Reversed Cherenkov-Transition Radiation by a Charge Crossing a Left-Handed Medium Boundary Sergey N. Galyamin, 1 Andrey V. Tyukhtin, 1 Alexey Kanareykin, 2 and Paul Schoessow 2 1 Physical Department, St. Petersburg State University, St. Petersburg 198504, Russia 2 Euclid Tech Labs, LLC, Solon, Ohio 44139, USA (Received 24 July 2009; published 2 November 2009) We analyze the radiation from a charged particle crossing the boundary between an ordinary medium and a ‘‘left-handed’’ metamaterial. We obtain exact and approximate expressions for the field components and develop algorithms for their computation. The spatial radiation in this system can be separated into three distinct components, corresponding to ordinary transition radiation having a relatively large magnitude, Cherenkov radiation, and reversed Cherenkov-transition radiation (RCTR). The last one is explained by reflection and refraction of reversed Cherenkov radiation at the interface. Conditions for generating of RCTR are obtained. We note properties of this radiation that have potential applications in the detection of charged particles and accelerator beams and for the characterization of metamaterial macroscopic parameters (", ). DOI: 10.1103/PhysRevLett.103.194802 PACS numbers: 41.60.Bq, 41.20.q In the 1960s, Veselago introduced the concept of ‘‘left- handed media’’ (LHM), i.e., media having simultaneously negative permittivity and permeability [1,2]. In LHM, the electric field vector, magnetic field vector, and wave vector form a left-handed orthogonal set. The direction of the energy flow and the direction of the phase velocity are opposite in LHM, resulting in very unusual properties of electromagnetic waves propagating in these media. Note that the ‘‘left-handed’’ properties can be realized only in a limited frequency range [1,2]. Therefore, it would be more correct to refer to a ‘‘left-handed frequency band’’ (LHFB) as opposed to a ‘‘right-handed frequency band’’ (RHFB) where the familiar properties of the medium oc- cur. However, the term LHM is widespread now in the sci- entific literature, and we will use it as well, with the under- standing that a LHM is a medium possessing a LHFB. Artificial materials possessing left-handed properties in the gigahertz frequency band have been demonstrated recently (see, for example, [3–6]). These metamaterials (MTMs) are composed of discrete conducting elements having their size and spacing much smaller than the wave- lengths of interest. Therefore, such media can be described by the macroscopic parameters "ð!Þ and ð!Þ. Radiation from a charge traversing the interface between two media is one of the principal problems of electro- dynamics. In the case of ordinary (‘‘right-handed’’) media (RHM), this question was investigated as early as 1946 [7,8]. The case of an interface between RHM and LHM was partially discussed in [9,10]. However, a quantitative investigation of the field was not performed. Cherenkov radiation (CR) in LHM has been investigated in more detail [11–14]. In particular, it was shown that the moving particle generates both ordinary (forward) and reversed (backward) CR [11]. We analyze the electromagnetic field generated by a small bunch with a charge q passing through the interface (located at z ¼ 0) separating two homogeneous isotropic frequency dispersive media described by permittivity and permeability: " 1 ð!Þ,  1 ð!Þ for z< 0 and " 2 ð!Þ,  2 ð!Þ for z> 0 (Fig. 1). There are no surface charges and currents located at z ¼ 0. The bunch moves uniformly along the z axis in accordance with z ¼ Vt ¼ ct. The dimensions of the bunch are assumed to be negligible. Therefore, the charge density  and the current density ~ j ¼ j~ e z can be written in the form  ¼ qðxÞðyÞðz  VtÞ; j ¼ V: (1) Further, we will assume that both media possess nonzero losses, resulting in small positive values of Im" 1;2 > 0 and Im 1;2 > 0. We will let these terms go to 0 in final results. The medium filling the volume z< 0 is right-handed, that is, Re" 1 > 0, Re 1 > 0 for all frequencies where Re" 1 Re 1 > 0. The medium filling the region z> 0 is supposed to have both RHFB and LHFB for propagating waves: (I) RHFB where Re" 1 > 0, Re 1 > 0 and (II) LHFB where Re" 1 < 0, Re 1 < 0. The conditions of continuity of tangential components of electric ( ~ E) and magnetic ( ~ H) strengths must be satisfied in the plane z ¼ 0. q V bunch z 0 ) ( ), ( 1 1 ω µ ω ε 1 st medium: 2 nd medium: ) ( ), ( 2 2 ω µ ω ε θ R FIG. 1. Geometry of the problem. PRL 103, 194802 (2009) PHYSICAL REVIEW LETTERS week ending 6 NOVEMBER 2009 0031-9007= 09=103(19)=194802(4) 194802-1 Ó 2009 The American Physical Society