Abstract

The plane-wave method is modified for tunneling evanescent solutions of Maxwell’s equations with complex k vectors (tunneling modes). In our formulation the imaginary part of the k vectors is not necessarily parallel to the real part of the k vectors. We present computed complex photonic band structures of the simple-cubic (sc) and face-centered-cubic (fcc) lattice structures at the symmetric points, such as X, M, and R for the sc and X, L, W, U, and K for the fcc with various directions of imaginary k vectors. In addition, relations between imaginary k vectors and tunneling modes are examined at points X, M, R, and K. Tunneling electromagnetic modes can mathematically exist in the photonic band gaps among propagating eigenmodes even in an infinite photonic crystal. With the concept of electromagnetic tunnelings in photonic crystals, we explain classically various kinds of important phenomena such as inhibition of the spontaneous emission and localized defect modes. The method developed here for solutions with complex eigenfrequencies and k vectors can be readily extended to media with loss or gain by adoption of complex dielectric constants.

© 1995 Optical Society of America

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