Beam Wander
RMS turbulence-induced wander of a collimated Gaussian beam, its angular jitter and the mean on-axis loss versus Cn², path length and beam radius.
Wander variance ⟨r_c²⟩
8.73e-5 m²
F = 0.979 at L/z_R = 1.23
RMS wander √⟨r_c²⟩
0.934 cm
Angular RMS wander
9.34 µrad
√⟨r_c²⟩ / L
Beam radius W(L)
3.18 cm
free-space Gaussian spread at the receiver
Mean on-axis loss
0.693 dB
10·log10(1 + 2⟨r_c²⟩/W²)
Rytov variance σ_R²
0.199
plane wave
Fried parameter r₀
7.85 cm
2W₀ = 4.00 cm
RMS wander vs Cn²
Wander and beam radius vs distance
Model
⟨r_c²⟩ = 2.42 Cn² L³ W₀^(−1/3) · F, F = 3∫₀¹ ξ² (1 + ((1 − ξ)L/z_R)²)^(−1/6) dξ, loss = 10·log10(1 + 2⟨r_c²⟩/W²)
Collimated beam, Kolmogorov spectrum with infinite outer scale (Andrews & Phillips, Laser Beam Propagation through Random Media, 2nd ed., 2005). Eddies larger than the beam at each point z deflect it, with lever arm ξ = 1 − z/L to the receiver. Their 2.42 Cn² L³ W₀^(−1/3) takes the beam radius as W₀ all along the path; keeping its diffraction, W(z) = W₀√(1 + (z/z_R)²) with z_R = πW₀²/λ, multiplies it by F (our derivation; the same integral as Berman, Chumak & Gorshkov, Phys. Rev. E 76, 056606, 2007). F ≈ 1 within a Rayleigh range and falls as 2.025 (L/z_R)^(−1/3) far beyond it, where the wander grows as L^(8/3). A finite outer scale reduces the wander, so the value is an upper bound.
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
Weak fluctuations (σ_R² < 1, 2W₀ < r₀), collimated Gaussian beam, constant Cn², infinite outer scale: ⟨r_c²⟩ = 2.42 Cn²L³W₀^(−1/3)·F, where F (our derivation) keeps the beam's diffraction along the path: F = 0.98 at L = 1.2 z_R, 0.74 at 12 z_R. Mean on-axis loss 10 log(1 + 2⟨r_c²⟩/W²), W the diffracted beam radius.
- Churnside J. H., Lataitis R. J. (1990). Wander of an optical beam in the turbulent atmosphere. Appl. Opt. 29(7), 926. doi:10.1364/ao.29.000926
- Andrews L. C., Phillips R. L. (2005). Laser Beam Propagation through Random Media, 2nd ed. SPIE. Beam wander of a Gaussian beam, ⟨r_c²⟩ = 2.42 C_n² L³ W₀^(−1/3) for a collimated beam.
- Berman G. P., Chumak A. A., Gorshkov V. N. (2007). Beam wandering in the atmosphere: the effect of partial coherence. Phys. Rev. E 76, 056606. Eqs. (12)–(14): the wander integral with the diffracting beam radius and its L^(8/3) far-field limit. doi:10.1103/PhysRevE.76.056606