Lensing system and Fourier transformation using epsilon-near-zero metamaterials

Metamaterials and metastructures with effective relative permittivity near zero exhibit unusual wave properties such as uniform phase distributions across such domains. Here we discuss the possibility of using εnear-zero (ENZ) metamaterials for lensing and Fourier transforming. Owing to the possibility of having ENZ metamaterials in different wavelength regimes, the concepts shown here can be utilized at any frequency bands in which such materials can be constructed. Comments Navarro-Cía, M., Beruete, M., Sorolla, M., & Engheta, N. (2012). Lensing system and Fourier transformation using epsilon-near-zero metamaterials. Physical Review B, 86(16), 165130. doi: 10.1103/PhysRevB.86.165130 ©2012 American Physical Society This journal article is available at ScholarlyCommons: http://repository.upenn.edu/ese_papers/632 PHYSICAL REVIEW B 86, 165130 (2012) Lensing system and Fourier transformation using epsilon-near-zero metamaterials M. Navarro-Cı́a,1,2,3,* M. Beruete,2,† M. Sorolla,2,‡ and N. Engheta3,§ 1Experimental Solid State Group, Department of Physics, Imperial College London, London SW7 2AZ, United Kingdom 2Millimeter and Terahertz Waves Laboratory, Universidad Pública de Navarra, E-31006 Pamplona, Spain 3Department of Electrical and Systems Engineering, University of Pennsylvania, 200 South 33rd Street, ESE 203 Moore, Philadelphia, Pennsylvania 19104, USA (Received 22 November 2011; revised manuscript received 23 September 2012; published 23 October 2012) Metamaterials and metastructures with effective relative permittivity near zero exhibit unusual wave properties such as uniform phase distributions across such domains. Here we discuss the possibility of using ε-near-zero (ENZ) metamaterials for lensing and Fourier transforming. Owing to the possibility of having ENZ metamaterials in different wavelength regimes, the concepts shown here can be utilized at any frequency bands in which such materials can be constructed. DOI: 10.1103/PhysRevB.86.165130 PACS number(s): 81.05.Xj, 42.30.Kq, 42.79.Bh, 78.67.Pt Extreme-parameters metamaterials,1–6 especially ε-nearzero (ENZ) metamaterials, have attracted a great deal of attention in recent years. Unlike other metamaterials possessing negative constitutive parameters that usually happen near resonances, the ENZ properties are displayed somewhat away from resonances, where losses are less.7 The ENZ regime may be obtained just slightly above the plasma frequency of materials, for example, above plasma frequency in metal and highly doped semiconductor at optics and THz, respectively. At lower frequencies, a simple yet effective approach is based on arrangements of hollow waveguides at cutoff frequencies.4,8 Some of the exciting features of ENZ metamaterials include the phenomenon of supercoupling, optical lumped insulator elements, and substrates for the optical displacement-current wires in metatronics.2–4,8 The ENZ metamaterials have played roles in tunneling, beam forming, energy squeezing, sensing, and cloaking.4–6,8–12 Fourier transforms are ubiquitous in modern society due to its broad utility in many branches of science and engineering, such as signal processing, antennas, imaging (e.g., magnetic resonance imaging, x rays, tomography, radar, etc.), holography, spatial distribution and spectral composition of radiation sources, and spectroscopy, just to name a few.13,14 The propagation and diffraction of electromagnetic waves may provide Fourier transformation. It is well known that a Fraunhofer diffraction pattern is directly related to the Fourier transform of the source distribution.15 However, the distances involving far-zone patterns are not usually practical, and instead converging lenses have solved this problem. It has been demonstrated decades ago that lenses bring the Fraunhofer diffraction pattern to its focal length, significantly reducing the space required for this spatial Fourier transform.13,14 In this paper we numerically demonstrate that anisotropic ENZ metamaterials and ENZ metastructures may hold the promise for the design of novel lensing systems and consequently Fourier transforming, expanding thus the range of potential applications of these specialized media. First, we consider a homogeneous isotropic ENZ material as a starting point. In the next stage we provide a thorough numerical analysis of a real implementation of the Fourier transform system. Finally, using the numerical simulation, the proposed anisotropic metastructure is compared with the structure with homogeneous isotropic ENZ materials. For simplicity we reduce all analyses to two dimensional with transverse-magnetic (TM) excitation. Numerical simulations have been conducted using the finite-integration-technique software CST Microwave StudioTM.16 Let us assume a homogenous isotropic material whose relative permittivity is near zero. The profile of a lens made of such ENZ material, which should be designed to perform the Fourier transform may be a plano-concave shape [Fig. 1(a), assuming without loss of generality that the input face is planar and the output face is a concave circular arc]. This is due to the fact that the phase front at the exit face of an ENZ material conforms to the shape of the exit face.5 This can also be easily explained using geometrical optics.17 Let us consider the effect of the plano-concave lens on a normally incident unit-amplitude (1 V/m) plane wave impinging on its planar input face. If the lens indeed operated as a Fourier transformer, one would expect a sinc function at its focal length rather than a δ function since the finite size of the lens’ front face limits the spatial range of the incident signal. This is mathematically represented as follows13: Uzf (x) = exp ( i k 2zf x2 )