Supermodes in Coupled Multi-Core Waveguide Structures
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.