Fraunhofer diffraction pattern of a random distribution of identical apertures in a plane screen

Abstract

A general expression is derived for the expectation value of the intensity in the Fraunhofer diffraction pattern of a plane screen of any shape containing any number of randomly positioned identical apertures. The special case of a rectangular screen is treated in more detail and the result is generalized for a screen of any shape. It is shown that the expectation value of the intensity has an intense narrow maximum in the centre, the shape of which resembles the Fraunhofer pattern of an aperture of the same size and shape as the screen. The effect of apertures overlapping is taken into account and it is shown that the effect can be corrected for by multiplying two terms in the expression for the expectation value of the intensity by different factors which depend on the number of apertures and the ratio of the area of one aperture to the area of the screen. Consideration is given to the failure of the scalar theory for apertures whose edges are separated by less than one wavelength and it is shown that in most cases the effects due to this cause are less serious than those due to overlapping except where the apertures are smaller than about twice the wavelength.