We develop a generalized formalism for describing the propagation of an electromagnetic wave along the z-direction of a dielectric medium. The derivation is achieved by casting the 2-D transverse part of the Maxwell equations in a Schrodinger-like form whose Hamiltonian is identified to be pseudo-Hermitian. The developed formalism is combined with the variational principle to derive a set of nonorthogonal coupled-mode theory which is slightly different from that derived using the same variational principle but with the 3-D Maxwell equations. By showing that the 3-D variational approach suffers from a mode-expansion incompatibility issue that is absent in our 2-D case, we conclude that our nonorthogonal coupled-mode theory is more rigorous. Owing to the complexity of the second-order error of the propagation constant as revealed by further analysis, it is found that our nonorthogonal coupled-mode theory may not necessarily be more accurate in practice. The developed pseudo-Hermitian formalism may provide a good framework for the analysis and design of various integrated optical devices.
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