On approximations of Helmholtz equation in the halfspace: Their relevance to inverse diffraction

The solution of Helmholtz equation with Dirichlet boundary conditions (BC) having compact support can be approximated along two basic paths, i.e., in the spatial frequency (reciprocal) domain and in the spatial (direct) one. The former leads to the paraxial equation, the latter to the Fresnel transform. We show that these results coincide. The paraxial approximation requires the Fourier transform (FT) of the boundary condition to have an adequately bounded support. This constraint applies therefore to Fresnel's formula and to the related ‘Fourier optics’ approximation. We discuss, by an example, the sensitivity of approximated solutions with respect to the BC type. Asymptotic expansions for the diffracted field are often used to solve diffraction and scattering problems. The above mentioned constraints must be taken into account when the BC must be identified starting from e.g. far-field data: although e.g. Fresnel's formula does not suggest any restriction on the BC's FT support, it will yield physically meaningful results only within the prescribed domain. The connection between physical constraints and regularisation of the inverse problem is also briefly discussed