Title: Hidden symmetries in $N$-layer dielectric stacks,
Abstract: The optical properties of a multilayer system of dielectric media with
arbitrary $N$ layers is investigated. Each layer is one of two dielectric
media, with thickness one-quarter the wavelength of light in that medium,
corresponding to a central frequency. 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 the
central frequency ($T_0$), there are either $(N/2+1)$ or $(N+1)$ discrete
values of $T_0$ for even or odd $N$, respectively. We explain the high
degeneracy in the $T_0$ values by defining new symmetry operations that do not
change $T_0$. Analytical formulae were derived for the $T_0$ values and their
degeneracy as functions of $N$ and an integer parameter for each sequence we
call "charge". Additionally, the bandwidth of the transmission spectra at $f_0$
is investigated, revealing some asymptotic behavior at large $N$.