An Unwrapped Phase Distribution Model for Speckle/Turbulence

Several models have been proposed for the probability density function (p.d.f.) of phase for a speckle field created by a coherent beam and a diffuse object1-5 and also for a coherent beam propagating through a random medium such as clear air turbulence6-7. The p.d.f. models for speckle as well as many of the p.d.f. models for turbulence are given for the wrapped phase since the models are based on extracting phase information from the complex field amplitudes representing the radiation. Wrapped phase refers to phase angles that are limited to principal values (0-2π radians). The phase fluctuations really represent the optical path length variations which can greatly exceed 2π when represented as phase changes. Path length variations are linearly related to unwrapped phase variations by the relationship ϕ = 2πΔL/λ where ΔL represents path length variations. Consequently in a realistic situation phase is not limited to principal values. In addition the wrapped p.d.f. expressions are mathematically complicated and it would be difficult to extend them to the case where a speckle field and a turbulence field are combined (ie. where a speckle field is propagating through turbulence). Consequently simple Gaussian unwrapped phase p.d.f. models for both speckle alone and turbulence alone and a model for the combined speckle field in turbulence using the simpler unwrapped models are proposed. It will be shown using a standard technique for calculating a wrapped p.d.f. from an unwrapped p.d.f5 that the wrapped p.d.f.’s obtained from unwrapped Gaussian models are approximately equivalent to the known wrapped p.d.f.’s for speckle phase. It will also be shown from our experimental work that the measured unwrapped phase is approximately Gaussian.