Inverse scattering problem in Hilbert space

Inverse scattering problems are considered for inhomogeneous dielectric structures with absorption. The retrieval of the complex permittivity is based on the solution of 3D complex-valued integral equations that are reduced to convolution equations with respect to lateral co-ordinates using the plane wave decomposition of Green functions. Then, the inverse problem is reduced to one-dimensional integral equation of the 1-st kind relative to the depth profile of the lateral spectrum of complex permittivity. To solve this ill-posed problem for each pair of spectrum components, a regularization method based on the Tikhonov´s theory has been worked out for complex-value functions in Hilbert spaces. Finally, using this method, the desired solution is obtained by the inverse Fourier transform. Numerical results of the algorithm application to the coherent scanning tomography of absorbing targets buried in multilayer media are demonstrated.