We calculated propagation constants of supermodes for two-dimensional two-slab waveguides, with small core gap, using second-order perturbation expansion from gapless slab waveguide system, and compared our results with the existing works. In the perturbation calculation, we used trapezoidal method to calculate the integral over the transverse direction in space and obtained second-order expansion of (core gap)/(core width) for propagation constants. Our result can explain the qualitative relationship between the propagation constants and the gap distance in the neighbor of (core gap)/(core width) being zero.

Herein, we analytically derive the effective index and field distribution of the LP01 mode of a step-index N-sided regular-polygonal-core fiber. To do this, we utilize the lowest-order non-anomalous approximation of the π/N expansion. These properties are also calculated numerically and the results are compared the with approximations.