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When an optical imaging system is treated as a linear system, for incoherently illuminated objects, a statistically homogeneous, isotropic, random medium placed between the object and the imaging system multiplies the free-space transfer function of the optical system by a correction factor. If a medium having a Gaussian-correlated refractive index is assumed, then the correction factor for an on-axis quasimonochromatic point source has the form a+b erf x/x for propagation distances that are in the near field of the correlation interval of the medium. This form is found by using the first Born approximation to the two coupled wave equations satisfied by a mutual coherence function. The ensemble-averaged effect of imaging through a random medium is that the medium acts as a low-pass filter, decreasing the resolution of the imaging system.

By assuming the random, time-varying boundary conditions for the scalar wave equation to be ergodic, we can associate with the time-varying boundary conditions an ensemble of strictly monochromatic boundary conditions. Formally solving and comparing the solutions for each type of boundary condition, we conclude that the time-averaged and ensemble-averaged powers (squares of the field) are the same at all points where the path difference to any two points on the boundary is small compared to c/Δν, where c is the free-space speed of light and Δν is the frequency spread of the time-varying boundary conditions. That is, if the boundary conditions are ergodic, the solutions are ergodic.

By assuming the random, time-varying boundary conditions for the scalar wave equation to be ergodic, we can associate with the time-varying boundary conditions an ensemble of strictly monochromatic boundary conditions. Formally solving and comparing the solutions for each type of boundary condition, we conclude that the time-averaged and ensemble-averaged powers (squares of the field) are the same at all points where the path difference to any two points on the boundary is small compared to c/Δv, where c is the free-space speed of light and Δv is the frequency spread of the time-varying boundary conditions. That is, if the boundary conditions are ergodic, the solutions are ergodic.

When an optical imaging system is treated as a linear system, for incoherently illuminated objects, a statistically homogeneous, isotropic, random medium placed between the object and the imaging system multiplies the free-space transfer function of the optical system by a correction factor. If a medium having a Gaussian-correlated refractive index is assumed, then the correction factor for an on-axis quasimonochromatic point source has the form a+b erf x/x for propagation distances that are in the near field of the correlation interval of the medium. This form is found by using the first Born approximation to the two coupled wave equations satisfied by a mutual coherence function. The ensemble-averaged effect of imaging through a random medium is that the medium acts as a low-pass filter, decreasing the resolution of the imaging system.