Friedparameter

The Fried parameter, usually denoted r0, is a length scale that characterizes the size of a coherent patch of a distorted wavefront caused by atmospheric turbulence. Introduced by David Fried in 1966, it serves as a measure of how much of an optical aperture can work coherently at a given wavelength. Physically, r0 is approximately the diameter over which the RMS phase aberration due to turbulence is about one radian.

Mathematically, r0 is defined by r0 = [0.423 k^2 ∫_0^∞ Cn^2(z) dz]^{-3/5}, where k = 2π/λ is the optical wavenumber and Cn^2(z) is the refractive-index structure constant of the atmosphere integrated along the line of sight. The integral reflects the strength and distribution of turbulence with altitude. In practice, r0 increases with wavelength as r0 ∝ λ^{6/5} and decreases with stronger turbulence or higher optical turbulence concentration.

Implications for imaging and telescope design: if a telescope’s diameter D is smaller than or comparable to r0, diffraction-limited performance is attainable at that wavelength; if D greatly exceeds r0, atmospheric turbulence limits the resolution. The approximate number of deformable-mirror actuators needed across the pupil scales like D/r0, guiding adaptive optics system design. The Fried parameter also relates to the atmospheric coherence time τ0 ≈ 0.31 r0 / v̄, with v̄ the effective wind speed, which sets how rapidly the wavefront changes.

Applications and context: r0 is a central concept in site testing, telescope design, and adaptive optics, providing a bridge between atmospheric statistics and achievable angular resolution. Typical r0 values at visible wavelengths on good sites range from a few tens of centimeters to a few decimeters, increasing at longer wavelengths.