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REPORTS the position, and t is the time. Bloch waves Cerenkov radiation in photonic rise from the resonance of a planewave source with the photon dispersion relation Cryst (1) Chiyan Luo, Mihai Ibanescu, Steven G. Johnson, where o is the frequency, n is the band index, the wave vector k can include an arbitrary 」D. Joannopoulos reciprocal lattice vector G, and o (k) w (k+ G. The CR consists of modes satis In a conventional material, the coherent Cerenkov radiation due to a moving fying Eq. 1(24), which can be found in k charged particle is associated with a velocity threshold, a forward-pointing adiation cone. and a forward direction of emission We describe different behavior for the Cerenkov radiation in a photonic crystaL. In particular, this A0.5 adiation is intrinsically coupled with transition radiation and is observable without any threshold within one particle-velocity range, we found a radiation pattern with a backward-pointing radiation cone. In another velocity range, backward-propagating Cerenkov radiation can be expected Potential applica tions include velocity-sensitive particle detection and radiation generation at 0.2 selectable frequencies 80.1 When a charged particle travels inside a medi- charge crosses a dielectric boundary or expe- um,it can drive the medium to emit coherent riences an inhomogeneous dielectric environ X electromagnetic energy called Cerenkov radia- ment, and the conventional Cr, in which B 0.5 tion(CR)(). Extensively used in particle de- coherence is preserved throughout the medi- tectors and counters(2), CR in a conventional um. Both effects are incorporated in our ap- material possesses three key characteristics: it proach. However, unlike the Smith-Purcell 3 X occurs only when the particles velocity ex- effect, in which radiation is generated via a ceeds the mediums phase velocity, the energy periodic grating but then propagates through 0 =a/ propagates only in the forward direction, and a uniform medium, this Cr is generated and there is a forward-pointing conical wavefront. propagates within the same crystal in the These characteristics remain qualitatively un- form of Bloch waves. The properties of these -0.5 hanged even in the presence of material dis- Bloch waves can be substantially different 15 persion (3-6). One possible source of unusual from waves in a uniform medium, leading to Kz(2/a) CR is in a medium with simultaneously nega- effects not previously anticipated. In one tive permittivity and permeability, commonly case, we can reverse the overall cone that C known as a negative-index material for its re- encloses all traveling electromagnetic energy versal of Snells law of refraction (7-12), inIn another situation, we demonstrate a back- which CR is predicted to flow backward; i.e., ward-propagating CR behavior reminiscent o033Eo opposite to the particle velocity (7). Another of that predicted in negative-index materials possibility exists near a periodic structure, These are very general results based on direct where simple Bragg scattering of light can give solutions of Maxwells equations and should rise to radiation without any velocity threshold. find applications in particle detection and This was first confirmed by Smith and Purcell wave production techniques 3)in early experiments with electrons travel For simplicity, we focus on a two-dimen ing near the surface of a metallic grating. Cr sional photonic crystal(Fig. 1). Let a charge has since been studied in one-dimensionally q move in the (01) direction of a square periodic multilayer stacks (14, 15), and the lattice of air holes in a dielectric, in the xz Smith-Purcell effect has been extended to near plane, with parameters as specified in the Fig. Fig. 1. Band structure and analysis of cr in a the surface of dielectric structures(16, 17).A 1 legend. Figure 1A shows the calculated tonic crystal.(A) Transverse electric band photonic crystal (18-20), where very complex transverse electric(the electric field in the xz structure of a two-dimensional square lattice of Bragg scattering is possible, presents a rich new plane, appropriate for CR) band structure of air columns in a dielectric dielectric constant e medium for unusual photon phenomena(21- this photonic crystal. We take the particle's lattice period. The crystal structure and the ent particle-velocity regime tric interfaces(Fig. IB, inset). As a reference. nter, and corner of the first Brillouin zone. B) CR in a photonic crystal arises ity(v)of Method of oherent excitation of its eigenmodes this photonic crystal is ve=0. 44c(where c is sects a photonic-crystal dispersion surface. Blu moving the speed of light). The excited radiation can arrows indicate the group velocities of CR transition radiation, which occurs when the be determined by treating the charge as a modes.(C)Method of obtaining CR cone source with space-time dependence 8(r-vr)= shapes. The group velocities for all modes ob- Department of Physics and Center for Materials e, eikr-ik'v, that is, as a superposition of tained in(B)form a contour. a is the cone angle anewaves with different wave vecto es the angle for the overall radiation cone Technology, Cambridge, MA 02139, USA frequencies k. v, where 8 is the Dirac delta ay dashed lines). The angular density of *To whom correspondence should be addressed. E- function, e is the base of the natural loga- the arrows roughly reflects the CR angular rithm, i is the unit of imaginary numbers, ris distribution 17JanUary2003Vol299ScieNcewww.sciencemag.orgCerenkov Radiation in Photonic Crystals Chiyan Luo, Mihai Ibanescu, Steven G. Johnson, J. D. Joannopoulos* In a conventional material, the coherent Cerenkov radiation due to a moving charged particle is associated with a velocity threshold, a forward-pointing radiation cone, and a forward direction of emission. We describe different behavior for the Cerenkov radiation in a photonic crystal. In particular, this radiation is intrinsically coupled with transition radiation and is observable without any threshold. Within one particle-velocity range, we found a radiation pattern with a backward-pointing radiation cone. In another velocity range, backward-propagating Cerenkov radiation can be expected. Potential applications include velocity-sensitive particle detection and radiation generation at selectable frequencies. When a charged particle travels inside a medium, it can drive the medium to emit coherent electromagnetic energy called Cerenkov radiation (CR) (1). Extensively used in particle detectors and counters (2), CR in a conventional material possesses three key characteristics: it occurs only when the particle’s velocity exceeds the medium’s phase velocity, the energy propagates only in the forward direction, and there is a forward-pointing conical wavefront. These characteristics remain qualitatively unchanged even in the presence of material dispersion (3–6). One possible source of unusual CR is in a medium with simultaneously negative permittivity and permeability, commonly known as a negative-index material for its reversal of Snell’s law of refraction (7–12), in which CR is predicted to flow backward; i.e., opposite to the particle velocity (7). Another possibility exists near a periodic structure, where simple Bragg scattering of light can give rise to radiation without any velocity threshold. This was first confirmed by Smith and Purcell (13) in early experiments with electrons traveling near the surface of a metallic grating. CR has since been studied in one-dimensionally periodic multilayer stacks (14, 15), and the Smith-Purcell effect has been extended to near the surface of dielectric structures (16, 17). A photonic crystal (18–20), where very complex Bragg scattering is possible, presents a rich new medium for unusual photon phenomena (21– 23). We reveal a variety of CR patterns that can occur in a single photonic crystal under different particle-velocity regimes. CR in a photonic crystal arises from a coherent excitation of its eigenmodes by the moving charge. Its origin lies in both the transition radiation, which occurs when the charge crosses a dielectric boundary or experiences an inhomogeneous dielectric environment, and the conventional CR, in which coherence is preserved throughout the medium. Both effects are incorporated in our approach. However, unlike the Smith-Purcell effect, in which radiation is generated via a periodic grating but then propagates through a uniform medium, this CR is generated and propagates within the same crystal in the form of Bloch waves. The properties of these Bloch waves can be substantially different from waves in a uniform medium, leading to effects not previously anticipated. In one case, we can reverse the overall cone that encloses all traveling electromagnetic energy. In another situation, we demonstrate a backward-propagating CR behavior reminiscent of that predicted in negative-index materials. These are very general results based on direct solutions of Maxwell’s equations and should find applications in particle detection and wave production techniques. For simplicity, we focus on a two-dimensional photonic crystal (Fig. 1). Let a charge q move in the (01) direction of a square lattice of air holes in a dielectric, in the xz plane, with parameters as specified in the Fig. 1 legend. Figure 1A shows the calculated transverse electric (the electric field in the xz plane, appropriate for CR) band structure of this photonic crystal. We take the particle’s motion to be in the z direction and consider a path where the particle does not cross dielectric interfaces (Fig. 1B, inset). As a reference, the long-wavelength phase velocity (vc) of this photonic crystal is vc 0.44c (where c is the speed of light). The excited radiation can be determined by treating the charge as a source with space-time dependence (r–vt) k eik r – ik vt , that is, as a superposition of planewaves with different wave vectors k and frequencies k v, where is the Dirac delta function, e is the base of the natural logarithm, i is the unit of imaginary numbers, r is the position, and t is the time. Bloch waves arise from the resonance of a planewave source with the photon dispersion relation n(k) k v (1) where is the frequency, n is the band index, the wave vector k can include an arbitrary reciprocal lattice vector G, and n(k) n(k G). The CR consists of modes satisfying Eq. 1 (24), which can be found in k Department of Physics and Center for Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA. *To whom correspondence should be addressed. Email: joannop@mit.edu Fig. 1. Band structure and analysis of CR in a photonic crystal. (A) Transverse electric band structure of a two-dimensional square lattice of air columns in a dielectric (dielectric constant ε 12) with column radii r 0.4a, a being the lattice period. The crystal structure and the irreducible Brillouin zone are shown as insets. , X, and M are, respectively, the center, edge center, and corner of the first Brillouin zone. (B) Method of solving Eq. 1 in k space. CR occurs when the kz /v plane (dashed line) intersects a photonic-crystal dispersion surface. Blue arrows indicate the group velocities of CR modes. (C) Method of obtaining CR cone shapes. The group velocities for all modes obtained in (B) form a contour. is the cone angle for the mode with group velocity u, and m gives the angle for the overall radiation cone (gray dashed lines). The angular density of the arrows roughly reflects the CR angular distribution. R EPORTS 368 17 JANUARY 2003 VOL 299 SCIENCE www.sciencemag.org on June 8, 2007 www.sciencemag.org Downloaded from