Optical downlink modeling for low-earth-orbit and medium-earth-orbit satellites under atmospheric turbulence with a quantum-state-tomography use case

Artur Czerwinski; Jakub J. Borkowski; Maciej Wi\'sniewski; Saeed Haddadi

REVIEW 2 major objections 6 minor 73 references

Optical downlink modeling for low-earth-orbit and medium-earth-orbit satellites under atmospheric turbulence with a quantum-state-tomography use case

T0 review · 2 major / 6 minor · reviewed 2026-08-04 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read This paper establishes a multiplicative link-budget model for satellite-to-ground optical downlinks in which aperture averaging strongly suppresses turbulence-induced photon-loss fluctuations for LEO missions but barely helps MEO missions,

desk verdict A competent, tutorial-style link-budget scan with a useful QST diagnostic, but the headline LEO/MEO aperture-averaging contrast is very likely an artifact of applying Andrews' formula with total slant range instead of the distance to the dominant turbulent layer. read the letter →

arxiv 2512.13828 v2 pith:L4D73J4M submitted 2025-12-15 quant-ph physics.comp-phphysics.opticsphysics.space-ph

Artur Czerwinski , Jakub J. Borkowski , Maciej Wiśniewski , Saeed Haddadi This is my paper

classification quant-phphysics.comp-phphysics.opticsphysics.space-ph

keywords free-spaceopticsopticaldownlinklinkbudgetatmosphericturbulenceapertureaveragingscintillationquantumstatetomographysatellitecommunication

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper is trying to establish a practical, quantitative method for predicting photon loss in free-space optical downlinks from LEO and MEO satellites, with turbulence treated as a multiplicative random factor whose fluctuations can be mitigated by aperture averaging. It applies the model to a quantum state tomography use case to show how many received photons are needed to reconstruct quantum states with high fidelity. The central quantitative claim is that aperture averaging is highly effective for LEO (420 km) but nearly ineffective for MEO (20,200 km), so that LEO tomography needs roughly 10^4–10^6 photons while MEO tomography requires 10^7–10^9. A sympathetic reader would care because these numbers provide concrete design targets for satellite QKD, quantum networking, and optical communication demonstrators.

What carries the argument

The central object is the multiplicative transmittance model of Eq. (1), where the turbulence-induced random factor I is described by a log-normal distribution whose variance is the intensity scintillation index (ISI) σ²_I, computed from the Rytov variance using the Hufnagel–Valley C_n² profile, and where aperture averaging is introduced through the power scintillation index (PSI) σ²_P = Av_A(D) σ²_I with Av_A(D) = [1 + 1.062 (kD²/4L)]^(−7/6). This aperture-averaging factor carries the argument: it depends only on the ratio D²/L, which is what makes it effective for LEO (short L) and ineffective for MEO (long L). The QST use case then connects this machinery to quantum performance by replaci

What would settle it

A direct falsifier is an experimental satellite-to-ground downlink at LEO and MEO altitudes that measures the power scintillation index and the variance of received photon counts for a range of aperture diameters; if for a MEO link a 1.0 m aperture reduces the scintillation variance by a factor close to the Andrews prediction (which for 20,200 km gives Av_A ≈ 1 even for 1.5 m), the model holds, but if strong turbulence or non-Kolmogorov effects yield a different dependence on D/L, the specific predictions of Eq. (18) would fail.

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Extended reading notes

Core claim

The central claim is that the total downlink transmittance factorizes into independent contributions, η = η_int η_atm(ζ) η_d(z) I, where η_int is internal loss, η_atm is Beer–Lambert absorption/scattering, η_d is diffraction-limited geometric collection, and I is a log-normal turbulence factor with mean unity. Using the Hufnagel–Valley turbulence profile and the Andrews aperture-averaging factor Av_A(D) = [1 + 1.062 (kD²/4L)]^(−7/6), the paper finds that for LEO downlinks the power scintillation index (PSI) is strongly reduced by aperture averaging, smoothing the photon-loss curves and stabilizing received power, whereas for MEO downlinks the same apertures yield only marginal averaging beca

Load-bearing premise

The quantitative results rest on the assumed validity of the Hufnagel–Valley C_n² profile and the weak-turbulence Rytov/aperture-averaging formulas applied at zenith angles up to 80° and for paths up to 20,200 km; if the turbulence profile is unrepresentative or strong-fluctuation effects set in, the computed scintillation indices, the LEO/MEO contrast, and the QST fidelity curves all shift.

Editorial extensions

If this is right

If the model is correct, LEO optical downlink systems can stabilize received power and enable high-fidelity quantum state tomography with moderate telescopes (50–100 cm) and photon budgets around 10^4–10^6, while MEO systems need much larger apertures or photon budgets of 10^7–10^9.
The results imply that aperture averaging should be a primary design lever for LEO links but is insufficient for MEO links, where adaptive optics or advanced signal processing would be needed to mitigate turbulence-induced fluctuations.
The fidelity curves as a function of zenith angle provide a practical scheduling constraint: quantum communication and tomography windows should be limited to zenith angles where the effective photon number stays above the required threshold.
The model offers a benchmark for comparing theoretical lower bounds of photon loss with experimental values, which tend to be higher, helping to identify missing loss factors in real satellite missions.
The distinction between ISI and PSI quantifies how much turbulence-induced variance can be suppressed by aperture size, offering a direct design trade-off between telescope diameter and transmitted photon number.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The paper's separation of losses into independent multiplicative factors suggests a modular design methodology: one could validate each term (atmosphere, diffraction, turbulence) separately with dedicated experiments before combining them, which is not stated but follows from the factorization.
The near-ineffectiveness of aperture averaging for MEO implies that for MEO quantum links, the turbulence term remains nearly at the ISI level, meaning that time-correlated intensity fluctuations could introduce correlated errors in quantum protocols — an extension the paper does not explicitly explore.
The QST use case could be extended to entanglement-based protocols: since the fidelity of a single-photon reconstruction degrades with photon loss, entangled-state distribution over MEO would likely face even stricter constraints, as the paper only addresses single-photon polarization states.
The Hufnagel–Valley profile with fixed parameters (v_rms = 26.25 m/s, C0 = 1.7×10^-14) is a single representative snapshot; a testable extension would be to run the same model under a range of C_n² profiles (e.g., different seasons or site altitudes) to map the spread in required photon budgets.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Read: the paper is a competent, tutorial-style link-budget study with a quantum tomography use case that makes for a useful reference scan; the headline claim — aperture averaging suppresses LEO turbulence noise but is nearly useless for MEO — is very likely an artifact of applying the Andrews aperture-averaging formula with the total slant range instead of the distance to the dominant turbulent layer.

What the paper does well: it carefully assembles the standard Beer-Lambert, Gaussian-diffraction, Rytov-scintillation, and log-normal framework, normalizes the turbulence factor correctly, and gives a rare clear comparison of the three aperture-averaging models (Andrews, Giggenbach, Yura). The parameterized LEO/MEO scan is genuinely handy for anyone planning a demonstrator, and the QST-as-diagnostic wrapper produces concrete photon budgets — roughly 10^4–10^6 photons for LEO, 10^7–10^9 for MEO. Those budgets are driven by geometric loss and are approximately right, so the QST feasibility conclusions mostly survive even if the turbulence part is wrong.

The soft spot is load-bearing. Eq. (18), Av_A(D) = [1 + 1.062(kD^2/4L)]^(-7/6), is a horizontal-path result, but the paper evaluates it with L equal to the full satellite range. For a downlink, the intensity correlation scale at the receiver is set by the distance to the turbulent layer, a few kilometers, not 20,200 km. The authors themselves introduce Giggenbach's L' ≈ 12 km (Eqs. 19–21), note it outperforms all-regime approximations, and then drop it for Andrews. Plugging L' into the same expression gives, for a 1 m aperture at MEO, Av ≈ 0.02 rather than ≈ 0.98 — i.e., strong aperture averaging, comparable to LEO. This also sits at odds with measured GEO downlink scintillation, which shows strong averaging on sub-meter apertures. So the central LEO/MEO contrast in Section III.A, and the design advice that follows, needs to be redone with the Giggenbach or Yura model the paper already cites.

Minor issues, in proportion: h0 is printed as 6600 km where the arithmetic needs 6.6 km; alpha0 is quoted for 800 nm while the simulations run at 1550 nm; the H_OGS-dependent term added to the Hufnagel-Valley profile is underived and shifts ground-level C_n^2 by roughly a factor of 1.7; and the QST section never states whether eta is sampled per measurement from the log-normal distribution or used as its mean. The overlap with the authors' own [25] and [11] is substantial — the new material is the scan itself, the model comparison, and the QST wrapper.

Recommendation: send it to a serious referee. It's a useful reference paper, but as it stands the aperture-averaging guidance is unsound, and that should be caught before anyone builds a budget on it. Major revision asked for: redo Section III.A with a downlink-appropriate formula, fix the typos, clarify the simulation, and moderate the novelty claims.

Referee Report

2 major / 6 minor

Summary. The paper develops a multiplicative transmittance model for satellite-to-ground optical links, combining internal receiver efficiency, atmospheric absorption/scattering along a slant path, diffraction loss, and turbulence-induced intensity fluctuations. Using the Hufnagel–Valley C_n^2 profile, the Rytov scintillation index, and the Andrews aperture-averaging factor, it simulates photon loss versus zenith angle for a LEO (420 km) and a MEO (20,200 km) satellite. It then feeds the resulting mean link transmittance into a quantum-state-tomography (QST) simulation with SIC-POVMs and Poisson shot noise, reporting average reconstruction fidelities for different apertures and photon budgets. The central claim is that aperture averaging is effective for LEO downlinks but nearly ineffective for MEO downlinks, so that MEO QST requires orders-of-magnitude larger photon budgets (about 10^7–10^9 photons) than LEO (about 10^4–10^6).

Significance. If correct, the numerical estimates would provide useful design guidance for satellite QKD/QST demonstrations, particularly for choosing receiver aperture and photon budgets as a function of orbital altitude and zenith angle. The paper's assembly of standard propagation formulas is mostly transparent, the parameter set in Table I is explicit, and the effective-pass-time analysis in Fig. 1 is a useful practical addition. The QST use case is also a sensible diagnostic framing. However, the central LEO-vs-MEO aperture-averaging contrast rests on a propagation-distance choice that the paper itself shows is questionable, and the QST simulations as written do not appear to include turbulent transmittance fluctuations. These issues are load-bearing and currently prevent the quantitative conclusions from being accepted.

major comments (2)

[Section III.A, Eq. (18)] The central LEO-vs-MEO contrast is an artifact of using the total slant range L in the Andrews aperture-averaging factor. Eq. (18) is a horizontal-path expression in which the Fresnel zone is evaluated at the total distance L; for a satellite downlink the intensity-correlation scale at the receiver is set by the dominant turbulent layers near the ground/tropopause, not by the satellite distance. The paper itself introduces the Giggenbach model in Eqs. (19)–(21), with L' ≈ 12 km at zenith, and states that it 'outperforms more general all-regime approximation functions', yet discards it for simplicity. Replacing L by L' changes the results dramatically: for D = 1 m, λ = 1550 nm and ζ = 0, Eq. (20) gives Av_G ≈ [1 + 1.062 × (4.05×10^6 m^{-1} × 1 m^2)/(9 × 1.2×10^4 m)]^{-7/6} ≈ 0.014, whereas Eq. (18) with L = 20,200 km gives Av_A ≈ 0.94. Thus a 1 m aperture would average MEO fluctuations al
[Section IV.A, Eqs. (37)–(38)] The QST simulations do not appear to include turbulence-induced intensity fluctuations. The effective photon number is set to eN = ηN with η from Eq. (1), which includes the stochastic factor I, but the simulation then samples m_k ~ Pois(n_k) with n_k determined by a single mean transmittance; no draw from the log-normal distribution in Eq. (36) is described. As written, Figs. 5 and 6 quantify only shot noise under a deterministic mean loss, not the effect of atmospheric turbulence on QST fidelity. Statements such as 'turbulence amplifies the poor SNR' in the MEO case are therefore unsubstantiated by the simulation. The authors should either sample the transmittance I (with σ_j^2 = σ_I^2 or σ_P^2) and rerun the QST, or explicitly restrict their claims to tomography under mean-link losses and remove the turbulence-related interpretation.

minor comments (6)

[Section II.B, Eq. (4)] The text states h0 = 6600 km, but the numerical value exp(−α0 h0) ≈ 0.9675 requires h0 ≈ 6.6 km. The 'km' unit is a typo; if h0 = 6600 km were used in Eq. (5), atmospheric extinction would be negligible. Please correct and ensure Eq. (5) uses the same scale height.
[Section II.B / Table I] The attenuation coefficient α0 = 5×10^-6 m^-1 is quoted for λ = 800 nm, but all simulations use λ = 1550 nm. Since γ(h) is wavelength-dependent, please either state the 1550 nm attenuation coefficient or justify using the 800 nm value in the link-budget calculations.
[Section II.D.2, Eq. (13)] The variable z is used both as the slant-path coordinate and, implicitly, as altitude in C_n^2(z). Please clarify the integration variable and its relation to height above sea level to avoid ambiguity in the path integral.
[Section IV.A, Eq. (38)] The notation 'eN = ηN' is used without specifying whether η is the random transmittance or its mean. Given the earlier definition of I, please define whether eN is the expected received photon number or an instantaneous one; this also relates to the major comment above.
[Conflict of Interest] The conflict-of-interest statement reads 'The author declares...' but the paper has three authors. Please change to 'The authors declare...'.
[References] References [55]–[57] appear not to be cited in the text; please add citations or remove them from the bibliography.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the link model is assembled from external standard formulas, and the QST curves are a simulated use case, not an independent validation of the link model.

full rationale

The quantitative inputs are fixed external parameter choices (H-V profile, Eq. 10, with v_rms = 26.25 m/s, C0 = 1.7e-14 m^-2/3, H_OGS = 65 m, λ = 1550 nm, η_int = 0.4) and standard published formulas: Beer-Lambert attenuation, Gaussian diffraction loss, Rytov scintillation, and the Andrews/Giggenbach/Yura aperture-averaging expressions. No parameter is fitted to the target quantities. The LEO/MEO contrast in Fig. 2 is a direct evaluation of the externally cited Andrews formula Eq. (18) with L set to the slant range; it is a computation from a chosen model, not a quantity defined in terms of the conclusion. The QST section is explicitly a simulated use case: eN = ηN is inserted into Poisson photon-count statistics and least-squares reconstruction, so the fidelity curves follow from the link model by construction, but the paper does not offer these simulations as independent verification of the link model and explicitly restricts validity to the parameters in Table I. The main scientific caveat—whether Eq. (18) should use total slant range rather than an effective turbulent-layer distance, as in Giggenbach's L' or Yura's h_s—is a modeling-validity question, not circularity, because the formula and its arguments are externally given and not chosen to force the LEO/MEO result. Self-citations [11,25] support the standard multiplicative transmittance decomposition jointly with external references such as [13]; they do not import a uniqueness theorem, forbid alternative models, or smuggle in an ansatz. Thus no circular step is present.

Assumptions & free parameters 7 free parameters · 7 assumptions · 0 invented entities

No new physical entities are introduced. The paper's inputs are standard atmospheric and turbulence parameters selected by hand from the literature rather than fitted to data; the load-bearing burden falls on the chosen parameter values, the H-V profile, and the weak-turbulence domain assumptions.

free parameters (7)

h0 (atmospheric scale height) = 6600 m (text mistakenly says 6600 km)
Chosen from Ref. [28] together with α0; enters Eq. (5) and the 0.143 dB zenith-loss estimate. The unit inconsistency affects quantitative results.
α0 (sea-level attenuation coefficient) = 5×10^-6 m^-1
Value quoted in Ref. [28] for λ=800 nm, but simulations are run at λ=1550 nm (Table I) with no wavelength correction.
η_int (internal receiver efficiency) = 0.4
Table I; chosen as a typical detector-system efficiency; shifts all loss curves by about 3.98 dB.
w0 (transmitter beam waist) = 1 cm
Table I; chosen for the simulation; controls diffraction loss through Eq. (8).
C0 (ground refractive-index structure parameter) = 1.7×10^-14 m^-2/3
Table I; standard H-V value, but site-dependent; directly affects σ_R² and the PSI.
v_rms (rms wind speed) = 26.25 m/s
Selected in Section II.D.1 as 'moderate wind'; scales the high-altitude term of C_n²(h).
H_OGS (ground station altitude) = 65 m
Table I; enters the H-V profile and the lower limit of the Rytov integral in Eq. (13).

assumptions (7)

domain assumption The four loss factors in Eq. (1) are statistically independent and multiply.
Invoked in Section II.A to justify η = η_int η_atm η_d I.
domain assumption Plane-parallel atmosphere and sec(ζ) scaling for atmospheric attenuation (Eq. 6).
Used to convert zenith transmittance into slant-path transmittance.
ad hoc to paper The Hufnagel-Valley C_n² profile, including the H_OGS-shifted ground term (Eq. 10), represents the turbulent atmosphere for LEO/MEO downlinks.
The H_OGS generalization is presented without derivation and is not standard in the cited HV references.
domain assumption Rytov weak-turbulence theory and the empirical saturation formula (Eqs. 13-14) remain applicable through zenith angle 80° and up to MEO distances.
The paper plots to ±80°, but weak-fluctuation validity degrades at large zenith angles, as acknowledged in the text.
domain assumption The Andrews aperture-averaging factor with ρ_I = sqrt(L/k) (Eq. 18) is valid for the studied regimes.
The paper selects Av_A over Giggenbach and Yura forms (Section II.D.3).
domain assumption Received intensity follows a log-normal distribution with E[I] = 1 (Eqs. 34-36).
Used to model turbulence-induced transmittance fluctuations in the weak-turbulence regime.
domain assumption Photon loss is polarization-independent and can be represented solely by the reduced mean photon number eN = ηN.
Implicit in Section IV.A; background counts, dark counts, and polarization-dependent losses are not modeled.

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Cite this review

Pith. "Pith review of Optical downlink modeling for low-earth-orbit and medium-earth-orbit satellites under atmospheric turbulence with a quantum-state-tomography use case." pith.science (2026). https://pith.science/paper/L4D73J4M

@misc{pith2026251213828,
  author       = {Pith},
  title        = {Pith review of: Optical downlink modeling for low-earth-orbit and medium-earth-orbit satellites under atmospheric turbulence with a quantum-state-tomography use case},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/L4D73J4M}},
  note         = {Machine review of arXiv:2512.13828}
}

read the original abstract

This paper presents a comprehensive analysis of the link budget for free-space optical systems involving Low Earth Orbit (LEO) and Medium Earth Orbit (MEO) satellites. We develop a detailed model of the satellite-to-ground channel that accounts for the primary physical processes affecting transmittance: atmospheric absorption and scattering, free-space diffraction, and turbulence-induced fluctuations. The study introduces a general method for computing transmittance along a slant path between a satellite and an optical ground station, incorporating zenith angle, slant range, and altitude-dependent attenuation. The proposed framework is intended to support the design and evaluation of space-based optical links and serves as a critical tool for defining technical specifications in satellite communication demonstrators and simulations. Numerical estimates are provided to illustrate the magnitude of losses under typical operational conditions, including the role of aperture averaging. In addition to the link budget analysis, we introduce a satellite-based quantum use case. We propose a scheme for quantum state tomography performed on states generated by an onboard photon source on an LEO or MEO satellite and transmitted to the optical ground station. This approach enables continuous verification of the quality of quantum resources that can be used to perform quantum protocols within quantum information networks.

Figures

Figures reproduced from arXiv: 2512.13828 by the authors.

**Figure 1.** FIG. 1. Comparison of total visibility time and effective time for optical communication as a function of satellite [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗

**Figure 2.** FIG. 2. Aperture averaging according to the Andrews’ model ( [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗

**Figure 3.** FIG. 3. Photon loss for an LEO satellite pass as a function of the zenith angle [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: FIG. 4. Photon loss for an MEO satellite pass as a function of the zenith angle [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]

**Figure 5.** Figure 5: FIG. 5. QST results for an LEO downlink (altitude of 420 km); each panel shows the reconstruction fidelity as a [PITH_FULL_IMAGE:figures/full_fig_p016_5.png]

**Figure 6.** Figure 6: FIG. 6. QST results for an MEO downlink (altitude of 20 200 km): each panel shows the reconstruction fidelity as [PITH_FULL_IMAGE:figures/full_fig_p017_6.png]

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Reference graph

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Reviewed August 4, 2026 · model on record in the stance chip above.

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Depending on the value ofC 2 n, turbulence is typically classified as weak, moderate, or strong

Refractive Index Structure Coefficient according to the Hufnagel-Valley (H-V) Model The strength of atmospheric turbulence is commonly characterized by the refractive index structure pa- rameterC 2 n (m−2/3), which directly influences the severity of scintillation effects. Depending on the value ofC 2 n, turbulence is typically classified as weak, moderat...

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Intensity Scintillation Index (ISI) The fluctuations in optical power,P, caused by atmospheric turbulence along an optical propagation path are characterized by the ISI, denoted byσ 2 I . It is defined as the normalized variance of the optical intensity [1, 6]: σ2 I = Var[P(t, p)] (E[P(t, p)])2 = ⟨P 2⟩ ⟨P⟩ 2 −1,(11) where⟨x⟩represents the expected value o...

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These power variations can lead to signal fading, affecting the reliability of optical communication links

Power Scintillation Index (PSI) The PSI,σ 2 P , characterizes the fluctuations in received optical power due to atmospheric turbulence by taking into account the properties of the optical equipment. These power variations can lead to signal fading, affecting the reliability of optical communication links. The relationship between the PSI,σ 2 P , and the I...

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Log-normal probability density function for turbulence modeling In this section, we discuss the use of the Log-normal (LN) probability distribution to model the effects of atmospheric turbulence on the intensity of the optical signal. Under the first-order Rytov approximation, the irradiance,P, can be given as [6]: P=A 2e2χ,(25) whereAis the amplitude of ...

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For small apertures, where aperture averaging has little effect,σ 2 j =σ 2 I (ISI-based model with Eq. (13))

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(15) and a specific framework for computing Av(D))

For larger apertures, where aperture averaging reduces the effects of scintillation,σ 2 j =σ 2 P (PSI-based model with Eq. (15) and a specific framework for computing Av(D)). The distribution Eq. (36) describes the relative intensity fluctuations in terms of a log-normal model, whereσ 2 j governs the width of the fluctuation. The transition from the origi...

2020