Abstract
We study the transmission of electromagnetic waves through layered structures of metallic and left-handed media. Resonant band structures of transmission coefficients are obtained as functions of the incidence angle, the geometric parameters, and the number of unit cells of the superlattices. The theory of finite periodic systems that we use is free of assumptions, the finiteness of the periodic system being an essential condition. We rederive the correct recurrence relation of the Chebyshev polynomials that carry the physical information of the coherent coupling of plasmon modes and interface plasmons and surface plasmons, responsible for the photonic bands and the resonant structure of the surface plasmon polaritons. Unlike the dispersion relations of infinite periodic systems, which at best predict the bandwidths, we show that the dispersion relation of this theory predicts not only the bands, but also the resonant plasmons’ frequencies, above and below the plasma frequency. We show that, besides the strong influence of the incidence angle and the characteristic low transmission of a single conductor slab for frequencies ω below the plasma frequency ω p , the coherent coupling of the bulk plasmon modes and the interface surface plasmon polaritons lead to oscillating transmission coefficients and, depending on the parity of the number of unit cells n of the superlattice, the transmission coefficient vanishes or amplifies as the conductor width increases. Similarly, the well-established transmission coefficient of a single left-handed slab, which exhibits optical antimatter effects, becomes highly resonant with superluminal effects in superlattices. We determine the space-time evolution of a wave packet through the λ / 4 photonic superlattice whose bandwidth becomes negligible, and the transmission coefficient becomes a sequence of isolated and equidistant peaks with negative phase times. We show that the space-time evolution of a Gaussian wave packet, with the centroid at any of these peaks, agrees with the theoretical predictions, and no violation of the causality principle occurs.
Highlights
For many years, photonic crystals (PCs) of metal-dielectric structures designed to control and manipulate the propagation of electromagnetic fields have been widely studied, both theoretically and experimentally[1,2,3,4,5,6,7,8,9]
We find, among others, the rather cumbersome transfer matrix method introduced by Pendry for cylindrical dielectric arrays and the transfer matrices defined in terms of reflection and transmission amplitudes introduced by Botten et al Others, like those in [19,20,21,22], start well, obtaining the unit-cell transfer matrices, but when they have to deal with a superlattice, their theoretical approach becomes extremely cumbersome or decide to follow P
We studied the transmittance and dispersion relations of electromagnetic waves through metallic and left-handed superlattices, based on the theory of finite periodic systems briefly reviewed here
Summary
Photonic crystals (PCs) of metal-dielectric structures designed to control and manipulate the propagation of electromagnetic fields have been widely studied, both theoretically and experimentally[1,2,3,4,5,6,7,8,9]. As will be seen below and was shown in [23], these results allow us to obtain accurate values for the resonant energies and wave functions (for open SLs) and accurate energy eigenvalues and eigenfunctions for bounded SLs. Given the matrix elements αn and β n , see Equations (16) and (17), the transfer matrix of a (time reversal invariant) superlattice with n unit cells is: Mn =. Chebyshev polynomials defined in terms of KΛ, the Bloch wavenumber K and the SL periodicity Λ His transfer-matrix approach describes infinite superlattices with dispersion relations that predict at best the widths of continuous bands. We present some specific results for these quantities
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