Phase-factor spectra of turbulent phase screens

The optical phase ϕ is a key quantity in the physics of light propagating through a turbulent medium. In certain respects, however, the statistics of the phase factor, ψ=exp⁡(iϕ), are more relevant than the statistics of the phase itself. Here, we present a theoretical analysis of the 2D phase-factor spectrum F_ψ(κ) of a random phase screen. We apply the theory to four types of phase screens, each characterized by a power-law phase structure function, D_ϕ(r)=(r/r_c)^γ (where r_c is the phase coherence length defined by D_ϕ(r_c)=1rad²), and a probability density function p_α(α) of the phase increments for a given spatial lag. We analyze phase screens with turbulent (γ=5/3) and quadratic (γ=2) phase structure functions and with normally distributed (i.e., Gaussian) versus Laplacian phase increments. We find that there is a pronounced bump in each of the four phase-factor spectra F_ψ(κ). The precise location and shape of the bump are different for the four phase-screen types, but in each case it occurs at κ∼1/r_c. The bump is unrelated to the well-known "Hill bump" and is not caused by diffraction effects. It is solely a characteristic of the refractive-index statistics represented by the respective phase screen. We show that the second-order ψ statistics (covariance function, structure function, and spectrum) characterize a random phase screen more completely than the second-order ϕ counterparts.