Plug-and-play paradigm for the analysis of scattering by "layer-cake" periodic systems

We investigate scattering of scalar plane waves by a heterogeneous layer that is periodic in the direction parallel to its boundary. On describing the layer as a union of periodic sub-laminae, we develop a solution of the scattering problem by combining the concept of propagator matrices and that of Bloch eigenstates featured by the unit cell of each sub-lamina. The Bloch eigenstates are obtained by solving the quadratic eigenvalue problem (QEP) that seeks a complex-valued wavenumber normal to the layer boundary given (i) the excitation frequency, and (ii) real-valued wavenumber parallel to the boundary – which is preserved throughout the system. Spectral analysis of the QEP reveals sufficient conditions for discreteness of the eigenvalue spectrum and the fact that all eigenvalues come in complex-conjugate pairs. By deploying the factorization afforded by the propagator matrix approach, we demonstrate explicitly that the contribution of individual eigenvalues (and affiliated eigenmodes) to the solution diminishes exponentially with absolute value of their imaginary part, which then forms a rational basis for truncation of the factorized Bloch-wave solution. The proposed developments cater for optimal design of the rainbow traps and metasurfaces, whose potency to manipulate waves is controlled not only by the individual dispersion and impedance characteristics of the component sub-laminae, but also by ordering and generally "fitting" of the latter into the composite layer. By deploying the Bloch propagator approach, evaluation of the trial configurations – as generated by (a) permutation and (b) window translation/stretching of the component sub-laminae – can be accelerated by decades.