Progressively Attenuated Multi-Branch Reception for Inter-HAPS Optical Links

Give each aperture a different operating range

The receiver is a spatial array of separate, nonoverlapping circular collection apertures. Each aperture feeds its own attenuator and identical photodetector/front end, followed by DC blocking. The configurations studied contain one to five apertures: a centered aperture, a pair, a three-aperture line, a four-aperture square, and a center with four surrounding apertures.

Branch 1 has unit transmission. Each subsequent branch has progressively stronger fixed attenuation, following αₖ = ρᵏ⁻¹ for 0 < ρ < 1. Strongly attenuated branches provide headroom when optical loading is high; lightly attenuated branches preserve sensitivity as received power falls. The architecture uses spatial collection rather than splitting one collected beam, and does not require continuously adjusted attenuation or detector-mode switching.

Manuscript · Sections I–II; Fig. 1; Eqs. (1)–(9)

Select the best branch that remains linear

Saturation depends on the peak optical load reaching the photodetector or analog front end, including desired DC bias, modulation and background light. Removing DC after photodetection does not undo detector saturation, and the background shot-noise fluctuations remain after DC blocking.

For every common two-dimensional pointing realization, the receiver first excludes saturated branches and then selects the remaining branch with the greatest instantaneous electrical SNR. Spatial coupling differs between apertures, so selecting by attenuation coefficient alone is insufficient. Operation is valid only when the selected branch also meets the minimum SNR; otherwise the model distinguishes all-branches-saturated outage from insufficient-SNR outage.

Manuscript · Sections II-B–III; Eqs. (10)–(16)

Keep the shared pointing error in the outage model

A Gaussian beam, finite-aperture coupling and Gaussian angular jitter determine each aperture’s collected signal. A usable branch must avoid a high-coupling saturation region while remaining within a sufficiently strong-coupling SNR region. The paper derives closed-form marginal saturation, insufficient-SNR, usability and outage probabilities using the Rician displacement distribution and Marcum-Q function.

Every aperture experiences the same random beam-center displacement, so its pointing gain is correlated with the other branches. System outage is therefore evaluated over the joint two-dimensional pointing plane. Multiplying individual branch outage probabilities would lose this dependence; the full-system expression is a joint probability integral, rather than the branch-level closed form.

Manuscript · Section III-A–C; Eqs. (17)–(30)

What the numerical comparisons show

The numerical baseline uses 1550 nm light, a 0.021 m beam waist, 4 W DC bias, a 0.2 W signal scale, optical efficiency 0.5, 0.10 m aperture radius and 0.45 m center spacing. Detector responsivity is 0.85 A/W, background power 50 nW, electrical noise bandwidth 10 GHz, per-axis pointing-jitter standard deviation 10 µrad, and minimum SNR 25 dB. Figures 2–3 use a 5 mA saturation threshold; Monte Carlo comparisons use 200000 common pointing realizations.

In the reported distance comparison at 50 km, total outage falls from approximately 0.92 with one aperture to 0.11 with five branches. At 100 km, the reported total outage is approximately 1.46 × 10⁻⁵, 1.27 × 10⁻⁵ and 9.53 × 10⁻⁶ for three, four and five branches respectively. More branches shift the minimum-outage region toward shorter separation; at longer distances, reduced coupling increasingly makes SNR the limiting factor.

For the four-aperture square, reducing the progression factor strengthens attenuation in the later branches and moves the saturation-to-reliable-operation transition toward shorter distances. The distance–attenuation sweep reports a minimum outage of approximately 3.7 × 10⁻⁶ under the modeled conditions.

A separate branch-usability comparison uses four branches, ρ = 0.6 and a tighter 2 mA saturation threshold. At 50 km, usability rises from about 0.15 for Branch 1 to 0.89 for Branch 4. At 300 km, the ordering reverses: the reported usable probabilities are approximately 0.953, 0.933, 0.905 and 0.865 for Branches 1–4. Analytical and Monte Carlo branch-usability curves agree with maximum absolute error below 1.9 × 10⁻³.

Manuscript · Section IV; Figs. 2–4

Engineering relevance and model boundaries

The contribution is a low-complexity receiver strategy for widening optical dynamic range across changing inter-HAPS distances. The analysis and simulation identify complementary operating regions for fixed attenuation levels; they do not establish a universal attenuation setting or a measured communications-throughput gain.

The adopted clear-air stratospheric model omits turbulence-induced scintillation and uses a background-shot-noise-limited baseline. Thermal noise, dark-current noise, desired-signal shot noise, relative-intensity noise, quantization noise and amplifier noise are excluded. The reported results are numerical, without a demonstrated airborne receiver or field deployment.

Manuscript · Sections II-A–B and V