The propagation of guided TE and TM modes through graded-index planar waveguides is reported in this paper. Both a real-valued and complex-valued position-dependent refractive index are supposed for a film layer. It is possible to set various refractive index profiles based on five distribution parameters. For the position-dependent refractive index, the governing equations are transformed to Heun’s differential equation, an exact local solution expressed in terms of local Heun functions. The general nature of these functions is demonstrated based on four degenerate cases of Heun’s equation. The calculation of guided modes requires evaluation of the general solution in the interval containing two regular singular points. For this purpose, the generalized Heun function is introduced and employed in general solutions to the governing equations. The applicability of the generalized solutions is demonstrated by the calculation of guided modes for both the real-valued and complex-valued refractive index.

中文翻译：

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渐变折射率平面波导中的电磁波

本文报道了导引的TE和TM模式通过梯度折射率平面波导的传播。对于膜层，均假定了实数值和复数值位置相关的折射率。可以基于五个分布参数设置各种折射率分布。对于与位置有关的折射率，将控制方程式转换为Heun微分方程，这是用局部Heun函数表示的精确局部解。这些函数的一般性质基于Heun方程的四个退化案例得到证明。引导模式的计算需要在包含两个规则奇异点的区间内评估一般解。为此，引入了广义Heun函数，并将其用于控制​​方程的一般解。