## Forward- and backward-propagating Cerenkov radiation in strong chiral media

Optics Express, Vol. 15, Issue 15, pp. 9793-9798 (2007)

http://dx.doi.org/10.1364/OE.15.009793

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### Abstract

The mathematical solution for Cerenkov radiation (CR) in lossless chiral media, which has the strong chiral parameters, is introduced in this paper. We reveal unique behavior for the CR in strong chiral medium under different particle-velocity regimes. Within one particle-velocity range, a radiation pattern with double cone of propagation can be expected, and the radiation is associated with forward and backward directions of emission.

© 2007 Optical Society of America

## 1. Introduction

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

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

5. J. B. Pendry, “A chiral route to negative refraction,” Science **306**, 1353–1355 (2004). [CrossRef] [PubMed]

8. Q. Cheng and T. J. Cui, “Negative refractions and backward waves in biaxially anisotropic chiral media,” Opt. Express **14**, 6322–6332 (2006). [CrossRef] [PubMed]

9. S. Tretyakov, A. Sihvola, and L. Jylha, “Backward-wave regime and negative refraction in chiral composites,” Photonics Nanostruct. Fundam. Appl. **3**, 107–117 (2005). [CrossRef]

10. Y. Jin and S. He, “Focusing by a slab of chiral medium,” Opt. Express **13**, 4974–4979 (2005). [CrossRef] [PubMed]

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

12. J. Lu, T. Grzegorczyk, Y. Zhang, J. Pacheco Jr., B. -I. Wu, J. A. Kong, and M. Chen, “Čerenkov radiation in materials with negative permittivity and permeability,” Opt. Express **11**, 723–734 (2003) [CrossRef] [PubMed]

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

14. B. Lastdrager, A. Tip, and J. Verhoeven, “Theory of Čerenkov and transition radiation from layered structures,” Phys. Rev. E **61**, 5767–5778 (2000). [CrossRef]

15. C. Y. Luo, M. Ibanescu, S. G. Johnson, and J. D. Joannopoulos, “Cerenkov radiation in Photonic Crystals,” Science **299**, 368–371 (2003). [CrossRef] [PubMed]

## 2. General formulations

*κ*indicates the chirality, which is assumed to be positive in this paper. Similar dual conclusions can be easily expanded to the negative chirality.

*ε*and

*µ*are the permittivity and permeability of the chiral medium, respectively.

_{+}and ν

_{-}, respectively:

17. S. Tretyakov, I. Nefedov, A. Sihvola, S. Maslovski, and C. Simovski, “Waves and energy in chiral nihility,” J. Electromagn. Waves Appl. **17**, 695–706 (2003). [CrossRef]

7. T. G. Mackay, “Plane waves with negative phase velocity in isotropic chiral mediums,” Microwave Opt. Technol. Lett. **45**, 120–121 (2005). [CrossRef]

9. S. Tretyakov, A. Sihvola, and L. Jylha, “Backward-wave regime and negative refraction in chiral composites,” Photonics Nanostruct. Fundam. Appl. **3**, 107–117 (2005). [CrossRef]

10. Y. Jin and S. He, “Focusing by a slab of chiral medium,” Opt. Express **13**, 4974–4979 (2005). [CrossRef] [PubMed]

18. C. Zhang and T. J. Cui, “Spatial dispersion and energy in strong chiral medium,” Opt. Express **15**, 5114–5119 (2007). [CrossRef] [PubMed]

*Ẽ*and magnetic field

*H̃*become

*Ẽ*and

*H̃*represent the Fourier transforms of

*E⃑*(

*t*) and

*H⃑*(

*t*), respectively. Since Eq. (5) is linear, its solution can be expressed as the following:

*V′*is the source region occupied by

*J̃*(

*r⃑′*), and

*r,r′*) is the dyadic Green’s function of the observation point

*r⃑*and of the source point

*r⃑′*in the unbounded chiral medium [19].

*φ*due to isotropy of the chiral media, we can get

*Ĝ*

_{±}(

*ρ*,

*φ*,

*z*;

*ρ′*,

*φ′*,

*ω*/

*ν*=(

*i*/4)

*H*

^{(1)}

_{0}(

*γ*

_{±}

*R*)

*e*,

^{iβz}*β*=

*ω*/

*ν*,

*H*() denotes the Hankel function

*is the three-dimensional unit dyadic.*

**U***=0 into (8), then after a lengthy mathematical manipulation,*

**ρ′***Ẽ*can be expressed as the following:

*τ*

_{±}=

*γ*

_{±}

*ρ*

*ρ,̂*and

*are the unit vectors along*φ ^

*ρ*and

*φ*directions, respectively. After a similar manipulation, the solution for Eq. (6) (the corresponding magnetic field

*H̃*) can be obtained:

## 3. Discussions

_{±}|<ν.

*Ẽ*

_{+}(for the wave number

*k*

_{+}) and

*Ẽ*

_{-}(for the wave number

*k*

_{-}). Since we are interested in radiation from the charge, we use the asymptotic values of

*H*

^{(1)}

_{0}(

*τ*

_{±}) and

*H*

^{(1)}

_{1}(

*τ*

_{±}) to find the far-field solutions.

*τ*

_{±}|≫1) asymptotic expression for the Hankel functions,

*Ẽ*

_{±},

*H̃*

_{±}in the following way:

*and*ρ ^

*ẑ*direction for the left- and right-polarized waves are written as:

*W*

_{+z}>0,

*W*

_{+ρ}>0,

*W*

_{-z}<0,

*W*

_{-ρ}>0. Then it is found that CR in this case is associated with forward and backward directions of emission. Somewhat Similar phenomenon is observed in photonic crystal [15

15. C. Y. Luo, M. Ibanescu, S. G. Johnson, and J. D. Joannopoulos, “Cerenkov radiation in Photonic Crystals,” Science **299**, 368–371 (2003). [CrossRef] [PubMed]

12. J. Lu, T. Grzegorczyk, Y. Zhang, J. Pacheco Jr., B. -I. Wu, J. A. Kong, and M. Chen, “Čerenkov radiation in materials with negative permittivity and permeability,” Opt. Express **11**, 723–734 (2003) [CrossRef] [PubMed]

*Ẽ*

_{+}and

*Ẽ*

_{-}form cones around the

*zẑ*direction. The direction

*θ*

_{±}that

*k⃑*

_{±}makes with

*zẑ*is determined from Eq. (15.1). Thus there are two cones of radiation for two cylindrical waves (

*Ẽ*

_{+}and

*Ẽ*

_{-}) of the radiated field. The line connecting points

*A*to

*B*forms the phase front of the radiation which is propagating with the wave vector

*k⃑*

_{+}, and the line connecting points

*A′*to

*B*forms the phase front of the radiation which is propagating with the wave vector

*k⃑*

_{-}. 2)

*ν*

_{+}<

*ν*<|

*ν*

_{-}|.

*γ*

_{+}is real and

*γ*

_{-}is imaginary. Then

*Ẽ*

_{+},

*H̃*

_{+},

*S̃*

_{+},

*W̃*

_{+}are the same with that in case 1, and the right-polarized waves are evanescent in the

*direction. We can find a forward-propagating CR. CR in this case is shown in Fig. 1(b). The constant phase front of*ρ ^

*Ẽ*

_{+}forms a cone around the

*zẑ*direction. The direction

*θ*

_{+}that

*k⃑*

_{+}makes with

*zẑ*is determined from cos

*θ*

_{+}=

*β*/

*k*

_{+}. Thus there is a single cone of radiation for the left-polarized waves (

*Ẽ*

_{+}) of the radiated field. The line connecting points

*A*to

*B*forms the phase front of the radiation which is propagating with the wave vector

*k⃑*

_{+}.

_{±}|.

*γ*

_{+}and

*γ*

_{-}are imaginary. The fields decrease exponentially in the

*direction, and there is no CR field in strong chiral media.*ρ ^

*ε,µ*and

*κ*are complex [7

7. T. G. Mackay, “Plane waves with negative phase velocity in isotropic chiral mediums,” Microwave Opt. Technol. Lett. **45**, 120–121 (2005). [CrossRef]

*k*

_{±}are complex. Considering the analysis in Ref. 20, the condition for Cerenkov radiation is now ν

^{2}>(

*ω*/Re(

*k*

_{±}))

^{2}. The argument of the Hankel functions is complex, but the solutions of Eq. (5) and Eq. (6) are unchanged. Using the analysis method similar to that in Ref. 20, we can see that the direction of power radiation is determined by the arguments of

*ε,µ*and

*κ*. There are still backward power for the right-polarized waves and forward power for the left-polarized waves in strong chiral medium. In addition we see that the directions of

*k⃑*

_{±}are different from that of

*S̃*

_{±}. If the losses are small, there is almost no difference between the direction of

*k⃑*

_{+}and the direction of

*S̃*

_{+}, and the direction of

*k⃑*

_{-}is almost opposite to that of

*S̃*

_{-}.

## 4. Conclusions

## References and links

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

2. | J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. |

3. | D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. |

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

5. | J. B. Pendry, “A chiral route to negative refraction,” Science |

6. | T. G. Mackay and A. Lakhtakia, “Plane waves with negative phase velocity in Faraday chiral mediums,” Phys. Rev. E. |

7. | T. G. Mackay, “Plane waves with negative phase velocity in isotropic chiral mediums,” Microwave Opt. Technol. Lett. |

8. | Q. Cheng and T. J. Cui, “Negative refractions and backward waves in biaxially anisotropic chiral media,” Opt. Express |

9. | S. Tretyakov, A. Sihvola, and L. Jylha, “Backward-wave regime and negative refraction in chiral composites,” Photonics Nanostruct. Fundam. Appl. |

10. | Y. Jin and S. He, “Focusing by a slab of chiral medium,” Opt. Express |

11. | L. D. Landau, E. M. Liftshitz, and L. P. Pitaevskii, |

12. | J. Lu, T. Grzegorczyk, Y. Zhang, J. Pacheco Jr., B. -I. Wu, J. A. Kong, and M. Chen, “Čerenkov radiation in materials with negative permittivity and permeability,” Opt. Express |

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

14. | B. Lastdrager, A. Tip, and J. Verhoeven, “Theory of Čerenkov and transition radiation from layered structures,” Phys. Rev. E |

15. | C. Y. Luo, M. Ibanescu, S. G. Johnson, and J. D. Joannopoulos, “Cerenkov radiation in Photonic Crystals,” Science |

16. | I. V. Lindell, A. H. Sihvola, S. A. Tretyakov, and A. J. Viitanen, |

17. | S. Tretyakov, I. Nefedov, A. Sihvola, S. Maslovski, and C. Simovski, “Waves and energy in chiral nihility,” J. Electromagn. Waves Appl. |

18. | C. Zhang and T. J. Cui, “Spatial dispersion and energy in strong chiral medium,” Opt. Express |

19. | S. Bassiri, N. Engheta, and C. H. Papas, “Dyadic Green’s function and dipole radiation in chiral media,” Alta Freq. LV-2, 83–88 (1986). |

20. | M. H. Saffouri, “Treatment of Cerenkov radiation from electric and magnetic charges in dispersive and dissipative media,” Nuovo Cimento |

21. | J. V. Jelly, |

**OCIS Codes**

(160.0160) Materials : Materials

(160.1890) Materials : Detector materials

**ToC Category:**

Physical Optics

**History**

Original Manuscript: June 1, 2007

Revised Manuscript: July 8, 2007

Manuscript Accepted: July 9, 2007

Published: July 20, 2007

**Citation**

Min Cheng, "Forward- and backward-propagating Cerenkov radiation in strong chiral media," Opt. Express **15**, 9793-9798 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-15-9793

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### References

- V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of ε and μ," Sov. Phys. Usp. 10, 509-514 (1968). [CrossRef]
- J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Magnetism from conductors and enhanced nonlinear phenomena," IEEE Trans. Microwave Theory Tech. 47, 2075-2084 (1999). [CrossRef]
- D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000) [CrossRef] [PubMed]
- R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79(2001). [CrossRef] [PubMed]
- J. B. Pendry, "A chiral route to negative refraction," Science 306, 1353-1355 (2004). [CrossRef] [PubMed]
- T. G. Mackay and A. Lakhtakia, "Plane waves with negative phase velocity in Faraday chiral mediums," Phys. Rev. E. 69,026602-026610 (2004). [CrossRef]
- T. G. Mackay, "Plane waves with negative phase velocity in isotropic chiral mediums," Microwave Opt. Technol. Lett. 45, 120-121 (2005). [CrossRef]
- Q. Cheng and T. J. Cui, "Negative refractions and backward waves in biaxially anisotropic chiral media," Opt. Express 14, 6322-6332 (2006). [CrossRef] [PubMed]
- S. Tretyakov, A. Sihvola, and L. Jylha, "Backward-wave regime and negative refraction in chiral composites," Photonics Nanostruct. Fundam. Appl. 3, 107-117 (2005). [CrossRef]
- Y. Jin and S. He, "Focusing by a slab of chiral medium," Opt. Express 13, 4974-4979 (2005). [CrossRef] [PubMed]
- L. D. Landau, E. M. Liftshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media (Pergamon, New York, ed. 2, 1984).
- J. Lu, T. Grzegorczyk, Y. Zhang, J. PachecoJr., B. -I. Wu, J. A. Kong, and M. Chen, "Èerenkov radiation in materials with negative permittivity and permeability," Opt. Express 11, 723-734 (2003) [CrossRef] [PubMed]
- S. J. Smith, E. M. Purcell, "Visible light from Localized surface charges moving across a grating," Phys. Rev. 92, 1069-1069 (1953). [CrossRef]
- B. Lastdrager, A. Tip, and J. Verhoeven, "Theory of Èerenkov and transition radiation from layered structures," Phys. Rev. E 61, 5767-5778 (2000). [CrossRef]
- C. Y. Luo, M. Ibanescu, S. G. Johnson, and J. D. Joannopoulos, "Cerenkov radiation in Photonic Crystals," Science 299,368-371 (2003). [CrossRef] [PubMed]
- I. V. Lindell, A. H. Sihvola, S. A. Tretyakov, and A. J. Viitanen, Electromagnetic Waves in Chiral and BiIsotropic Media (Artech House, Boston, 1994)
- S. Tretyakov, I. Nefedov, A. Sihvola, S. Maslovski, and C. Simovski, "Waves and energy in chiral nihility, " J. Electromagn. Waves Appl. 17, 695-706 (2003). [CrossRef]
- C. Zhang and T. J. Cui, "Spatial dispersion and energy in strong chiral medium," Opt. Express 15, 5114-5119 (2007). [CrossRef] [PubMed]
- S. Bassiri, N. Engheta, and C. H. Papas, "Dyadic Green’s function and dipole radiation in chiral media," Alta Freq. LV-2, 83-88 (1986).
- M. H. Saffouri, "Treatment of Cerenkov radiation from electric and magnetic charges in dispersive and dissipative media," Nuovo Cimento 3D, 589-622 (1984). [CrossRef]
- J. V. Jelly, Cerenkov radiation and Its Applications (Pergamon, London, 1958).

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