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By the use of phase perturbation theory we show that if a single realizationof a one-dimensional randomly rough interface between two dielectric media isilluminated at normal incidence from either medium by a broadband Gaussianbeam, it produces a scattered field whose differential reflection coefficientclosely matches the result produced by averaging the differential reflectioncoefficient produced by a monochromatic incident beam over the ensemble ofrealizations of the interface profile function.

Replacement of ensemble averaging by the use of a broadband source in  scattering of light from a one-dimensional randomly rough interface between  two dielectric media