On the exact Kirchhoff and Rayleigh-Sommerfeld theories for the focusing of an infinite scalar spherical wave-field

The first and second Rayleigh-Sommerfeld diffraction theories, together with the Kirchhoff diffraction theory, are widely used in the formulation of optical diffraction problems [ 1 ]. They arise in the context of the half-space boundary-value problem of scalar wave theory; all sources of the optical field are assumed to be located in the Z~< 0 half-space, and the Z > 0 half-space is taken to be free space. The field in the positive halfspace is to be determined, given some boundary-condition on the Z = 0 plane [2 ]. If the field on the Z = 0 plane is specified, then the exact solution to the boundary-value problem is given by the Rayleigh-Sommerfeld first diffraction integral. Alternatively, if the normal derivative of the field on the Z = 0 plane is specified, then the exact solution is given by the Rayleigh-Sommerfeld second diffraction integral. The Kirchhoff diffraction integral requires that both the field and its normal derivative be specified on the Z = 0 plane. All three diffraction integrals give the exact solution for the diffracted field, provided the correct boundary-conditions are used [ 1 ]. In practice, the correct boundary-conditions are rarely known. When applying diffraction theory to a physical problem, one often uses geometrical optics to obtain approximate boundary-conditions (this is the well-known physical-optics or Kirchhoff approximation). Because th e boundary-conditions are obtained from a field which is not the correct solution to the problem, the three diffraction integrals lead to different results. Stamnes [ 1 ] refers to the combination of the exact diffraction integrals with boundary conditions obtained using the physical-optics approximation, as the Kirchhoffand the first and second Rayleigh-Sommerfeld diffraction theories. In effect, the three diffraction theories give the exact solutions to three different boundary value problems, all of which approximate the same physical problem. Recently, the exact solution was obtained for the Rayleigh-Sommerfeld first diffraction theory describing the focusing of a converging monochromatic spherical incident wave in an infinite-aperture system [ 3 ]. In this communication, the same physical problem is solved using the Rayleigh-Sommerfeld second diffraction theory, and the Kirchhoffdiffraction theory. The exact analytic solutions are very simply related, and provide the only example to date, of a focusing problem which can be solved exactly using all three diffraction theories. A direct analytic comparison of the results of all three theories over the entire positive half-space, has not been possible in the past. It should be noted that the analysis presented here is based on scalar theory, and taken alone, cannot adequately describe the focusing of electromagnetic waves. The theory is strictly valid when applied to true scalar