## Imitating the Cherenkov radiation in backward directions using one-dimensional photonic wires

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

*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]

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]

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]

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]

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]

*m*∈

**Z**. 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. 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]

*ω*=

*vk*, called

_{z}*v*-line. This line is outside the light cone

*ω*=

*c*|

*k*| 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

_{z}*k*. The resulting radiation, whose dispersion relation is given by

_{z}*ω*=

*v*(

*k*+

_{z}*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*=

*ω*(

_{m}*k*). The photonic bands extend in the entire first Brillouin zone (−

_{z}*π*/

*a*<

*k*<

_{z}*π*/

*a*). If the shifted

*v*-line

*ω*=

*v*(

*k*+

_{z}*G*) intersects the dispersion curve, the charged particle excites resonantly the photonic band mode at the intersection point [16]. 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

*θ*can be greater than 90°, in a striking contrast to the ordinary Cherenkov radiation with

_{G}*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.

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]

*ε*(

*ω*) = 1−

*ω*

^{2}

*/*

_{p}*ω*/(

*ω*+

*iη*), and the radius of the wire is taken to be 0.5

*a*. The air-hole radius is fixed to be 0.3

*a*.

*ω*=

*ω*/√2 (we choose

_{p}*ω*/2

_{p}a*πc*= 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.

*v*=0.7

*c*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

*ω*)

*dω*represents the probability of finding the propagating photon in the frequency interval between

*ω*and

*ω*+

*dω*, over all solid angles in far-field.

*ωa*/2

*πc*=

*v*/(

*c*+

*v*) and decreases exponentially from

*ωa*/2

*πc*= 0.75 to 1.0, owing to the absence of the photonic band modes. The spectra above

*ωa*/2

*πc*= 1.0 (

*ω*=

*ω*) are less than 10

_{p}^{−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,

*n*is dispersion-free. Here,

*µ*

_{0},

*e*, and

*h*(= 2

*πh̄*) 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*/(2

*πc*) = 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*/(2

*πc*) = 0.01 and height Γ = 0.01

*µ*

_{0}

*e*

^{2}/(2

*h*). This small probability can be increased with the multiple factor of

*N*

^{2}

*by using bunched electrons of number*

_{e}*N*. Another way to increase the probability is to decrease impact parameter

_{e}*b*.

*d*Γ/

*dϕ*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*(

*θ*−

*θ*), where

_{G}*θ*~ 144°,123°, and 107° for the lowest three peaks. It is remarkable that the angular distribution at

_{G}*ωa*/2

*πc*= 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

**. However, the cross term in the radiation intensity |**

*E***|**

*E*^{2}between

*m*and −

*m*gives rise to the

*ϕ*-dependence of exp(±2

*imϕ*), having the 2|

*m*|-fold symmetry.

*imϕ*). 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.

*l*, and there is a 2

*l*+1-fold degeneracy concerning the azimuthal angular momentum

*m*. This degeneracy is lifted by the coupling, resulting in a rather dense band structure of lifted bands. The dense bands tend to mix with each other if the dissipation is non-negligible. The previous study on the Smith-Purcell radiation in aligned nano-particles indicates the above features, but nearly circular-symmetric angular distribution was obtained at a resonance frequency [12

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

*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

## References and links

1. | P. A. Cherenkov, “Visible Emission of Clean Liquids by Action of Radiation,” Dokl. Akad. Nauk SSSR |

2. | L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, |

3. | V. Veselago, “The electrodynamics of substances with simultaneously negative values of |

4. | C. Luo, M. Ibanescu, S. G. Johnson, and J. D. Joannopoulos, “Cerenkov radiation in photonic crystals,” Science |

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. |

6. | T. Ochiai and K. Ohtaka, “Electron energy loss and Smith-Purcell radiation in two- and three-dimensional photonic crystals,” Opt. Express |

7. | C. Kremers, D. N. Chigrin, and J. Kroha, “Theory of Cherenkov radiation in periodic dielectric media: Emission spectrum,” Phys. Rev. A |

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. |

9. | R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science |

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. |

11. | S. J. Smith and E. M. Purcell, “Visible light from localized surface charges moving across a grating,” Phys. Rev. |

12. | F. J. García de Abajo, “Smith-Purcell radiation emission in aligned nanoparticles,” Phys. Rev. E |

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. |

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. |

15. | K. O. Hill and G. Meltz, “Fiber Bragg grating technology fundamentals and overview,” J. Lightwave Technol. |

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 |

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 |

**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

- P. A. Cherenkov, “Visible Emission of Clean Liquids by Action of Radiation,” Dokl. Akad. Nauk SSSR 2, 451 (1934).
- L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media (Butterworth-Heinemann, Oxford, 1985).
- V. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ,” Sov. Phys. Usp. 10(4), 509–514 (1968). [CrossRef]
- C. Luo, M. Ibanescu, S. G. Johnson, and J. D. Joannopoulos, “Cerenkov radiation in photonic crystals,” Science 299(5605), 368–371 (2003). [CrossRef] [PubMed]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- F. J. García de Abajo, “Smith-Purcell radiation emission in aligned nanoparticles,” Phys. Rev. E 61(5), 5743–5752 (2000). [CrossRef]
- 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]
- 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]
- K. O. Hill, and G. Meltz, “Fiber Bragg grating technology fundamentals and overview,” J. Lightwave Technol. 15(8), 1263–1276 (1997). [CrossRef]
- A similar mechanism works in photonic crystals if they are periodic in the direction of the particle trajectory [4,6,7].
- K. Ohtaka, “Energy-band of photons and low-energy photon diffraction,” Phys. Rev. B 19(10), 5057–5067 (1979). [CrossRef]
- 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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