New analytic and computational formalism for the band structure of N-layer photonic crystals

This Letter presents a new analytic and computational formalism for the eigenfrequency spectra of arbitrary, one-dimensional, N-layer photonic band gap (PBG) materials. The secular equation is formulated in terms of tangents only, a form that has the following beneficial attributes: (a) a compact, algorithmically simple, N×N Hermitian eigenvalue-eigenvector problem (real symmetric at symmetry points) that can be diagonalized once to find both the eigenfrequencies and associated wave amplitudes, and (b) a transparent analytical structure that can be exploited to gain additional insights such as physically appealing, geometric representations of the eigenfrequency condition and analytic forms not otherwise available. The formalism is demonstrated on the example of an eighth-wave/quarter-wave/half-wave PBG stack.