# Adjustable spatial filter for optimal free-space quantum communication round
the clock

Andrej<EMAIL_ADDRESS>Fraunhofer Institute for Applied
Optics and Precision Engineering, Albert-Einstein-Str. 7, 07745 Jena, Germany
Friedrich Schiller University Jena, Faculty of Physics and Astronomy, Max-
Wien-Platz 1, 07743 Jena, Germany  Nico Döll  Fraunhofer Institute for Applied
Optics and Precision Engineering, Albert-Einstein-Str. 7, 07745 Jena, Germany
Uday Chandrashekara  Fraunhofer Institute for Applied Optics and Precision
Engineering, Albert-Einstein-Str. 7, 07745 Jena, Germany Christopher Spiess
Fraunhofer Institute for Applied Optics and Precision Engineering, Albert-
Einstein-Str. 7, 07745 Jena, Germany Friedrich Schiller University Jena,
Faculty of Physics and Astronomy, Max-Wien-Platz 1, 07743 Jena, Germany
Fabian<EMAIL_ADDRESS>Fraunhofer
Institute for Applied Optics and Precision Engineering, Albert-Einstein-Str.
7, 07745 Jena, Germany Abbe Center of Photonics, Friedrich Schiller University
Jena, Albert-Einstein-Str. 6, 07745 Jena, Germany

###### Abstract

Free-space quantum communication in daylight relies crucially on spatial
filtering. The optimal filter configuration, however, depends on ever-changing
link conditions. To this end, we developed an adjustable spatial filter that
can be used to change the system field of view on the fly. We demonstrate its
use in quantum key distribution over a 1.7-km free-space link. Furthermore, we
compare it to filtering with multi-mode fibre coupling. Finally, we
extrapolate our results to a broader range of realistic link conditions and
show that active field-of-view optimization has the potential to substantially
improve the overall secure key output of the system.

##  Introduction

Quantum communication with light is a rapidly developing field [1]. Its most
technologically mature form is quantum key distribution (QKD) [2]. On the path
towards large-scale QKD networks [3], the research focus has been increasingly
moving from fundamental towards the more practical aspects.

One of the remaining practical challenges in quantum communication over free-
space channels is operation in daylight [4]. During daytime, a vast amount of
sunlight enters the receiver telescopes and may end up completely masking the
weak quantum signal upon detection. In order to reduce this background noise
to acceptable levels, strong filtering is necessary in all the available
degrees of freedom. Implementation of temporal filtering is relatively
straightforward with well-established methods. Similarly, spectral filtering
can be effortlessly implemented with off-the-shelf components. Efficient
spatial filtering, on the other hand, remains a very challenging task due to
atmospheric turbulence, which randomly distorts the signal wavefront at the
receiver on millisecond time scales [5]. This makes the light very difficult
to focus through a pinhole or couple into an optical fibre. The majority of
free-space quantum communication experiments to date were thus limited to low
ambient light conditions, such as in nighttime.

Nevertheless, several proof-of-principle QKD experiments in daylight have been
reported over terrestrial free-space links. The first successful demonstration
was performed in 2000 with a faint pulse source, using multi-mode (MM) fibre
coupling as a means of spatial filtering [6]. The first entanglement-based
implementation followed in 2009, where a pinhole was used as a spatial filter
[7]. These experiments were done without any mitigation of the effects of
atmospheric turbulence. More recently, it was demonstrated that having only
tip-tilt wavefront correction suffices for QKD in daylight through 53 km of
atmosphere [8]. Soon after, the first demonstrations of daylight QKD with
adaptive optics for higher-order wavefront correction have emerged [9, 10].
All these employed a fixed spatial filter.

For optimal spatial filtering, the field of view (FOV) of the system needs to
be reduced to roughly match the average angular extent of the signal focal
spot. Finding the exact optimum, however, is far from straightforward due to
the intricate interplay between received signal, noise, and the performance
metric of the particular quantum communication protocol (e.g. secure key
rate), and in fact even depends on the ever-changing link conditions.
Moreover, there is an interplay between the different filtering degrees of
freedom – weaker filtering in one can be compensated by stronger filtering in
another. Gruneisen et al. comprehensively studied the concept of spatial
filtering for space-to-Earth QKD with simulations [11]. It can be seen from
their results that the optimal spatial filter FOV, particularly in the case of
tip-tilt-only correction, depends on the link characteristics, such as solar
radiance, satellite altitude angle, and distance. Although not explicitly
pointed out by Gruneisen et al., their plots indicate that having an
adjustable spatial filter that could impose optimal filtering at all times
would yield a considerable boost to the total generated key. We note that
adaptive spatial filtering in their work refers to a fixed FOV filter with
adaptive optics and not to a filter with an adaptive or adjustable FOV. Ko et
al. made the first attempts to experimentally address the question of optimal
filtering and show an interplay between temporal and spatial degrees of
freedom [12]. While they clearly find the optimum in the temporal domain,
measurements with only two different fixed spatial filters did not offer much
insight into optimal spatial filtering. Furthermore, their experiment was done
over a short 275-m link, for which they reported a negligible effect of
atmospheric turbulence.

Here, we introduce the concept of an adjustable spatial filter (SF) for
applications in quantum communication, which can be adjusted to the optimum
size under different link conditions. We built a prototype SF module that can
seamlessly change the system FOV with $\upmu$rad precision. We integrated it
into a free-space QKD system and experimentally demonstrated its use over a
1.7-km link in different link conditions. We report on our findings and
compare optimal spatial filtering to simpler filtering with MM fibre coupling.
Furthermore, we develop a theoretical model of our system and use it to
investigate the potential of an adjustable spatial filter over a broader range
of realistic link conditions.

##  Methods

###  Experimental system

To study the experimental impact of variable spatial filtering, we used the
entanglement-based free-space quantum key distribution system reported in
[13], which is schematically shown in Fig. 1. The two communicating parties,
Alice and Bob, are separated by a 1.7-km free-space channel, which is spanned
by two identical telescopes with a primary aperture diameter of 200 mm [14].
To mitigate the effects of atmospheric turbulence, we use a 1064 nm beacon
laser and tip-tilt correction at the receiver terminal. The quantum signal is
generated at Alice with an entangled pair source (EPS), which generates
polarization-entangled photon pairs at 810 nm by the means of spontaneous
parametric down-conversion. One photon of each entangled pair is guided with a
single-mode fibre to a local polarization analysis module (PAM). Here,
polarization is measured in one of the two mutually unbiased bases, which
without the loss of generality we call horizontal-vertical and diagonal-
antidiagonal, and are chosen at random with 50:50 probability. For detection,
we use commercial single-photon avalanche detectors (SPADs) with detection
efficiency of $>$ 60% and a dark count rate of $<$ 500 cps. Each detection
event is given a precise arrival time by a rubidium-clock-driven time tagger.
The other photon of each pair is emitted into free space, combined with the
beacon beam, and sent to Bob, where it is similarly measured with a
combination of a PAM, 4 detectors, and a time tagger. The data generated by
both time taggers was stored and later processed offline, as explained below
in Section 2.3.

Figure 1:  Schematic representation of the experimental system. Here, EPS is
the entangled pair source, PAM-A is the polarization analysis module at Alice,
PAM-B is the polarization analysis module at Bob, SPADs are single-photon
avalanche diodes, and H, V, D, and A stand for horizontal, vertical, diagonal,
and antidiagonal polarization, respectively.

In addition to polarization analysis optics, Bob’s PAM further incorporates
dedicated spatial and spectral filters to prevent large amounts of background
noise from reaching the detectors. The latter is realized using a stack of
commercial interference filters, resulting in a passband with a full width at
half maximum (FWHM) of 2.95 nm around 810 nm. For comparison, the EPS has a
spectral bandwidth of 0.45 nm (FWHM), meaning that we do not utilize the full
potential of spectral filtering here. The spatial filter with an adjustable
FOV was developed specifically for the experiments here and is discussed in
detail in Section 2.2 below. Furthermore, Bob’s PAM also includes a camera for
imaging the focal plane of the spatial filter, which is used for alignment and
characterization of the spatial filter and the incoming beam, and a waveplate-
based polarization controller, which is used to align polarization frame of
reference between Alice and Bob. When using the camera, a mirror is
temporarily inserted into the beam path (dashed rectangle in Fig. 1). Alice’s
PAM is a much simpler version of Bob’s, since it does not require dedicated
filtering. This is due to the fact that it is directly connected to the EPS
with a single-mode fibre and thus not exposed to the free-space link. Both
PAMs were sealed, so that the light from the outside could enter Alice’s PAM
only through the input fibre and Bob’s PAM only through a small aperture.
Bob’s PAM further incorporated an inner wall to prevent any stray light that
was rejected by the filters to reach the polarization analysis part.

Finally, note that after polarization analysis, the photons are coupled into
MM fibres, which may also contribute to spatial filtering and is crucial for
interpretation of our experimental results. Although having a dedicated
spatial filtering module makes MM fibre coupling obsolete from the spatial
filtering point of view, it is still practical for guiding light from the PAM
to the detectors, allowing the PAM to be much more compact.

###  Spatial filtering

We developed a dedicated adjustable spatial filter (SF) that allowed us to
vary the system FOV. It is realized with a lens-based telescope and a small
opening in its focal plane, acting as a field stop, which was provided by Carl
Zeiss Microscopy GmbH. It has a near-square shape and its side length is
electronically adjustable between zero and about 400 $\upmu$m with a precision
of a few microns. For our optical system, 1 $\upmu$m in the focal plane of the
SF translates to about 0.344 $\upmu$rad in the receiver telescope primary
aperture plane. In terms of the system FOV, the SF therefore has a dynamical
range of about 0-140 $\upmu$rad and a precision of $\sim$1 $\upmu$rad. MM
fibre coupling in PAM also acts as a field stop – the nominal MM fibre core
diameter expressed in the receiver telescope primary aperture is about 51.6
$\upmu$rad. In the following, unless explicitly stated otherwise, all the
field values are given in angular units in the receiver telescope primary
aperture plane.

Aligning the SF field stop and the signal focal spot is of critical
importance. For this purpose, we imaged the SF focal plane with the camera in
Bob’s PAM. To observe the field stop, we illuminated the SF with diffuse
incoherent light. To observe the signal focal spot, we fully opened the SF and
replaced the EPS by a laser of the same wavelength. Due to atmospheric
turbulence, the short-term focal spot can be heavily distorted and its shape
fluctuates randomly on millisecond timescales [5]. However, what matters here
is the long-term average of the spot, which dictates the mean signal
throughput and can typically be well approximated with a Gaussian [5]. Note
that the size of the long-term focal spot varies with the strength of
turbulence. While this effect is significantly reduced by the tip-tilt
correction, the long-term size may still change considerably with time. At
various occasions, we measured values ranging from 11.4 $\upmu$rad to 39.4
$\upmu$rad (FWHM). For comparison, the estimated FWHM of an ideal
(diffraction-limited) spot for our system is 4.2 $\upmu$rad. An example long-
term focal spot measurement (exposure time $>$20 s), superimposed with images
of the SF field stop at different sizes, is shown in Fig. 2. MM fibre cores
are also drawn for reference (dashed circle).

Figure 2:  Average signal distribution, the adjustable field stop, and the
multi-mode fibre cores, as seen in the focal plane of the spatial filter. The
signal spot was measured while the tip-tilt correction was on. (a)-(c) Signal
with the field stop of various sizes and a schematic representation of ideal
alignment of the fibres. (d) Schematic representation of fibre misalignment.

We then characterized the signal throughput dependence on the SF setting,
which was done directly with the quantum signal from the EPS. At night, when
the background noise was negligible, we scanned the SF size and recorded the
total photon rate at Bob. The measurements are shown in Fig. 3(a). Since the
mean focal spot can be well approximated with a Gaussian, we model the signal
transmitted through the square-shaped field stop by a function of the form

$f_{1}(\theta_{\text{SF}};S_{0},\Delta)=S_{0}\,\text{erf}^{2}\biggl{[}\frac{\sqrt{\text{ln}(2)}\theta_{\text{SF}}}{\Delta}\biggr{]}\,,$
(1)

where erf is the error function, $\theta_{\text{SF}}$ is the spatial filter
FOV, corresponding to the side length of the field stop, $\Delta$ is the FWHM
of the focal spot, and $S_{0}$ is the normalization factor, corresponding to
the signal photon rate with fully open spatial filter. We can see that the
chosen model (solid line) shows a good agreement with the measurements.

Figure 3:  Characterization of the spatial filter. (a) Signal count rate
dependence on the spatial filter field of view. (b) Background noise count
rate dependence on the spatial filter field of view. The dashed line is the
corresponding estimate for the case if there was no multi-mode (MM) fibre
coupling. The vertical dotted line indicates the point where the diagonal of
the spatial filter matches the MM fibre core diameter.

Similarly, we characterized the background noise dependence on the SF setting.
These measurements were performed when the link was exposed to direct sunlight
and the EPS turned off. One such set of measurements is shown in Fig. 3(b).
The noise that is transmitted by the SF scales quadratically with
$\theta_{\text{SF}}$ [11]. At a certain value of $\theta_{\text{SF}}$,
however, the MM fibres start limiting the system FOV, which breaks the
quadratic dependence on $\theta_{\text{SF}}$ and the level of detected noise
eventually saturates. In the case of perfect alignment of the MM fibres, all
the four fibres would act as one and the noise would be given by an overlap
integral of a square and a circle. Our measurements, however, showed a slower
convergence towards the saturation than such a model would predict. We believe
this is due to a slight misalignment of the MM fibres in our system, as
schematically shown in Fig. 2(d). The four fibres then effectively act as a
single one with a smoother profile than a sharp circle, which results in a
slower convergence of the overlap integral. We therefore looked for another
function, which would still increase quadratically in the limit of small
$\theta_{\text{SF}}$, while having a slower convergence towards saturation for
larger $\theta_{\text{SF}}$. As it turns out, the measured data can be well
described with a function of the form

$f_{2}(\theta_{\text{SF}};B_{0},\gamma)=B_{0}\left(1-\exp{\left[-\frac{\theta_{\text{SF}}^{2}}{2{\gamma}^{2}}\right]}\right)\,,$
(2)

where $B_{0}$ is the normalization factor and parameter $\gamma$ is the
function turning point, i.e. the point at which the slope of the curve starts
decreasing with $\theta_{\text{SF}}$.

The turning point $\gamma$ is expected to be a system constant that depends on
the MM fibre coupling configuration. As $\theta_{\text{SF}}$ increases, the MM
fibre coupling should start affecting the background rate when the diagonal of
the SF field stop, which relates to $\sqrt{2}\theta_{\text{SF}}$, matches the
diameter of the MM fibre core image in the SF field stop plane, as
schematically shown in Fig. 2(b). We measured the noise dependence on the SF
setting on 9 different occasions throughout a day, each time finding a good
agreement with Eq. (2). As predicted, all the fits resulted in very similar
$\gamma$ values, having the mean and standard deviation of 36.8 $\upmu$rad and
1.4 $\upmu$rad, respectively. Moreover, $\sqrt{2}\gamma$ yields
$52.0~{}\upmu\text{rad}$, which is indeed very close to the nominal MM fibre
FOV of $51.6~{}\upmu\text{rad}$. This confirms $\gamma$ is a system constant
and allows us to set it to a fixed value in our model. We are then left only
with one free parameter to describe the amount of background noise, namely
$B_{0}$, which represents the amount of noise in the case when the SF is
completely open and the MM fibre coupling performs all the spatial filtering
(i.e. MM fibre coupling limit).

The corresponding fit of the data is shown in Fig. 3(b) (solid line). The
point at which the SF field stop diagonal matches the MM fibre core diameter
in the focal plane of the SF is also shown (vertical dotted line). We can
clearly see that it matches the turning point of the fit function and marks
the boundary between the two regimes: when spatial filtering is dominated by
the adjustable spatial filter and when it is dominated by MM fibre coupling.
To estimate background rates for the case if there was no MM fibre coupling
(e.g. free-space detectors inside the PAM), we perform Taylor expansion of Eq.
(2) and keep only the terms up to the quadratic one (dashed line), arriving at

$\widetilde{f}_{2}(\theta_{\text{SF}};B_{0},\gamma)=B_{0}\frac{\theta_{\text{SF}}^{2}}{2{\gamma}^{2}}\,.$
(3)

This allows us to assess the role of MM fibres in our system as well as to
generalize our results to a system without MM fibre coupling, as shown later
in Section 3.1. This approximation will eventually break down, because the
noise will not increase indefinitely with $\theta_{\text{SF}}$ but would
saturate at the FOV limit imposed by the optical system before the SF.
However, any practical detection system would most likely saturate well before
that limit is reached. Note that the fibre-dependent $\gamma$ is part of the
fibre-free model Eq. (3) only because we defined $B_{0}$ in terms of noise
collected by the fibres in the MM fibre coupling limit.

###  Data acquisition and post-processing

For each data point, we recorded time tags of single-photon detection events
at Alice and Bob for 8 seconds. The two sets of time tags were combined and
processed offline, whereby differences in the local clocks were compensated
for by exploiting the inherent time correlations of entangled photon pairs
[15]. We then extracted coincidences, which we define as events when a single-
photon detection at one site happens within $\pm\tau/2$ of a detection time at
the other site, where $\tau$ is called coincidence window [16]. For our
experiments, we chose $\tau=800\ \text{ps}$. Note that the extraction of
coincidences essentially performs the function of temporal filtering.

With the coincidence rates for all detector combinations, we estimated the
asymptotic secure key as

$R=C_{\text{sift}}\left[1-f\text{H}_{2}(E_{\text{HV}})-\text{H}_{2}(E_{\text{DA}})\right]\,$
(4)

where $C_{\text{sift}}$ is the total coincidence rate after sifting, i.e.
coincidences in the same measurement basis at Alice and Bob, $f$ is the error
correction efficiency for which we for simplicity take a constant value of
1.22 [17], $\text{H}_{2}(x)$ is the binary entropy function defined as
$\text{H}_{2}(x)=-x\log_{2}(x)-(1-x)\log_{2}(1-x)$, and $E_{\text{HV}}$ and
$E_{\text{DA}}$ are the measured error rates in horizontal-vertical and
diagonal-antidiagonal basis, respectively [16].

###  Model for key rate dependence on spatial filtering

We now integrate the spatial filtering model above into the QKD model from
[16]. Assuming the error rate to be the same in both bases, it is given by

$E=E_{\text{HV}}=E_{\text{DA}}=\frac{{\eta}^{c}C^{\text{t}}E^{\text{pol}}+\frac{1}{2}C^{\text{acc}}}{{\eta}^{c}C^{\text{t}}+C^{\text{acc}}}\,,$
(5)

where $C^{\text{t}}$ is the true coincidence rate, $C^{\text{acc}}$ is the
accidental coincidence rate, ${\eta}^{c}$ is the coincidence-window-dependent
detection efficiency, and $E^{\text{pol}}$ is the baseline error rate due to
polarization mismatch. Furthermore,

$C^{\text{t}}=\eta_{\text{A}}S_{\text{B}}^{\text{t}}\,,$ (6)
$C^{\text{acc}}=\frac{\left(1-\text{e}^{-S_{\text{A}}^{\text{m}}\tau}\right)\left(1-\text{e}^{-S_{\text{B}}^{\text{m}}\tau}\right)}{\tau}\,,$
(7)

and

${\eta}^{c}=\text{erf}\left[\sqrt{\text{ln}(2)}\frac{\tau}{t_{\Delta}}\right]\,,$
(8)

where $\eta_{\text{A}}$ is the total single photon efficiency in Alice’s
channel, $S_{\text{B}}^{\text{t}}$ is the true (signal) photon rate at Bob,
$S_{\text{A}}^{\text{m}}$ and $S_{\text{B}}^{\text{m}}$ are the total measured
photon rates at Alice and Bob, respectively, and $t_{\Delta}$ is the FWHM of
the total system timing jitter. $C^{\text{t}}$ is related to the measured
coincidence rate $C^{\text{m}}$ through

$C^{\text{m}}={\eta}^{c}C^{\text{t}}+C^{\text{acc}}\,.$ (9)

We adapt the model to our experimental system by expressing
$S_{\text{B}}^{\text{t}}$ with Eq. (1) and defining

$S_{\text{B}}^{\text{m}}=S_{\text{B}}^{\text{t}}+S_{\text{B}}^{\text{b}}+S_{\text{B}}^{\text{d}}\,,$
(10)

where $S_{\text{B}}^{\text{d}}$ is the dark count rate and
$S_{\text{B}}^{\text{b}}$ is the background photon count rate at Bob according
to Eq. (2) or Eq. (3), depending on whether the MM fibres are considered or
not, using the empirically determined $\gamma$.

For the experiments here, most of the remaining model parameters can be set to
a fixed value. $S_{\text{B}}^{\text{d}}=761\ \text{cps}$ and $t_{\Delta}=710\
\text{ps}$ are system constants and were measured in advance.
$\eta_{\text{A}}$ consists of intrinsic EPS efficiency, the fibre-only channel
efficiency, and detector efficiency, all of which are sufficiently stable to
be considered constant over the course of our experiments. We estimated
$\eta_{\text{A}}=12.2\%$, which was done by measuring $C^{\text{m}}$,
$S_{\text{A}}^{\text{m}}$, and $S_{\text{B}}^{\text{m}}$ at night, when
$S_{\text{B}}^{\text{b}}\approx 0$, and using Eqs. (6)-(10). Similarly,
$S_{\text{A}}^{\text{m}}$ was near-constant at 1.4 Mcps. We determined
$E^{\text{pol}}$ as

$E^{\text{pol}}=\frac{C^{\text{wrong}}-\frac{1}{4}C^{\text{acc}}}{C^{\text{sift}}}\,,$
(11)

where

$C^{\text{right}}=C^{\text{HV}}+C^{\text{VH}}+C^{\text{DA}}+C^{\text{AD}}\,,$
(12)
$C^{\text{wrong}}=C^{\text{HH}}+C^{\text{VV}}+C^{\text{DD}}+C^{\text{AA}}\,,$
(13) $C^{\text{sift}}=C^{\text{right}}+C^{\text{wrong}}\,,$ (14)

and $C^{\text{ij}}$ is the coincidence rate between i-th channel at Alice and
j-th channel at Bob. We estimated $E^{\text{pol}}=2.6\%$, which we also assume
to be constant throughout the day of experiments.

Finally, using Eqs. (1)-(3) and (5)-(10) with Eq. (4), together with the fixed
parameters from above, we arrive at a secure key rate model
$R(\theta_{\text{SF}};S_{0},\Delta,B_{0})$ that is a function of the SF
setting and has three link-conditions-dependent free parameters. In Section 3,
we use this model for fitting the experimentally determined key rates and for
extrapolation to other potential link conditions. Note, that $B_{0}$ could be
independently characterized at the time of each experiment by turning the EPS
off and measuring and fitting the noise-only dependence on
$\theta_{\text{SF}}$, as in Fig. 3(b). On the other hand, removing the
background noise to measure only signal was not possible. Moreover, with the
quantum signal, it was also not possible to directly measure the focal spot.
However, fitting the model to the measurements not only allowed us to
interpolate and extrapolate the data but also provided us with an estimate for
$\Delta$ and $S_{0}$, which offered further insights into the link conditions
at the time of the experiments.

##  Results

###  Experimental demonstration of adjustable spatial filtering

We now experimentally demonstrate the use of the adjustable spatial filter for
quantum key distribution. On 15 June 2022, we performed two QKD experiments.
The first experiment was done at about 8 pm, right before sunset, when the Sun
was still fully above the horizon and the link was exposed to direct sunlight.
At this time, we recorded the highest noise rates during the entire day. This
is most likely due to the fact that Bob’s receiver was facing north-west,
meaning that the angle between the receiver telescope orientation and the Sun
was the smallest then. The second experiment was done at about 11 pm, long
enough after the sunset, so that the background noise was at a negligible
level. Each experiment consisted of a simultaneous recording of time tags at
both sites for a range of SF settings, which lasted several minutes in total.
Right after, we also measured the background-only dependence on the SF setting
by turning the EPS off. During the measurements within each experiment, we
assume the link conditions, i.e. the amount of sunlight and the strength of
atmospheric turbulence, did not considerably change. The recorded time tags
were later synchronized and post-processed to calculate the secret key rates,
as explained in Section 2.3. The results are shown in Fig. 4. From the
background-only measurements, we extracted the background noise normalization
factor $B_{0}$, as explained in Section 2.2. Using this $B_{0}$, we then fit
the experimentally determined key rates with the model
$R(\theta_{\text{SF}};S_{0},\Delta,B_{0})$ described in Section 2.4 (solid
lines), thus extracting the signal spot size $\Delta$ and the signal
normalization factor $S_{0}$. The corresponding estimated secure key rate in
the absence of the MM fibres is also shown (dashed lines).

The extracted parameters are given in Table 1. Since $\Delta$ is mostly
influenced by the residual wavefront aberrations uncorrected by the tip-tilt
system, it may give us an idea about the atmospheric turbulence strength
during the experiments – the larger the spot size, the stronger the
turbulence. The extracted values therefore indicate that the turbulence was
weaker right before sunset than later at night. This is not surprising, since
the surface and air temperatures around sunset are typically nearly identical,
resulting in lower wind speeds and weaker turbulence at that time [5].
Turbulence strength not only influences the focal spot size but also link
efficiency by spreading the beam size at the receiver telescope aperture, in
turn reducing receiver collection efficiency. Moreover, stronger turbulence
makes the link alignment more difficult, thus limiting the attainable
alignment precision and further lowering the link efficiency. This explains
why the extracted $S_{0}$ was larger at 8 pm, despite the EPS output being
almost the same during both experiments. The difference in $S_{0}$ was thus
mostly due to the difference in link efficiency. Consequently, higher key
rates could be achieved in direct sunlight than in the night. However, if we
had the same turbulence at 11 pm as at 8 pm, the key rate would saturate at a
value larger than 6 kbps.

Figure 4:  Experimentally determined asymptotic secure key rate dependence on the spatial filter field of view (points) and the corresponding fit (solid line) for two experiments with different link conditions. The dashed lines show the corresponding estimates for the case if there was no multi-mode (MM) fibre coupling at the receiver.  Table 1: Extracted model parameters for the experiments shown in Fig. 4. |  $B_{0}$ [Mcps] |  $\Delta$ [$\upmu$rad] |  $S_{0}$ [kcps]
---|---|---|---
8 pm |  1.59 $\pm$ 0.03 |  18.5 $\pm$ 0.7 |  237 $\pm$ 5
11 pm |  $\sim$ 0 |  23.0 $\pm$ 1.2 |  117 $\pm$ 3

The experiments clearly show that under certain conditions, spatial filtering
by the means of MM fibre coupling alone can be sufficient for QKD. However,
further reducing the FOV to the optimal value still results in a considerable
increase of the achievable secure key rate in daytime. The improvement is even
more pronounced in the case without MM fibre coupling. In nighttime, such
strong levels of spatial filtering are not needed anymore, therefore fully
opening the SF field stop becomes the optimal strategy, with or without MM
fibre coupling (the two curves overlap). These results already indicate that
the strategy to be actively adjusting the system FOV to its optimum, as the
link conditions change, has the potential to improve the total secure key
output of the system. The two experiments, however, do not cover a wide range
of possible link conditions. Moreover, since we cannot change the link-
conditions-dependent parameters at will, we cannot experimentally observe the
effect of each one of them individually over the actual free-space link.
Additionally optimizing also temporal filtering, which was not done here,
would improve the key rates and might further affect the optimal spatial
filtering configuration. In the next section, we thus use the developed model
to extrapolate our experimental results to a broader range of possible link
conditions, changing one parameter at a time, while also optimizing the
coincidence window for each data point.

###  Extrapolation to different link conditions

To extend the study to a wider range of terrestrial free-space link
conditions, we take the model and parameters extracted from the 8 pm
experiment and vary one parameter at a time. We limit ourselves to the more
general case without MM fibre coupling. Furthermore, we numerically optimize
coincidence window for each data point. We thus extrapolate the key rate
dependence on the spatial filter FOV to a set of evenly spaced values over the
following intervals: $B_{0}\in[0,1.59]$ Mcps, $\Delta\in[11.4,39.4]$
$\upmu$rad, and $S_{0}\in[0,237]$ kcps. The bounds correspond to the largest
and the smallest values we have recorded with our system in various
experiments over several months.

Figure 5:  Extrapolation of the model extracted from the 8 pm experiment to
different link conditions, assuming no fibre coupling and optimal coincidence
window. In each plot, one of the link-conditions-dependent parameters is
varied: (a) background noise count rate, (b) mean focal spot size, and (c)
signal count rate.

The results are shown in Fig. 5. We may clearly see that the largest influence
on the optimal FOV is due to the change of $\Delta$. To quantitatively
benchmark the potential of the adjustable spatial filter to improve the
overall output of the system, let us now consider a hypothetical scenario,
where only $\Delta$ would change while $B_{0}$ and $S_{0}$ would stay
constant. Suppose we would be experiencing weak turbulence, corresponding to
the smallest $\Delta$ considered in our extrapolations, and the system would
be operating with the optimal FOV (point A). The corresponding key rate would
be 5.9 kbps. If the conditions would then worsen to the point where $\Delta$
would be the largest considered here, the key rate would drop to 0.85 kbps
(point B). With the adjustable spatial filter, we could then optimize the
spatial filter FOV and increase the key rate to 1.7 kbps (point C), which is a
2-fold improvement. Should the conditions improve again to the starting ones,
the key rate would increase to 3.8 kbps (point D). We could then again
optimize the FOV and increase the key rate back to 5.9 kbps (point A), which
is a 1.5-fold improvement. For a system like ours, an adjustable spatial
filter therefore has the potential to improve the overall key output by a
substantial amount.

$B_{0}$ and $S_{0}$ also affect the optimal FOV, but to a smaller extent. We
may also note that worsening of $B_{0}$ and $S_{0}$ (i.e. increase and
decrease, respectively) shifts the optimal FOV to smaller values, while
worsening of $\Delta$ (increase) shifts it to larger values. $S_{0}$, on the
other hand, has a strong influence on the secure key rate. This explains why
we could achieve higher key rates at 8 pm in direct sunlight than at 11 pm,
when the background noise was negligible. In real QKD, an interplay between
these effects is therefore taking place.

##  Conclusion

We developed an adjustable spatial filter for quantum communication in
daylight and introduced the idea of active optimization of the system field of
view with the ever-changing link conditions. We demonstrated the filter in a
real QKD system with tip-tilt-only wavefront correction over a 1.7-km free-
space channel and showed it has the potential to substantially improve the
total secure key rate output.

The exact factor of improvement, however, heavily depends on the free-space
channel conditions, in particular turbulence strength and the amount of
daylight noise. Furthermore, it depends on the underlying system, such as the
level of filtering in other degrees of freedom and the level of compensation
of atmospheric turbulence. For systems without any wavefront correction at the
receiver, an adjustable spatial filter would be of even higher practical
relevance, since the focal spot size would vary to a much greater extent with
the varying turbulence strength. On the other hand, if the system incorporated
full adaptive optics, the benefit of having it adjustable would be much less
pronounced, since the focal spot size would not significantly change with link
conditions, as it is evident from the simulations in [11]. High-performance
adaptive optics would also enable efficient single-mode fibre coupling, which
is the extremest form of spatial filtering. However, our work shows that
intermediate levels of spatial filtering can also offer considerable benefits
for quantum communication in scenarios where only low-order wavefront
correction is desirable or possible.

While the experiments and extrapolations in this work were done for a
terrestrial free-space link, the approach can be readily extended to satellite
links, where the optimal field of view would further depend on the satellite
orbit and change with the satellite altitude angle [11]. This means that an
adjustable spatial filter in an optical ground station would allow for optimal
performance with a single system, not only at any time of the day but also for
a link with any satellite. Finally, we stress that although we demonstrated
the adjustable spatial filter with polarization-based QKD, the approach is
suitable for improving performance of other encodings and protocols in free-
space optical communication – quantum as well as classical.

## Funding

German Federal Ministry of Education and Research - BMBF (16KIS1263K);
European Space Agency - ESA (4000125842/18/NL/MH/mg).

## Acknowledgments

We thank Herbert Gross for fruitful discussions and help with the spatial
filter design; Mirko Liedtke and Carl Zeiss Microscopy GmbH for providing the
adjustable field stop; Sakshi Sharma for help with the entangled pair source;
Matthias Goy and Daniel Heinig for experimental support; Stadtwerke Jena for
giving us access to their rooftop for experiments. This research was conducted
within the scope of the project QuNET, funded by the German Federal Ministry
of Education and Research (BMBF) in the context of the federal government’s
research framework in IT-security “Digital. Secure. Sovereign.” AK and CS are
part of the Max Planck School of Photonics supported by the BMBF, the Max
Planck Society, and the Fraunhofer Society. AK is supported by ESA through the
Networking/Partnering Initiative.

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