Perturbation Analysis with Approximate Integration for Propagation Mode in Two-Dimensional Two-Slab Waveguides

On the basis of perturbation expansion from a gapless system, we calculate the propagation constant and propagation mode wave function in two-dimensional two-slab waveguides with a core gap small enough that there is only one propagation mode. We also perform calculations without the approximation for comparison. Our result shows that first-order perturbation contains the first-order Taylor expansion of (core gap)/(core width), and when the integration of the perturbation is suitably approximated, the result of the first-order perturbation is the same as that of the first-order Taylor expansion of (core gap)/(core width). 1. Introduction When monochromatic light enters into one of the cores of a waveguide array in which each core has the same width and the same gap, the light propagates to adjacent cores in turn, and its trajectory becomes V-shaped. Such optical behavior has been theoretically and numerically analyzed. Theoretical analysis includes matrix method [1], the coupled mode equation, and the coupled power equation [2–9]. In the coupling mode equations, it is assumed that there is at least one propagation mode for each core and that all of the propagation modes are independent from each other. This assumption is invalid in the case that the number of independent propagation modes is less than the number of cores. Such a situation can occur when the core gap becomes small enough that the index distribution is approximated by a gapless core. When such a situation occurs, analysis based on the coupled mode or the coupled power equation is invalid. Our interest is finding a simple method to analyze an optical behavior in such a situation. An optical behavior for parallel slab waveguides with small core gap was theoretically analyzed using even and odd mode analysis, which was called supermode analysis later [3–5]. In this analysis, the exact solution for the Maxwell’s equations with the index distribution for two parallel slab waveguides, which becomes essentially one for Schr？dinger equation for double finite wells in the field of quantum mechanics, is used to analyze the optical behavior. However, the method using supermodes is not useful when the number of cores increases, because the solution must be individually expressed in every core and clad. In the field of quantum mechanics, perturbation theory is widely used as an approximation [10, 11]. In the field of electromagnetics and optics, it is also explained in [12], and applied to explain bend losses for a fiber [13, 14]. Coupled mode equation is also based on the perturbation theory [4–7].