Exact analytical solution for fields in gradient index metamaterials with different loss factors in negative and positive refractive index segments

Mariana Dalarsson,Nebojša Dončov,Tatjana Asenov,Martin Norgren,Zoran Jakšić

doi:10.1117/1.jnp.7.073086

Abstract

Gradient refractive index metamaterials are of interest for various applications of transformation optics. Wave propagation through gradient index metamaterials using an exact analytical approach is investigated. Composite materials containing constituents with negative real and positive real indexes of refraction are considered. An exact analytical solution for the field distribution is obtained for the sinusoidal spatial variation of complex effective permittivity and permeability along a fixed direction, under the assumption that the wave impedance remains spatially uniform across the structure. Loss factors in the constituent materials can be different from each other corresponding to the realistic situations. Temporal dispersion can be arbitrary subject to the physical limitations imposed by the Kramers-Kronig relations. A numerical model based on the Z -transform is developed to verify the analytical results. The approach can be applied to arbitrary periodic refractive index profiles using the Fourier series method.

Highlights

Electromagnetic metamaterials (MM) are artificial composites with electromagnetic properties not readily found in nature
A dispersive transmission line matrix (TLM) Z-transform model of the lossy MM-composite, described in Ref. 37, is used here to verify the analytical solution for gradient index metamaterials with arbitrary loss factor in positive refractive index media (PRM) and negative refractive index metamaterials (NRM) presented in Secs. 2 and 3
We have presented a simple exact analytical solution to Helmholtz equation for periodic structures with graded permittivity and permeability profile changing according to a cosine function along the direction of propagation, with arbitrary loss factors in PRM and NRM

Summary

Introduction

Electromagnetic metamaterials (MM) are artificial composites with electromagnetic properties not readily found in nature. The properties of NRM, such as the negative index of refraction (and negative phase velocity), inverse Doppler effect, radiation tension instead of pressure, etc.[11,12] resulted in a number of proposed applications Among those we mention superlenses and hyperlenses that enable imaging far below the diffraction limit,[13,14] resonant cavities, and waveguides with geometrical dimension orders of magnitude smaller than the operating wavelength[15] as well as invisibility cloaks and generally transformation optics.[16]. In this paper, we present an exact analytical solution of Helmholtz equations for the propagation of electromagnetic waves through a periodic gradient-index PRM–NRM composite with most general loss factors in the two materials and with sinusoidal periodicity for the case of constant impedance throughout the structure. A comparison of the obtained analytical solution to the results of numerical simulation using a Z-transform based model is given

Problem Formulation

Analytical Solutions of the Field Equations

Numerical Model of Metamaterials

Results and Discussion

Conclusion