Hidden symmetries in N-layer dielectric stacks

Abstract

The optical properties of a multilayer system with arbitrary N layers of dielectric media are investigated. Each layer is one of two dielectric media, with a thickness one-quarter the wavelength of light in that medium, corresponding to a central frequency f0. Using the transfer matrix method, the transmittance T is calculated for all possible 2N sequences for small N. Unexpectedly, it is found that instead of 2N different values of T at f0 (T0), there are only discrete values of T0, for even N, and (N + 1) for odd N. We explain this high degeneracy in T0 values by finding symmetry operations on the sequences that do not change T0. Analytical formulae were derived for the T0 values and their degeneracies as functions of N and an integer parameter for each sequence we call ‘charge’. Additionally, the bandwidth at f0 and filter response of the transmission spectra are investigated, revealing asymptotic behavior at large N.