resolution can be further improved by a
factor of 3 on the event camera (TPX3CAM) using centroiding techniques Kim2020
. QCLFM only utilizes the strong correlations inherent in spontaneous
parametric downconversion, and does not leverage photon entanglement. Thus it
is possible to imagine combining QCLFM with other quantum sensing techniques
which directly utilize entanglement, such as quantum lithography Boto2000 ;
DAngelo2001 , to achieve resolution beyond the diffraction limit.
Additionally, the current QCLFM design is still in its most basic form, we
believe significant performance gains can be made by incorporating more
sophisticated techniques already developed for the conventional LFM design
such as 3D deconvolution and wavefront shaping.

One limitation of the current QCLFM design and refocusing algorithm is that it
can only be used for targets with a known or constant phase profiles. A
potential method to solve this is to also capture the beam in an intermediate
plane in addition to just the near and far field. With this extra information
one may be able to determine both the amplitude and phase of the diffraction
pattern of the sample at two different planes from which the phase and
amplitude of the sample can be retrieved. The feasibility of this approach
will require further investigation. Another limitation to the technique’s
performance (and all single photon quantum imaging techniques in general) is
in its slow data acquisition speed, which is in the order of a few hundred
seconds per frame as compared to the 10-100 Hz frame rate of classical LFM
designs. This is mainly limited by the detection efficiency and timing
resolution of the detection camera. The single photon detection efficiency of
the camera used here is $\sim 7\%$ with an effective timing resolution of
$\sim 8$ ns Vidyapin2022 . Since the pair detection efficiency is quadratic
with respect to the single photon detection efficiency, with a camera
efficiency of 50$\%$, we could expect a $(50/7)^{2}\approx 50$ times
improvement in the data acquisition speed. Moreover, the signal-to-noise ratio
of temporal correlation measurement scales as the square-root of the timing
resolution, so if a timing resolution of 0.1 ns can be achieved, a further 10
times reduction to the data acquisition time is possible. Single-photon
avalanche photodiode array cameras meeting these specifications are currently
in development and may become commercially available soon Canon . We expect
this makes real time imaging and microscopy with QCLFM (and many other quantum
imaging techniques) a possibility in the near future.

##  IV Methods

### Experimental Setup

Figure 5: Experimental setup of QCLFM (a) Experimental setup of the QCLFM.
PBS: polarizing beamsplitter, LP filter: longpass filter. (b) A typical image
captured on the camera, before time correlation analysis, accumulated over 25
s, of a 1951 USAF resolution target placed 200 um away from the objective
focus. The signal beam covers the whole sensor area, while the idler is the
bright region on the top-right corner. (c) The modified experimental setup for
performing momentum correlation measurement. (d) Image captured on the camera,
before time correlation analysis, accumulated over 10 s, for performing
momentum correlation measurement. The darker patch seen in the idler beam in
both (b) and (d) is due to a damaged region on the camera system resulting in
lower detection efficiency.

The schematic of our QCLFM setup is shown in Fig. 5(a). A fiber coupled 405 nm
continuous wave (CW) laser, at $\sim 30$ mW, is used to pump a $1\times
2\times 1$ mm (H$\times$W$\times$L) Type II periodically-poled potassium
titanyl phosphate (ppKTP) crystal to produce entangled photon pairs through
SPDC, at a rate of approximately $15\times 10^{6}$ photon pairs per second or,
after accounting for the $7\%$ camera detection efficiency, about 15 photons
per second per pixel detected on the camera. The photon pairs are orthogonal
in polarization, correlated in position, momentum and time. The SPDC photons
are then separated by a polarizing beam splitter (PBS) into two paths. In the
path of the signal photon, the plane of the crystal is first imaged onto the
sample placement region and then onto a time-tagging event camera (TPX3CAM
Nomerotski2019 ; ASI ) to record the signal photon’s position information and
arrival time. In the path of the idler photon, the Fourier plane of the
crystal is projected onto a corner of the camera to record the idler photons’
momentum information and the subsequent arrival time. Thus, the position
information of the signal photon is captured directly and its momentum
information can be inferred from the idler partner. The photon pairs are
identified through a time correlation measurement, with a coincidence gating
time of 10 ns, on all photons detected between the idler beam region and the
rest of the camera. A typical image captured, before time correlation
analysis, can be seen in Fig. 5(b). The idler beam on the corner is seen much
brighter compared to the signal beam covering the whole sensor, this is due to
a similar amount of photons being concentrated onto a smaller area.

The TPX3CAM has $256\times 256$ pixels with a pixel pitch of 55 $\mu$m. The
pixels can individually time the arrival time of a light pulse with 1.6 ns
accuracy. The camera is made single photon sensitive with an attached image
intensifier in which a single photon is converted into a flash of light bright
enough to be registered by the camera. The flash of light will illuminate a
small cluster of pixels on the camera, on which a clustering identification
and centroiding algorithm must be implemented to regroup each cluster into a
single event. A consequence of the clustering is a slight blurring of the raw
image data, resulting in it being difficult to find the objective focus
precisely. This is what resulted in the slight decrease in resolution at 0
$\mu$m seen in Fig. 3(d). The timing of the camera pixels is intensity
dependent, in which brighter pixels of a cluster will be registered as an
earlier event than dimmer pixels of the same cluster, even though they are
created from the same single photon event. This intensity-dependent timing
must also be corrected and will result in losing some timing accuracy giving
an effective timing accuracy of $\sim 8$ ns. The quantum efficiency of the
camera system (camera + intensifier) was measured to be $\sim 7\%$. More
details on the camera and the post-processing required can be found in
Zhao2017 ; Vidyapin2022 .

The lenses in the signal beam were chosen such that the signal beam is not too
tightly focused onto the sample region, allowing the field of view to stay
roughly constant in the region of $\sim\pm 2$ mm from the objective focus.
When magnified onto the camera, the beam size should also roughly match the
sensor area of the camera to ensure most photons will be captured. Likewise,
in the idler beam, the lenses were chosen such that the idler beam will cover
roughly a region of $100\times 100$ pixels of the camera.

In order to achieve the highest degree of momentum correlation possible, a
pair of lenses with focal length 100 mm and 150 mm were used to collimate and
demagnify the laser beam to $\sim 1$ mm in diameter when hitting the ppKTP
crystal. The expected degree of momentum correlation can be calculated as
Defienne2019

$\sigma_{k_{1}+k_{2}}=\sqrt{1/l_{c}^{2}+1/(4\omega_{p}^{2})},$ (1)

where $\sigma_{k_{1}+k_{2}}$ is the standard-deviation in the momentum
correlation between two photons with momentum $k_{1}$ and $k_{2}$, $l_{c}$ is
the coherence length of the pump laser and $\omega_{p}$ is the pump beam waist
radius at the crystal. For our pump laser, $l_{c}\approx 200$ $\mu$m and
$\omega_{p}\approx 500$ $\mu$m, thus giving $\sigma_{k_{1}+k_{2}}\approx
5\times 10^{-3}$ $\mu$m-1.

The experimental setup to perform momentum correlation measurement is shown in
Fig. 5(b). Here, the Fourier plane of the crystal is imaged by both the signal
and idler photons, each onto a separate region of the camera. The common 200
mm lens is moved along the beam path until the highest degree of momentum
correlation is observed. This setup is designed such that alteration of the
idler path is kept to a minimum when converting to the QCLFM setup of Fig.
5(a). Momentum correlation measurement should be performed first in order to
ensure that when converted to QCLFM, the camera is imaging the exact Fourier
plane of the crystal which would contain the highest degree of momentum
correlation.

Figure 6: Momentum correlation measurement (a) Joint probability distribution
of photon momentum in the x-direction $k_{x}$. (b) Joint probability
distribution of photon momentum in the y-direction $k_{y}$. (c) Sum coordinate
projection of the joint probability distribution with the fitted Gaussian to
the cross-section of the $k_{x}$ and $k_{y}$ direction shown in the inset.
Fitted Gaussian gives $\sigma_{k_{x1}+k_{x2}}=6.1\times 10^{-3}$ $\mu$m-1
(0.90 pixels) and $\sigma_{k_{y1}+k_{y2}}=6.4\times 10^{-3}$ $\mu$m-1 (0.93
pixels).

The measured momentum correlation, determined by fitting a Gaussian of the
form $f(k)=a\exp\left(-\frac{(k-b)^{2}}{2\sigma_{k_{1}+k_{2}}^{2}}\right)$ to
the central peak of the sum coordinate projection of the momentum joint
probability distribution seen in Fig. 6(c), is
$\sigma_{k_{x1}+k_{x2}}=6.1\times 10^{-3}$ $\mu$m-1 (0.90 pixels) and
$\sigma_{k_{y1}+k_{y2}}=6.4\times 10^{-3}$ $\mu$m-1 (0.93 pixels) in the x and
y direction respectively. This is in agreement with the expected $5\times
10^{-3}$ $\mu$m-1 as calculated using Eq. (1).

The achievable DOF of the current experimental design is mostly limited by the
degree of momentum correlation. Although approximately $100\times 100$ pixels
have been allocated for momentum measurement, effectively, the momentum
resolution is only $50\times 50$ as the FWHM of the momentum correlations is
2.1 pixels wide. Using a laser with a longer coherence length of $l_{c}=2$ mm
will be able to improve $\sigma_{k_{1}+k_{2}}$ by a factor of 5 which would
allow the full sensor area of the camera ($256\times 256$ pixels) to be
effectively used for momentum measurement when two cameras are used as the
experiment was initially designed.

### Digital Refocusing Procedure

Using the position and momentum information of each photon, the operation for
digital refocusing of a sample placed out of focus by a distance $z$ can be
achieved using two steps. First, the trajectory of the photons are determined
through ray tracing using a ray transfer matrix

$\begin{bmatrix}\overrightarrow{r_{2}}\\\
\overrightarrow{\theta_{2}}\end{bmatrix}=\begin{bmatrix}A&B\\\
C&D\end{bmatrix}\begin{bmatrix}\overrightarrow{r_{1}}\\\
\overrightarrow{\theta_{1}}\end{bmatrix},$ (2)

where $\overrightarrow{r_{1}}$ and $\overrightarrow{r_{2}}$ are the positions
of each photon pair detected at the two camera image planes,
$\overrightarrow{\theta_{1}}$ and $\overrightarrow{\theta_{2}}$ are the angles
at which the photons hit each plane and the ray transfer (ABCD) matrix is
determined by the optical components placed between the two image planes.
Since $\overrightarrow{r_{1}}$ and $\overrightarrow{r_{2}}$ are measured
directly by the two cameras and the ABCD matrix is also known,
$\overrightarrow{\theta_{1}}$ and $\overrightarrow{\theta_{2}}$ can be
determined from Eq. (2). Knowing $\overrightarrow{r}$ and
$\overrightarrow{\theta}$, an appropriate ABCD matrix is then used to
digitally propagate each photon to the location of the sample. The image
obtained after this first step is the sample’s diffraction pattern amplitude
at a distance $z$ when illuminated by a plane wave propagating in the
z-direction (this is theoretically shown in the Supplementary Materials).

Next, a Gerchberg-Saxton type algorithm can be used to recover the amplitude
of the sample if the sample has a uniform or known phase profile. The
traditional Gerchberg-Saxton algorithm Saxton1972 ; Fienup1982  uses the
amplitude information obtained at the image plane and Fourier plane to
retrieve the phase information of the two through the Fourier transform
relation between the two planes. Here, we use the amplitude of an arbitrary
plane and the phase of another to retrieve the corresponding missing phase and
amplitude of the two planes. The relationship between the two planes is no
longer a direct Fourier transform but that of wave diffraction where
techniques for approximating wave diffraction, such as the angular spectrum
method or Fresnel diffraction method, must be used. The algorithm works as
follows:

  1. 1.

Guess an initial phase profile for the diffraction pattern amplitude obtained
after ray tracing (a flat initial phase was used here).

  2. 2.

Apply the angular spectrum method to the diffraction pattern to reverse
diffraction and obtain an amplitude and phase profile of the sample (Fig. 2 b1
to b2).

  3. 3.

Replace the phase profile of the sample obtained from 1. with the known phase
profile (which is zero for the targets used in this experiment) and then apply
the angular spectrum method to obtain the amplitude and phase of its
diffraction pattern (Fig. 2 b2 to b3).

  4. 4.

Update the diffraction pattern obtained from ray tracing with the new phase
profile (Fig. 2 b3 to b4).

  5. 5.

Repeat the process from step 2.

The angular spectrum method is applied as follows, for a beam with an initial
amplitude and phase profile of $U_{i}(x,y)e^{i\phi_{i}(x,y)}$, the profile
after propagating a distance $z$ is

$U_{f}(x,y)e^{i\phi_{f}(x,y)}=\mathscr{F}^{-1}\left[\mathscr{F}\left(U_{i}(x,y)e^{i\phi_{i}(x,y)}\right)e^{ik_{z}z}\right],$
(3)

where $\mathscr{F}$ ($\mathscr{F}^{-1}$) denotes the (inverse) Fourier
transform operation and $k_{z}=\sqrt{k^{2}-k_{x}^{2}-k_{y}^{2}}$ with
$k=2\pi/\lambda$ being the photon wavenumber.

## Acknowledgements

The authors are grateful to Denis Guay, and Doug Moffatt for technical
support. The authors acknowledge the support of Canada Research Chair, NRC-
uOttawa Joint Centre for Extreme Quantum Photonics (JCEP), Quantum Sensors
Challenge Program at the National Research Council of Canada and Defence
Research and Development Canada.

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## Supplementary Materials

### Refocusing Operation

For a plane wave traveling in an arbitrary direction
$U(x,y,z)=Ae^{i(ux+vy+wz)}$ (with $2\pi/\lambda=k=\sqrt{u^{2}+v^{2}+w^{2}}$)
illuminating a target with complex spatial profile
$f(x,y)=\mathscr{F}^{-1}\left[F(k_{x}/2\pi,k_{y}/2\pi)\right]$ ($\mathscr{F}$
denoting the Fourier transform operation) placed at $z=0$, the transmitted
wave can then be written as a superposition of plane waves (also known as the
angular spectrum method) given by

$\displaystyle U_{T}(x,y,z)$
$\displaystyle=\frac{1}{4\pi^{2}}\int\int^{\infty}_{-\infty}F(k_{x}/2\pi,k_{y}/2\pi)e^{i(k_{x}x+k_{y}y)}U(x,y,0)e^{i\tilde{k}_{z}z}dk_{x}dk_{y}$
$\displaystyle=\mathscr{F}^{-1}\left[F(k_{x}/2\pi,k_{y}/2\pi)e^{i(\tilde{k}_{z}z)}\right]U(x,y,0),$
(4)

with $\tilde{k}_{z}=\sqrt{k^{2}-(k_{x}+u)^{2}-(k_{y}+v)^{2}}$.

In the case of small angles approximation where the x and y components of the
k-vector is much smaller than the z-component, we can approximate
$\tilde{k}_{z}$ using a Taylor series expansion

$\displaystyle\tilde{k}_{z}$
$\displaystyle=\sqrt{k^{2}-k_{x}^{2}-k_{y}^{2}-2k_{x}u-2k_{y}v-u^{2}-v^{2}}$
$\displaystyle\approx k_{z}-\frac{k_{x}u+k_{y}v}{k_{z}}+O(u^{2},v^{2}),$ (5)

where $k_{z}=\sqrt{k^{2}-k_{x}^{2}-k_{y}^{2}}$.

Now substituting back into eq.4 we have

$\displaystyle U_{T}(x,y,z)$
$\displaystyle\approx\mathscr{F}^{-1}\left[F(k_{x}/2\pi,k_{y}/2\pi)e^{i\left(k_{z}-\frac{k_{x}u+k_{y}v}{k_{z}}\right)z}\right]U(x,y,0)$
$\displaystyle=\mathscr{F}^{-1}\left\\{\mathscr{F}\left[f(x-\Delta x,y-\Delta
y)\right]e^{ik_{z}z}\right\\}U(x,y,0)$ $\displaystyle=\tilde{f}(x-\Delta
x,y-\Delta y,z)U(x,y,0),$ (6)

where we have used the Fourier transform property $\mathscr{F}\left[f(x-\Delta
x\right]=F(k_{x}/2\pi)e^{-ik_{x}\Delta x}$ with $\Delta x=\frac{uz}{k_{z}}$
and $\Delta y=\frac{vz}{k_{z}}$ and
$\tilde{f}(x,y,z)\equiv\mathscr{F}^{-1}\left\\{\mathscr{F}\left[f(x,y)\right]e^{ik_{z}z}\right\\}$
is the diffracted field of a target $f(x,y)$ illuminated by a plane wave
traveling in the z-direction.

For this experiment, there are an infinite number of plane waves illuminating
the target of which we are able to measure a total of $N\times M$, the total
number of pixels used to capture the idler photons in the far-field plane,
Thus we will denote the transmitted wave associated with a particular far-
field pixel, with pixel index $n$ and $m$ in the x and y direction
respectively, as $U_{T}{{}_{nm}}(x,y,z)=\tilde{f}(x-\Delta x_{n},y-\Delta
y_{m},z)U_{nm}(x,y,0)$ with $U_{nm}(x,y,0)=A_{nm}e^{i(u_{n}x+v_{m}y)}$. The
captured image in the near-field plane $I_{0}(x,y,z_{0})$ is thus given by the
sum of all the transmitted waves squared, i.e.

$I_{0}(x,y,z_{0})=\left|\sum_{n,m}\tilde{f}(x-\Delta x_{n},y-\Delta
y_{m},z_{0})U_{nm}(x,y,0)\right|^{2}.$ (7)

Since we can track each photon in forming $I_{0}(x,y,z_{0})$ from the far-
field to the near-field plane, we can break up $I_{0}(x,y,z_{0})$ into a total
of $N\times M$ individual images $\tilde{I}_{nm}(x,y,z_{0})$ for each
$(u_{n},v_{m})$ pixel in the far-field thus

$I_{0}(x,y,z_{0})=\sum_{n,m}\tilde{I}_{nm}(x,y,z_{0})=\left|\sum_{n,m}\tilde{f}(x-\Delta
x_{n},y-\Delta y_{m},z_{0})U_{nm}(x,y,0)\right|^{2}.$ (8)

Making a coordinate shift gives

$\displaystyle\sum_{n,m}\tilde{I}_{nm}(x+\Delta x_{n},y+\Delta y_{m},z_{0})$
$\displaystyle=\left|\sum_{n,m}\tilde{f}(x,y,z_{0})U_{nm}(x+\Delta
x_{n},y+\Delta y_{m},0)\right|^{2}$
$\displaystyle=\left|\tilde{f}(x,y,z_{0})\right|^{2}\left|\sum_{n,m}U_{nm}(x+\Delta
x_{n},y+\Delta y_{m},0)\right|^{2}.$ (9)

On the left side of eq. 9, the operation of shifting each image
$\tilde{I}_{nm}$ by a distance of $\Delta x_{n}=\frac{u_{n}z_{0}}{k_{z}}$ and
$\Delta y_{m}=\frac{v_{m}z_{0}}{k_{z}}$ in the x-y plane is equivalent to the
ray-tracing operation of shifting the position of each photon by the same
factor based on the direction the photon is traveling given by
$\theta_{x}\approx\frac{u_{n}}{k_{z}}$ and
$\theta_{y}\approx\frac{v_{m}}{k_{z}}$ after propagating a distance $z_{0}$.
On the right hand side, the first term is the intensity of the diffraction
pattern of the target $\left|\tilde{f}(x,y,z_{0})\right|^{2}$ as if obtained
when illuminated by a plane wave traveling in the z-direction. The second term
is the intensity of the illuminating beam at $z=0$,
$\left|\sum_{n,m}U_{nm}(x+\Delta x_{n},y+\Delta y_{m},0)\right|^{2}$, with
each plane wave also shifted by $\Delta x_{n}$ and $\Delta y_{m}$ in the x-y
plane. If the intensity of the illuminating beam is uniform across the target
then,

$\sum_{n,m}\tilde{I}_{nm}(x+\Delta x_{n},y+\Delta
y_{m},z_{0})\propto\left|\tilde{f}(x,y,z_{0})\right|^{2}.$ (10)

Thus, with the amplitude of the diffraction pattern known and if we also know
the phase profile of the target, the amplitude of the target can be retrieved
using a Gerchberg-Saxton type algorithm whose implementation is detailed in
the main text.

##  V Depth of field of a conventional microscope

The depth of field for a conventional optical microscope is given by

$\text{DOF}=\frac{n\cdot\lambda}{\text{NA}^{2}}+\frac{n\cdot
e}{\text{M}\cdot\text{NA}},$ (11)

where, for this experiment, $n=1$ is the refractive index in air,
$\lambda=810$ nm is the photon wavelength, $\text{NA}=0.45$ is the objective
numerical aperture, $\text{M}=20$ is the magnification and $e$ the distance
resolved. So for $e=5$ $\mu$m the DOF is 4.6 $\mu$m and for $e=10$ $\mu$m the
DOF is 5.1 $\mu$m, thus QCLFM has a DOF that is at least 2 orders of magnitude
larger compared to a conventional microscope.

### Images of full dataset

All images for the dataset shown in Fig. 3(d) of the main manuscript is shown
below in Fig. 7.

Fig. 8 shows the dataset when using a 35 mm focal length condenser lens
instead of the 100 mm condenser lens as used for Fig. 7. This configuration
was chosen such that the smallest resolvable feature at each distance is close
to the resolution limit of the camera. With this configuration, at the
objective focus, the field of view (FOV) is approximately 1/3 of that when
using a 100 mm condenser lens. However, the FOV expands much more rapidly with
the 35 mm lens allowing larger target feature to better fit into the FOV when
far away from the objective focus.

In Fig. 8, we can see that beyond -200 $\mu$m away from the objective focus,
the presence of the target can no longer be observed in the near field of the
crystal plane imaged by the signal beam. From -2.8 mm away from the objective
focus, the presence of the target can once again be observed through ghost
imaging in the far field of the crystal plane imaged by the idler beam. A plot
showing the smallest resolvable features vs. the distance from focus is shown
in Fig. 9.

Figure 7: Full dataset of Fig. 3 of the manuscript showing digitally refocused
images of groups 6 and 7 of the 1951 USAF resolution target placed at various
distances from the objective focus. The smallest resolvable features after
refocusing are boxed in blue. Figure 8: Digitally refocused images of a 1951
USAF resolution target placed at various distances from the objective focus
when using a 35 mm focal length condenser lens. The centred resolution target
group from 0 to -750 $\mu$m away from the focus is group 7, -1 mm to -2.3 mm
is group 6, -2.8 mm to -4.4 mm is group 5, -5 mm to -7 mm is group 4 and -10
mm to -17.5 mm is group 3. The smallest resolvable features after refocusing
are boxed in blue. Images of the near field crystal plane, imaged by the
signal beam, and far field crystal plane, imaged by the idler beam, before
refocusing are shown below each refocused image.  Figure 9: Plot showing the
smallest resolvable features vs. the distance from focus for data shown in
Fig. 8.