On the Theory of Diffraction by an Aperture in an Infinite Plane Screen. I

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Abstract
The diffraction of a scalar plane wave by an aperture in an infinite plane screen is examined theoretically. The wave function at an arbitrary point in space is expressed in terms of its values in the aperture, and constructed so as to vanish on the screen, in accordance with the assumed boundary condition. An integral equation to determine the aperture field is obtained from the continuity requirement for the normal derivative of the wave function on traversing the plane of the aperture. Utilizing the integral equation (whose solution is generally unobtainable), the amplitude of the diffracted spherical wave at large distances from the aperture is exhibited in a form which is stationary with respect to small variations (relative to the correct values) of the aperture fields arising from a pair of incident waves. This expression is independent of the scale of the aperture fields. The transmission cross section of the aperture for a plane wave is found to be simply related to the diffracted amplitude observed in the direction of incidence. The variational formulation is applied in detail for a wave incident normally on a circular aperture. By comparison with the exact results available for this problem, it appears that the use of suitable trial aperture fields in the variational formulation yields approximate, yet accurate, expressions for the diffracted amplitude and transmission cross section over a wide range of frequencies.

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