We consider a smooth perturbation δ (x, y, z) of a constant background permittivity = 0 that varies periodically with x, does not depend on y, and is supported on a finite-length interval in z. We investigate the theoretical and numerical determination of such perturbation from (several) fixed frequency y-invariant electromagnetic waves.By varying the direction and frequency of the probing radiation a scattering matrix is defined. By using an invariant-imbedding technique we derive an operator Riccati equation for such scattering matrix. We obtain a theoretical uniqueness result for the problem of determining the perturbation from the scattering matrix.We also investigate a numerical method for performing such reconstruction using multi-frequency information of the truncated scattering matrix. This relies on ideas of regularization and recursive linearization. Numerical experiments are presented validating such approach.

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We have applied the first-order non-degenerate adiabatic (FONDA) theory to electron - scattering with exchange treated rigorously. The pure rotational body-frame and the laboratory-frame FONDA cross sections are calculated and compared with our body-frame fixed-nuclei (BF-FN) and adiabatic-nuclei (AN) results, respectively. For consistency, all the scattering calculations performed are based on identical potential models and values of input-scattering parameters. These comparisons reveal significant differences between the pure rotational FONDA and BF-FN and AN results near threshold energy range. For instance, the laboratory-frame FONDA (LF-FONDA) and AN cross sections differ by 54, and 117% for to j = 2 and to j = 4 excitations, respectively at 0.01 eV. The body-frame FONDA (BF-FONDA) and BF-FN results are also different by 53, 81, 10 and 96% at 0.0027 eV in the , , and symmetries, respectively. These differences stem from the breakdown of the target-state degeneracy assumption near threshold energy range. We also compare our cross sections for the - system with previously published results. We conclude that stable rotational excitation cross sections require significantly higher values of convergence parameters than previously used.