A spectral inversion scheme in cylindrical coordinates, applying the moment method procedure is suggested to reconstruct permittivity profiles of inhomogeneous dielectric objects. Angular spectral domain reconstruction is shown to be less sensitive to the ill-posedness due to the noise in the measured scattered field than the configuration domain reconstruction.
Scattered field of inhomogeneous dielectric objects may be obtained from the moment method, by discretizing an inhomogeneous dielectric scatterer into small cells much smaller than one wave length. The size of the cell is small enough that the polarization current (or relative dielectric constant multiplied by the total electric field) inside each cell is taken as a constnat and the integral for the scattered field may be obtained by the summation over the total number of cells
Moment method used in the direct scattering problem may be applied in reverse sequence to obtain the polarization currents up to the numerical accuracy inherent in the moment method. From the polarization currents, one may obtain the total field inside the scatterer and finally the distribution of dielectric constants. This inverse scattering scheme using the moment method does not require plane-wave illumination nor the first order Born or Rytov approximation, it may be used even in the case of strong scatterers. While this inversion scheme provides superresolution in the reconstruction of complex permittivity distributions, its reconstructed profile suffer from the extremely large error even if very small error or noise in the measured fields. This difficulty is the ill-posedness inherent in the inverse source problems.
Previous works of the inverse scattering using the moment method is twofold; One in the configuration domain and the other in the spectral domain. The merit of spectral inversion scheme is that the contributions from the measurement location, geometry of objects, and types of basis function are separated explicitly by its formulation. Furthermore, the ill-posed characteristics in the spectral scheme may be clearly explained by the exponential decaying behavior of high spectral components of the scattered field.
Angular spectral inversion scheme is formulated and shown to give less sensitive ill-posedness compared with that of the configuration domain and spectral domain. This formulation gives explicitly various contributions which help us to diagnose the actual mechanism of the ill-posedness to be the evanescent behavior of the angular spectrum of the higher spectral modes other than the propagation modes. Angular spectral formulation shows that the number of propagation modes are determined from the size of scatterer. Another characteristics of angular spectral scheme is that the scattering fields out of scatter can be recovered effectively, using finite number of angular spectra.
By performing numerical simulations for various types of dielectric objects when the noise contaminates the scattered field, it is demonstrated that the angular spectral inversion scheme is more compatible with noisy situation than the configuration domain approach. When the number of discretized homogeneous scatterers is less then the number of propagating angular spectral modes, it is demonstrated that ill-posed characteristics does not appear. In the case of hollow cylindrical scatterer, the ill-posed characterestics may be reduced significantly by filtering the higher order non propagating spectral modes.
