DOI link for Three-dimensional Waves
Three-dimensional Waves book
This chapter examines the focusing properties of three-dimensional scalar waves, and begins by comparing various approximate theories of focusing. When the Kirchhoff approximation is combined with the Kirchhoff integral or either of the Rayleigh-Sommerfeld integrals, we obtain the diffraction formulae. The so-called Debye integral representation for focused wave fields (Debye 1909) is obtained by evaluating the angular-spectrum integral asymptotically and retaining only the contribution of the interior stationary point. The classical result for the three-dimensional field distribution around a focus is due to Lommel (1885). He considered the diffraction of a converging spherical wave by a circular aperture, and was able to express the field in terms of a convergent series of Bessel functions, now known as Lommel functions. In the Kirchhoff approximation the angular spectrum is given by the Fourier transform of the unperturbed incident field inside the aperture. To demonstrate the reciprocity principles, the authors consider the focusing of a perfect wave.