Rayleigh’s Integral in the Near Fresnel Region

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Abstract
An approximation, called the RF approximation, is derived for evaluating Rayleigh’s integral in the near-Fresnel region for cases in which an arbitrarily inclined, monochromatic wavefront is incident upon a finite aperture having simply periodic, complex transmittance. This approximation is identical with the result obtained by integrating Rayleigh’s integral over infinite, simply periodic apertures. The RF approximation corresponds physically to a set of plane waves emanating from the aperture together with a set of surface waves bound to the aperture. The coefficients of these wave forms are found to be the product of the Fourier coefficients of the complex transmittance of the aperture and of exponential frequency responses that are easily computed. The character of the plane waves and surface waves is consistent with that known from more elementary theory and experiment.

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