Bloch waves in an infinite, periodically-perforated sheet

Bloch waves in infinite periodic structures - much in vogue in the present metamaterial age [1, 2] - can be conveniently studied by COMSOL Multiphysics®. This is demonstrated by a simple, yet rich two-dimensional example: a perforated sheet with square symmetry (Figure 1). Its implementation in COMSOL is quite simple [3], the essential feature being "Solid Mechanics > Connections > Periodic Condition > Periodicity Settings > Floquet Periodicity" in both x- and y directions. Instead of plane waves in homogeneous media, one has to deal with their generalizations: the Bloch waves. The frequencies of such Bloch waves are obtained from the solutions of an eigenvalue problem with prescribed wave vector resulting in dispersion relations which make up the band structure (Figure 2). The characteristic features of such (running or standing) Bloch waves can be examined with the COMSOL plotting and animation tools. E.g., it is apparent in Figure 3 that the deformation of the top and bottom sides of the unit cells is not exactly sinusoidal as it would be the case with a plane wave.