The concept of an angular spectrum of plane waves, and its relation to that of polar diagram and aperture distribution

A critical examination is made of the somewhat loose and incomplete statement that a polar diagram is the Fourier transform of an aperture distribution. By aperture distribution it is necessary to understand, in the two-dimensional case, distribution across the aperture of the component along the aperture plane of the electromagnetic field in the plane of propagation. Furthermore, the concept of the polar diagram has to be replaced by that of an angular spectrum, except in the common case when the aperture may be considered more or less limited in width, and the field is being evaluated at a point whose distance from the aperture is large compared with the width of the aperture (and the wavelength). For example, it is convenient for some purposes to regard the problem of diffraction of a plane wave by a semi-infinite plane screen, with a straight edge, as a problem about an aperture distribution in the plane of the screen. This is a case for which the concept of a polar diagram is not in general applicable, and has to be replaced by that of an angular spectrum. The field at all points in front of a plane aperture of any distribution may be regarded as arising from an aggregate of plane waves travelling in various directions. The amplitude and phase of the waves, as a function of their direction of travel, constitutes an angular spectrum, and this angular spectrum, appropriately expressed, is, without approximation, the Fourier transform of the aperture distribution. If the aperture distribution is of such a nature that the concept of the polar diagram is applicable at sufficiently great distances, then the polar diagram is equal to the angular spectrum. But the angular spectrum is a concept that is always applicable, whereas the polar diagram is one that is liable to be invalid (for example, in the Sommerfeld theory of propagation over a plane, imperfectly reflecting earth).