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Generalized Transformation Optics of Linear Materials Home Search Collections Journals About Contact us My IOPscience Generalized transformation optics of linear materials This article has been downloaded from IOPscience. Please scroll down to see the full text article. 2011 J. Opt. 13 055105 (http://iopscience.iop.org/2040-8986/13/5/055105) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 152.3.193.67 The article was downloaded on 08/04/2011 at 16:57 Please note that terms and conditions apply. IOP PUBLISHING JOURNAL OF OPTICS J. Opt. 13 (2011) 055105 (12pp) doi:10.1088/2040-8978/13/5/055105 Generalized transformation optics of linear materials Robert T Thompson1, Steven A Cummer2 and Jorg¨ Frauendiener1,3 1 Department of Mathematics and Statistics, University of Otago, PO Box 56, Dunedin 9054, New Zealand 2 Department of Electrical and Computer Engineering and Center for Metamaterials and Integrated Plasmonics, Duke University, Durham, NC 27708, USA 3 Centre of Mathematics for Applications, University of Oslo, PO Box 1053, Blindern, NO-0316 Oslo, Norway E-mail: [email protected] Received 4 February 2011, accepted for publication 8 March 2011 Published 7 April 2011 Online at stacks.iop.org/JOpt/13/055105 Abstract We continue the development of a manifestly four-dimensional, completely covariant, approach to transformation optics in linear dielectric materials begun in a previous paper. This approach, which generalizes the Plebanski based approach, is systematically applicable for all transformations and all general linear materials. Importantly, it enables useful applications such as arbitrary relative motion, transformations from arbitrary non-vacuum initial dielectric media, and arbitrary space–times. This approach is demonstrated for a resulting material that moves with uniform linear velocity, and in particular for a moving cloak. The inverse problem of this covariant approach is shown to generalize Gordon’s ‘optical metric’. Keywords: transformation optics, optical metric, electrodynamics in materials, metamaterials 1. Introduction properties, and demonstrated how it could be used to create novel devices. Closely related work by Greenleaf The emerging field of transformation optics, where useful et al [9] developed a similar concept for electric current arrangements of man-made ‘metamaterials’ [1–3] are designed flow and applied it to impedance tomography. The initial via transformations of electromagnetic fields, has theoretical approaches to transformation optics relied on purely spatial roots stretching back nearly a century, to the early days of transformations [8, 10, 11]. Leonhardt and Philbin generalized general relativity. The idea that the behavior of light in a these to transformations involving both space and time by gravitational field can be replicated by a suitable distribution using the explicit equivalence given by De Felice, thus linking of refractive media appears to have been first postulated transformation optics to differential geometry [12]. Another by Eddington [4]. Subsequently, Gordon [5] studied the approach [13], based on field-transforming metamaterials, inverse problem, that of representing a refractive medium considered more general transformations in the Fourier as a vacuum space–time described by an ‘optical metric’. domain, but at the expense of the intuitive and appealing Later, Plebanski [6] found effective constitutive relations geometric interpretation. For more recent reviews see [14, 15]. for electromagnetic waves propagating in vacuum space– In examining the details of the Plebanski–De Felice times, but, while recognizing the formal equivalence of these approach, we find some limitations that are addressed here. equations to those in a macroscopic medium in flat space–time, As pointed out by Plebanski himself [6], the equation now did not exploit this equivalence to actually describe such a bearing his name is not strictly covariant, because its derivation medium. This was first done by De Felice [7], who used the requires a matrix inversion that is not a true tensor operation. Plebanski equations to describe the equivalent medium of both One consequence of this, in the context of transformation a spherically symmetric gravitational system and Friedmann– optics, is that magneto-electric coupling terms cannot always Robertson–Walker space–time. be simply interpreted as a material velocity, as is frequently More recently, Pendry [8] pointed out the specific done. This is because the Plebanski equations can only relationship between spatial transformations and material be identified with stationary media or with slowly moving 2040-8978/11/055105+12$33.00 1 © 2011 IOP Publishing Ltd Printed in the UK & the USA J. Opt. 13 (2011) 055105 R T Thompson et al (i.e. nonrelativistic), isotropic media [16]. This prompts the a relatively self-contained and more detailed description of question of whether the approach of [12] may be generalized covariant electrodynamics in both vacuum and linear materials. to clearly distinguish magneto-electric couplings from material This development relies on the geometric language and tools velocity, and allow for specially relativistic corrections. Just of differential geometry, such as exterior derivative, wedge such an approach was outlined in [16], and was demonstrated product, and the pullback of a tensor. These aspects of to recover several results obtained through other means. differential geometry are described in a myriad of excellent Here we provide a complete derivation of the approach sources, such as [20–23]. The most important of these for our outlined in [16], further generalizing the result obtained purpose, the pullback map, is described in appendix B.The there. A physically realistic scenario for transformation optics development and notation, in particular the sign convention, designed devices is that the device moves with arbitrary follow that of [24], while more information, particularly for velocity. We demonstrate that the approach described here may electrodynamics in materials, can be found in [25, 26]. We be applied to find the material properties of a transformation use the Einstein summation convention, indices are lowered αβ when the resultant material is constrained to move with (raised) by the metric tensor gαβ (its inverse g ), and the speed arbitrary uniform velocity with respect to the frame in which of light and Newton’s constant are set to c = G = 1. the transformations are given. This represents a departure from most previous examples in transformation optics, where either 2.1. Field strength tensor the resulting material is stationary or the velocity is dictated by the transformation itself. While a few examples exist of In free space, classical electrodynamics is modeled as a nonrelativistic moving dielectrics in transformation optics for principal U(1) fiber bundle over a space–time manifold special cases [13, 17], we provide a systematic approach that M (which we assume to be equipped with a metric) with is widely applicable for any velocity. connection one-form A = Aμ (frequently called a ‘gauge Another limitation of the Plebanski–De Felice approach is field’, A is the covariant version of the four-vector potential). that the resulting material must reside in vacuum Minkowski The field strength F = Fμν is the curvature two-form of the space–time. The approach described here relaxes this U(1) fiber bundle, equal to the exterior derivative of A, condition, allowing for physically realistic scenarios such as transformations in arbitrary non-vacuum initial dielectric F = dA ⇒ Fμν = Aν,μ − Aμ,ν, (1) media [18], or in arbitrary space–times—thus providing general relativistic corrections for transformations in arbitrary where the comma indicates a derivative. The components of space–times, such as the weakly curved space–time near F can be represented as a matrix, that in a local orthonormal Earth [19]. Lastly, we show that this covariant approach is frame (or Minkowski space–time with Cartesian coordinates) consistent with, and generalizes, Gordon’s optical metric as have values essentially the inverse problem of transformation optics. ⎛ ⎞ 0 −E −E −E The paper is organized as follows: in section 2 we review x y z ⎜ Ex 0 Bz −By ⎟ the completely covariant theory of vacuum electrodynamics Fμν = ⎝ ⎠ . (2) E −B 0 B using modern language, for which the necessary ideas y z x E B −B 0 and notation from differential geometry may be found in z y x appendix B. This review is presented in some detail, because With this choice of values for the components of F, the four- in section 3 the covariant theory of section 2 is extended force vector on a particle moving with four-velocity uν and to describe electrodynamics in macroscopic linear dielectric charge q is α αμ ν materials. This section presents a slight departure from the f = qg Fμνu , (3) usual description of electrodynamics in dielectric materials in order to clearly distinguish material effects from space–time which is just the Lorentz force. For example, a particle at rest ν = ( , , , ) effects. Section 4 describes the concept of transformation with respect to this system has four-velocity u 1 0 0 0 , μ = ( , , , ) optics and presents an interpretation consistent with the for which f q 0 Ex Ey Ez , recovering the usual notion geometric picture of the preceding sections. The main result that a charged particle at rest feels only the electric part of the ν x y z for applications in transformation optics is equation (30). field. For a particle moving with u = γ(1,v ,v ,v ),the μ Section 4.1 examines a particular transformation both when the spatial components of f are f = qγ(E +v × B), while the resulting material is at rest and when it is in motion relative time component is the change in energy per unit time, or the to the frame in which the fields have been measured. As power. expected, it is found that the results for the material in motion The E and B fields are now tightly intertwined, simply smoothly recover, in the limit v → 0, the results for the representing different components of a single object, F.
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