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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 13 — Jun. 21, 2010
  • pp: 14165–14172
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Imitating the Cherenkov radiation in backward directions using one-dimensional photonic wires

Tetsuyuki Ochiai  »View Author Affiliations


Optics Express, Vol. 18, Issue 13, pp. 14165-14172 (2010)
http://dx.doi.org/10.1364/OE.18.014165


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Abstract

A novel radiation emission from traveling charged particles in vacuum is theoretically demonstrated. This radiation is conical as in the Cherenkov radiation, but emerges in backward directions of the particle trajectories. The basic mechanism of the radiation is the Smith-Purcell effect via the interaction between the charged particles and a circular-symmetric photonic wire with a one-dimensionally periodic dielectric function. The wire exhibits the photonic band structure characterized with angular momentum. The charged particle can resonantly excite the photonic band modes with particular angular momentum, depending on the particle velocity. A simple kinetics of the Smith-Purcell effect enables us to design the conical radiation emitted in backward directions. Numerical results of the backward radiation are also presented for a metallic wire with aligned air holes.

© 2010 Optical Society of America

The Cherenkov radiation is a radiation emission induced by external charged particles traveling in transparent medium [1

1. P. A. Cherenkov, “Visible Emission of Clean Liquids by Action of Radiation,” Dokl. Akad. Nauk SSSR 2, 451 (1934).

, 2

2. L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media (Butterworth-Heinemann, Oxford, 1985).

]. This radiation takes place when the velocity of the particles is faster than light velocity in the medium. It has a conical distribution (so-called Cherenkov cone) in the emission angle and has a broad spectrum. Since its first experimental observation in 1934, the Cherenkov radiation has played a major role in detection of high-energy charged particles. Its playgrounds range from subatomic physics to cosmology and biology.

In ordinary medium, the Cherenkov cone, which is formed by relevant wave number vectors, emerges with angle θ CR = cos−1(c/(nv)) in the forward direction of the particle trajectory. Here, n,c, and v refer to the refractive index of the medium, light velocity in vacuum, and the particle velocity, respectively. In 1968, Veselago argued that the Cherenkov cone appears in the backward direction, in a medium with simultaneously negative permittivity and permeability, called double-negative material [3

3. V. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and µ,” Sov. Phys. Usp. 10(4), 509–514 (1968). [CrossRef]

]. This prediction has stimulated theoretical studies of the Cherenkov radiation in artificially photonic structures [4–8

4. C. Luo, M. Ibanescu, S. G. Johnson, and J. D. Joannopoulos, “Cerenkov radiation in photonic crystals,” Science 299(5605), 368–371 (2003). [CrossRef] [PubMed]

], after an experimental realization of the double-negative material [9

9. R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292(5514), 77–79 (2001). [CrossRef] [PubMed]

]. In particular, certain two-dimensional photonic crystals are shown to exhibit the backward Cherenkov radiation [4

4. C. Luo, M. Ibanescu, S. G. Johnson, and J. D. Joannopoulos, “Cerenkov radiation in photonic crystals,” Science 299(5605), 368–371 (2003). [CrossRef] [PubMed]

]. Experimental evidences of the backward Cherenkov radiation were reported recently [10

10. S. Xi, H. Chen, T. Jiang, L. Ran, J. Huangfu, B. I. Wu, J. A. Kong, and M. Chen, “Experimental Verification of Reversed Cherenkov Radiation in Left-Handed Metamaterial,” Phys. Rev. Lett. 103(19), 194801 (2009). [CrossRef]

].

Let us consider a photonic wire structure with one-dimensional periodicity with period a in the dielectric function. We assume that the photonic wire has the circular symmetry with respect to the wire axis. Therefore, the photonic band modes are classified according to the angular momentum mZ. There is a double degeneracy between m and −m, except for m = 0, unless optical activity is negligible. Such photonic wires include linear chain of spherical nano-particles [12

12. F. J. García de Abajo, “Smith-Purcell radiation emission in aligned nanoparticles,” Phys. Rev. E 61(5), 5743–5752 (2000). [CrossRef]

, 13

13. Y. Hara, T. Mukaiyama, K. Takeda, and M. Kuwata-Gonokami, “Heavy photon states in photonic chains of resonantly coupled cavities with supermonodispersive microspheres,” Phys. Rev. Lett. 94(20), 203905 (2005). [CrossRef] [PubMed]

], distributed Bragg reflector (DBR) of circular pillar [14

14. J. M. Gérard, B. Sermage, B. Gayral, B. Legrand, E. Costard, and V. Thierry-Mieg, “Enhanced spontaneous emission by quantum boxes in a monolithic optical microcavity,” Phys. Rev. Lett. 81(5), 1110–1113 (1998). [CrossRef]

], and fiber Bragg grating [15

15. K. O. Hill and G. Meltz, “Fiber Bragg grating technology fundamentals and overview,” J. Lightwave Technol. 15(8), 1263–1276 (1997). [CrossRef]

]. Then, we consider a charged particle traveling outside the wire, in parallel to the wire axis. A schematic illustration of the system under study is shown in Fig. 1, for a wire with aligned spherical particles inside.

Fig. 1. Schematic illustration of the system under study. It consists of a periodic arrangement of spherical air holes with lattice constant a, embedded in a infinitely-long circular cylinder. A charged particle travels with impact parameter b, outside the cylinder. The particle trajectory is parallel to the cylindrical axis (taken to be the z axis). The charged particle can induce the radiation emission that can be detected at a far-field observation point of solid angle Ω = (θ,ϕ).

The traveling charged particle accompanies the evanescent radiation wave whose dispersion relation is given by ω = vkz, called v-line. This line is outside the light cone ω = c|kz| in the frequency- momentum space, and thus the charged particle cannot emit propagating radiation by itself. However, the evanescent wave is scattered by the photonic wire, acquiring the Umklapp shift to the initial momentum kz. The resulting radiation, whose dispersion relation is given by ω = v(kz+G) being G a reciprocal lattice, can enter inside the light cone. Therefore, the induced radiation can be propagating. This is the so-called Smith-Purcell effect which provides a different mechanism of charged-particle-induced radiation. Besides, the scattering by the photonic wire can show a significant enhancement owing to a resonance in the photonic wire. Let us express the dispersion curve of the photonic band modes with angular momentum m by ω = ωm(kz). The photonic bands extend in the entire first Brillouin zone (−π/a < kz < π/a). If the shifted v-line ω = v(kz + G) intersects the dispersion curve, the charged particle excites resonantly the photonic band mode at the intersection point [16

16. A similar mechanism works in photonic crystals if they are periodic in the direction of the particle trajectory [4, 6, 7].

]. When the point is inside the light cone, a strong signal of the Smith-Purcell radiation emerges. The emission (polar) angle of the Smith-Purcell radiation is determined kinetically by

θG=cos1(cνcGω).
(1)

We should stress here that the polar angle θG can be greater than 90°, in a striking contrast to the ordinary Cherenkov radiation with θ CR < cos−1(1/n) < 90°. Hence, the emission angle can be in the backward direction. Moreover, the radiation has the dominant angular momentum of ±m, depending on the mode to be excited.

As a numerical demonstration, we consider a metallic wire with periodic arrangement of spherical air holes inside. We choose this system simply because numerical calculation of high accuracy is available by employing vector cylindrical- and spherical-wave expansions along with the photonic Korringa-Kohn-Rostoker formalism [17

17. K. Ohtaka, “Energy-band of photons and low-energy photon diffraction,” Phys. Rev. B 19(10), 5057–5067 (1979). [CrossRef]

, 18

18. K. Ohtaka, T. Ueta, and K. Amemiya, “Calculation of photonic bands using vector cylindrical waves and reflectivity of light for an array of dielectric rods,” Phys. Rev. B 57(4), 2550–2568 (1998). [CrossRef]

]. The dielectric constant of the wire is modeled with the free-electron metal, ε(ω) = 1−ω 2 p/ω/(ω + ), and the radius of the wire is taken to be 0.5a. The air-hole radius is fixed to be 0.3a.

Figure 2 shows the angular-momentum-resolved photonic band structure of the photonic wire. These bands are basically the zone-folded ones of the surface plasmon polariton of the homogeneous metal wire, and thus the band diagram becomes dense around ω = ωp/√2 (we choose ωpa/2πc = 1). In reality, the dense region is not favored because if we take account of dissipation, photonic bands there are readily merged, turning into a broad peak of the photonic density of states. Therefore, we are forced to utilize the bands at low frequencies.

Fig. 2. The photonic band diagram of the metallic wire with periodic arrangement of air holes inside. The radius and the dielectric constant of the wire are given by 0.5a and ε = 1−(ωp/ω)2, respectively, where ωpa/(2πc) is fixed to be 1. The air-hole radius is taken to be 0.3a. The photonic bands are classified according to angular momentum m (indicated in the legend) with respect to the wire axis. The light line ω = c|kz| and the shifted v-line ω = v(kz + G) with v = 0.7c are also shown by dashed and solid lines, respectively.

Let us consider the radiation spectrum induced by an electron traveling with velocity v=0.7c and impact parameter b = a. By overlaying the shifted v-line to the photonic band diagram, we can find a sequence of the intersection points, which represent the excitation of the photonic band modes. Figure 3 shows the resulting radiation spectra Γ(ω) with and without dissipation in the metal wire. To be precise, the net radiation energy W obtained in far-field over all solid angles is expressed as

W=0dωh¯ωΓ(ω),
(2)

per unit trajectory length. The quantity Γ(ω) represents the probability of finding the propagating photon in the frequency interval between ω and ω + , over all solid angles in far-field.

In Fig. 3 the spectra have the low-frequency cut off at ωa/2πc = v/(c+v) and decreases exponentially from ωa/2πc = 0.75 to 1.0, owing to the absence of the photonic band modes. The spectra above ωa/2πc = 1.0 (ω = ωp) are less than 10−7 of the vertical axis (not shown). It is remarkable that a sequence of sharp peaks are obtained in the radiation spectrum even if the dissipation is taken into account. The peaky spectra show a striking contrast to the flat spectrum of the Cherenkov radiation,

ΓCR(ω)=μ0e22h(1(cnv)2),
(3)
Fig. 3. The Smith-Purcell radiation spectra Γ(ω) for dissipation-less (η = 0, black curve) and dissipative (ηa/(2πc) = 0.01, red curve) metallic wire. The following parameters are used: v = 0.7c and b = a. For comparison, the Cherenkov radiation spectrum ΓCR(ω) is also shown (by blue curve) for a medium with refractive index n = 2.

provided that n is dispersion-free. Here, µ 0,e, and h(= 2πh̄) stand for the vacuum permeability, electron charge, and Planck’s constant, respectively. The lowest (in frequency) three peaks, for instance, are attributed to m = 0 (ωa/(2πc) = 0.447), m = ±1 (0.508) and m = ±2 (0.580). Each peak has the certain height and width, from which we can evaluate the probability of finding photon around the peak frequency. For instance, in the lowest peak, the probability is evaluated as 1.52×10−4 per electron and per spherical hole. Here, we approximated the peak by a rectangular one of width δωa/(2πc) = 0.01 and height Γ = 0.01µ 0 e 2/(2h). This small probability can be increased with the multiple factor of N 2 e by using bunched electrons of number Ne. Another way to increase the probability is to decrease impact parameter b.

The azimuthal-angle distribution dΓ/ of the induced radiation at the peak frequencies is shown in Fig. 4. Here, the polar-angle distribution is given by a delta-function δ (θθG), where θG ~ 144°,123°, and 107° for the lowest three peaks. It is remarkable that the angular distribution at ωa/2πc = 0.447 is almost circular symmetric, while the other two exhibits two-fold and four-fold rotational symmetries. In general, a 2|m|-fold symmetric angular distribution is obtained by exciting the mode with ±m. These two angular-momentum components dominate in the radiation field E. However, the cross term in the radiation intensity |E|2 between m and −m gives rise to the ϕ-dependence of exp(±2imϕ), having the 2|m|-fold symmetry.

Finally, the energy density and the Poynting vector flow of the radiation field at the lowest peak frequency (ωa/(2πc) = 0.447) are shown in Fig. 5. We can find that the Poynting vector flow exhibits the radiation emission in backward directions in far-field. In near-field, however, it is directed to forward directions (not-shown because of a strong scale contrast of the Poynting vector between near- to far-field). This is reasonable because the relevant photonic band mode has the positive group velocity in the z direction. In the intermediate region 1.4 < x/a < 2.0 an optical eddy is formed connecting from the near- to the far-field configurations. Besides, the maximal energy density is found on the surface of the metal wire. This property indicates that the photonic band mode concerned comes from the surface plasmon polariton of the homogeneous metal wire. Note that the above configuration is not of a time-snapshot, but of a particular time-Fourier component. Therefore, the wavefront is hidden there. We will see a clear wavefront in a time-domain calculation.

Fig. 4. The azimuthal angle distribution of the induced radiation emission at the lowest three peaks in Fig. 3. Black, red, and blue curves stand for ωa/(2πc) = 0.447,0.508, and 0.580, respectively.

Let us discuss in detail our radiation in comparison to the ordinary Cherenkov radiation. The spectrum of our radiation is peaky, practically regarded as a set of monochromatic waves having different frequencies. Each monochromatic wave has the fixed polar angle, and the intensity distribution of the azimuthal angle oscillates with exp(±2imϕ). Two extreme cases should be noted: m = 0 and m→±∞. Both of the cases give rise to a conical radiation with the circular symmetry. We should comment, however, that such circular-symmetric distribution can take place even at finite m, when the degeneracy between m and −m is lifted. The lifting occurs if the substances have nonzero magneto-optical effect and a static magnetic field is applied parallel to the wire axis. The polar angle changes with changing the velocity of the particle and the physical parameters of the photonic wire. The physical parameters, such as the air-hole radius and wire radius in Fig. 1, affect the photonic band structure and thus the resonance frequencies. In this way, we can control the emission angle.

Fig. 5. The radiation energy density and the Poynting vector flow in the zx plane (y = 0) at ωa/2πc = 0.447. The Poynting vector flow below x/a = 1.4 is omitted and the maximal energy density is normalized to be 1. The electron trajectory is at (x,y) = (a,0), and is indicated by the green arrow.

In summary, we have presented a simple model to imitate the Cherenkov radiation in backward directions. This radiation is caused by the Smith-Purcell effect via the interaction between charged particles and a photonic wire structure with a one-dimensionally periodic dielectric function. The key ingredients are the circular symmetry and the one-dimensional periodicity of the photonic wire. The excitation of the photonic band modes of zero angular momentum can yield a conical radiation emitted in backward directions. The excitation of the modes with nonzero angular momentum m results in peculiar radiation patterns with 2|m|-fold rotational symmetries. However, they can be conical if optical activity is non-negligible. We have also discussed possible experimental realizations using more realistic photonic wires.

Acknowledgements

This work was partially supported by Ministry of Education, Culture, Sports, Science and Technology, Japan (MEXT) Grant No. 20560042.

References and links

1.

P. A. Cherenkov, “Visible Emission of Clean Liquids by Action of Radiation,” Dokl. Akad. Nauk SSSR 2, 451 (1934).

2.

L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media (Butterworth-Heinemann, Oxford, 1985).

3.

V. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and µ,” Sov. Phys. Usp. 10(4), 509–514 (1968). [CrossRef]

4.

C. Luo, M. Ibanescu, S. G. Johnson, and J. D. Joannopoulos, “Cerenkov radiation in photonic crystals,” Science 299(5605), 368–371 (2003). [CrossRef] [PubMed]

5.

F. J. García de Abajo, A. G. Pattantyus-Abraham, N. Zabala, A. Rivacoba, M. O. Wolf, and P. M. Echenique, “Cherenkov effect as a probe of photonic nanostructures,” Phys. Rev. Lett. 91(14), 143902 (2003). [CrossRef]

6.

T. Ochiai and K. Ohtaka, “Electron energy loss and Smith-Purcell radiation in two- and three-dimensional photonic crystals,” Opt. Express 13(19), 7683–7698 (2005). [CrossRef] [PubMed]

7.

C. Kremers, D. N. Chigrin, and J. Kroha, “Theory of Cherenkov radiation in periodic dielectric media: Emission spectrum,” Phys. Rev. A 79(1), 013829 (2009). [CrossRef]

8.

S. N. Galyamin, A. V. Tyukhtin, A. Kanareykin, and P. Schoessow, “Reversed Cherenkov-Transition Radiation by a Charge Crossing a Left-Handed Medium Boundary,” Phys. Rev. Lett. 103(19), 194802 (2009). [CrossRef]

9.

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292(5514), 77–79 (2001). [CrossRef] [PubMed]

10.

S. Xi, H. Chen, T. Jiang, L. Ran, J. Huangfu, B. I. Wu, J. A. Kong, and M. Chen, “Experimental Verification of Reversed Cherenkov Radiation in Left-Handed Metamaterial,” Phys. Rev. Lett. 103(19), 194801 (2009). [CrossRef]

11.

S. J. Smith and E. M. Purcell, “Visible light from localized surface charges moving across a grating,” Phys. Rev. 92(4–15), 1069 (1953). [CrossRef]

12.

F. J. García de Abajo, “Smith-Purcell radiation emission in aligned nanoparticles,” Phys. Rev. E 61(5), 5743–5752 (2000). [CrossRef]

13.

Y. Hara, T. Mukaiyama, K. Takeda, and M. Kuwata-Gonokami, “Heavy photon states in photonic chains of resonantly coupled cavities with supermonodispersive microspheres,” Phys. Rev. Lett. 94(20), 203905 (2005). [CrossRef] [PubMed]

14.

J. M. Gérard, B. Sermage, B. Gayral, B. Legrand, E. Costard, and V. Thierry-Mieg, “Enhanced spontaneous emission by quantum boxes in a monolithic optical microcavity,” Phys. Rev. Lett. 81(5), 1110–1113 (1998). [CrossRef]

15.

K. O. Hill and G. Meltz, “Fiber Bragg grating technology fundamentals and overview,” J. Lightwave Technol. 15(8), 1263–1276 (1997). [CrossRef]

16.

A similar mechanism works in photonic crystals if they are periodic in the direction of the particle trajectory [4, 6, 7].

17.

K. Ohtaka, “Energy-band of photons and low-energy photon diffraction,” Phys. Rev. B 19(10), 5057–5067 (1979). [CrossRef]

18.

K. Ohtaka, T. Ueta, and K. Amemiya, “Calculation of photonic bands using vector cylindrical waves and reflectivity of light for an array of dielectric rods,” Phys. Rev. B 57(4), 2550–2568 (1998). [CrossRef]

OCIS Codes
(050.1960) Diffraction and gratings : Diffraction theory
(290.4210) Scattering : Multiple scattering
(230.5298) Optical devices : Photonic crystals

ToC Category:
Physical Optics

History
Original Manuscript: May 25, 2010
Revised Manuscript: June 9, 2010
Manuscript Accepted: June 9, 2010
Published: June 16, 2010

Citation
Tetsuyuki Ochiai, "Imitating the Cherenkov radiation in backward directions using one-dimensional photonic wires," Opt. Express 18, 14165-14172 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-13-14165


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References

  1. P. A. Cherenkov, “Visible Emission of Clean Liquids by Action of Radiation,” Dokl. Akad. Nauk SSSR 2, 451 (1934).
  2. L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media (Butterworth-Heinemann, Oxford, 1985).
  3. V. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ,” Sov. Phys. Usp. 10(4), 509–514 (1968). [CrossRef]
  4. C. Luo, M. Ibanescu, S. G. Johnson, and J. D. Joannopoulos, “Cerenkov radiation in photonic crystals,” Science 299(5605), 368–371 (2003). [CrossRef] [PubMed]
  5. F. J. García de Abajo, A. G. Pattantyus-Abraham, N. Zabala, A. Rivacoba, M. O. Wolf, and P. M. Echenique, “Cherenkov effect as a probe of photonic nanostructures,” Phys. Rev. Lett. 91(14), 143902 (2003). [CrossRef]
  6. T. Ochiai, and K. Ohtaka, “Electron energy loss and Smith-Purcell radiation in two- and three-dimensional photonic crystals,” Opt. Express 13(19), 7683–7698 (2005). [CrossRef] [PubMed]
  7. C. Kremers, D. N. Chigrin, and J. Kroha, “Theory of Cherenkov radiation in periodic dielectric media: Emission spectrum,” Phys. Rev. A 79(1), 013829 (2009). [CrossRef]
  8. S. N. Galyamin, A. V. Tyukhtin, A. Kanareykin, and P. Schoessow, “Reversed Cherenkov-Transition Radiation by a Charge Crossing a Left-Handed Medium Boundary,” Phys. Rev. Lett. 103(19), 194802 (2009). [CrossRef]
  9. R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292(5514), 77–79 (2001). [CrossRef] [PubMed]
  10. S. Xi, H. Chen, T. Jiang, L. Ran, J. Huangfu, B. I. Wu, J. A. Kong, and M. Chen, “Experimental Verification of Reversed Cherenkov Radiation in Left-Handed Metamaterial,” Phys. Rev. Lett. 103(19), 194801 (2009). [CrossRef]
  11. S. J. Smith and E. M. Purcell, “Visible light from localized surface charges moving across a grating,” Phys. Rev. 92(4–15), 1069 (1953). [CrossRef]
  12. F. J. García de Abajo, “Smith-Purcell radiation emission in aligned nanoparticles,” Phys. Rev. E 61(5), 5743–5752 (2000). [CrossRef]
  13. Y. Hara, T. Mukaiyama, K. Takeda, and M. Kuwata-Gonokami, “Heavy photon states in photonic chains of resonantly coupled cavities with supermonodispersive microspheres,” Phys. Rev. Lett. 94(20), 203905 (2005). [CrossRef] [PubMed]
  14. J. M. Gérard, B. Sermage, B. Gayral, B. Legrand, E. Costard, and V. Thierry-Mieg, “Enhanced spontaneous emission by quantum boxes in a monolithic optical microcavity,” Phys. Rev. Lett. 81(5), 1110–1113 (1998). [CrossRef]
  15. K. O. Hill, and G. Meltz, “Fiber Bragg grating technology fundamentals and overview,” J. Lightwave Technol. 15(8), 1263–1276 (1997). [CrossRef]
  16. A similar mechanism works in photonic crystals if they are periodic in the direction of the particle trajectory [4,6,7].
  17. K. Ohtaka, “Energy-band of photons and low-energy photon diffraction,” Phys. Rev. B 19(10), 5057–5067 (1979). [CrossRef]
  18. K. Ohtaka, T. Ueta, and K. Amemiya, “Calculation of photonic bands using vector cylindrical waves and reflectivity of light for an array of dielectric rods,” Phys. Rev. B 57(4), 2550–2568 (1998). [CrossRef]

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