# Photonic Transmittance in Metallic and Left Handed Superlattices

Departmento de Ciencias Básicas, Universidad Autónoma Metropolitana, 02120 Ciudad de México, Mexico

Received: 1 February 2020 / Revised: 3 April 2020 / Accepted: 9 April 2020 / Published: 18 April 2020

(This article belongs to the Special Issue Metamaterials for Advanced Photonic and Plasmonic Applications – Selected Papers from Metamaterials’2019)

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 $\omega $ below the plasma frequency ${\omega}_{p}$ , 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 $\lambda /4$ 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.