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Satellite Quantum Communications: Fundamental Bounds and Practical Security

Stefano Pirandola

arXiv:2012.01725v4quant-phphysics.ao-phphysics.app-phphysics.opticsphysics.space-ph

TL;DR

Ground-based fiber rates decline with distance, motivating satellite links as a potential alternative for long-distance quantum communication. The paper derives satellite communication bounds and studies achievable finite-size CV-QKD rates across operating configurations; it reports that a sun-synchronous satellite can outperform a ground chain of capacity-level repeaters.

  • Problem

    Ground-fiber transmissivity and communication rates decay exponentially with distance, while repeaters introduce trust or demanding long-coherence-time and distillation requirements.

    Source: §I · p. 1

  • Method

    The paper extends free-space analysis to satellite links, establishing upper limits and analyzing achievable finite-size, composable secret-key rates across configurations.

    Source: §IV · pp. 21–22

  • Results

    A sun-synchronous satellite can distribute substantially more secret-key bits per day than a standard fiber connection with capacity-level repeaters.

    Source: §I · p. 2

  • Takeaways & Limitations

    The reported comparison supports the potential of satellite links for large-scale quantum communications against ground networks with repeaters.

    Source: §I · p. 2

    Achievability of the stated upper bound assumes fixed satellite geometry or a communication timescale faster than orbital motion; realistic clocks require modifying the lower bound.

    Source: §4 · p. 11

Abstract

from arXiv · show

Satellite quantum communications are emerging within the panorama of quantum technologies as a more effective strategy to distribute completely-secure keys at very long distances, therefore playing an important role in the architecture of a large-scale quantum network. In this work, we apply and extend recent results in free-space quantum communications to determine the ultimate limits at which secret (and entanglement) bits can be distributed via satellites. Our study is comprehensive of the various practical scenarios, encompassing both downlink and uplink configurations, with satellites at different altitudes and zenith angles. It includes effects of diffraction, extinction, background noise and fading, due to pointing errors and atmospheric turbulence (appropriately developed for slant distances). Besides identifying upper bounds, we also discuss lower bounds, i.e., achievable rates for key generation and entanglement distribution. In particular, we study the composable finite-size secret key rates that are achievable by protocols of continuous variable quantum key distribution, for both downlink and uplink, showing the feasibility of this approach for all configurations. Finally, we present a study with a sun-synchronous satellite, showing that its key distribution rate is able to outperform a ground chain of ideal quantum repeaters.

I. INTRODUCTION

Ground-based fiber links face distance-dependent rate limits, motivating satellite links as a possible way to simplify quantum networks. This work studies satellite communication limits and achievable security across uplink and downlink configurations, then compares satellite distribution with ground repeater chains.

I. INTRODUCTION

  • Satellite links may avoid ground-fiber rate limits and simplify networks, motivating this study of ultimate bounds and practical security across satellite configurations.The analysis covers uplink and downlink, variable altitudes and zenith angles, and atmospheric and background-noise effects.Source: §I · p. 1

A. Structure of the paper

  • The paper first develops satellite-link bounds, then studies composable finite-size CV-QKD rates with fading and orbital dynamics, and finally compares satellite rates with ground repeater chains.The first part derives upper and lower bounds; the second part analyzes achievable key rates.Source: A. Structure of the paper · p. 2

A. Geometric considerations

  • The geometry uses satellite altitude and zenith angle to determine slant distance; formulas are most accurate within about 60° of zenith, and achievable-rate analysis is restricted to that window.Uplink and downlink share diffraction and extinction effects, while turbulence-induced fading and background thermal noise differ between them.Source: A. Geometric considerations · pp. 2–3

B. Free-space diffraction

  • A finite receiver aperture captures only part of a diffracting Gaussian beam, yielding a diffraction-based bound; the plotted horizon rate is optimistic because refraction elongation is neglected.The figure compares bounds across satellite altitudes and shows zenith and horizon positions; its assumed beam is collimated.Source: B. Free-space diffraction · p. 3, Figure 1 · p. 3

C. Atmospheric extinction

Atmospheric extinction and receiver inefficiency reduce satellite-link bounds, while beam wandering adds configuration-dependent fading. The model distinguishes turbulence effects in uplink and downlink and develops them for satellite altitudes and slant paths.

C. Atmospheric extinction

  • The extinction model adapts Beer–Lambert attenuation to satellite altitude and zenith angle; its zenith atmospheric transmissivity is approximately 0.967.The small-zenith-angle approximation is already good at 30 km and nearly exact beyond 100 km.Source: C. Atmospheric extinction · p. 4
  • About one order of magnitude: diffraction, extinction, and non-ideal quantum efficiency together reduce the ultimate bounds relative to diffraction-only bounds.The bounds decrease further when fading and background noise are included.Source: C. Atmospheric extinction · p. 4

D. Fading process induced by beam wandering: Turbulence and pointing errors

  • Beam wandering combines transmitter pointing error with atmospheric turbulence, whose relative importance differs between uplink and downlink.Downlink turbulence is negligible after diffraction spreads the beam, whereas uplink turbulence acts near beam generation.Source: D. Fading process induced by beam… · pp. 4–5
  • The fading model separates fast turbulent beam broadening from slower centroid wandering and additional pointing-error motion.It assumes weak turbulence, appropriate for zenith angles θ ≲ 1.Source: D. Fading process induced by beam… · pp. 4–5
  • In uplink, the short-term spot size exceeds the diffraction-limited spot size by an order of magnitude; the model derives spot-size and centroid-wandering parameters using satellite-appropriate turbulence profiles.At λ = 800 nm, the reported gap is nearly constant in the far field; the H-V model uses different profiles for night and day.Source: D. Fading process induced by beam… · p. 5

E. Bounds for the satellite fading channels in uplink and downlink

The section extends fading-channel bounds to satellite geometry and distinguishes uplink from downlink effects. It compares those bounds across altitude, zenith angle, and operating time, while noting when the bounds are achievable and the costs of slow detection.

E. Bounds for the satellite fading channels in uplink and downlink

  • The bound combines the perfectly aligned-link rate −log2(1 − η) with a correction for beam wandering from turbulence and/or pointing errors.The satellite expression extends the analytical form of the free-space bound in Ref. to satellite communications.Source: E. Bounds for the satellite fading… · p. 6
  • The satellite bound uses geometry-dependent transmissivity and fading parameters, with uplink additionally including turbulence-induced beam spreading and wandering.Downlink includes centroid wandering from pointing error; uplink includes both that effect and turbulence.Source: E. Bounds for the satellite fading… · p. 6
  • About one–two orders of magnitude: uplink turbulence further lowers the bound relative to downlink, and daytime conditions lower the uplink bound further.The comparison covers zenith and θ = 1 configurations at multiple altitudes; pointing error already reduces the downlink rate from the LEO region.Source: E. Bounds for the satellite fading… · p. 6, Figure 2 · p. 7
  • For typical satellite altitudes at night, the pure-loss fading assumption makes the bound achievable as both secret-key and entanglement-distribution capacities.With non-trivial daytime background noise, the same quantity remains an upper bound but is not guaranteed achievable.Source: E. Bounds for the satellite fading… · p. 7
  • Slow detection averages centroid wandering over acquisition times of about 100 ms or more, potentially increasing detection efficiency but reducing clock rate and collecting more noise.The averaging is over times beyond the typical turbulence and pointing-error timescales.Source: E. Bounds for the satellite fading… · p. 7

F. Satellite bounds with background noise

The section models receiver-collected background noise alongside loss and fading, accounting for operational and receiver settings. It distinguishes the practical significance of that noise between night and daytime operation.

F. Satellite bounds with background noise

  • The ultimate-bound calculations neglect setup noise, while the later composable QKD-rate analysis explicitly includes a realistic estimate of it.The total thermal noise is decomposed into receiver-collected background noise and extra setup noise.Source: F. Satellite bounds with background… · p. 7
  • The background-noise estimate depends on link direction and time of day as well as receiver aperture, field of view, detection time, wavelength, and spectral filter.The illustrative receiver calculation assumes Δλ = 1 nm, Δt = 10 ns, Ω_fov = 10^-10 sr, and a_R = 40 cm.Source: F. Satellite bounds with background… · pp. 7–8
  • Background noise is practically negligible at night but important during the day, when satellite links require a thermal-loss channel model combining loss, fading, and receiver-collected thermal noise.The supplied table summarizes mean background thermal photons per mode for uplink and downlink under different conditions.Source: F. Satellite bounds with background… · p. 8, Table I · p. 8

1. Noise filtering

The section connects receiver filtering and noisy-channel bounds to practical secure-key range limits. Its results show daytime uplink is especially constrained, while narrower filtering can expand feasible ranges.

1. Noise filtering

  • At 800 nm, a 1 nm filter spans ≃470 GHz; LEO use may require Doppler compensation, while receiver interferometry can reduce effective bandwidth.Adaptive optics can track the beam’s central frequency during satellite flyby.Source: §1 · p. 8
  • A 0.1 pm filter is 10^-4 narrower than 1 nm and suppresses thermal noise by 10^-4, making daytime noise values almost negligible.The paper includes this narrower-filter regime in its subsequent analysis.Source: §1 · p. 8
  • A larger receiver aperture increases transmissivity but also collects more background noise, while oversized apertures can aggravate turbulence effects.The aperture can be optimized over the rate; turbulence effects include off-axis scintillation and more short-term speckles.Source: §1 · pp. 8–9

2. Accounting for the untrusted noise

  • The noisy satellite link is treated as a thermal-loss channel; a collective entangling-cloner attack supplies a key-capacity upper bound, averaged over fading.The upper bound is applied to instantaneous channels and averaged over the fading process.Source: §2 · p. 9

3. Maximum ranges

  • Daytime uplink range is roughly limited to the Kármán line; downlink reaches LEO in cloudy conditions and MEO in clear skies.These limits apply to the regime considered.Source: §3 · p. 10, Table III · p. 10
  • For day-uplink, the maximum-range bound is zmax ≲ 340 km, reaching LEO; an effective 0.1 pm filter around 800 nm sensibly increases day-time ranges.Source: §3 · p. 10

4. Analysis of the thermal-loss bounds

With narrow filtering, the thermal-loss bounds can collapse to the loss-limited capacity in low-noise satellite scenarios; the paper then develops finite-size composable CV-QKD rates that account for fading and orbital dynamics.

4. Analysis of the thermal-loss bounds

  • At LEO altitudes, night-time downlink and uplink bounds coincide with B, establishing E ≃ K ≃ B; at MEO altitudes, a gap prevents claiming achievability for the stated parameters.The thermal correction is practically negligible at night, with small deviations at GEO altitudes.Source: §4 · p. 10, Figure 3 · p. 11
  • For clear-day downlink with a 1-nm filter, the bounds separate from the LEO region onward, and increasing altitude widens the gap and moves the upper bound farther from B.The passage identifies background thermal noise as limiting rate performance in day-time operation.Source: §4 · p. 10, Figure 3 · p. 11
  • With a 0.1-pm filter, the thermal-loss bounds collapse to the loss-limited bound B at all altitudes in low-background conditions, giving E ≃ K ≃ B.The figure caption identifies the narrow-filter case as coinciding with the loss-limited bounds at the zenith and 1 radiant.Source: §4 · p. 10, Figure 3 · p. 11

III. COMPOSABLE FINITE-SIZE SECURITY FOR SATELLITE CV-QKD

  • The practical-rate analysis extends a pilot-based coherent-state CV-QKD protocol to satellite physical models and includes finite-size and composable security effects.It accounts for fading and orbital dynamics in deriving achievable rates.Source: §III · p. 11, A. Overview · p. 12

4. Analysis of the thermal-loss bounds

  • Cloudy-day downlink reaches B throughout the LEO region, while day-time uplink does so only in its first part; reaching B at all altitudes requires a narrower filter.These results concern the stronger-background scenarios of cloudy-day downlink and day-time uplink.Source: §4 · p. 11, Figure 4 · p. 12

A. Overview

  • The adopted post-selection strategy is described as more robust for collecting parameter-estimation statistics than clusterization, which may suffer from poor statistics.Signals are grouped by transmissivity and processed slot-by-slot.Source: A. Overview · pp. 11–12

B. Composable key rate at fixed transmissivity

  • At fixed transmissivity, parameter estimation bounds loss and noise for rate calculation, while error-correction and privacy-amplification imperfections enter the composable security accounting.The security parameter combines parameter-estimation, correctness, and secrecy errors.Source: B. Composable key rate at fixed… · p. 13

C. Pilots, post-selection, lattice allocation, and defading

Bright pilot pulses track link transmissivity so the parties can post-select and bin signal data, then map the bins to a worst-case fixed-transmissivity thermal-loss channel for security analysis.

C. Pilots, post-selection, lattice allocation, and defading

  • Bright, phase-fixed pilot pulses are interleaved with signals to estimate the satellite link’s instantaneous transmissivity, with the pilot noise negligible relative to the signal amplitude.The paper states that the pilots enable almost perfect transmissivity estimation.Source: C. Pilots, post-selection, lattice… · p. 14
  • The paper estimates that 10^6 pilot photons can yield about 10^3 collected photons even at τ ≃ 10^-3, supporting near-perfect transmissivity estimation in its stated worst-case LEO example.The example is a 2000-km-high satellite at 1 radiant with approximately 30 dB loss.Source: C. Pilots, post-selection, lattice… · p. 14
  • The parties retain pulses with transmissivity above ηth, assign the retained interval to lattice slots, and associate each slot’s data with its minimum transmissivity.The threshold is set as ηth = fthη for fixed fth < 1.Source: C. Pilots, post-selection, lattice… · p. 14
  • Rather than process slots independently, the parties can map them to the minimum-transmissivity slot, producing a worst-case thermal-loss Gaussian channel with fixed ηth and noise n̄*.The method assumes Gaussian noise to overestimate Eve’s strategy and underestimate the parties’ performance.Source: C. Pilots, post-selection, lattice… · p. 14

D. Parameter estimation and composable key rate

The protocol estimates worst-case channel parameters and uses them to lower-bound the key rate; its composable accounting also addresses signal-pilot discrepancies and extends heterodyne security to general coherent attacks.

D. Parameter estimation and composable key rate

  • After reducing fading to a fixed worst-case channel, Alice and Bob estimate ηth and noise using sacrificed data, with confidence bounds used to form a lower-bound key rate.The passage states that each worst-case estimator is correct up to error εpe.Source: D. Parameter estimation and composable… · pp. 14–15
  • Estimating ηth from signals can detect and account for Eve’s potential discrimination between signals and pilots or errors in slot allocation.The formalism also applies when signal-pilot interactions differ by forming worst-case estimators from the signals.Source: D. Parameter estimation and composable… · pp. 14–15
  • The worst-case total thermal noise uses an upper bound on noise across the post-selection interval, and the practical setup-noise expression is needed to compute it.The passage defines n̄wc as at least n̄(τ) for every τ in the interval.Source: D. Parameter estimation and composable… · p. 15
  • For heterodyne detection, the paper extends security from collective to general coherent attacks using a modification of the rate construction and energy-test accounting.The rate uses the stated key-generation pulse count after accounting for parameter estimation and pilot pulses.Source: D. Parameter estimation and composable… · p. 15

E. Setup noise and observations about the LO

The section compares TLO and LLO setup-noise tradeoffs across channel transmissivities and examines orbital variation in achievable CV-QKD key rates. It estimates secret-key yields per zenith-crossing passage and finds CV-QKD feasible for downlink and uplink in both daytime and nighttime conditions, despite fading and fast orbital dynamics.

E. Setup noise and observations about the LO

  • TLO avoids phase error but incurs attenuation-dependent electronic noise, whereas LLO reduces electronic noise but adds phase noise; TLO is preferable at high transmissivity and LLO at low transmissivity.For the illustrated parameters, the LLO phase-noise contribution can be kept low with a laser linewidth of about 1.6 KHz.Source: E. Setup noise and observations about… · p. 15, Figure 5 · p. 16
  • LLO interleaves bright reference pulses between signal and pilot pulses, using half the link time for synchronization; using both polarizations compensates the resulting throughput factor of 1/2.Bob uses the received references to reconstruct Alice’s rotating frame and rotate measurement outcomes.Source: E. Setup noise and observations about… · p. 16

G. Orbital slicing

  • Orbital slicing estimates a block’s rate by taking the minimum rate at the largest zenith angle in each slice, then averages those slice rates over the orbit.The 1-radiant rate uses the border value at θ = ±1 as a conservative estimate.Source: G. Orbital slicing · pp. 17, 19

F. Key rate analysis with orbital dynamics

  • Positive key rates are achievable across configurations in LEO/sub-LEO: setup 2 improves downlink into LEO, while setup 3 enables uplink only in sub-LEO; daytime downlink matches night-time rates.Daytime uplink rates remain lower because ground-level turbulence reduces transmissivity more strongly during the day.Source: F. Key rate analysis with orbital… · p. 17, Figure 6 · p. 18

G. Orbital slicing

  • At 530 km, the 1-radiant window lasts about 200 seconds, providing about 2 × 10^9 pulses; after the 10° mask, 131 seconds are available for processing, estimated to take about 120 seconds.The satellite is visible for about 131 seconds in the post-window sector, allowing classical error correction and privacy amplification before the flyby ends.Source: G. Orbital slicing · pp. 18–19
  • A zenith-crossing downlink passage distributes ≃6.13 × 10^7 secret bits at night and ≃6.08 × 10^7 by day, lower bounds on daily totals because non-zenith passages are also exploitable for QKD.The estimates account for 200 s of quantum communication; at 10 MHz, daytime downlink is about 304 kbits/s.Source: G. Orbital slicing · p. 20
  • At 10 MHz, uplink yields about 424 kbit/s at night and 273 kbit/s by day, or ≃1.69 × 10^7 and ≃1.09 × 10^7 secret bits per zenith-crossing passage, respectively.These passage totals use a quantum transit time of 40 s.Source: G. Orbital slicing · p. 20
  • CV-QKD is feasible for LEO/sub-LEO satellite key distribution with turbulence- and pointing-error fading and fast orbital dynamics, for night- and daytime downlink and uplink.The analysis includes the added challenges fast orbital dynamics create for transmission using reasonably large block sizes.Source: G. Orbital slicing · p. 20

H. Satellites versus ground-based repeaters

For sufficiently separated stations, a sun-synchronous satellite can outperform fiber-based links, including chains of ideal repeaters. The comparison assumes particular orbital paths, operating conditions, and link configurations.

H. Satellites versus ground-based repeaters

  • A near-polar sun-synchronous orbit passes over a given Earth location at roughly the same local mean solar time, supporting stable day- or night-time operating conditions.At 530 km altitude, the paper gives an orbital period of about 95 minutes and 15 orbits per day.Source: H. Satellites versus ground-based… · p. 20
  • The model assumes stations lie along the satellite’s orbital path and interact only during zenith crossings, with the same day- or night-time conditions at both stations.Close stations may communicate with the satellite simultaneously; the zenith-crossing-only assumption is described as a worst case.Source: H. Satellites versus ground-based… · p. 21
  • The fiber comparison uses the same 10 MHz clock and includes repeaterless and ideal-repeater links, while satellite rates are modeled for sun-synchronous downlink and uplink.The satellite’s day- or night-time operation is considered for each link direction.Source: H. Satellites versus ground-based… · p. 21, Figure 11 · p. 21
  • Beyond 215 km, a satellite beats repeaterless fiber; at about 6,675 km, downlink matches 30 ideal repeaters by day or night, while uplink performs worse.The figure compares secret-key bits per day against ground distance, using a 10 MHz clock; the modeled downlink and uplink altitudes are 530 km and 103 km, respectively.Source: H. Satellites versus ground-based… · p. 21, Figure 11 · p. 21

IV. CONCLUSIONS

The conclusions combine satellite-link upper bounds with practical finite-size CV-QKD rate analysis across operating configurations. They identify link-specific physical effects and report feasibility over a broad range of satellite altitudes and conditions.

IV. CONCLUSIONS

  • The satellite analysis extends prior free-space methods to slant propagation at variable altitudes, with atmospheric turbulence and background-noise models that differ from ground free-space cases.Source: §IV · p. 21
  • The upper-bound analysis covers secret and entanglement bits while accounting for diffraction, atmospheric extinction, setup inefficiency, pointing error, turbulence, thermal background, and setup noise.Source: §IV · pp. 21–22
  • Uplink turbulence matters near the transmitter, where the beam spot is small; in downlink, turbulence is negligible and pointing error is the relevant effect.Source: §IV · p. 22
  • Narrow frequency filters can greatly suppress background noise from sky brightness and Moon and Earth albedos in homodyne-like setups.Source: §IV · p. 22
  • Finite-size, composable key rates were evaluated across satellite altitudes and zenith angles, and optimal protocols yield secure-key and entanglement-distribution rates over a broad altitude range.Source: §IV · p. 22
  • The practical-rate study uses pilot-guided post-selected heterodyne CV-QKD while accounting for satellite orbital dynamics.Source: §IV · p. 22

Appendix A: Basic geometry for satellite communications

The appendix relates slant distance, satellite altitude, and zenith angle, then extends atmospheric-loss calculations to account for refraction. Refraction is negligible near the zenith but increasingly relevant toward the horizon.

Appendix A: Basic geometry for satellite communications

  • Slant distance z, altitude h, and zenith angle θ describe the ground-station–satellite geometry; z equals h only at the zenith.The geometry also allows the zenith angle or slant distance to be expressed from the other parameters.Source: Appendix A · p. 25, Figure 12 · p. 25
  • For θ ≤ 1 and satellite altitudes around 20 km or less, h ≃ z cos θ and z ≃ h sec θ are excellent approximations.The flat-Earth approximation is introduced for small zenith angles.Source: Appendix A · p. 25

Appendix B: Refraction effects

  • Refraction changes the zenith angle to an apparent angle and elongates the optical path; at the horizon, the refraction angle exceeds one degree.The surface refractive index is approximately 1.00027.Source: Appendix B · p. 26
  • Refraction is negligible within about 1 radian of the zenith but increasingly relevant near the horizon.For a 780 km satellite at the maximum apparent zenith angle, the passage reports atmospheric loss of about 7.1 dB with refraction versus about 3.4 dB without it.Source: Appendix B · p. 26

Appendix C: Atmospheric turbulence

The appendix characterizes atmospheric turbulence using altitude-dependent refractive-index structure profiles and distinguishes turbulence regimes by their physical effects. Its adopted profiles decrease rapidly with altitude, making roughly 20 km an effective atmospheric thickness.

1. Weak turbulence

  • In weak turbulence, small eddies broaden the beam while larger eddies cause slower random centroid wandering; stronger turbulence can deform and break the beam into patches.The wandering evolves on a timescale of about 10–100 ms.Source: §1 · pp. 26–27
  • The refractive-index structure constant C_n^2 underlies measures of turbulence strength, including scintillation and spherical-wave coherence length.The coherence length enters the expressions for beam spot sizes.Source: §1 · p. 27

2. Refraction index structure constant

  • The Hufnagel–Valley profile uses atmospheric-condition parameters that vary by time of day; the study adopts H-V5/7 for night and specified H-V parameters for day and worst-case day.The night model uses A = 1.7 × 10^-14 m^-2/3 and v = 21 m/s; the worst-case daytime model uses A = 2.75 × 10^-14 m^-2/3 and v = 57 m/s.Source: §2 · p. 27
  • The Hufnagel–Valley and simplified Hufnagel–Stanley profiles agree well, and C_n^2 decreases exponentially at higher altitudes.The structure constant is negligible beyond about 20 km; the single-layer average departs from the profiles after 15 km and differs by about two orders of magnitude at 20 km.Source: §2 · p. 27
  • For zenith angles up to about 1 radian, the atmosphere crossed by the beam is modeled using an effective thickness of about 20 km; at 1 radian, the slant section is about 37 km.For larger angles, refraction lengthens the slant distance through the atmosphere.Source: §2 · pp. 27–28

3. Scintillation index and Rytov variance

The appendix uses the scintillation index and Rytov variance to characterize turbulence, then compares coherence lengths and their effects on downlink and uplink beams. Turbulence is nearly negligible for downlink under stated conditions but remains consequential for uplink.

3. Scintillation index and Rytov variance

  • The scintillation index measures normalized irradiance variance, and Rytov variance serves as a weak-fluctuation measure and check for whether turbulence is weak.The weak-turbulence condition is a scintillation index below 1 throughout the beam profile.Source: §3 · p. 28
  • For zenith angles within 1 radian, the Rytov variance remains below unity, but exceeds unity beyond about 1.2 radians under standard conditions.With worst-case daytime parameters, it is about 0.6 at the zenith and about 2 already at 1 radian.Source: §3 · p. 28, Figure 15 · p. 29

4. Coherence length

  • At 100 km slant distance and 800 nm, downlink coherence length is about 1.8 m at zenith and 68 cm at 1 radian, versus about 2.9 cm for uplink at 1 radian.Coherence length increases with distance and wavelength, scaling as λ^6/5.Source: §4 · p. 29
  • For receiver apertures up to 1 m in the stated downlink examples, turbulence-induced beam spreading and wandering are negligible, leaving spot sizes approximately diffraction-limited.For a 530 km satellite, the estimated speckle count is at most 1.01 at zenith and 1.07 at 1 radian.Source: §4 · p. 29
  • Unlike downlink, uplink turbulence cannot be neglected even at zenith.The appendix treats the uplink coherence length with a plane-wave approximation for slant distances at least 160 km and zenith angles up to about 1 radian.Source: §4 · pp. 29–30
  • The plane-wave coherence-length approximation closely matches the exact integrals, practically overlapping at zenith; its asymptotic approximation is excellent for slant distances of at least 160 km and zenith angles up to about 1 radian.The daytime model has a different asymptotic prefactor from the night-time model.Source: §4 · p. 30

5. Spot sizes for uplink

For uplink, the appendix develops analytical spot-size approximations under weak turbulence and a collimated-beam assumption, then evaluates their behavior with altitude. Turbulence substantially enlarges both the short-term spot and centroid wandering relative to diffraction-limited propagation.

5. Spot sizes for uplink

  • For a collimated uplink Gaussian beam in weak turbulence, Yura’s condition permits analytical expressions for long- and short-term spot sizes.At 800 nm, the stated condition is compatible with typical satellite communications for spot sizes much larger than 1.4 mm.Source: §5 · p. 30
  • The appendix simplifies uplink spot-size and centroid-wandering formulas using the asymptotic coherence length for zenith angles up to about 1 radian.The simpler analytical expressions closely match values computed from the exact formulas at typical optical frequencies.Source: §5 · p. 31
  • Centroid-wandering standard deviation rises from about 0.5–1 m near the Kármán line to about 200–300 m at GEO altitude, depending on zenith angle.For the studied 800 nm, 20 cm collimated beam, the short-term spot size is considerably larger than the diffraction-limited spot size.Source: §5 · p. 31, Figure 17 · p. 31

Appendix D: Background noise in satellite communications

The appendix estimates satellite background noise for uplink and downlink, accounting for reflected sunlight, sky brightness, and Earth thermal radiation. It treats the Earth and Moon as Lambertian disks, making received intensity dependent on the receiver’s field of view.

Noise model

  • The Earth and Moon are approximated as Lambertian disks, so the intensity detected by the satellite receiver depends on its angular field of view.The approximation treats surface scattering as uniform, with radiance independent of angle.Source: Appendix D · p. 31

Uplink

  • In uplink, daytime noise is mainly sunlight reflected by Earth; at night, it is sunlight reflected by the Moon and then Earth.The model uses Earth albedo AE ≃0.3 and, for full-Moon night, κnight ≃7.36 × 10^-7, roughly 10^-6 of the daytime value.Source: Appendix D · pp. 31–32

Thermal radiation

  • Earth black-body radiation contributes ≃4.8 × 10^-13 mean photons, negligible relative to the uplink background-noise values reported in Table I.The estimate uses λ = 800 nm, an average Earth surface temperature of ≃288 K, and the stated receiver parameter ΓR.Source: Appendix D · p. 32

Downlink

  • Downlink background noise is estimated at ≃3 × 10^-6 to ≃0.3 mean photons, for sky conditions from full-Moon clear night to cloudy daytime.The estimate assumes the receiver’s field of view excludes the Moon and Sun.Source: Appendix D · p. 32

Estimate precision

  • The background thermal-noise estimates can be refined with dedicated programs such as MODTRAN, LibRadtran, or 6SV.The passage specifically notes that these programs can provide more detailed sky-brightness calculations.Source: Appendix D · p. 32

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