FSO Fade Probability
Gamma-gamma and log-normal fade probability versus fade threshold for an FSO link in turbulence, with aperture averaging and N uncorrelated receivers.
Rytov variance σ_R²
0.0199
Scintillation index σ_I²(D)
0.0199
Gamma-gamma α, β
103, 98.5
P(fade > 10.0 dB), gamma-gamma
8.73e-43
primary result, weak to strong
P(fade), log-normal
3.73e-60
log-normal (weak-turbulence comparison)
Fade probability vs threshold
Model
I = XY, α = 1/(e^σ²lnX − 1), β = 1/(e^σ²lnY − 1)
Gamma-gamma irradiance with α and β set by the aperture-averaged large- and small-scale log variances (Al-Habash, Andrews & Phillips, Opt. Eng. 40, 1554, 2001). It is valid from weak to strong turbulence. The log-normal curve is the weak-turbulence comparison and underestimates deep fades in moderate to strong turbulence.
P(I < t⟨I⟩) with t = 10^(−F/10) for a fade of F dB below the mean. For N receivers whose separation is far larger than both √(λL) and L/(kρ₀) (the scattering-disk size that sets the correlation width in strong turbulence; ρ₀ is the spatial coherence radius, k = 2π/λ), the fades are independent and selection combining fails only when all fail: P^N.
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
Gamma-gamma irradiance with α, β from the aperture-averaged plane-wave log variances (weak to strong turbulence); log-normal for comparison. P^N assumes N uncorrelated receivers with selection combining.
- Al-Habash M. A., Andrews L. C., Phillips R. L. (2001). Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media. Opt. Eng. 40(8), 1554–1562. doi:10.1117/1.1386641
- Andrews L. C., Phillips R. L. (2005). Laser Beam Propagation through Random Media, 2nd ed. SPIE. Gamma-gamma model with aperture averaging.