REPORTS persion at v=0.6c becomes so small that a these tails can exhibit a backward radiation centimeter, which should be amenable to strong intensity peak cone is formed and effect as well. rect experimental observation. almost overlaps with the overall cone. a clear The effects presented here can be extend- A number of applications also appear pos distinction between the Cr behaviors for v< ed to three-dimensional photonic crystals sible. Particles traveling at speeds below the ve and v>vs is that for v >vs the measured with little change. As a complete photonic phase-velocity threshold cannot be detected by flux in the z direction is positive over the band gap in three dimensions is not required, conventional CR counters, and currently their emission band, but for v <v certain frequen- the crystal structure can be flexibly chosen. observation relies on other devices, such as cy regions appear where the flux can be For experimental studies, an appropriate scintillation counters, proportional counters, or negative(Fig 3C). Because positive and neg- structure that quantitatively approximates our cloud chambers. These other devices, however, ative flux values can occur at the same fre- calculations could be a square lattice of air lack the unique advantages of strong velocity quency [e.g, at(oa)/(2Tc)=0.2], this pho- holes perforating a finite-height silicon slab, sensitivity and good radiation directionality onic crystal does not behave as a uniform interacting with fast electrons, and operated as in conventional CR(2). with a photonic negative-index medium, as noted before An- near the communication wavelength. The crystal, one should be able to achieve velocity other distinction lies in the field pattern and is same physics should apply to dielectric-in- selectivity and distinctive radiation pattern similar to that reported in (5) at a phonon air-type crystals as well. a practical issue is without any velocity threshold. Moreover, on resonance frequency: The near-static, nonra- whether the radiation intensity is sufficiently the high-energy side, CR with a sharp radia diating field extends beyond the overall cone strong to be observable for small v. Our tion wavefront is possible for particles trav. when v ve, whereas for v ve, the field numerical simulation indicates that, across a eling through an all-air path inside a photonic outside the cone is strictly zero. These near- bandwidth of around 40%, the average radi- crystal, allowing complete absence of the static fields create an artificial peak around ation energy is roughly in a ratio of 1: 1. 5: 4: impurity scattering and random ionization w=0 in Fig 3C for v vc, (as in v=0.1c) 20 for v/c=0. 1, 0.15, 0.3, and 0.6. Further- losses inherent in a dense medium. This which we have verified to reduce to O for w+ more, the radiation intensity at small v may should improve the performance of present 0 with increasing computational cell sizes even be much larger than these numbers in detectors. Finally, the Cr frequency is set b and time steps. Finally, there are high- narrow bandwidths around specific frequen- the photonic crystal and is thus selectively higher-order radiation, behind the charge in states in a photonic crystal. In conventional ing up the possibility of flexible radiation o all cases of Fig 3B. The higher order modes CR, for v>v an electron can emit hundreds sources for frequencies that are otherwise typically have smaller group velocities and of photons per centimeter of its path. Thus, difficult to access. thus form dense forward-pointing cones for the velocities studied here. the emission smaller angles, as first predicted in(6)(for rates correspond to a range starting from coherently driven slow-light media). Here roughly 10 and ranging up to 200 photons per 1.L D. Landau, E M. Liftshitz, L p. Pitaevski, Electro- 0.1c v=0.15c V=0.3c v=0.6c Pergamon, London, 1958). 3. L M. Frank, Nuc Instrum. Methods Phys. Res. Sect. A 248.7(1986 4. G. N. Afanasiev, V. G. Kartavenko, E. N. Magar. 5. T. E. Stevens, I. K Wahlstrand, ]. KuhL R. Merlin, o033Eo Science291.627(2001) 6. I Carusotto, M. Artoni, G.C. L Rocca, F. Bassani, Phys no cone a.>/2 1/2 J. B. Pendry, A. J. Holden, w.J. Stewart, L Youngs, B IEEE Trans. Microwave Theory Tech. 10. D. R. Smith, W.J. Padilla, D. C. Vi Nasser, S.Schultz, Phys. Rev. Lett. 84, 4184(2000 11. R A Shelby, D. R. Smith, S. Schultz, Science 292, 77 13.5.」smth,E 15. B. Lastdrager, A. Tip, J. Verhoeven, Phys. Rev. E 61 5767( Abajo, Phys. Rev. Lett. 82, 2776(1999). 235(2002) 80 S. John, Phys. Rev. Lett. 58, 2486(198 20. ). D. Joannopoulos, R. D. Meade, J. N. winn, Photonic 6020316 oa/2Ic 0a/2c oa/2Tc ca/2c 22 M: Not c. Phbs, e , 62, 10696 (2000: Fig. 3. FDTD simulation results for CR in the photonic crystal of Fig. 1. Each column represents the n Phys. Rev. 865, 201104(R)(2 M os, Appl. Phys. Its for the value of v shown on the top. (A)Overall radiation cone shapes(dashed lines deduced from the group velocity contours in Fig. 2C.(B)Distribution of the radiated magnetic field Lett81.2352(2002) HyBlue, white, and red represent negative, zero, and positive field values, respectively. The color 24. The CR modes are excited with a strengt tional to the density of bles are chosen separately for best illustration in each case. (C) The frequency spectrum of the electromagnetic flux along z through a line perpendicular to v, in arbitrary units(a u. ). Brillouin zone: G=k-k; and e"rnc is the G-Fourier 370 17JanUary2003Vol299ScieNcewww.sciencemag.orgpersion at v
0.6c becomes so small that a strong intensity peak cone is formed and almost overlaps with the overall cone. A clear distinction between the CR behaviors for v vc and v  vc is that for v  vc the measured flux in the z direction is positive over the emission band, but for v vc certain frequen￾cy regions appear where the flux can be negative (Fig. 3C). Because positive and neg￾ative flux values can occur at the same fre￾quency [e.g., at (a)/(2 c)
0.2], this pho￾tonic crystal does not behave as a uniform negative-index medium, as noted before. An￾other distinction lies in the field pattern and is similar to that reported in (5) at a phonon resonance frequency: The near-static, nonra￾diating field extends beyond the overall cone when v vc, whereas for v  vc, the field outside the cone is strictly zero. These near￾static fields create an artificial peak around 
0 in Fig. 3C for v vc, (as in v
0.1c) which we have verified to reduce to 0 for   0 with increasing computational cell sizes and time steps. Finally, there are high￾frequency radiation “tails,” corresponding to higher-order radiation, behind the charge in all cases of Fig. 3B. The higher order modes typically have smaller group velocities and thus form dense forward-pointing cones of smaller angles, as first predicted in (6) (for coherently driven slow-light media). Here these tails can exhibit a backward radiation effect as well. The effects presented here can be extend￾ed to three-dimensional photonic crystals with little change. As a complete photonic band gap in three dimensions is not required, the crystal structure can be flexibly chosen. For experimental studies, an appropriate structure that quantitatively approximates our calculations could be a square lattice of air holes perforating a finite-height silicon slab, interacting with fast electrons, and operated near the communication wavelength. The same physics should apply to dielectric-in￾air–type crystals as well. A practical issue is whether the radiation intensity is sufficiently strong to be observable for small v. Our numerical simulation indicates that, across a bandwidth of around 40%, the average radi￾ation energy is roughly in a ratio of 1:1.5:4: 20 for v/c
0.1, 0.15, 0.3, and 0.6. Further￾more, the radiation intensity at small v may even be much larger than these numbers in narrow bandwidths around specific frequen￾cies, because of the unusual photon density of states in a photonic crystal. In conventional CR, for v  vc an electron can emit hundreds of photons per centimeter of its path. Thus, for the velocities studied here, the emission rates correspond to a range starting from roughly 10 and ranging up to 200 photons per centimeter, which should be amenable to di￾rect experimental observation. A number of applications also appear pos￾sible. Particles traveling at speeds below the phase-velocity threshold cannot be detected by conventional CR counters, and currently their observation relies on other devices, such as scintillation counters, proportional counters, or cloud chambers. These other devices, however, lack the unique advantages of strong velocity sensitivity and good radiation directionality as in conventional CR (2). With a photonic crystal, one should be able to achieve velocity selectivity and distinctive radiation patterns without any velocity threshold. Moreover, on the high-energy side, CR with a sharp radia￾tion wavefront is possible for particles trav￾eling through an all-air path inside a photonic crystal, allowing complete absence of the impurity scattering and random ionization losses inherent in a dense medium. This should improve the performance of present detectors. Finally, the CR frequency is set by the photonic crystal and is thus selectively scalable beyond optical wavelengths, open￾ing up the possibility of flexible radiation sources for frequencies that are otherwise difficult to access. References and Notes 1. L. D. Landau, E. M. Liftshitz, L. P. Pitaevskii, Electro￾dynamics of Continuous Media (Pergamon, New York, ed. 2, 1984). 2. J. V. Jelly, Cerenkov Radiation and Its Applications (Pergamon, London, 1958). 3. I. M. Frank, Nucl. Instrum. Methods Phys. Res. Sect. A 248, 7 (1986). 4. G. N. Afanasiev, V. G. Kartavenko, E. N. Magar, Physica B 269, 95 (1999). 5. T. E. Stevens, J. K. Wahlstrand, J. Kuhl, R. Merlin, Science 291, 627 (2001). 6. I. Carusotto, M. Artoni, G. C. L. Rocca, F. Bassani, Phys. Rev. Lett. 87, 064801 (2001). 7. V. G. Veselago, Sov. Phys. Usp. 10, 509 (1968). 8. J. B. Pendry, A. J. Holden, W. J. Stewart, I. Youngs, Phys. Rev. Lett. 76, 4773 (1996). 9. J. B. Pendry, A. J. Holden, D. J. Robbins, W. J. Stewart, IEEE Trans. Microwave Theory Tech. 47, 2075 (1999). 10. D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat￾Nasser, S. Schultz, Phys. Rev. Lett. 84, 4184 (2000). 11. R. A. Shelby, D. R. Smith, S. Schultz, Science 292, 77 (2001). 12. R. A. Shelby, D. R. Smith, S. C. Nemat-Nasser, S. Schultz, Appl. Phys. Lett. 78, 489 (2001). 13. S. J. Smith, E. M. Purcell, Phys. Rev. 92, 1069 (1953). 14. K. F. Casey, C. Yeh, Z. A. Kaprielian, Phys. Rev. 140, B768 (1965). 15. B. Lastdrager, A. Tip, J. Verhoeven, Phys. Rev. E 61, 5767 (2000). 16. F. J. Garcia de Abajo, Phys. Rev. Lett. 82, 2776 (1999). 17. K. Ohtaka, S. Yamaguti, Opt. Quantum Electron. 34, 235 (2002). 18. E. Yablonovitch, Phys. Rev. Lett. 58, 2059 (1987). 19. S. John, Phys. Rev. Lett. 58, 2486 (1987). 20. J. D. Joannopoulos, R. D. Meade, J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton Univ. Press, Princeton, NJ, 1995). 21. M. Notomi, Phys. Rev. B 62, 10696 (2000). 22. C. Luo, S. G. Johnson, J. D. Joannopoulos, J. B. Pendry, Phys. Rev. B 65, 201104 (R) (2002). 23. C. Luo, S. G. Johnson, J. D. Joannopoulos, Appl. Phys. Lett. 81, 2352 (2002). 24. The CR modes are excited with a strength propor￾tional to the density of radiation states and qv
e*knG, where k is the Bloch-reduced k in the first Brillouin zone; G
k – k; and e*knG is the G-Fourier Fig. 3. FDTD simulation results for CR in the photonic crystal of Fig. 1. Each column represents the results for the value of v shown on the top. (A) Overall radiation cone shapes (dashed lines) deduced from the group velocity contours in Fig. 2C. (B) Distribution of the radiated magnetic field Hy. Blue, white, and red represent negative, zero, and positive field values, respectively. The color tables are chosen separately for best illustration in each case. (C) The frequency spectrum of the electromagnetic flux along z through a line perpendicular to v, in arbitrary units (a.u.). R EPORTS 370 17 JANUARY 2003 VOL 299 SCIENCE www.sciencemag.org on June 8, 2007 www.sciencemag.org Downloaded from