2.1.

Atmospheric Scattering-Induced Noise

Figure 1(a) shows the basic elements of an optical receiver including a primary optic of diameter $D_R$ that defines the entrance pupil, an FS of diameter $d$, a collimating lens, and a spectral filter with bandpass $\mathrm{\Delta }\lambda$. The FS limits the chief ray of the system to define the linear angle FOV, $\mathrm{\Delta }\theta \approx d/f$, where $f$ is the focal length of the primary optic. Reducing the size of the FS reduces the FOV and, correspondingly, the number of sky noise photons transmitted by the FS.

Fig. 1

Schematics illustrating (a) the basic optical components of an optical receiver including a primary optic of diameter $D_R$ and focal length $f$, an FS of diameter $d$ that defines the linear and solid-angle fields of view $\mathrm{\Delta }\theta$ and ${\mathrm{\Omega }}_{\mathrm{FOV}}$, and a spectral filter with spectral bandpass $\mathrm{\Delta }\lambda$, and (b) the effects of turbulence on the signal distribution at the FS.

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 expression²⁰

2.2.

Atmospheric-Turbulence-Induced Aberrations

Figure 1(b) shows the effects of turbulence on signal transmission at the FS. In the absence of aberrations, the primary optic focuses an incident signal to a diffraction-limited irradiance distribution. For a signal photon derived from an attenuated laser pulse, the irradiance distribution represents the photon probability function. For the case of a planar wavefront with uniform amplitude incident upon a circular aperture, the diffraction-limited irradiance distribution at focus is described by the Airy function with a central disk of diameter $2.44\lambda f/D_R$. Reducing the FS to this diameter passes 84% of the signal while blocking sky noise associated with larger field angles. Reducing the FS further decreases both the transmitted sky noise and the signal.

Aberrations increase the size of the signal irradiance distribution at the FS. For atmospheric-turbulence-induced aberrations, the strength of turbulence integrated over a propagation path is characterized by Fried’s coherence length $r_0$.⁴² Sarazin and Roddier^43,44 computed the angular FWHM 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 focus is found to be approximately $2\lambda f/r_0$. Relative to the diffraction-limited case, high transmission requires the diameter of the FS to be increased from $2.44\lambda f/D_R$ to $2\lambda f/r_0$. This increases the linear FOV from $\mathrm{\Delta }\theta \approx 2.44\lambda /D_R$ to $\mathrm{\Delta }\theta \approx 2\lambda /r_0$ and, correspondingly, increases the number of noise photons transmitted by the FS by a factor of ${(D_R/1.22r_0)}^2$.

In principle, an AO system can restore the aberrated wavefront to near-diffraction-limited quality. Within the boundaries established by diffraction and turbulence, AO could play a significant role in preserving the transmission of signal photons at reduced FOVs that would substantially reduce background sky noise. The potential benefit of AO to sky noise reduction can be estimated as follows. Atmospheric turbulence is characterized by standard altitude-dependent turbulence profiles. The strength of turbulence is both angle-dependent and wavelength-dependent. For the commonly used ${\mathrm{HV}}_{5/7}$ turbulence profile,⁴⁵ henceforth referred to as $1\times {\mathrm{HV}}_{5/7}$, 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, the corresponding range of turbulence-limited FOVs is 18 to $40\mu \mathrm{rad}$. In the absence of turbulence, the diffraction-limited FOV is $\sim 2\mu \mathrm{rad}$. At 75 deg from zenith, reducing the FOV from the $40\mu \mathrm{rad}$ turbulence-limited FOV to the $2\mu \mathrm{rad}$ diffraction-limited FOV would reduce sky noise by a factor of 400. Assuming a perfect AO system, this reduction in optical noise could be achieved without increasing the signal loss at the FS.

3.1.

Quantum Key Distribution Receiver with Adaptive Optics

Figure 2 shows a conceptual schematic of an optical receiver that includes both a QKD receiver and an AO system. The AO system architecture is based on systems previously demonstrated.^36,37 AO systems require light from either a natural or artificial beacon to probe the atmospheric turbulence.^36,37,46 Control signals for the AO system are generated from measurements performed on the aberrated beacon wavefront. In the analysis that follows, it is assumed that the satellite includes both an AO beacon laser and a QKD photon source. The two sources generate copropagating optical pulses that are synchronized in time, but at different wavelengths to allow chromatic separation at the receiver.

Fig. 2

Schematic illustrating the integration of an AO system with a telescope and QKD receiver. Components include a primary mirror (PM), FSM, DM, DBS, BCSF, BS, FPTS, SHWFS, CPU, M, QCSF, PBS, Geiger-mode APD, and HWP. Control signals are shown as dashed lines.

Wavefront errors caused by atmospheric turbulence consist of a tilt component that causes an image to jitter and higher-order spatial components that cause the point spread function to enlarge. Correspondingly, the AO system consists of a fast steering mirror (FSM) that tracks the tilt component of the wavefront error and a deformable mirror (DM) that compensates higher-order wavefront errors. A dichroic beam splitter (DBS) diverts the beacon wavefront to the wavefront sensing system. A beacon-channel spectral filter (BCSF) transmits the beacon wavelength while blocking other spectral components. A beam splitter (BS) transmits a portion of the beacon light to a focal plane tracking sensor (FPTS) that measures the tilt component of the wavefront error and generates control signals for the FSM. The reflected light propagates to a Shack–Hartmann wavefront sensor (SHWFS) that determines the higher-order aberrations and generates control signals for the DM. In a closed-loop AO system, dynamic feedback control reduces residual wavefront errors. The AO system’s dynamic performance limit is characterized by the system bandwidth. Beyond this frequency, the amplitude response of the system is insufficient to be considered useful or stable.

The quantum channel wavelength is transmitted by the DBS, reflected by a mirror (M), and brought to a focus at the FS of the system. A quantum-channel spectral filter (QCSF) transmits the quantum-channel wavelength to the QKD receiver while blocking other spectral components. Within the QKD receiver, a 50/50 BS randomly directs photons to the two mutually unbiased measurement bases of the BB84 protocol. In the reflected path, a polarizing beam splitter (PBS) and two gated avalanche photodiodes (APDs) measure the polarization state of the photon in the rectilinear basis. In the transmitted path, a half-wave plate (HWP) rotates the polarization states by 45 deg for measurement in the diagonal polarization basis.

3.2.

Numerical Simulation of Propagation Through Turbulence and an Adaptive Optics System

The photon capture efficiency associated with turbulence, ${\eta }_{\text{spatial}}$, is defined to include the turbulence-related losses associated with both transmitter-to-receiver aperture coupling and propagation through the FS of the receiver. Aperture-to-aperture coupling losses due to diffraction are accounted for separately. The term ${\eta }_{\text{spatial}}$ is quantified with a simulation code that includes the effects of turbulence on wave propagation and the effects of a closed-loop AO system within the receiver as shown in Fig. 2. Wave-optics simulations are performed with Atmospheric Compensation Simulation, a simulation code developed by Science Applications International Corporation.^40,41 The wave-optics propagation code has been anchored to experimental data and used to anchor other simulation codes.⁴⁷ The code is based on the principles of scalar diffraction theory. The simulated AO system includes an FPTS and FSM for tilt estimation and correction and an SHWFS and DM for higher-order aberration correction. The hardware emulations include models for the wavefront sensor cameras that include real world effects such as noise, pixel diffusion, and latencies.

Simulations are performed for receiver pointing angles ranging from zenith to 75 deg from zenith. For each elevation angle, 20 realizations of atmospheric turbulence are simulated. For each realization of turbulence, the atmosphere is simulated by 10 phase screens distributed throughout the atmospheric path. 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^48,49 propagate the optical field from the transmitter through the phase screens⁵⁰ to the receiving aperture. From there, the optical field is reflected from an FSM and then a DM. These components are simulated in a closed-loop for iterative feedback control.

3.3.

Adaptive Optics System Parameters

The FPTS is modeled as a lens and focal plane quadrant detector. The SHWFS is modeled as a $32\times 32$ 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 $33\times 33$ array of actuators. The diameter $d_l$ of the individual lenslets in pupil space is such that $d_l/r_0$ is less than unity for the range of turbulence strengths encountered in a $1\times {\mathrm{HV}}_{5/7}$ atmospheric profile. In stronger turbulence, where $d_l/r_0$ exceeds unity, the system performance is degraded.⁴⁶ In the simulations, the FPTS centroid, SHWFS centroids, and residual wavefront errors are updated at 10 kHz. The FSM and DM are also updated at 10 kHz. The tracking bandwidth is 200 Hz and the bandwidth for higher-order correction is 500 Hz. These system parameters are considered to be within the state of the art.

It is assumed that the cooperative beacon on the satellite provides light at 810-nm wavelength for the FPTS and SHWFS. The quantum channel wavelength is assumed to be 780 nm, allowing separation of the two wavelengths. It is further assumed that any beacon light transmitted by the DBS and QCSF is insignificant compared to other noise sources. 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.

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. In order to consider the worst-case scenario, the wind direction is assumed to be opposite to the slew direction, producing the highest Greenwood frequency for a particular turbulence profile.

3.4.

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 $f_G$, is dependent upon the slew rate of the telescope 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 $r_0$. Greenwood frequencies exceeding the correction bandwidth and isoplanatic angles smaller than the angular subtense of the source result in a degraded performance of the AO system. 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 leads to degradation in 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.

Table 1 shows turbulence parameters calculated at each of the five elevation angles for the two turbulence profiles and two orbit altitudes considered in the analysis that follows. For a 10-cm transmitter aperture, the isoplanatic angles are larger than the angular subtense of the source for all cases. For the 800-km orbit, the Greenwood frequency remains within the 500 Hz AO system bandwidth for all cases shown except 75 deg in $3\times {\mathrm{HV}}_{5/7}$ turbulence. For the 400-km orbit, the higher slew rates cause the Greenwood frequency to exceed the AO system bandwidth at 75 deg from zenith in $1\times {\mathrm{HV}}_{5/7}$ turbulence and at all angles shown below zenith in $3\times {\mathrm{HV}}_{5/7}$ turbulence. 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 $d_l=3.1\mathrm{cm}$. In $3\times {\mathrm{HV}}_{5/7}$ turbulence, $r_0$ is smaller than the subaperture size at the 60 deg and 75 deg elevation angles leading to under-resolved local wavefront tilts.

Table 1

Turbulence parameters for five elevation angles relative to zenith with 400- and 800-km altitude circular orbit altitudes. Parameters include Fried’s coherence length r0, the isoplanatic angle θ0, Rytov, and the Greenwood frequency fG. Parameters are shown for 1×HV5/7 and 3×HV5/7 turbulence profiles.

3.5.

Results: Photon Capture Efficiency Probability Distributions

The turbulence-related photon capture efficiency ${\eta }_{\text{spatial}}$ is calculated for each elevation angle from the 20 realizations of atmospheric turbulence. For each realization, the simulated AO system iterates to steady state minimizing the wavefront error at the SHWFS. Figure 3 shows sample results presented as probability density functions (PDFs) calculated for $1\times {\mathrm{HV}}_{5/7}$ turbulence, a $D_R=1\text{m}$ receiver aperture size, and a pointing angle at zenith. The transmitter is treated as an unresolved point source relative to a 16-cm grid spacing at the 400-km orbit. Capture efficiencies calculated with tilt correction only are shown in red. Capture efficiencies calculated with the addition of higher-order AO are shown in blue. Since tilt correction is required for tracking the satellite, the case without atmospheric tilt correction is not considered here.

Fig. 3

Examples of PDFs for the turbulence-related photon capture efficiency ${\eta }_{\text{spatial}}$ calculated for a $1\times {\mathrm{HV}}_{5/7}$ turbulence profile and a pointing angle at zenith. Results obtained with a $20\text{-}\mu \mathrm{rad}$ FOV are shown for (a) a 400-km altitude circular orbit and, (b) an 800-km altitude circular orbit. Results obtained with a $2\text{-}\mu \mathrm{rad}$ FOV are shown for (c) a 400-km altitude circular orbit and (d) an 800-km altitude circular orbit. Results obtained with tilt correction only are shown in red. Results obtained with the addition of higher-order AO are shown in blue.

Figures 3(a) and 3(b) show results calculated for a $20\text{-}\mu \mathrm{rad}$ FOV with 400- and 800-km altitude circular orbits, respectively. At this FOV, the mean capture efficiencies increase from 88% with tilt correction alone to 93% with the addition of higher-order AO. Figures 3(c) and 3(d) show the corresponding results calculated for a $2\text{-}\mu \mathrm{rad}$ FOV. At the reduced FOV, turbulence leads to significant losses at the FS. The mean capture efficiencies now increase from only about 5.5% with tilt correction alone to about 73% with the addition of higher-order wavefront compensation. In both cases, the capture efficiency improves measurably with the addition of higher-order AO, but does not reach unity. This is due to the imperfect nature of the AO system and the fact that the small turbulence-related losses that occur in aperture-to-aperture coupling cannot be recovered by any AO system implemented in the receiver. The relative benefit of AO is more significant at the smaller FOV where losses due to turbulence are greater. Increasing the orbit altitude from 400 to 800 km decreases the Greenwood frequency. However, this has a negligible effect on the results since, for the examples shown, all turbulence parameters are within the correction tolerances of the AO system modeled.

4.1.

Secure Key Rate Equations for the Vacuum-Plus-Weak-Decoy-State Quantum Key Distribution Protocol

This section reviews the rate equations for the vacuum-plus-weak-decoy-state QKD protocol implemented via polarization encoding of photons from a Poissonian light source as presented by Ma et al.⁵⁵ The calculation includes the effects of sky noise, detector dark counts, polarization crosstalk, mean photon number, and photon losses. The SKG rate per signal state, or secret bit yield, is given by

The overall QBER associated with signal photons is given by

The background detection probability is calculated including contributions from sky radiance and detector dark counts

4.2.

Satellite-to-Earth Quantum Channel Parameters

SKG rates are calculated with the following parameters assumed: The quantum channel wavelength is $\lambda =780\mathrm{nm}$ for low beam divergence, high atmospheric transmission, and high detector efficiency.¹⁴ The beacon channel wavelength is 810 nm. The radial extent of the receiver primary optic is $R=0.5\mathrm{m}$. The spectral filter bandpass and transmission are $\mathrm{\Delta }\lambda =0.2\mathrm{nm}$ and ${\eta }_{\text{spectral}}=0.9$, consistent with a commercially available dielectric interference filter.⁵⁶ The detector efficiency, dark count rate, and gate duration are ${\eta }_{\text{detector}}=0.6$, $f_{\text{dark}}=250\mathrm{Hz}$, and $\mathrm{\Delta }t=1\mathrm{ns}$ consistent with the operation of a commercially available Geiger-mode APD.⁵⁷ The collective optical efficiency of the remaining components in the receiver is ${\eta }_{\text{receiver}}=0.5$.

It is assumed that the output of the quantum channel source is expanded to uniformly illuminate the transmitter exit pupil and that all losses incurred by vignetting at the transmitter aperture contribute to the attenuation that is required to achieve the mean photon numbers, $\mu$ and $\nu$. Under this assumption, it is not necessary to account for losses within the transmitter. For the aperture sizes and propagation ranges considered, the angle-dependent aperture-to-aperture coupling efficiency ${\eta }_{\mathrm{geo}}$ can be approximated by the Friis equation, ${\eta }_{\mathrm{geo}}={(\pi D_T{D}_R/4\lambda z)}^2$, which assumes a uniformly illuminated transmitter aperture.^58,59 The diameters of the transmitter and receiver apertures are $D_T=10\mathrm{cm}$ and $D_R=1\mathrm{m}$, respectively. The propagation distance $z$ is calculated as a function of the receiver pointing angle and satellite altitude. For a 400-km altitude circular orbit, propagation distances range from 400 km at zenith to 1175 km at 75 deg from zenith. For an 800-km orbit, the propagation distances range from 800 km at zenith to 2033 km at 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. 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 background detection probability $Y_0$ is calculated from Eq. (8) with the number of noise photons $N_b$ calculated from Eq. (1). The gain of the signal state $Q_{\mu }$ is calculated from Eq. (3) with the signal transmission efficiency $\eta$ calculated from Eq. (9), including contributions from ${\eta }_{\mathrm{geo}}$, ${\eta }_{\mathrm{trans}}$, and ${\eta }_{\text{spatial}}$. The signal QBER $E_{\mu }$ is calculated from Eq. (7) with the error rate due to noise, $e_0$, taken to be 1/2 under the assumption the background is random. It should be noted that the polarization distribution of sky radiance has been studied in detail,⁶⁰ but is not included in this analysis. The probability of a projective polarization measurement in a matched polarization basis yielding an incorrect result due to depolarization in propagation and improper alignment has been measured experimentally in a satellite-Earth optical link.⁶¹ The following analysis assumes the reported measured value of $e_{\text{detector}}=2.8\%$. The gain of the single-photon states $Q_1$ is calculated from Eq. (4), assuming a decoy-state mean photon number $\nu =0.05$. In order to specify an optimized mean photon number for the signal state, Eq. (2) was evaluated against $\mu$ as a free parameter. The mean photon number $\mu =0.45$ was determined to be an optimum value for the parameters assumed here.²⁹ The single-photon error rate $e_1$ is calculated from Eq. (5) with $Y_1$ calculated from Eq. (6). The efficiency of error correction $f(E_{\mu })$ is assumed to be a constant value of 1.22, the commonly used value associated with Cascade error correction.

4.3.

Results: Secure Key Generation Rate Probability Distributions

The statistical nature of turbulence leads to statistical distributions of SKG rates. SKG rates are calculated by evaluating Eq. (2) in Sec. 4.1 for statistical distributions of ${\eta }_{\text{spatial}}$ similar to those described in Sec. 3.5. Figures 4(a) and 4(b) show examples of SKG rate PDFs calculated from the capture efficiencies ${\eta }_{\text{spatial}}$ shown in Figs. 3(c) and 3(d), respectively, for the case of a $2\mu \mathrm{rad}$ FOV at zenith. This example assumes a $1\times {\mathrm{HV}}_{5/7}$ turbulence profile and a sky radiance of $25\mathrm{W}/({\mathrm{m}}^2\mathrm{sr}\mu \mathrm{m})$. In order to present SKG rates in units of secure key bits per second, a system rate of 10 MHz is assumed as a multiplicative factor to Eq. (2).

Fig. 4

Examples of SKG rate PDFs calculated in the presence of turbulence assuming a receiver with a $2\text{-}\mu \mathrm{rad}$ FOV pointing to zenith, a sky radiance of $25\mathrm{W}/({\mathrm{m}}^2\mathrm{sr}\mu \mathrm{m})$, a turbulence profile of $1\times {\mathrm{HV}}_{5/7}$, a system rate of 10 MHz, and a satellite in a circular orbit at an altitude of (a) 400 km and (b) 800 km. Results obtained with tilt correction alone are shown in red. Results obtained with the addition of higher-order AO are shown in blue.

Figure 4(a) shows results calculated for the 400-km altitude orbit. SKG rates with tilt correction alone are shown in red with a distribution around a mean value of 406 Hz. Note that a significant probability exists for a key rate of zero. Introducing higher-order AO results in a significant increase in SKG rates as shown in blue with a distribution around a mean value of 8.3 kHz. Figure 4(b) shows results calculated for the 800-km orbit. With tilt correction alone, the mean SKG rate is only 0.1 Hz, with zero being the most probable key rate. Introducing higher-order AO increases the mean SKG rate to 1.9 kHz. The SKG rates are significantly lower for the 800 km orbit as a consequence of the reduced geometrical capture efficiency ${\eta }_{\mathrm{geo}}$ at the longer propagation distances.