Numerical simulation of 3D Fox–Li integral equation described by Rayleigh–Sommerfeld diffraction for MEMS-VCSEL

Abstract

The Fox–Li integral equation is famous as a simulation method for Fabry–Perot cavities. Three-dimensional and higher-order modes can be easily and precisely calculated using the equation and an ordinary personal computer. Fresnel diffraction is generally used as an optical propagation equation in such approaches. However, when considering an optical microcavity, Fresnel diffraction may not be appropriate because it includes the approximation that the propagation distance is larger than the wavelength. In this study, we applied Rayleigh–Sommerfeld diffraction because it is known to be the exact solution while considering optical diffraction for optical cavities comprising plane mirrors. We calculated the 3D Fox–Li integral equation using the Rayleigh–Sommerfeld diffraction, when the cavity condition differs from the approximation. We obtained the diffraction loss and mode profile and compared them with the numerical solution obtained using the Fresnel diffraction formula. Consequently, the diffraction loss was found to be different when the condition did not match the approximation. However, it was similar in the range with the approximation. We also confirmed that the diffraction loss can be determined by the Fresnel number under all conditions when using Rayleigh–Sommerfeld diffraction. In addition, we investigated the applicability of this method to an actual device such as MEMS-VCSEL. We demonstrate that the Fox–Li method described by the Rayleigh–Sommerfeld diffraction can be applied to various applications for micro-optical devices.