Investigation of the photonic structure using different analytical and numerical methods

In this study, the eigenvalues and eigenstates of photonic crystals and micro-resonators, which have a wide working area among photonic structures (Fabry-Perot interferometer, photonic crystals and dielectric micro-resonators), were studied in detail. For the morphology-dependent resonance modes (MDRs) of dielectric microsphere resonators, as an alternative to traditional methods based on Mie scattering theory and offering a transcendental equation set for solution, the WKB (Wentzel-Kramers-Brillouin) approach, which is mostly used in quantum mechanics, is used and two quantization conditions are obtained for both transverse electric (TE) and transverse magnetic (TM) modes. In addition, as an alternative to the finite difference (FD) method, which is one of the traditionally used numerical methods and based on the lattice mesh design, the resonance modes and field distributions are obtained with very good accuracy using the RBF (Radial Basis Function) method, which uses random point sets on the grid structure in the discrete space rather than the lattice mesh design. The differential equations that need to be solved in order to obtain the eigenvalues and the field distributions of such photonic structures are not easy ones. However, these equations encountered in electromagnetic theory and the differential equations for massive particles are very similar. Therefore, the equations for photonic crystals and micro-resonators can be solved by methods, which are used extensively in quantum mechanics. On the other hand, it is predicted that the methods used for perfect spherical micro-resonators in this study, can also be used for other geometries such as prolate and oblate.