Exactly Solvable Diffraction-Grating Scattering Problems for TE and TM waves

Traditional scattering theory for diffraction gratings often relies on the paraxial approximation, which assumes small incidence angles and specific grating parameters. This paper addresses the challenge of finding exact analytical solutions for diffraction gratings—including those with magnetic properties—for both Transverse Electric (TE) and Transverse Magnetic (TM) waves, without restricting the analysis to the paraxial regime.

The authors utilize a dynamical formulation of stationary scattering (DFSS). This technique maps the spatial scattering problem onto a quantum mechanical time-evolution problem. By treating the coordinate along the grating's thickness as an effective time variable, the scattering process is described by an effective non-Hermitian Hamiltonian. This formulation allows the authors to derive the fundamental transfer matrix for the system, which determines the scattering amplitudes for any incidence angle.

The study establishes that a generalized class of diffraction gratings, defined by periodic relative permittivity and permeability profiles, is exactly solvable. The authors provide a systematic procedure to compute diffracted beam amplitudes using a finite number of algebraic operations and integrals. By avoiding the paraxial approximation, the derived expressions are robust for arbitrary incidence angles. The authors also provide a concrete example of their method applied to a generalization of Berry's grating, confirming the validity of their results through the reciprocity principle.

Exactly solvable models are rare in scattering theory and serve as essential benchmarks for validating numerical methods and developing physical intuition. By extending exact solvability to TM waves and magnetic gratings, this work provides a powerful analytical framework for designing and analyzing complex optical structures that go beyond the limitations of traditional diffraction theories.