2.1.

Effects of Turbulence-Induced Aberrations in the Plane of a Field Stop

Figure 2 shows conceptually the effects of turbulence on the photon wavefront and probability distribution at either an FS or an SMF. A planar optical wavefront propagates through atmospheric turbulence. The resulting aberrated wavefront is incident upon a primary optic and brought to a focus within the system.

Fig. 2

Block diagram illustrating a planar wavefront propagating through atmospheric turbulence, incident upon a primary optic of diameter $D_R$, and brought to a focus at either (a) a circular aperture of diameter $d$ or (b) the entrance of an SMF. An AO system is included to compensate for turbulence-induced wavefront errors.

Figure 2(a) shows the case where the efficiency of propagation through a circular aperture FS is considered both with and without AO. Optimizing the quantum channel performance requires the FS diameter, $d$, be adjusted to maximize the transmission of signal photons while minimizing the transmission of noise photons. Maximizing signal photon transmission requires $d$ to be sufficiently large to pass most of the photon probability distribution. Minimizing channel noise, however, requires $d$ to be sufficiently small to block noise photons that are not blocked by spectral and temporal filtering techniques. Figure 2(b) shows the case where the FS is replaced by an SMF. Note that the SMF could be placed in any image plane of the FS. For the case of SMF-coupled quantum detectors, this could be in the focal plane where the APDs are located in Fig. 1.

In this analysis, the classical irradiance distribution is assumed to represent the probability function associated with the transverse momentum of an individual photon. The minimum spot size at the focus occurs in the absence of aberrations. For the case of a planar wavefront with uniform amplitude incident upon a circular aperture, the diffraction-limited irradiance distribution at the focus is described by the Airy function

The effect of atmospheric turbulence on the size of the irradiance distribution can be estimated as follows. The strength of turbulence integrated over a propagation path is characterized by Fried’s coherence length $r_0$.¹⁹ Sarazin and Roddier^20,21 computed the angular full width at half maximum of the long-exposure irradiance distribution with turbulence to be approximately $\lambda /r_0$. In analogy to the Airy disk, we define the turbulence-induced spot size to be that which captures about 84% of the power. For primary optic diameters larger than $r_0$, this aberrated spot size at the focus is found to be $\sim 2\lambda f/r_0$. High signal transmission efficiency requires the diameter of the FS to be increased by a factor of $D_R/1.22r_0$ relative to the diffraction-limited case. In a noisy channel, this increases the transmitted noise photons by a factor of ${(D_R/1.22r_0)}^2$.

Atmospheric turbulence is characterized by standard altitude-dependent turbulence profiles. The strength of turbulence is both angle and wavelength dependent. For the commonly used ${\mathrm{HV}}_{5/7}$ turbulence profile,²² henceforth referred to as $1{\mathrm{xHV}}_{5/7}$, and a wavelength of 780 nm, $r_0$ ranges from about 9 to 4 cm for pointing angles ranging from 0 deg to 75 deg from zenith, respectively. For a $D_R=1\mathrm{m}$ diameter receiver aperture, turbulence will increase the size of the signal distribution by a factor of about 9 to 20, respectively. This blurring of the signal distribution requires the FS to be significantly larger than the Airy disk to avoid signal losses and, in this example with moderate turbulence, can increase noise by factors ranging from about 80 to 400.

2.2.

Optical Fiber Coupling Efficiency

In the case where the FS is replaced by the SMF, the fiber mode characteristics must also be considered. The efficiency with which an optical field, $F_{\text{optical}}$, couples into the mode, $F_{01}$, of an SMF is given by the normalized projection of the optical field onto the fiber mode

In the simulations that follow, the fiber mode is assumed to be Gaussian with radius ${\omega }_0$. For an aperture of dimension $D$ and focal length $f$, the efficiency of mode matching is characterized by the parameter $\beta =\pi D{\omega }_0/2f\lambda$. The central obscuration of the telescope receiver is assumed to be 20%, and the fiber mode radius is assumed to satisfy $\beta =1.07$. This value for $\beta$ yields optimum coupling efficiency for a diffraction-limited field in accordance with the analysis of Ruilier, reproduced in Appendix A.

3.1.

Numerical Methods for Calculating the Effects of Atmospheric Turbulence and an Adaptive-Optics System under Closed-Loop Control

In principle, an AO system can restore an aberrated wavefront to near-diffraction-limited quality. In practice, however, AO does not completely compensate for turbulence-induced aberrations. Limitations occur due to the finite spatial resolution and finite temporal response of the AO system. For a given set of atmospheric parameters and AO system specifications, the performance of an optical channel can be evaluated using numerical methods.

Numerical simulations are performed with the Atmospheric Compensation Simulation (ACS) code developed by Leidos.^23,24 ACS has been anchored to experiment and used to anchor other simulation codes.²⁵ The software generates statistical realizations of atmospheric turbulence in the form of two-dimensional phase functions as shown in Fig. 3. Each phase screen is a random realization of turbulence consistent with Kolmogorov statistics and the specified turbulence strength profile. Numerical methods based on scalar Fresnel integrals^26,27 propagate the optical field from the transmitter through the phase screens²⁸ to the receiving aperture. Simulations are performed for receiver pointing angles ranging from zenith to 75 deg from zenith. For each elevation angle, 10 realizations of atmospheric turbulence are simulated. For each realization of turbulence, the atmosphere is simulated by 10 phase screens distributed throughout the atmospheric path. The simulated AO system includes an FPTS and FSM for tilt estimation and correction and an SHWFS and DM for higher-order aberration correction. These components are simulated in a closed loop for iterative feedback control. The ACS hardware emulations include models for the wavefront sensor cameras that include real-world effects, such as noise, pixel diffusion, and latencies. The simulations reported here assume one frame of latency and a very high signal-to-noise ratio but omit pixel diffusion. The complex optical field resulting from propagation through turbulence and an AO system is then used to calculate the transmission efficiency through a circular FS or into an SMF. In the latter case, the transmission efficiency is calculated via the normalized projection of the optical field onto the fiber mode.

Fig. 3

Schematic of a ground-station quantum receiver illustrating elements of the numerical simulations, including wave propagation through statistical realizations of turbulence, a closed-loop AO system, and transmission either through an FS or into an SMF.

3.2.

Adaptive-Optics System Parameters

Table 1 summarizes the AO system parameters assumed in the simulations. The FPTS is modeled as a lens and focal plane quadrant detector. The SHWFS is modeled as a $16\times 16$ element array of lenslets and quadrant detectors. The SHWFS is assumed to be shot-noise limited as is typically the case for systems with cooperative beacons. The DM is modeled as a continuous face sheet driven by a $19\times 19$ array of actuators comprised of a $17\times 17$ array of active actuators with an additional ring of slaved actuators. In the simulations, the FPTS centroid, SHWFS centroids, and residual wavefront errors are updated at 2 kHz. The FSM and DM are also updated at 2 kHz. The tracking bandwidth is 150 Hz, and the bandwidth for higher-order correction is 200 Hz. These system parameters are considered to be within the state of the art.

Table 1

AO system parameters assumed in the simulations.

It is assumed that the cooperative beacon on the satellite provides light at 810-nm wavelength for the FPTS and SHWFS. The signal wavelength is assumed to be 780 nm allowing separation of the two wavelengths. Applying wavefront correction at a wavelength that is shorter than the beacon wavelength can lead to residual wavefront errors. While these errors are accounted for in the simulations, they are negligible due to the small separation in wavelengths. Similarly, the beacon and quantum channel pulses are separated in time but on a timescale over which the atmosphere is static in the simulations.

3.3.

Atmospheric Turbulence Parameters

The effects of turbulence on an optical field are characterized by temporal, angular, and spatial coherence parameters. The temporal coherence, given by the Greenwood frequency $f_G$, depends on the telescope slew rate and, therefore, the altitude of the satellite. The angular coherence is given by the isoplanatic angle ${\theta }_0$. The spatial coherence is given by Fried’s coherence length $r_0$. Residual wavefront errors will occur when Greenwood frequencies exceed the AO correction bandwidth, isoplanatic angles become smaller than the angular subtense of the source, or $r_0$ becomes smaller than the SHWFS subaperture size, $d_l$. Rytov is a direct measure of scintillation experienced by the optical field at the receiver entrance pupil. Rytov values greater than about 0.4 indicate deep turbulence where scintillation degrades the performance of the AO system. In the presence of scintillation, the SHWFS is unable to accurately measure wavefront errors due to intensity nulls in the field.

For the purpose of analysis, it is assumed that the satellite travels in either a 400- or 800-km-altitude circular orbit. The telescope slews to follow the motion of the satellite. The altitude-dependent wind speed is described by the Bufton wind profile. To consider the worst-case scenario, the wind direction is assumed to be opposite the slew direction producing the highest Greenwood frequency for a particular turbulence profile.

Table 2 shows turbulence parameters calculated at 780-nm wavelength for five elevation angles, three turbulence profiles, and two orbit altitudes. The three turbulence profiles, designated as $1{\mathrm{xHV}}_{5/7}$, $2{\mathrm{xHV}}_{5/7}$, and $3{\mathrm{xHV}}_{5/7}$, are based on the ${\mathrm{HV}}_{5/7}$ model with the altitude-dependent structure constant, $C_n^2$, values multiplied by a scaling factor of 1, 2, or 3. The AO system characteristics assumed in this analysis are less than ideal for complete compensation of turbulence-induced wavefront errors in an LEO-Earth channel. Partial compensation, however, can still provide a benefit, and such a system may be of interest based on cost and availability. For cases where the turbulence characteristics exceed the design specifications of the AO system assumed in the analysis, values are shown in bold.

For a 10-cm transmitter aperture, the angular subtense of the source lies within the isoplanatic angles for all cases. For the 400-km orbit, Greenwood frequencies exceed the 200-Hz AO system bandwidth for all cases shown. For the 800-km orbit, lower slew rates result in lower Greenwood frequencies and, within 45 deg of zenith in $1{\mathrm{xHV}}_{5/7}$ turbulence, $f_G$ lies within the AO system bandwidth. At the 75-deg elevation angle, Rytov values indicate deep turbulence conditions. For $D_R=1\mathrm{m}$, the SHWFS subaperture size in pupil space is $d_l=6.25\mathrm{cm}$. In $3{\mathrm{xHV}}_{5/7}$ turbulence, $r_0$ is smaller than the subaperture size for all angles considered. This leads to underresolved local wavefront tilts and reduced AO performance.^28,29 In $1{\mathrm{xHV}}_{5/7}$ turbulence, wavefront tilts are well resolved within 45 deg of zenith.

5.1.

Quantum Bit Error Rates

For applications based on the detection of polarization states of individual photons, such as discrete-variable QKD or measurements of Bell inequalities, quantum bit errors will occur due to noise in the channel resulting from background photons, detector dark counts, and cross talk in polarization measurements. The overall QBER associated with signal photons in a QKD protocol can be written as³⁰

The number of sky-noise photons, $N_b$, entering the receiver in a detection window is proportional to the sky radiance according to the radiometric expression³¹

5.2.

Satellite-to-Earth Quantum Channel Parameters

QBERs are calculated for the AO parameters given in Table 1, the atmospheric turbulence parameters given in Table 2, and the remaining channel parameters given in Table 3. For the purpose of this example, the quantum source is assumed to emit a weak coherent pulse with a mean photon number of $\mu =0.45$, a value determined to be optimum for the decoy-state QKD protocol.³² Background sky radiance levels considered include a nighttime value of $H_b=1.5\times {10}^{-2}\mathrm{W}/({\mathrm{m}}^2\mathrm{sr}\mu \mathrm{m})$ and daytime values of 25 and $100\mathrm{W}/({\mathrm{m}}^2\mathrm{sr}\mu \mathrm{m})$.^31,32 The detector dark count rate is 250 Hz. Polarization cross talk is taken to be $e_{\text{detector}}=2.8\%$, an experimentally measured value in a satellite-Earth optical link.³³ Spatial filtering of sky noise is included via a $\mathrm{\Delta }\theta =2\mu \mathrm{rad}$ receiver FOV for the case with an FS and a $\mathrm{\Delta }\theta =1\mu \mathrm{rad}$ FOV for the case with an SMF. This latter value was determined to yield the optimum SMF-coupling efficiency in the absence of aberrations. Spectral and temporal filtering are included via a $\mathrm{\Delta }\lambda =0.2\mathrm{nm}$ spectral filter bandpass and a $\mathrm{\Delta }t=1\mathrm{ns}$ detector gate duration.

Table 3

Quantum channel parameters assumed in the simulations.

Parameters contributing to optical efficiency include the transmitter and receiver aperture diameters, $D_T=10\mathrm{cm}$ and $D_R=1\mathrm{m}$, respectively. The angle-dependent aperture-to-aperture coupling efficiency ${\eta }_{\mathrm{geo}}$ is approximated by the Friis equation, ${\eta }_{\mathrm{geo}}={(\pi D_T{D}_R/4\lambda z)}^2$, which assumes a uniformly illuminated transmitter aperture.^34,35 The propagation distance $z$ is calculated for 400- and 800-km satellite orbit altitudes with elevation angles ranging from zenith to 75 deg from zenith. The angle-dependent transmission efficiencies ${\eta }_{\mathrm{trans}}$ are calculated with MODTRAN assuming clear sky conditions. Assumed values range from ${\eta }_{\mathrm{trans}}=0.92$ at zenith to ${\eta }_{\mathrm{trans}}=0.74$ at 75 deg from zenith. Efficiencies associated with turbulence-related losses, ${\eta }_{\text{spatial}}$, are calculated in the simulation. In the absence of turbulence-related losses, ${\eta }_{\text{spatial}}=1$ and the signal transmission efficiencies $\eta$, expressed in dB of loss, are $\sim 18$ to 28 dB for the 400-km orbit and 24 to 33 dB for the 800-km orbit. The spectral filter transmission is ${\eta }_{\text{spectral}}=0.9$ and the detector efficiency is ${\eta }_{\text{detector}}=0.6$. The collective optical efficiency of the remaining components in the receiver is ${\eta }_{\text{receiver}}=0.5$.

Strengths of turbulence include $1{\mathrm{xHV}}_{5/7}$, $2{\mathrm{xHV}}_{5/7}$, and $3{\mathrm{xHV}}_{5/7}$. Cloud-free sky conditions are assumed, and the error rate due to noise, $e_0$, taken to be ½ under the simplifying assumption of the background is random.

5.3.

Quantum Bit Error Rate Simulation Results

Numerical results for the QBER, $E_{\mu }$, are calculated from Eq. (3) for each turbulence frame. The average values over all frames across the 10 realizations of turbulence are summarized in Figs. 6Fig. 7–8 as a function of the ground-station pointing angle. QBER values are calculated for 400- and 800-km satellite altitudes and $1{\mathrm{xHV}}_{5/7}$, $2{\mathrm{xHV}}_{5/7}$, and $3{\mathrm{xHV}}_{5/7}$ atmospheric turbulence profiles and are displayed in correspondence to the fiber coupling efficiencies shown in Fig. 4. The maximum acceptable value for the QBER will depend on the requirements for a given quantum communication application. Previously, the QBER was considered a threshold metric for the decoy-state QKD protocol with a satellite-Earth quantum channel.³² QBERs $>7\%$ precluded secure key generation. In Figs. 6–8, regions of 7% and higher are shaded in red for illustrative purposes. Because the plotted QBERs represent averages, some frames within the time series will deviate from the average values. In some cases where the average QBER is $>7\%$, there will be atmospheric frames where the QBER falls sufficiently below 7% to produce nonzero secure key rates.

Fig. 6

Average QBERs versus ground-station pointing angle calculated assuming a nighttime sky radiance of $H_b=1.5\times {10}^{-2}\mathrm{W}/({\mathrm{m}}^2\mathrm{sr}\mu \mathrm{m})$ and either a circular FS, shown as open circles, or an SMF, shown as solid circles. Results calculated with tracking only are shown as dashed lines. Those calculated with higher-order AO are shown as solid lines. Combinations of turbulence strength and circular orbit altitude are given by (a) $1{\mathrm{xHV}}_{5/7}$ and 400 km, (b) $1{\mathrm{xHV}}_{5/7}$ and 800 km, (c) $2{\mathrm{xHV}}_{5/7}$ and 400 km, (d) $2{\mathrm{xHV}}_{5/7}$ and 800 km, (e) $3{\mathrm{xHV}}_{5/7}$ and 400 km, and (f) $3{\mathrm{xHV}}_{5/7}$ and 800 km.

Fig. 7

Average QBERs versus ground-station pointing angle calculated assuming a daytime sky radiance of $H_b=25\mathrm{W}/({\mathrm{m}}^2\mathrm{sr}\mu \mathrm{m})$ and either a circular FS, shown as open circles, or an SMF, shown as solid circles. Results calculated with tracking only are shown as dashed lines. Those calculated with higher-order AO are shown as solid lines. Combinations of turbulence strength and circular orbit altitude are given by (a) $1{\mathrm{xHV}}_{5/7}$ and 400 km, (b) $1{\mathrm{xHV}}_{5/7}$ and 800 km, (c) $2{\mathrm{xHV}}_{5/7}$ and 400 km, (d) $2{\mathrm{xHV}}_{5/7}$ and 800 km, (e) $3{\mathrm{xHV}}_{5/7}$ and 400 km, and (f) $3{\mathrm{xHV}}_{5/7}$ and 800 km.

Fig. 8

Average QBERs versus ground-station pointing angle calculated assuming a daytime sky radiance of $H_b=100\mathrm{W}/({\mathrm{m}}^2\mathrm{sr}\mu \mathrm{m})$ and either a circular FS, shown as open circles, or an SMF, shown as solid circles. Results calculated with tracking only are shown as dashed lines. Those calculated with higher-order AO are shown as solid lines. Combinations of turbulence strength and circular orbit altitude are given by (a) $1{\mathrm{xHV}}_{5/7}$ and 400 km, (b) $1{\mathrm{xHV}}_{5/7}$ and 800 km, (c) $2{\mathrm{xHV}}_{5/7}$ and 400 km, (d) $2{\mathrm{xHV}}_{5/7}$ and 800 km, (e) $3{\mathrm{xHV}}_{5/7}$ and 400 km, and (f) $3{\mathrm{xHV}}_{5/7}$ and 800 km.

Figure 6 shows results calculated assuming a nighttime sky radiance value of $H_b=1.5\times {10}^{-2}\mathrm{W}/({\mathrm{m}}^2\mathrm{sr}\mu \mathrm{m})$. For the case of the SMF, shown as solid circles, QBERs are generally large with tracking only. However, with AO, QBERs remain below 3% for pointing angles within 50 deg of zenith even for the case of strong turbulence. For the case of the FS, shown as open circles, the larger FOV permits low QBERs with tracking only. It should be noted that in the relatively benign nighttime sky-noise scenario, small diffraction-limited FOVs are not necessarily required for spatial filtering of sky noise.

Figure 7 shows results calculated assuming a daytime sky radiance value of $H_b=25\mathrm{W}/({\mathrm{m}}^2\mathrm{sr}\mu \mathrm{m})$. This value for the background radiance is more than 3 orders of magnitude larger than the nighttime scenario summarized in the preceding figure. The small field angles assumed for both the FS and the SMF provide a substantial level of noise filtering but can also introduce substantial signal losses as a consequence of atmospheric turbulence. With the spectral and temporal filtering conditions assumed, QBERs calculated with tracking only are unacceptably large for nearly all cases shown. With AO however, QBERs remain in relatively benign territory for a range of ground-station pointing angles. Under moderate $1{\mathrm{xHV}}_{5/7}$ turbulence conditions, QBERs $<7\%$ occur within 60 deg of zenith. Under strong $3{\mathrm{xHV}}_{5/7}$ turbulence conditions, QBERs $<7\%$ occur within about 45 deg of zenith. In many cases, QBER values for the FS and SMF are similar. This is most likely due to the competing effects of the increased noise filtering and reduced transmission introduced by the SMF.

Figure 8 shows results calculated assuming a daytime sky radiance value of $H_b=100\mathrm{W}/({\mathrm{m}}^2\mathrm{sr}\mu \mathrm{m})$, a factor of 4 larger than the daytime scenario summarized in the preceding figure. QBERs calculated with tracking only are unacceptably large for all cases shown. With AO however, QBERs are observed in relatively benign territory within 60 deg of zenith under moderate $1{\mathrm{xHV}}_{5/7}$ turbulence conditions and within about 40 deg under stronger $2{\mathrm{xHV}}_{5/7}$ turbulence conditions.

5.4.

Secure Key Rates for the Vacuum-Plus-Weak-Decoy-State QKD Protocol

This section considers secure key generation rates as a metric for quantum channel performance. Secure key rates are calculated for a decoy-state QKD protocol implemented over a satellite-Earth channel.¹ The secure key generation rate per signal state, or secret bit yield, is given by³⁰

5.5.

Secure Key Rate Simulation Results

Secure key rates are calculated assuming a 10-MHz source and a decoy-state mean photon number of $\nu =0.05$. The efficiency of error correction $f(E_{\mu })$ is taken to be a constant value of 1.22, the commonly used value associated with Cascade error correction. Rates are calculated from Eq. (7) with the source rate applied as a multiplicative factor. Results are summarized in Figs. 9Fig. 10–11 as average values over all frames across the 10 realizations of turbulence. Average rates of secure key generation are calculated for 400- and 800-km satellite altitudes and $1{\mathrm{xHV}}_{5/7}$, $2{\mathrm{xHV}}_{5/7}$, and $3{\mathrm{xHV}}_{5/7}$ atmospheric turbulence profiles. Plots (a) through (f) show the results as a function of ground-station pointing angle in correspondence to the QBER plots in Figs. 6–8.

Fig. 9

Secure key generation rates versus ground-station pointing angle calculated assuming a nighttime sky radiance of $H_b=1.5\times {10}^{-2}\mathrm{W}/({\mathrm{m}}^2\mathrm{sr}\mu \mathrm{m})$ and either a circular FS, shown as open circles, or an SMF, shown as solid circles. Results calculated with tracking only are shown as dashed lines. Those calculated with higher-order AO are shown as solid lines. Combinations of turbulence strength and circular orbit altitude are given by (a) $1{\mathrm{xHV}}_{5/7}$ and 400 km, (b) $1{\mathrm{xHV}}_{5/7}$ and 800 km, (c) $2{\mathrm{xHV}}_{5/7}$ and 400 km, (d) $2{\mathrm{xHV}}_{5/7}$ and 800 km, (e) $3{\mathrm{xHV}}_{5/7}$ and 400 km, and (f) $3{\mathrm{xHV}}_{5/7}$ and 800 km.

Fig. 10

Secure key generation rates versus ground-station pointing angle calculated assuming a daytime sky radiance of $H_b=25\mathrm{W}/({\mathrm{m}}^2\mathrm{sr}\mu \mathrm{m})$ and either a circular FS, shown as open circles, or an SMF, shown as solid circles. Results calculated with tracking only are shown as dashed lines. Those calculated with higher-order AO are shown as solid lines. Combinations of turbulence strength and circular orbit altitude are given by (a) $1{\mathrm{xHV}}_{5/7}$ and 400 km, (b) $1{\mathrm{xHV}}_{5/7}$ and 800 km, (c) $2{\mathrm{xHV}}_{5/7}$ and 400 km, (d) $2{\mathrm{xHV}}_{5/7}$ and 800 km, (e) $3{\mathrm{xHV}}_{5/7}$ and 400 km, and (f) $3{\mathrm{xHV}}_{5/7}$ and 800 km.

Fig. 11

Secure key generation rates versus ground-station pointing angle calculated assuming a daytime sky radiance of $H_b=100\mathrm{W}/({\mathrm{m}}^2\mathrm{sr}\mu \mathrm{m})$ and either a circular FS, shown as open circles, or an SMF, shown as solid circles. Results calculated with tracking only are shown as dashed lines. Those calculated with higher-order AO are shown as solid lines. Combinations of turbulence strength and circular orbit altitude are given by (a) $1{\mathrm{xHV}}_{5/7}$ and 400 km, (b) $1{\mathrm{xHV}}_{5/7}$ and 800 km, (c) $2{\mathrm{xHV}}_{5/7}$ and 400 km, (d) $2{\mathrm{xHV}}_{5/7}$ and 800 km, (e) $3{\mathrm{xHV}}_{5/7}$ and 400 km, and (f) $3{\mathrm{xHV}}_{5/7}$ and 800 km.

Figure 9 shows results calculated assuming a nighttime sky radiance of $H_b=1.5\times {10}^{-2}\mathrm{W}/({\mathrm{m}}^2\mathrm{sr}\mu \mathrm{m})$. With tracking only, secure key generation is possible over a range of elevation angles and turbulence conditions. For the case of the SMF shown as solid circles, this includes nonzero key rates where the average QBERs shown in Fig. 6 exceed 7%. This results from the fact that the average QBER includes atmospheric frames where the QBER is lower than the mean. For the 800-km orbit, secure key generation is possible but over a more limited range of sky angles and at lower rates due to increased propagation losses. Introducing AO improves the channel performance substantially. Secure key generation becomes possible within 60 deg of zenith for both orbit altitudes and all strengths of turbulence shown. Where secure key generation is possible with tracking, AO increases key rates by an order of magnitude or more.

Figure 10 shows results calculated assuming a daytime sky radiance value of $H_b=25\mathrm{W}/({\mathrm{m}}^2\mathrm{sr}\mu \mathrm{m})$. With the spectral and temporal filtering conditions assumed, secure key generation is possible with tracking but primarily for the 400-km-altitude orbit and over a range of sky angles that decreases with the increasing strength of turbulence. Introducing AO improves the channel performance substantially. For the 400-km-altitude orbit, secure key generation is possible within 60 deg of zenith for all strengths of turbulence shown. For the 800-km-altitude orbit, secure key generation is possible within 40 deg of zenith for all strengths of turbulence shown. Where secure key generation is possible with tracking, AO increases key rates by 1 to 2 orders of magnitude.

Figure 11 shows results calculated assuming a daytime sky radiance value of $H_b=100\mathrm{W}/({\mathrm{m}}^2\mathrm{sr}\mu \mathrm{m})$. Under the conditions of $1{\mathrm{xHV}}_{5/7}$ turbulence and the lower 400-km orbit, secure key generation is possible with tracking but only within zenith angles of about 30 deg. Otherwise, secure key generation is only possible with AO. For the 400-km-altitude orbit, AO makes secure key generation possible within 40 deg to 60 deg of zenith depending on the strength of turbulence. For the 800-km-altitude orbit, AO makes secure key generation possible but over a more restricted range of sky angles. In strong $3{\mathrm{xHV}}_{5/7}$ turbulence, the reduced FOV of the SMF permits secure key generation within 45 deg of zenith where it is not possible with the FS that is matched to the diffraction-limited field angle.