Hidden symmetries in N-layer dielectric stacks

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 f ₀. Using the transfer matrix method, the transmittance T is calculated for all possible 2^N sequences for small N. Unexpectedly, it is found that instead of 2^N different values of T at f ₀ (T ₀), there are only discrete values of T ₀, for even N, and (N + 1) for odd N. We explain this high degeneracy in T ₀ values by finding symmetry operations on the sequences that do not change T ₀. Analytical formulae were derived for the T ₀ values and their degeneracies as functions of N and an integer parameter for each sequence we call 'charge'. Additionally, the bandwidth at f ₀ and filter response of the transmission spectra are investigated, revealing asymptotic behavior at large N.