三维光波导中的波传播
Wave Propagation in a 3-D Optical Waveguide
AI总结：
本文针对三维圆柱对称开放光波导中的时谐波传播问题，推广二维变换方法至三维，处理特征值问题的奇异性，构建完整求解方案，为后续特例研究奠定基础。
摘要：
本文研究三维光纤中的波传播问题。目标是获得由源在圆柱对称波导中引起的时谐场的解。该问题的几何结构对应于开放波导，这使得问题具有挑战性。为了解决该问题，我们构建了一种变换理论，该理论是Magnanini和Santosa给出的求解该问题二维版本的方法的非平凡推广。\n扩展到三维因定义变换核的特征值问题在原点及无穷远处均具有奇异性而变得复杂。这些奇异性要求研究特征值问题解的行为。此外，推导求解波传播问题所需的变换公式涉及非平凡的计算。\n本文完整描述了如何构造具有圆柱对称性的三维光波导中波传播问题的解。后续文章将研究阶跃折射率光纤和同轴波导的特例。在这些情况下，我们将获得场的具体公式和数值示例。
英文摘要：
In this paper we study the problem of wave propagation in a 3-D optical fiber. The goal is to obtain a solution for the time-harmonic field caused by a source in a cylindrically symmetric waveguide. The geometry of the problem, corresponding to an open waveguide, makes the problem challenging. To solve it, we construct a transform theory which is a nontrivial generalization of a method for solving a 2-D version of this problem given by Magnanini and Santosa.\cite{MS} The extension to 3-D is made complicated by the fact that the resulting eigenvalue problem defining the transform kernel is singular both at the origin and at infinity. The singularities require the investigation of the behavior of the solutions of the eigenvalue problem. Moreover, the derivation of the transform formulas needed to solve the wave propagation problem involves nontrivial calculations. The paper provides a complete description on how to construct the solution to the wave propagation problem in a 3-D optical waveguide with cylindrical symmetry. A follow-up article will study the particular cases of a step-index fiber and of a coaxial waveguide. In those cases we will obtain concrete formulas for the field and numerical examples.
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