光子晶体传输特性的时域精细积分法分析

将精细积分法应用于时域有限差分法中,提出了一种求解光子晶体传输特性的时域精细积分法,并对其计算精度及稳定性进行了分析.从一阶麦克斯韦方程出发,在空间上采用Yee元胞进行差分离散,结合吸收边界条件及激励源表达式将方程整理为标准的一阶常微分方程组形式.通过时间步长的精细划分和指数矩阵的加法定理,在时间上利用精细积分法对齐次微分方程进行积分求解,并结合激励向量的特解得到空间离散的场分量,最终通过傅里叶变换求得方程的解.利用时域精细积分法对光子晶体进行了实例计算,并将其结果分别与时域有限差分法和四阶龙格库塔法在精度、稳定性等方面进行了比较,结果表明时域精细积分法具有更高的计算精度,并且克服了时域有限差分法以及四阶龙格库塔法在计算稳定性上对时间步长的限制.提出的方法具有精确、稳定的特点,为光子晶体传输特性的研究提供了一种新的有效的分析方法.

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

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.