Analysis of photonic crystal transmission properties by the precise integration time domain

Abstract

Photonic crystals are materials patterned with a periodicity in the dielectric constant, which can create a range of forbidden frequencies called as a photonic band gap. The photonic band gap of the photonic crystal indicates its primary property, which is the basis of its application. In recent years, photonic crystals have been widely used to design optical waveguides, filters, microwave circuits and other functional devices. Therefore, the study on the transmission properties in photonic crystal is significantly important for constructing the optical devices. The finite difference time domain (FDTD) is a very useful numerical simulation technique for solving the transmission properties of the photonic crystals. However, as the FDTD method is based on the second order central difference algorithm, its accuracy is relatively low and the Courant stability condition must be satisfied when this method is used, which may restrict its application. To increase the accuracy and the stability, considerable scientific interest has been attracted to explore the schemes to improve the performance of the FDTD. The fourth order Ronge-Kutta (RK4) method has been applied to the FDTD method, which improves the accuracy and eliminates the influence of accumulation errors of the results, but the stability remains very poor if the time step is large. An effective time domain algorithm based on the high precision integration is proposed to solve the transmission properties of photonic crystals. The Yee cell differential technique is used to discretize the first order Maxwell equations in the spatial domain. Then the discretized Maxwell equations with the absorption boundary conditions and the expression of excitation source are rewritten in the standard form of the first order ordinary differential equation. According to the precise division of the time step and the additional theorem of exponential matrix, the high precision integration is used to obtain the homogeneous solution. To obtain the discretized electric and magnetic fields, the particular solution must be solved based on the excitation and then be added to the homogeneous solution. The transmission properties of photonic crystals are obtained by the Fourier transform. Practical calculation of photonic crystals is carried out by the precise integration time domain, and the accuracy and the stability are compared with those from the FDTD and the RK4 methods. The numerical results show that the precise integration time domain has a higher calculation precision and overcomes the restriction of stability conditions on the time step, which provides an effective analytical method of studying the transmission properties of photonic crystals.