2.1

Quantum Key Distribution Source and Protocol

During the pass, the QKD source generates entangled photon pairs, which by convention are said to each contain one signal photon and one idler photon. It transmits the signal photon to the optical terminal and locally performs a polarization measurement of the idler photon using polarizing beam splitters and single photon detecting avalanche photodiodes in geiger-mode. It randomly measures in either the horizontal-vertical, or diagonal-antidiagonal basis. The time of each detection event is time-stamped and the data is stored in compressed form. On the ground, the telescope receives the quantum signal from the satellite and routes it to a detector where the same polarization measurement is performed. The synchronization signal, also transmitted by the quantum payload, is detected and time-stamped on both ends.

On a subsequent ground station pass, the quantum payload transmits the timestamps and which of the two polarization tests were made for each timestamp (but not the result of that polarization measurement!) via the laser communications channel to the ground station. This information contains the timing signal which is used to synchronize detection events recorded on the satellite to those on the ground with an accuracy of less than 1 nanosecond (limited by detector jitter). This enables the time windows where there was a coincident detection event of signal and idler photons to be identified, this also enables identification of which of those the same polarization test was made on both the satellite and ground station. This information is sent up to the quantum payload via the telemetry and telecommand link and detection events for which the signal photon did not make it to the ground station are discarded. The end result is a shared list of coincident signal and idler detections tested in the same basis. If quantum entanglement has been maintained then the results of the polarization tests will be correlated for signal and idler allowing the results to be used as a string of private, random, ones and zeros. This process is called ‘sifting’, or basis reconciliation, and the string is called the ‘raw key’.

Because of the losses in the space-to-ground channel there are fewer detection events on the ground (by a factor equal to the link loss) so it would naively seem most economical to upload detection events to the satellite and perform the processing steps for sifting there, instead. However, this process is computationally intensive and best performed where computational resources are not limited. Furthermore, the necessity of the optical channel for transmission of the quantum channel provides an opportunity to download the data via laser communication. This also enables simultaneous transmission of the quantum signal and data transmission for basis reconciliation, another advantage of this configuration.

After the sifting step, additional communication rounds over the telemetry link are required to compare samples of the raw key to perform error correction and verify the entanglement. Privacy amplification processing is then performed to ensure that only a minimal fraction of information could be determined by an eavesdropper (typically 1E-10 or smaller). Errors appear in the quantum signal due to dark counts, background light, and from imperfect entanglement quality of the quantum signal, which manifest as noise. If an eavesdropper is attempting to perform a man-in-the-middle attack then the entanglement is lost on the intercepted photons and these will create errors in processing. With no eavesdropper, the final result, after privacy amplification, is a shared string of random bits, which can be used as an encryption key with very strong security guarantees as to its privacy. If an eavesdropper is present the error correction and privacy amplification steps fail and there is no key. For an open-source QKD software stack see qcrypto.¹⁰

2.2

Optical Communication Transmitter

Since the entangled photon link relies on single-photon detection on the ground and necessitates a very high-gain spacecraft optical antenna to close successfully, it is possible to take advantage of this high-gain optical frontend and use it simultaneously for traditional laser communications. Such an approach allows not only quantum key exchange, but potentially also encrypted high-bandwidth optical downlinks using the same front end. An additional advantage of this architecture is that both the entangled photon (QKD) link and the data link can share the same internal fine-pointing system.

A simple solution to combine the two channels is by wavelength multiplexing. Adding a seed laser along the internal optical train with a different wavelength allows for easy routing of the channels both on the spacecraft and when received on ground using dichroic optics. Since the data seed laser can have significantly higher output power in comparison with the QKD source, detection on ground can be simplified by using a direct-detect chain with a single avalanche photodiode (APD). Similarly, for the beacon uplink channel that is used as feedback for pointing and tracking, a different wavelength will be used and routed onto a tracking sensor onboard the spacecraft. The uplink channel and the approach for pointing, acquisition, and tracking are further described in the Uplink Beacon section.

The lasercom transmitter is based on directly modulated laser diode at 915 nm. The modulator maximum data rate is 100 Mbps, using On/Off Keying (OOK) modulation. It reuses some of the electronics developed for the CLICK-BC mission.⁵ The optical communication link is further discussed in the Data Downlink section.

3.1

Optical Configuration

The optical design is targeting low complexity and the fewest number of surfaces necessary to accomplish signal routing, beacon tracking, and the highest possible transmit optical gain given the CubeSat size constraints. On the spacecraft transmit side, both the QKD and the data sources are assumed to be fiber-coupled and multiplexed using a wavelength division multiplexer (WDM). The WDM output will be collimated to fit onto a MEMS fast steering mirror (FSM), which will perform fine pointing adjustments to compensate for the residual spacecraft body pointing error. The multi-wavelength high-gain architecture favors a reflective telescope for beam expansion to avoid aberrations that come with a large refractive beam expander design. Consequently, the collimated beam corrected by the FSM will be magnified using an on-axis folded telescope to achieve the necessary optical gain. On the receive side, the same telescope will be used to collect light from the ground beacon, and route it onto a tracking sensor using a dichroic mirror.

Since the transmit beams are much narrower (12-15 μrad) than the expected maximum LEO point-ahead angle (54 μrad), the point-ahead has to be accounted for. In order to compensate for point-ahead, the FSM has to maintain an additional pointing offset between the received beacon and the transmit beams, using the instantaneous point-ahead angle between the spacecraft and the ground station. To account for potential mechanical misalignment and thermal drift, a calibration path is added by retro-reflecting a small fraction of the seed laser output onto the beacon tracking sensor. This enables tracking of the transmit beam simultaneously with the beacon, which in turn allows maintaining the point-ahead offset even if the alignment between the transmit fiber and the sensor drifts post-launch. A high-level diagram of this architecture is depicted in fig. 1.

Figure 1:

High-level optical layout diagram for a hybrid QKD/lasercom front end. Note the retroreflector for self calibration

3.2

Beam Expander Requirements

A critical element of the optical frontend is the beam expander which will provide the high optical gain necessary to close the QKD link. To meet our requirements, the expander needs to resize the transmit Gaussian beam waist from roughly 1-2 millimeters (constrained by the FSM surface area) such that the output beam at the QKD wavelength has a Gaussian full-width half maximum (FWHM) divergence of 12 μrad.

Since the expanded beam might experience truncation and obstruction within the telescope, it is more convenient to convert the requirement into an optical transmit gain for analysis. For an assumed 12 μrad freespace Gaussian beam, the corresponding far-field on-axis transmit gain can be calculated as follows:¹¹

where θ is the half-width beam divergence. As discussed previously, for low aberration and high gain, a reflective beam expander is preferred. In terms of volume and ease of alignment, a traditional on-axis folded telescope is likely the best solution. Given the volume constraints of CubeSats, we set a 95 mm upper limit on the telescope primary mirror diameter, so the aperture is limited to a single CubeSat unit face, of 10 × 10 cm. However, a potential problem with a traditional on-axis telescope is the central obstruction present due to the secondary mirror. For a Gaussian beam impinging on a centrally-obstructed telescope, the on-axis far-field gain can be calculated as:¹²

where R₁ is the radius of the primary mirror, is the ratio of the primary mirror to the Gaussian beam waist radius, and is the ratio of the secondary mirror radius to the primary mirror radius – the obstruction ratio. Unsurprisingly, it can be seen that an increasing obstruction ratio will degrade the gain of the telescope. Additionally, for each γ, there is an optimal α, or beam waist, that will maximize the transmit gain by minimizing the power lost due to truncation of the Gaussian beam. For high obstruction ratios, it has been shown that redistributing the power from a Gaussian to a ring or doughnut-shaped distribution can yield significant improvement in terms of transmit gain.¹³ The simplest way to accomplish such redistribution is by using optical axicons. However, using the axicons comes with additional challenges, especially in terms of alignment and precise manufacturing.

To find a suitable γ and α for our system, we analyze the maximum achievable telescope gain for a 95 mm diameter primary mirror at the QKD wavelength (780 nm) in terms of γ and compare it with our 108.8dB requirement. The result of this analysis can be seen in fig. 2. It can be seen that a traditional design with a 95 mm primary mirror can meet the gain requirement as long as the obstruction ratio is below roughly 35%, which is not a very difficult requirement to meet. If axicons are used to optimally redistribute power, the 95 mm telescope could theoretically meet the gain requirement all the way up to a γ of about 65%. However, due to the difficulties when dealing with axicons, this approach is not preferred since the gain improvement for smaller γ becomes negligible. Consequently, a traditional approach with a Gaussian input beam and a smaller γ is preferred.

Figure 2:

Theoretical maximum far-field on-axis transmit gain for a 95 mm centrally-obstructed telescope at 780 nm.

To leave margin for additional telescope implementation loss, such as obstruction loss due to spider vanes for mounting of the secondary mirror or transmitted wavefront error, a γ lower than 35% is desired. A γ of 25% leaves exactly a 1 dB margin in terms of the telescope gain, and will be considered as a baseline in the following calculations.

By maximizing eq. (2) as a function of α, the optimal Gaussian beam waist for a given γ can be found. Using a 95 mm diameter and γ = 0.25 yields α = 1.0522, which corresponds to an expanded beam waist ω = 45.14 mm. Finally, we can use ω to determine the necessary telescope magnification such that the beam fits onto the FSM with minimal truncation. The magnification calculation is summarized in Table 1 below for a few commercially available MEMS FSMs.¹⁴

Table 1:

Calculation of the necessary beam expander magnification.

3.3

Optical Modeling

As part of the preliminary optical design, a sequential Zemax optical model was developed to help guide the placement of the optics in the available volume. Given the primary mirror diameter limit (95 mm) and the determined magnification (18.3× for a 6.4mm FSM with 1mm margin) an on-axis all-reflective two-mirror telescope was modeled to fit within a roughly 150×100×100 mm³ (1.5U) with additional 100×100×100 mm³ (1U) available for the frontend optics. This preliminary 2U optical design leaves plenty of space for other support hardware, such as polarization calibration unit, electronic boards, fiber components, or extensions of the optical chain planned in future iterations. The overall optical configuration and layout is shown in Fig. 3 with the on-axis ray tracing overlay.

Figure 3:

Side view of a ray-trace of the preliminary optical model. The whole structure fits in 300×100×100 mm³ (i.e., 3U). The telescope backend includes the FSM and fold mirrors routing the signals behind the primary mirror. Also, the dichroic mirror splits the received light onto a tracking sensor, and the transmit signal coming out of a polarization unit.

The telescope, as shown in fig. 3, is of a modified Ritchey–Chrétien (R-C) configuration, which is a variant of the traditional Cassegrain telescope design, with a concave elliptical 95 mm primary mirror (easier to perform an interferometric null test compared to a hyperbolic mirror) and a convex hyperbolic secondary mirror, yielding a 20% obstruction ratio. The Ritchey–Chrétien design is chosen as it provides a wide Field of View (FoV) performance relaxing the satellite pointing jitter requirement significantly. The preliminary design has a superb on-axis performance of 0.99 Strehl ratio (Fig. 5 (top)), which is of most significance for the optical frontend. In the meantime, the off-axis performance (+/- 0.2 degree) also reaches 0.94 Strehl ratio (Fig. 5 (bottom)) as the R-C telescope optical design (combined with the tertiary collimating lens) controls the off-axis aberrations (e.g., coma). Commercially available CubeSat attitude control systems support precision pointing to within a few hundredths of a degree,¹⁵ and the wide FoV performance will enable a robust optical system with good safety margin and tolerances. The telescope has an intermediate focus near the location of the primary mirror, after which the aspheric tertiary lens (higher order plano-convex asphere) is placed to collimate and correct the beam, completing the afocal telescope configuration. All the ray incidence angles are minimized in order to preserve the polarization state across the pupil throughout the afocal system. Also, the on-axis design provides an axially symmetric polarization aberration, which is already minimized, so any residual effect can be more simply calibrated.

Figure 4:

Computer Aided Design rendering of the telescope and the optical bench.

Figure 5:

Huygens Point Spread Function (PSF) showing high Strehl ratio of 0.99 for on-axis (top) and 0.94 for 0.2 degree off-axis field of view (bottom).

The 6.4 mm Mirrorcle MEMS FSM is placed at the exit pupil of the afocal telescope at a fold angle of 15° in order to maximize the projected surface area and its angular steering range. An additional FSM is placed after the polarization unit to route the signals upward behind the primary mirror so that the 0.5U volume can be utilized efficiently. Next, the dichroic mirror is placed to reflect received light onto a tracking sensor and transmit the collimated transmit beam through. The dichroic and fold mirrors are angled such that the two chains continue diagonally to the corners of the structure, maximizing the available optical path length. This allows using a focusing lens with a longer focal length for the sensor, decreasing its field of view and increasing its angular sensitivity/resolution for more precise beacon tracking using FSM. The design utilizes higher order aspheric lenses for collimation, focusing onto the sensor, as well as for the telescope tertiary. The retro-reflector, as well as any filtering optics were omitted from this preliminary design.

Using the Zemax model, the angular transfer function (effective focal length) of the tracking sensor was determined to be roughly 775 mm/rad, taking the angular telescope magnification into account. Several Huygens point-spread functions with different spot diameters were also exported for various telescope incidence angle, to be used in analysis of the tracking sensor noise sensitivity described in the following sections.

4.1

Range Loss

We are considering a satellite at an altitude h. The Earth radius is noted R_E, and we assume R_E = 6378 km. The elevation ξ for a given angle of incidence from nadir at the payload θ_inc is given by the following equation:

The slant range L is then*:

This can be simplified to:

Assuming a transmitter with FWHM divergence of θ_TX, the 1/e² transmitter beam radius W₀ is

With a pointing error θ_PT, and assuming a Gaussian beam, the range loss G_range is:

Scintillation

Atmospheric turbulence has an impact on the laser beams, for both uplink and downlink. Turbulence creates variations in the received power, called scintillation, and for uplink in particular, it introduces beam wander. Our link budget is based on the model of Ref. 16. First, we define the atmospheric turbulence strength, using the Hufnagel-Valley 5/7 model:¹⁶

For both uplink and downlink, the model includes a small scale and a large scale variance of the received irradiance noted σ_x,up, σ_y,up and σ_x,down, σ_y,down. The calculations for the scintillation variances are detailed in appendix B and appendix C.

For each link, the resulting variances and can be plugged to a Gamma-Gamma distribution:¹⁶

The Gamma-Gamma Cumulative Density Function (CDF) is difficult to compute. One can either integrate the Probability Density Function (PDF) of the distribution, or fit another distribution whose CDF can be directly computed. We chose to fit another using the α-μ distribution:

In eq. (11) and eq. (12), Q is the regularized upper incomplete gamma function, and Q^–1 is its inverse such that y = Q(μ, x), x = Q^–1(μ.y). The two distributions are fitted by matching the expected value and two other moments.^{17, 18}

In particular, since Γ(z + 1) = zΓ(z),

We can get a system of 2 equations with logarithmic scaling, which helps with the large range of values due to the Gamma function. For n ≥ 2:

A closed-form equation exists for the Jacobian, again with a logarithmic scale, and with ψ as the digamma function:

The system can be solved by using an iterative least-squares algorithm. The α-μ distribution can be computed efficiently for a large array, and its inverse is also well-behaved. Thanks to the inverse distribution function, we can specify a desired fractional fade time and obtain a derated optical power satisfying this condition.

Uplink Beacon

Several design options are available for the pointing detector in closed-loop tracking. The most common ones are image sensors and four quadrant detectors or “quadcells”. Image sensors have several advantages: the readout circuit is integrated and offers performances close to the shot noise floor with little integration effort. Also, the estimated spot position is a function of the pixel pitch, which is well known, contrary to quadcells where a transfer function needs to be estimated in order to find the spot position. On the other end, quadcells allow more flexible implementations, with a wider wavelength range available and the opportunity to customize filtering. This is the reason why a quadcell-based detector has been selected for this project. The payload is intended to be used with a ground station at night in an urban environment. Frequency filtering along with a modulated beacon source allow the tracking system to effectively remove background light from both the Earth and the Sun. It also enables the simultaneous tracking of the distant beacon source and of the internal position of the payload transmitter, so that continuous alignment calibration is performed.

The detector electronics for this payload are identical to the one in the CLICK-BC payload.⁵ It is composed of a silicon quadcell, four Transimpedance Amplifiers (TIAs), a frequency filtering stage and a 24 bit four-channel Analog to Digital Converter (ADC). The model for the detector and the TIA is shown in fig. 6. The digital signal processing section is similar to Ref. 5, however, two signals at two different frequencies are extracted in parallel in order to track the distant and the local source on the same detector. Contrary to the CLICK-BC payload, the transmit beam for this optical front-end is tight enough to require the introduction of a point-ahead angle between the received and the the transmitted beam. If a single FSM is used for both pointing error correction and point-ahead, the spot on the quadcell must be defocused in order to maintain a good enough pointing estimation error, unlike CLICK-BC, where a spot as small as possible was desired. For this reason, we first modeled the spot as a generic 2D Gaussian Point Spread Function (PSF) in order to get a rough estimate of the desired pointing parameters, such as the spot size and the quadcell effective focal length (EFL).

Figure 6:

Circuit for the quadcell transimpedance amplifier.

The free-space link parameters are listed in table 2. The resulting power distribution at the quadcell is plotted in fig. 7 against beacon divergence with a constant pointing error of 5 μrad, which is the ground station 3σ pointing requirement. A divergence of 500 μrad is selected as it offers sufficient link margin and is consistent with the optical ground station hardware and pointing capabilities. In these conditions, the power at the quadcell is P₀ = 668 pW. With a spot radius of W, we assume the PSF has form:

Table 2:

Link Budget Parameters

Case	Beacon Uplink	Data Downlink	Key Downlink
Altitude	500 km
Wavelength	978 nm	915 nm	760 nm to 790 nm
Transmitter divergence	500 μrad FWHM	20 μrad FWHM	12 μrad FWHM
Transmitter power	2.5 W	100 mW	25 × 106 photon/s
Transmitter optical loss	3dB	3dB	3dB
Pointing error	5 μrad
Receiver aperture	95 mm	600 mm	600 mm
Aperture obstruction (area)	5%	40%	40%
Receiver optical loss	3dB	3dB	1 dB
Minimum elevation	20 deg
Turbulence	HV 5/7, 95%	10 × HV 5/7	None
Atmospheric absorption	MODTRAN Urban	MODTRAN Urban	Urban, Maritime
Detector bandwidth	200 Hz	10 MHz to 1 GHz	Single photon

Figure 7:

Received power probability distribution at the payload quadcell versus transmitter divergence. The selected divergence for the uplink beacon is shown in green. The color indicates the probability that the measured power is higher than y axis power threshold I_th.

The PSF integrated over a square window is then:

The offset of the spot on the detector, noted x_spot and y_spot for its component on each axis, depends on the EFL, f_PT, and the pointing error, θ_еrr, with . The spot offset as well as the geometry of the quadcell detector are used to get the integration bounds x₁, x₂, y₁, and y₂ relative to the spot center, assumed to be 0 in eq. (18). The voltage signal for each quadrant of the quadcell after the TIA, as well as its noise, is calculated:

The voltage noise of the signal σ_PD is assumed to be composed of shot noise σ_shot and amplifier noise σ_TIA. I_PD is the current of an individual quadrant, R_PD is the quadcell responsivity, R_TIA is the transimpedance of the amplifier and ΔF is the detector bandwidth. The amplifier noise is found by simulating the circuit in fig. 6.

The quadcell position signal is a combination of the four quadrants:¹¹

The signal-to-noise ratio (SNR) is estimated for each spot position. The estimated spot position is numerically differentiated to find the signal slope on each axis.

With the local slope, we can convert the quadcell position error back to angular error at the payload telescope, called the Noise Equivalent Angle (NEA). The NEA for a W = 90 μm spot and a focal length of f_PT = 700mm is shown in fig. 8.

Figure 8:

Noise Equivalent Angle (NEA) in prad with a Gaussian PSF, plotted as a function of pointing error at the aperture of the payload. The largest expected Point-Ahead (PA) angle is shown as a dotted line, and the contour where NEA is better than the required 1 μrad is shown in solid black. The point-ahead margin, between the two, is 39.71 μrad.

Once an initial estimate for the spot size and EFL have been found, we can perform a new NEA analysis using the PSF obtained by ray-tracing from Zemax. The PSF is exported as a cross-section. This cross-section is linearly interpolated to form a 2D image of the spot with a resolution of 1000 by 1000 points. The cumulative sum of this array is then taken against each axis and normalized, to obtain the 2D integral of the PSF over a rectangular section. This rectangular section emulates the corner of one of the active elements of the quadcell. Unlike eq. (18), this calculation does not take into account the other two edges of the active element, however since the PSF is assumed to be of finite dimension and always smaller than the size of the active elements (1.5 × 1.5mm²), there is no impact from the quadcell active elements far edges for spots near the center.

The NEA for an in-focus spot is plotted in fig. 9a, and in fig. 9b for a wider defocused spot. The NEA is improved close to the quadcell center for the small spot, however, a wider spot enables larger pointing deviations with a small NEA, which is necessary to maintain the point-ahead offset. Figure 11a shows margins for both the point-ahead angle and the noise equivalent angle. As the spot widens, so does the margin for the maximum deviation angle, until the NEA drops below the required value over the entirety of the quadcell. The impact of the secondary mirror obstruction is limited, primarily degrading the link budget by reduction of the collection area, rather than having a negative impact on the PSF itself. Based on the results of fig. 11a, a spot size of around 100 μm is desired, with an acceptable deviation of 20 μm. The performance of the detector predicted by this analysis is compatible with the results of a similar design; Ref. 19 reports a NEA better than 0.1 μrad at a power of 200pW, with a focal length of 4 m.

Figure 9:

Noise Equivalent Angle (NEA) in prad with a Zemax-simulated PSF. The Point-Ahead (PA) margin is in red.

Figure 10:

Error in pointing angle estimation due to the PSF change at different telescope angles of incidence.

Figure 11:

Poiting performance for a single FSM and a 2 FSMs optical layout. The dashed line is the requirement for all quantities.

Unlike an imaging detector, we are evaluating the position of the beam relatively far away from a fixed position feature, like the boundaries of a pixel. Therefore, if using a single FSM, it is critical that the quadcell transfer function is well-known. A source of a variation of the PSF is the change in angle of incidence on the telescope, as seen in fig. 5. In order to estimate the error due to PSF changes, we evaluate the quadcell response with a 0.2 deg angle of incidence, and then use interpolation to inverse the quadcell transfer function at an incidence of 0 deg. The resulting error Δθ is then:

fig. 10 shows Δθ for two different spot size. With any amount of defocus, the error due to PSF varaitions are too large. To solve this issue, a second FSM is used to introduce the point-ahead angle. The quadcell must now track to the uplink beacon spot on its center, a well-defined location. In that configuration, the NEA is only relevant near the center of the quadcell plan, within an area defined by the FSM actuation noise. The FSM closed-loop control residual has been measured at 0.137 μrad. We selected a region of interest with a radius of 1 μrad. The resultin NEA is ploted in fig. 11b. In a two FSM configuration, spot sizes of 10 μm to 100 μm are optimal, with a wide acceptable range.

4.2

Data Downlink

The payload lasercom transmitter uses the same telescope as the key distribution link, with a higher divergence to account for chromatic aberrations in the optics. Data is modulated at 100 Mbps, with a 100mW average power. The power is limited to 100mW to avoid over-heating of the fast steering mirrors. The link parameters are listed in table 2.

Figure 12a shows the distribution of the optical power at the ground detector, for elevations ranging from 0 to 90 deg. Once the received power is known, we can compute the Bit Error Ratio (BER). First, we need to find the SNR. On the ground, an APD is used to receive the signal, with a responsivity of R_APD = 50A/W and an excess noise factor of F_APD = 3.2. The APD dark current is 1.5 nA. For an avalanche photodetector, the SNR can be expressed as:

ΔF is the bandwidth of the detector. For a data rate of 100 Mbps, a bandwidth of 100MHz is sufficient. The BER is obtained with eq. (24), and is plotted in fig. 12b for three different bandwidth values.

Figure 12:

Data downlink results for worst case scintillation, , with parameters summarized in table 2. The vertical red line indicates the elevation angle for which the link budgets must close.

With a 100MHz bandwidth at the detector output, the BER is better than 1 × 10^–5 once the elevation is higher than 15 deg. The relatively low range loss of the free-space optical channel could support higher data rates, for example 1 Gbps for elevations higher than 22 deg. The APD would also support the higher bandwidth required, however the data rate is limited by the modulator design, to 200 to 400Mbps. Higher rates would limit the laser diode options and require a more complex modulation circuit, or a change of modulation scheme.

4.3

Key Downlink

The quantum key distribution link is driving the optical front-end divergence and pointing requirements, and because of that, the optics are optimized to obtain the lowest divergence for this beam – 12 μrad FWHM at a center wavelength of 780 nm. For this analysis, we assume that the entangled photon source generates photon pairs at a rate comparable to the Micius mission.²⁰ Other link parameters are listed in table 2.

The overall loss of the optical channel, from the photon source to the ground single photon detector, is show in fig. 13a. The loss is plotted for a direct-overhead pass, with the time axis centered on the highest elevation at the ground station, when the satellite is directly above the ground telescope. The link is limited to elevations above 20 deg. There is currently no maximum elevation, but it is possible that the optical ground station telescope mount may have limitations. Also, at low elevation the Quantum Bit Error Rate (QBER) is too low to ensure privacy. With the free-space channel losses, and the specifications of the ground detectors, the Quantum Bit Error Rate can be computed.²¹ The single photon detectors and polarization controllers characteristics are listed in table 3. With the QBER, the asymptotic key rate, as well as the asymptotic key size for a single pass are obtained. The resulting key rate and QBER are shown in fig. 13b, for either a maritime or an urban aerosol model. For the urban case, the asymptotic key size is 16 603 bit, and for the maritime case, it is 39 572 bit long. The asymptotic key rate provides an estimate of the secret key that can be extracted assuming that the raw key is of infinite length. Claiming information theoretic security requires considering the finite size of the raw key and involves integration over the satellite pass and then optimization over the error correction parameters.²² The parameters are ϵ_cor, the probability that the key contains an error after error correction, and ϵ_sec, the probability that an attacker knows any part of the key without being detected. The finite key size are 1862 bit for the urban model, and 17 677 bit for maritime. The urban case is marginal for single pass key exchange, but can still support QKD demonstrations. The maritime case is sufficient for a single pass secure key exchange.

Figure 13:

Key downlink results

Table 3:

Quantum detection and privacy amplification and parameters

Table 4:

Single pass key exchange performances