The PML method for calculating the propagative wave numbers of electromagnetic wave in periodic structures

When the electromagnetic wave is incident on the periodic structures, in addition to the scattering field, some guided modes that are traveling in the periodic medium could be generated. In the present paper, we study the calculation of guided modes. We formulate the problem as a nonlinear eigenvalue problem in an unbounded periodic domain. Then we use perfectly matched layers to truncate the unbounded domain, recast the problem to a quadratic eigenvalue problem, and prove the approximation property of the truncation. Finally, we formulate the quadratic eigenvalue problem to a general eigenvalue problem, use the finite element method to discrete the truncation problem, and show numerical examples to verify theoretical results.

💡 Research Summary

This paper addresses the challenging problem of computing propagative wave numbers (also called exceptional or guided‑mode wave numbers) for electromagnetic waves interacting with periodic structures. When an incident wave impinges on a Λ‑periodic medium, the total field consists of the incident field, a scattered field, and possibly a guided mode that propagates along the periodic direction. The existence of such guided modes destroys the uniqueness of the classical scattering problem and is signaled by special values of the quasi‑periodic phase α, called propagative wave numbers.

The authors first formulate the problem as a nonlinear eigenvalue problem: find α∈