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Negative Refraction of Modulated Electromagnetic Waves D APPLIED PHYSICS LETTERS VOLUME 81, NUMBER 15 7 OCTOBER 2002 Negative refraction of modulated electromagnetic waves D. R. Smitha) and D. Schurig Physics Department, University of California, San Diego, La Jolla, California 92093 J. B. Pendry The Blackett Laboratory, Imperial College, London, SW7 2BU, United Kingdom ͑Received 3 June 2002; accepted 12 August 2002͒ We show that a modulated Gaussian beam undergoes negative refraction at the interface between a positive and negative refractive index material. While the refraction of the beam is clearly negative, the modulation interference fronts are not normal to the group velocity, and thus exhibit a sideways motion relative to the beam—an effect due to the inherent frequency dispersion associated with the negative index medium. In particular, the interference fronts appear to bend in a manner suggesting positive refraction, such that for a plane wave, the true direction of the energy flow associated with the refracted beam is not obvious. © 2002 American Institute of Physics. ͓DOI: 10.1063/1.1512828͔ Since the experimental demonstrations of a structured ␻2 p n͑␻͒ϭͱ⑀␮ϭ1Ϫ . ͑1͒ metamaterial with simultaneously negative permittivity and ␻͑␻ϩi⌫͒ negative permeability1,2—often referred to as a left-handed material—the phenomenon of negative refraction, predicted Note that the refractive index is negative below the plasma 3,4 ␻ Ϫ to occur in left-handed materials, has become of increasing frequency p , and has the value of approximately 1 when ␻ϭ␻ & ⌫ ␻Ӷ interest. The consequence of a modulated plane wave inci- p / and losses are small, / 1. dent on an infinite half space of negative refractive index A modulated plane wave has the form material has been considered by Valanju, Walser, and E͑r,t͒ϭeik"reϪi⍀t cos͑⌬k"rϪ⌬␻t͒, ͑2͒ Valanju,5 who have noted that the interference fronts due to the modulation appear to undergo positive refraction, despite where ⍀ is the carrier frequency of the wave, ⌬␻ is the the underlying negative refraction of the component waves. modulation frequency and the wave vector, k, is given in Valanju and co-workers further associate the direction of the terms of the incident angle advancing interference pattern with that of the group veloc- kϭsin͑␪͒kxˆϩcos͑␪͒kzˆ, ͑3͒ ity, and draw a broad set of conclusions regarding the valid- ity of negative refraction. where kϭ␻/c. Equation ͑3͒ can be equivalently viewed as In this letter, we examine the effect of a finite-width the superposition of two plane waves with different frequen- beam refracting at the interface between two semi-infinite cies, or media, one with positive refractive index and the other with E͑r,t͒ϭ 1 eikϩ"reϪi␻ϩtϩ 1 eikϪ"reϪi␻Ϫt, ͑4͒ negative refractive index. We find that the transmitted beam 2 2 does indeed refract in the negative direction, although the where the frequencies of the component waves are ␻Ϯϭ␻ interference fronts do not necessarily point in the same di- Ϯ⌬␻ with corresponding propagation vectors kϮϭkϮ⌬k. rection. We first describe the refraction of a modulated plane ⌬k is parallel to k and is given by wave incident on a half space with negative index, which ⌬␻ demonstrates in a straightforward manner that the interfer- ⌬kϭ k. ͑5͒ ence fronts of the transmitted wave display interesting be- ␻ havior. Because the negative index medium is assumed to be fre- We assume the geometry shown in Fig. 1, in which a quency dispersive, the transmitted waves will refract at modulated electromagnetic wave is incident from vacuum slightly different angles, thus introducing a difference in the onto a material with frequency dependent, negative index of phase versus interference propagation directions. The trans- refraction, n(␻). We choose the z axis normal to the inter- mitted wave can be expressed in the form face, the x axis in the plane of the figure, and the y axis out ϭ iq"r Ϫi⍀t ⌬ " Ϫ⌬␻ ͑ ͒ of the plane of the figure. While our arguments will apply to E͑r,t͒ e e cos͑ q r t͒, 6 a wave with arbitrary polarization, we assume here that the which again is composed of two propagating plane waves ͑ ͒ electric field is polarized along the y axis out of plane . For having propagation vectors qϩ and qϪ , the values of which convenience, we use identical causal plasmonic forms for the can be found using the conservation of the parallel wave permittivity ͑⑀͒ and the permeability ͑␮͒, so that the refrac- vector across the interface tive index is given by ϭ ͑ ͒ qx kx 7 a͒Electronic mail: [email protected] and the constancy of frequency 0003-6951/2002/81(15)/2713/3/$19.002713 © 2002 American Institute of Physics Downloaded 25 Jun 2003 to 128.97.11.57. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/aplo/aplcr.jsp 2714 Appl. Phys. Lett., Vol. 81, No. 15, 7 October 2002 Smith, Schurig, and Pendry FIG. 2. ͑Color͒ Refraction of a Gaussian beam into a NIM. The angle of incidence is 30°, n(␻Ϫ)ϭϪ1.66ϩ0.003i and n(␻ϩ)ϭϪ1.00ϩ0.002i. ⌬␻/␻ϭ0.07. Eq. ͑10͒ shows that the spacing between interference fronts FIG. 1. ͑Color͒ Normal incidence ͑a͒. The group velocity and interference wave vector are parallel. Incidence at 45° angle ͑b͒. On the incident side is one third that in free space, as seen in the figure. ͑left͒ the component wave vectors, interference wave vector and group ve- When a monochromatic wave is incident on the interface locity all point in the same direction. In the NIM ͑right͒ the interference between positive and negative media at an angle, previous wave vector has refracted positively and the component wave vectors and calculations and theory1,3 have shown that the phase velocity group velocity have refracted negatively with the component wave vectors approximately antiparallel to the group velocity. The dashed lines indicate is negative, and the transmitted ray in the medium is directed conservation of the transverse component of the various wave vectors. toward the interface and negatively refracted in accordance with Snell’s law. Equation ͑10͒, however, indicates that the interference fronts associated with a modulated plane wave q"qϭn2k2 ͑8͒ should undergo positive refraction. A calculation of the re- resulting in fraction of a modulated plane wave shows that this is indeed the case ͓Fig. 1͑b͒; see also EPAPS8͔: the incident interfer- ϭ Ϯͱ 2 2Ϫ 2 ͑ ͒ q kxxˆ n k kx zˆ. 9 ence pattern refracts to the opposite side of the normal, con- 5 This expression for individual plane waves is equivalent to sistent with Valanju and co-workers. Snell’s law in predicting the direction of refraction. The cor- The question thus naturally arises: what is the relation- rect sign of the square root is well established elsewhere.6,7 ship between the interference fronts associated with a modu- ⌬ ϭ⌬ ͑ ͒ lated wave and its group velocity? The group velocity is Using qx kx and Eq. 8 , the wave vector of the inter- 9 ference pattern is obtained defined as ͒ ͑ ͒ ϵٌ ␻͑ 1 ⌬␻ vg q q , 12 ⌬ ϭͫ ϩ ͑ 2 Ϫ 2͒ ͬ ͑ ͒ q kxxˆ k nng kx zˆ ⍀ . 10 qz which, for isotropic, low loss materials becomes The group index, given by q d␻͑q͒ q c ϭ ϭ ͑ ͒ ͑ ͒ vg ␻ sign n . 13 n␻͒ q d q ng͑ץ ϵ у ͑ ͒ ng 1 11 ␻ The scalar part of this expression is a material property andץ is constrained by causality to be greater than unity regardless the unit vector pre-factor is determined by the direction of of the sign of the refractive index of the medium. Thus, we the incoming wave. Note that when the index is negative, Eq. ͑ ͒ see that for a negative refractive index the z component of 13 predicts that the direction of the group velocity is anti- the interference wave vector has an opposite sign to the z parallel to the phase velocity, and thus refracts negatively— component of the individual wave vectors, so though the unlike the interference wave vector. This would seem to be a paradox, as the interference velocity of a modulated beam is individual wave vectors refract negatively, the interference 9 pattern refracts positively. Furthermore, the apparent velocity frequently equated with the group velocity. To resolve this of the interference fronts is directed away from the interface, dilemma, we examine the manner in which a point of con- as would be expected if the direction of the interference pat- stant phase on the interference patterns moves. Setting the ͑ ͒ tern were coincident with the direction of the group velocity. argument of the cosine in Eq. 6 equal to a constant, we find Note that Eq. ͑10͒ also implies that the spacing between that the velocity of such a point, vint , must satisfy interference fronts ͑or beats͒ in the negative index medium is ⌬q ϭ ͑ ͒ set by the dispersion characteristics of the medium rather ⌬␻ •vint 1. 14 than just the frequencies of the two component waves. This effect is illustrated in Fig. 1͑a͒ for a wave normally incident The smallest possible such velocity is the one parallel to ⌬q from vacuum on a medium with a frequency dispersive nega- ͑the interference direction͒, which has been interpreted as the 5 tive refractive index. In this example, the carrier frequency is group velocity by Valanju and co-workers; however, vint can assumed to be at the frequency where nϭϪ1. For our choice have an arbitrary component perpendicular to ⌬q. For non- .␻ ϭ3, and normal incidence, the group velocity has such a componentץ/(n␻)ץ ,of frequency dispersion ͓Eq.
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