Adaptive Optics for FSO
Fried parameter r₀, Greenwood frequency, isoplanatic angle and Strehl ratio with and without adaptive optics for a free-space optical link.
Fried parameter r₀
7.85 cm
D/r₀ = 1.27
Greenwood frequency f_G
54.4 Hz
0.427 v/r₀
Isoplanatic angle θ₀
44.4 µrad
Strehl (no AO)
0.514
σ² = 1.03 (D/r₀)^5/3 = 1.54 rad²
Strehl (with AO)
0.981
pitch 7.86 mm; σ²_fit = 6.05e-3, σ²_servo = 0.0248 rad²
AO improvement
1.91×
Rytov variance σ_R²
0.199
Fried parameter vs Cn²
Model
r₀ = (0.423 k² Cn² L)^(−3/5), θ₀ = (2.91 · 3/8 · k² Cn² L^(8/3))^(−3/5), f_G = 0.427 v/r₀
d = D√(π/4N), σ²_fit = 0.28 (d/r₀)^(5/3), σ²_servo = (f_G/f₃dB)^(5/3)
S ≈ e^(−σ²) + (1 − e^(−σ²)) / (1 + (D/r₀)²)
Plane-wave r₀ and θ₀ for a horizontal path with constant Cn² (Andrews & Phillips, Laser Beam Propagation through Random Media, 2nd ed., 2005). The uncorrected variance 1.03 (D/r₀)^(5/3) is Noll's (JOSA 66, 207, 1976). The N actuators of a continuous-facesheet mirror fill the pupil on a square grid of pitch d, and its fitting error is 0.28 (d/r₀)^(5/3) (Hudgin, JOSA 67, 393, 1977; Hardy, Adaptive Optics for Astronomical Telescopes, 1998). κ = 0.28 is for pyramid-shaped influence functions; Gaussian ones give ≈ 0.23 (Tyson's table in Berdeu et al., A&A 674, A112, 2023). The zonal law sits about 10 % below Noll's residual after the first N Zernike modes and needs many actuators (N ≥ 10 here); a hexagonal grid has a 7.5 % larger pitch. The servo-lag variance of a first-order loop depends only on f_G/f₃dB (Greenwood, JOSA 67, 390, 1977). The Strehl ratio adds the seeing halo to the Maréchal core (Parenti & Sasiela, JOSA A 11, 288, 1994), so it tends to (r₀/D)² without correction instead of underflowing.
Not modelled: wavefront-sensor noise, anisoplanatism, scintillation (which degrades wavefront sensing once σ_R² ≳ 0.5) and a turbulence profile other than a uniform path with one wind speed.
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
One uniform turbulent layer with one wind speed; the actuators fill the pupil on a square grid and set the fitting error of a continuous-facesheet mirror; the servo residual is capped at the open-loop variance. Not modelled: wavefront-sensor noise, anisoplanatism and scintillation.
- Andrews L. C., Phillips R. L. (2005). Laser Beam Propagation through Random Media, 2nd ed. SPIE. Plane-wave Fried parameter and isoplanatic angle for constant Cn².
- Noll R. J. (1976). Zernike polynomials and atmospheric turbulence. J. Opt. Soc. Am. 66, 207. Uncorrected phase variance. doi:10.1364/josa.66.000207
- Hudgin R. H. (1977). Wave-front compensation error due to finite corrector-element size. J. Opt. Soc. Am. 67, 393–395. Fitting error 0.28 (d/r₀)^(5/3) for a continuous mirror of actuator pitch d. doi:10.1364/josa.67.000393
- Greenwood D. P. (1977). Bandwidth specification for adaptive optics systems. J. Opt. Soc. Am. 67, 390. Greenwood frequency and servo-lag variance of a first-order loop. doi:10.1364/josa.67.000390
- Parenti R. R., Sasiela R. J. (1994). Laser-guide-star systems for astronomical applications. J. Opt. Soc. Am. A 11, 288. Semi-empirical Strehl ratio with the seeing halo. doi:10.1364/josaa.11.000288