Meng Yi-Chao, Guo Qi-Zhi, Tan Wei-Han, Huang Zhao-Ming
Institute of Fiber Optics, Shanghai University, Jiading Campus, Shanghai 201800, China.
J Opt Soc Am A Opt Image Sci Vis. 2004 Aug;21(8):1518-28. doi: 10.1364/josaa.21.001518.
A novel approach is proposed for obtaining the analytical solutions of the coupled-mode equations (CMEs); the method is applicable for an arbitrary number of coupled waveguides. The mathematical aspects of the CMEs and their solution by use of Chebyshev polynomials are discussed. When mode coupling between only adjacent waveguides is considered (denoted weak coupling), the first and second kinds of the usual Chebyshev polynomials are appropriate for evaluating the CMEs for linearly distributed and circularly distributed multiwaveguide systems, respectively. However, when one is considering the coupling effects between nonadjacent waveguides also (denoted strong coupling), it is necessary to use redefined generalized Chebyshev polynomials to express general solutions in a form similar to those for the weak-coupling case. As concrete examples, analytical solutions for 2 x 2, 3 x 3, and 4 x 4 linearly distributed directional couplers are obtained by the proposed approach, which treats the calculation as a nondegenerate eigenvalue problem. In addition, for the 3 x 3 circularly distributed directional coupler, which gives rise to a degenerate eigenvalue problem, an analytical solution is obtained in an improved way. Also, for comparison and without loss of generality, to clarify the difference between the two coupling cases, analytical solutions for a 5 x 5 circularly distributed directional coupler are obtained by use of the usual and the redefined generalized Chebyshev polynomials.
本文提出了一种获得耦合模方程（CMEs）解析解的新方法；该方法适用于任意数量的耦合波导。讨论了CMEs的数学方面及其利用切比雪夫多项式的求解方法。当仅考虑相邻波导之间的模式耦合（称为弱耦合）时，第一类和第二类通常的切比雪夫多项式分别适用于评估线性分布和圆形分布的多波导系统的CMEs。然而，当也考虑非相邻波导之间的耦合效应（称为强耦合）时，有必要使用重新定义的广义切比雪夫多项式以类似于弱耦合情况的形式来表达通解。作为具体示例，通过将计算视为非简并特征值问题的所提出方法，获得了2×2、3×3和4×4线性分布定向耦合器的解析解。此外，对于产生简并特征值问题的3×3圆形分布定向耦合器，以改进的方式获得了解析解。同样，为了进行比较且不失一般性，为了阐明两种耦合情况之间的差异，通过使用通常的和重新定义的广义切比雪夫多项式获得了5×5圆形分布定向耦合器的解析解。
