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 Photonic crystal fiber (PCF) is a microstructured fiber with two-dimensional periodic dielectric constant distribution composed of quartz and air holes. Some theoretical methods have been proposed for the study of its mode characteristics. Taking total reflection photonic crystal fiber as an example, the refractive index structure is decomposed into a pure center defect structure and a superposition of a perfect two - dimensional photonic crystal, and the periodic refractive index expansion is selected by Hermite function and cosine function respectively. At the same time, the x, y components of the transverse electric field distribution are expanded by the normalized Hermite Gaussian function. Based on the wave equation of electromagnetic field, the coupling between transverse electric fields is neglected. According to the expansion of the refractive index and the electric field, the eigenvalues ​​of the matrix and mode of each expansion coefficient are obtained. Each matrix element involved in the eigenvalue equation can be obtained by using orthogonal normalization properties of Hermitian function and some other identities. Solving the eigenvalue equation can get the propagation constant and mode field distribution of photonic crystal fiber. Using this algorithm, we can further study the photonic crystal fiber mode characteristics, dispersion characteristics, polarization characteristics