Nonlinear inverse scattering and three-dimensional near-field optical imaging

Abstract

The nonlinear inverse scattering problem for electromagnetic fields with evanescent components is considered. A solution to this problem is obtained in the form of a functional series expansion. The first term in the expansion corresponds to the pseudoinverse solution to the linearized inverse problem. The higher order terms represent nonlinear corrections to this result. Applications to the problem of three-dimensional optical imaging with subwavelength resolution are described and illustrated with numerical simulations.

Nanoscale optical tomography using volume-scanning near-field microscopy

“…Means to interpret NSOM images have been developed, 8,9 including the solution of a three-dimensional near-field inverse scattering problem. [10][11][12][13][14][15][16][17][18] The resulting methods are known as near-field optical tomography ͑NFOT͒. NFOT experiments require the acquisition of multiple complete NSOM data sets for varying directions of illumination or observation of the field.…”

Information content of the near field: three-dimensional samples

“…The model employed here included a weakly interacting probe, a linearized scattering model, and for the specific examples, Gabor-type holographic measurements. Future work will include nonlinear reconstructions for strongly scattering samples [19] and strongly interacting probes [36,37]. APPENDIX A 1.…”

“…Nonlinear reconstructions, i.e., beyond the Born approximation, have been formulated, but present a serious challenge to implementation, namely, regarding the stability. The study of the linearized problem also provides a starting point for the analysis [16] of the nonlinear problem [17][18][19][20]. Important questions remain: what are the fundamental limits on information or resolution in near-field microscopy?…”

On the convergence of the Born series in optical tomography with diffuse light

“…However, note that the influence of boundaries can be exponentially small, as is discussed in Section 6 below. Next, we specialize to the case of absorbing inhomogeneities, V = V α , where V α is defined by (8). Assuming that δα(r) = 0 if r is outside of a sphere of radius a, we have…”