Resolution limits in inverse source problem for strip currents not in Fresnel zone.

Abstract

This paper deals with the classical question of estimating achievable resolution in terms of configuration parameters in inverse source problems. In particular, the study is developed for two-dimensional prototype geometry, where a strip source (magnetic or electric) is to be reconstructed from its radiated field observed over a bounded rectilinear domain parallel to the source. Resolution formulas are well known when the field is collected in the far field or in the Fresnel zone of the source. Here, the plan is to expand those results by removing the geometrical limitations due to the far field or Fresnel approximations. To this end, the involved radiation operators are recast as Fourier-type integral operators upon introducing suitable variable transformations. For magnetic sources, this allows one to find a closed-form approximation of the singular system and hence to estimate achievable resolution, the latter given as the main beam width of the point-spread function. Unfortunately, this does not happen for electric currents. In this case, the radiation operator is inverted by a weighted adjoint inversion method (a back-propagation-like method) that directly allows one to find an analytical expression of the point-spread function and hence of the resolution. The derived resolution formulas are the same for magnetic and electric currents; they clearly point out the role of geometrical parameters and coincide with the one pertaining to the Fresnel zone when the geometry verifies the Fresnel approximation. A few numerical examples are also enclosed to check the theory.