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