Optical trimer: a theoretical physics approach to waveguide couplers

We study electromagnetic field propagation through an ideal, passive, three-dimensional, triangular three-waveguide coupler using a symmetry-based approach that capitalizes on the underlying su(3) symmetry. The planar version of this platform has already demonstrated its utility in photonic circuit design, enabling optical sampling, filtering, modulating, multiplexing, and switching. We aim to provide a practical tutorial on using group theory for the analysis of photonic lattices for those less familiar with abstract algebra methods. This approach serves as a powerful tool for optical designs. To illustrate this, we focus on the equilateral trimer, connected to the discrete Fourier transform, and the isosceles trimer, related to the golden ratio, providing stable single waveguide output. We also explore a scenario where the coupling in an equilateral coupler changes linearly with propagation distance. Going beyond the standard optical-quantum analogy, we show that coupled-mode equations for intensity and phase allows us to calculate envelopes for inputs within an intensity class, as well as individual input field amplitudes. This approach streamlines the design process by eliminating the need for point-to-point propagation calculations, highlighting the power of group theory in the field of photonic design.