The inverse problem of scattering is solved for 3D inhomogeneities of the complex permittivity in multilayer media. The proposed solution involves an analysis of the 2D lateral distributions of the scattered field with the imposed condition of the probing field invariability relative to the receiver. A generalization of the solution beyond the Born approximation is given. Necessary k-space representations of Green functions for a multilayer medium with an arbitrary located source are obtained analytically in closed forms. Based on this solution, methods of multifrequency and multilevel subwavelength tomography are developed; their feasibility is shown for multifrequency tomography in a three-layered absorbing medium and for multilevel tomography with a perfect lens that can enhance the sounding depth. The developed approach is valid not only in electromagnetic theory, but also in similar problems of quantum mechanics and acoustics.

The scanning tomography method is developed for electromagnetic sounding of a 3D structure of an inhomogeneous dielectric half-space. It is shown that known methods of physical diagnostics are suitable for this tomography with the depth of analysis from nanometers at optical frequencies up to several kilometers at ultralow frequencies. The areas of application include nanophysics, biological and medical diagnostics, subsurface remote sensing in geophysics and geology, etc. This approach is realized in the microwave scanning tomography of living tissues where a subwavelength resolution is achieved.

Decisive success has been achieved in developing the subsurface near-field scanning tomography that overcomes the Rayleigh diffraction limit of a resolution. It is related to the transformation of the multifrequency inverse scattering problem to that for a complex-valued synthesized pulse (pseudopulse). It leads to the integral equation that has maxima in the depth dependence of its kernel and, hence, to the much better depth resolution of tomography. Moreover, the noise related to surface scattering is mainly suppressed in such an approach. This idea is realized here in the microwave subsurface tomography of 3D inhomogeneous dielectric structures. For homogeneous dielectric targets, this approach is applied to obtain holography images of their shape.

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