FSO Geometric Loss
Geometric spreading loss of a divergent FSO beam: top-hat footprint from exit aperture and divergence versus the exact Gaussian-beam capture by the receiver aperture.
Inputs
Top-hat loss
20.04 dB
beam 1.00 m wide, 0.990 % caught
Gaussian loss
17.03 dB
w = 0.500 m, 1.98 % caught
Gaussian waist w₀ = max(d_t/2, λ/(πθ/2))
2.50 mm
Top-hat − Gaussian
3.01 dB
→ 3.01 dB only for θR ≫ d_t, small receiver
Model
Top-hat: L = 20 log₁₀((d_t + θR)/d_r) ≥ 0 Gaussian: η = 1 − exp(−2a²/w(R)²), w(R)² = w₀² + (θR/2)²
The top-hat form (Sheikh Muhammad et al. 2005, eq. 1) spreads the power evenly over a disc that starts at the exit aperture d_t and grows with the full divergence θ; once that disc is no wider than the receiver, nothing is lost. A real laser beam is closer to Gaussian. Here the Gaussian leaves the exit aperture with w₀ = max(d_t/2, λ/(πθ/2)) (an embedded Gaussian, M² ≥ 1) and its 1/e² radius grows as w(R)² = w₀² + (θR/2)², so the far-field half-angle is θ/2. The receiver of radius a catches 1 − exp(−2a²/w²). The Gaussian peak is twice the level of a top-hat that spreads all the power over the 1/e² disc, so the top-hat − Gaussian difference is 3.01 dB only when θR ≫ d_t and the receiver is small.
Perfect pointing, no turbulence. Pointing errors are in the pointing-loss calculator.
Model and references
Textbook approximation. A standard approximation from the literature (for example paraxial, plane-wave or Gaussian-beam), valid within the range the page states. About the tiers
Top-hat d_t + θR footprint (≥ 0 dB) beside the exact capture 1 − exp(−2a²/w²) of a Gaussian beam with half-angle θ/2 leaving the exit aperture (w² = w₀² + (θR/2)², w₀ = max(d_t/2, λ/(πθ/2))), 3.01 dB better for a small receiver once θR ≫ d_t. Perfect pointing, no turbulence.
- Sheikh Muhammad S., Köhldorfer P., Leitgeb E. (2005). Channel modeling for terrestrial free space optical links. Proc. ICTON 2005, Tu.B3.5, 407–410. Eq. (1): P_geo = 10 log₁₀(d_r/(d_t + lθ))². doi:10.1109/ICTON.2005.1505832
- Kim I. I. et al. (1998). Wireless optical transmission of fast ethernet, FDDI, ATM, and ESCON protocol data using the TerraLink laser communication system. Opt. Eng. 37(12), 3143. Eq. (1): the area form SA_R/(SA_T + π(θR)²/4), θ full angle. doi:10.1117/1.601981
- Degnan J. J., Klein B. J. (1974). Optical antenna gain. 2: Receiving antennas. Appl. Opt. 13(10), 2397. doi:10.1364/ao.13.002397
- Siegman A. E. (1990). New developments in laser resonators. Proc. SPIE 1224, 2. Embedded Gaussian of a beam with M² ≥ 1. doi:10.1117/12.18425