Photonic band structures of one- and two-dimensional periodic systems with metallic components in the presence of dissipation

Abstract

We present an approach that allows calculating photonic band structures of electromagnetic waves propagating in periodic systems containing dispersive and highly absorptive materials characterized by a dielectric function, which is frequency dependent and has a non-negligible imaginary part. This method, which provides a complementary approach to that of the transfer-matrix method, is based on the use of a position-dependent dielectric function and the plane-wave technique. The use of the complex form of the dielectric function transforms Maxwell's equations into a generalized nonlinear eigenvalue problem. At low filling fractions of the dispersive and absorptive component (f\ensuremath{\leqslant}1%), the generalized eigenvalue problem is reduced to a problem of solving sets of simultaneous nonlinear equations which correspond to the diagonal terms of the matrix equation in the plane-wave representation, with the nondiagonal elements taken into account perturbatively. The resulting complex band structure yields, in addition to the dispersion curves, the attenuation of each mode as it propagates through the system. We first consider a model system represented by a one-dimensional (1D) periodic array of alternating layers of vacuum, and a metal characterized by the complex frequency-dependent dielectric function. To calculate the photonic band structure of this system we employ, in addition to the transfer-matrix method and our perturbative plane-wave approach, a standard linearization technique which solves the general nonlinear eigenvalue problem by the diagonalization of an equivalent, enlarged, matrix. We then apply both our perturbative plane-wave approach and the linearization scheme to obtain the photonic band structures of an infinite array of parallel, infinitely long metallic rods whose intersections with a perpendicular plane form a simple square lattice. The interesting features associated with the presence of dissipation displayed by the photonic band structures, such as an asymmetric behavior of the absorption coefficient and the lifetime of each electromagnetic wave for wave vectors near the Brillouin-zone boundaries, the splitting of the lifetimes of degenerate modes, and the different dependences of the real and imaginary parts of the complex photonic band structure on the polarization of the electromagnetic waves in 2D systems, are discussed.