The classical theory of diffraction, according to which the distribution of light at the focal plane of a lens is the Fourier transform of the distribution at its entrance pupil, is applicable to lenses of moderate numerical aperture (NA). The incident beam, of course, must be monochromatic and coherent, but its polarization state is irrelevant since the classical theory is a scalar theory (see Chapter 2, “Fourier optics”). If the incident beam happens to be a plane wave and the lens is free from aberrations then the focused spot will have the well-known Airy pattern. When the incident beam is Gaussian the focused spot will also be Gaussian, since this particular profile is preserved under Fourier transformation. In general, arbitrary distributions of the incident beam, with or without aberrations and defocus, can be transformed numerically, using the fast Fourier transform (FFT) algorithm, to yield the distribution in the vicinity of the focus.

There are two basic reasons for the applicability of the classical scalar theory to systems of moderate NA. The first is that bending of the rays by the focusing element(s) is fairly small, causing the electromagnetic field vectors (E and B) before and after the lens to have more or less the same orientations. A scalar amplitude assigned to each point on the emergent wavefront from a system having low to moderate values of NA is sufficient to describe its electromagnetic state, whereas in the high-NA regime one can no longer ignore the vectorial nature of light.