# Hidden symmetries in $N$-layer dielectric stacks

## TL;DR

This paper uncovers hidden symmetries in multilayer dielectric stacks that lead to surprisingly few distinct transmittance values at the central frequency, revealing high degeneracy and new symmetry operations.

## Contribution

It introduces a novel symmetry framework explaining the degeneracy of transmittance in multilayer dielectric stacks with arbitrary layers.

## Key findings

- Number of distinct transmittance values is reduced from $2^N$ to approximately $N/2+1$ or $N+1$.
- Derived analytical formulas for transmittance values and their degeneracies.
- Observed asymptotic behavior of bandwidth at large number of layers.

## 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 $N$.