We propose a hybrid finite-difference frequency-domain method to study the perpendicular crossing waveguide, dielectric and microwave, TE and TM modes, by exploiting built-in structural symmetries in these waveguide devices. In the plus (+) symmetry model, the complete solution is obtained by solving two rectangular-shaped quarter structures each with two transparent boundaries and two symmetry boundaries. For the cross (x) symmetry model, solutions of four triangular-shaped quarter structures are needed but each with only one transparent boundary. Numerical results are verified by comparison between these two models and with the power conservation test. We show the total and the fundamental-mode, coupling coefficients of the reflected, cross and through power in the crossing waveguide as functions of the normalized frequency.
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