光子晶体光纤的Galerkin算法

光子晶体光纤(PCF)是由石英和空气孔构成的二维周期性介电常数分布的微结构光纤，人们已经提出了一些理论方法用于其模式特征的研究。以全反射光子晶体光纤为例。将其折射率结构分解为一个纯粹的中心缺陷结构和一个完美二维光子晶体的叠加，并分别选取厄米-高斯函数和余弦函数对其周期性折射率展开；同时将横向电场分布的x，y分量用正交归一化厄米-高斯(Hermite-Gauss)函数展开。从电磁场的波动方程出发，忽略横向电场之间的耦合。根据折射率和电场的展开式。得到关于各展开系数的矩阵和模式的特征方程。特征方程中涉及的各矩阵元素都可以利用厄米-高斯函数的正交归一化性质及其他一些恒等式解析求得。求解该特征方程，可得到光子晶体光纤的传播常数和模场分布。利用此算法，可以进一步研究光子晶体光纤的模式特性、色散特性、偏振特性等。

Galerkin Algorithm for Investigating the Photonic Crystal Fibers

The photonic crystal fiber is a kind of microstructured optical fiber with a 2 D periodial dielectric profile constructed by air/silica. There are a few models to investigate the mode characteristics. For the TIR PCF, as an example, the index profile can be considered as the construction of a pure center defect and a perfect 2 D photonic crystal with the periodic air hole lattice, which can be expanded with the normalized Hermite Gaussian basis set and cosine function respectively. From the electromagnetic wave equation, the mode eigen equation can be obtained with the algorithm when the transverse electric field is also expanded as the series of the Hermite Gaussian basis. All elements in the eigen equation are evaluated analytically based on the orthonormality and some identities of the Hermite Gaussian function. The propagation constant and the mode field distribution of the PCF can be calculated, hence the mode characteristics, the dispersion properties and the polarization properties can also be investigated with this algorithm.