Rayleigh-Sommerfeld and Helmholtz-Kirchhoff integrals: application to the scalar and vectorial theory of wave propagation and diffraction

Approximations for known integral solutions of the scalar Helmholtz equation (the Rayleigh-Sommerfeld (RS) integrals of type I and II) are discussed. The validity range of the approximated RS I integral for large, but finite distances from the boundary plane is calculated as a new result. The scalar Kirchhoff boundary conditions leading to a rough approximation of an aperture diffraction problem are inspected, and a new spatial frequency solution in the aperture plane is given. Published measurements and known approximate vectorial boundary values formulated with the Hertz vector are compared with the outcome of the traditional Kirchhoff theory. For the field propagating from light waveguide endfaces into a homogeneous medium, the polarization effects in the far-field are considered by approximating the near-field for weakly guiding fibers with a Hertz vector ansatz. Comparison with measurements shows that the polarization induced asymmetry of the far-field is small in comparison with the experimental uncertainties. Finally, the proper applications for the RS I; II integrals are clarified as opposed to using the scalar or vectorial Helmholtz-Kirchhoff integral