Matrix method for two-dimensional waveguide mode solution

In this paper, we show that the transfer matrix theory of multilayer optics can be used to solve the modes of any two-dimensional (2D) waveguide for their effective indices and field distributions. A 2D waveguide, even composed of numerous layers, is essentially a multilayer stack and the transmission through the stack can be analysed using the transfer matrix theory. The result is a transfer matrix with four complex value elements, namely A, B, C and D. The effective index of a guided mode satisfies two conditions: (1) evanescent waves exist simultaneously in the first (cladding) layer and last (substrate) layer, and (2) the complex element D vanishes. For a given mode, the field distribution in the waveguide is the result of a ‘folded’ plane wave. In each layer, there is only propagation and absorption; at each boundary, only reflection and refraction occur, which can be calculated according to the Fresnel equations. As examples, we show that this method can be used to solve modes supported by the multilayer step-index dielectric waveguide, slot waveguide, gradient-index waveguide and various plasmonic waveguides. The results indicate the transfer matrix method is effective for 2D waveguide mode solution in general.