Bessel functions come into wave optics because many optical elements – lenses, apertures, mirrors – are circular. We have met Bessel functions in several places (§8.3.4, §8.7, §12.2, §12.6.4 for example), although since most students are not very familiar with them (and probably becoming less so with the ubiquity of computers) we have restricted our use of them as far as possible. The one unavoidable meeting is the Fraunhofer diffraction pattern of a circular aperture, the Airy pattern, which is the diffraction-limited point spread function of an aberration-free optical system (§12.2). Another topic that involves the use of Bessel functions is the Fourier analysis of phase functions, in which the function being transformed contains the phase in an exponent. We met such a situation when we studied the acousto-optic effect, where a sinusoidal pressure wave affects directly the phase of the optical transmission function.
In this appendix we simply intend to acquaint the reader with the results that are necessary for elementary wave optics. The proofs can be found in the treatise by Watson (1958) and other places.