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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 15, Iss. 15 — Jul. 23, 2007
  • pp: 9793–9798
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Forward- and backward-propagating Cerenkov radiation in strong chiral media

Min Cheng  »View Author Affiliations


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

Negative refraction metamaterial, commonly known for its reversal of Snell’s law of refraction, has aroused great interest in scientific communities [1

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

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]

]. It is found that negative refraction can be realized using an chiral material [5

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

8

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]

]. In some recent literatures [9

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

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

], it is shown that negative refraction and backward propagation can be supported for one of internal eigenwaves in the isotropic chiral medium with the sufficiently strong chirality.

When a charged particle travels inside a medium, it can drive the medium to emit coherent electromagnetic energy called Cerenkov radiation (CR) [11

11. L. D. Landau, E. M. Liftshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media (Pergamon, New York, ed. 2, 1984).

]. There are three key characteristics for CR in a conventional material: 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. One possible source of unusual CR is in a negative refraction metamaterial in which CR is predicted to flow backward; i.e., opposite to the particle velocity [1

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

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]

]. Another possibility exists near a periodic structure. For a metallic grating [13

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]

] and one-dimensionally periodic multilayer stacks [14

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]

], simple Bragg scattering of light can give rise to radiation without any velocity threshold. A variety of CR patterns can occur in a single photonic crystal under different particle-velocity regimes [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]

], such as backward-propagating CR in one velocity range.

In this paper we analyze the CR emitted from a charged particle traveling with a constant velocity in strong lossless chiral media. A formal solution will be obtained for the electromagnetic field components. We reveal a variety of CR patterns that can occur in strong chiral medium under different particle-velocity regimes.

2. General formulations

According to Maxwell equations and the constitutive relation for isotropic chiral media [16

16. I. V. Lindell, A. H. Sihvola, S. A. Tretyakov, and A. J. Viitanen, Electromagnetic Waves in Chiral and BiIsotropic Media (Artech House, Boston, 1994)

]

D=εE+iκH,
(1)
B=μHiκE,
(2)

we obtain the following dispersion relation for the wavenumber k±=ω(με±κ), where “+” and “-” represent two different eigenwaves. In above expression, 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.

It has been shown that light wave propagation in such media possesses two circularly polarized eigenmodes, a left- and a right- circularly polarized (LCP and RCP) wave with two unequal characteristic phase velocities ν+ and ν-, respectively:

v±=ωk±=(με+κ)1,
(3)

Traditionally, it is regarded as a natural limit to all chiral media for κ>με, however in the recent research, we know that strong chiral medium with κ<με can exist at least at or near the resonant frequency of the permittivity of a chiral medium (called chiral nihility [17

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]

]). When κ>με, we have k=ω(μεκ)<0, and then one of the two circularly polarized eigenwaves will be backward wave [7

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

, 9

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

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

]. It is worth noting that strong chirality does not result in a strong spatial dispersion, strong chirality parameter leads to positive energy without any frequency-band limitation in the weak spatial dispersion [18

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

].

J˜(r,ω)=12πdtJ(t)eiωt=zq4π2ρδ(ρ)eiωzv,
(4)

Considering Eq. (4), the constitutive relations of Eq. (1) and Eq. (2), and Maxwell’s equations, the governing equations for the emitted electric field and magnetic field become

μ×(1μ×E˜)(εμκ2)ω2c2E˜ωc[κ×E˜+μ×(κμE˜)]=iωμJ˜,
(5)
ε×(1ε×H˜)(εμκ2)ω2c2H˜ωc[κ×H˜+ε×(κεH˜)]=iωμJ˜,
(6)

where and represent the Fourier transforms of E⃑(t) andH⃑(t), respectively. Since Eq. (5) is linear, its solution can be expressed as the following:

E˜(r)=iωμVdVJ˜(r)·Γ̂(r,r),
(7)

where V′ is the source region occupied by (r⃑′), and Γ^ (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

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).

].

Substituting (4) into (7), using Γ^(ρ,ϕ,z;ρ,ϕ,ω/v)=+dz'Γ(r,r)eiωz/v and knowing that the field is independent of the coordinate φ due to isotropy of the chiral media, we can get

E˜(ρ,z)=i(ωμq2π)ẑ·Γ̂(ρ,z;ρ',ωv).
(8)

Through calculating, Γ^ (ρ,φ,z;ρ′,φ′,ω/ν) can be expressed as [19

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).

]:

Γ̂(ρ,φ,z;ρ,φ,ωv)=χ+M+(k+)Ĝ+(ρ,φ,z;ρ,φ,ωv)
+χM(k)Ĝ(ρ,φ,z;ρ,φ,ωv),
(9)

where

χ±=k2k±24ω2μεκ,M±(k±)=[(1k±2)+U±(1k±)U±],

Ĝ ±(ρ,φ,z;ρ′,φ′,ω/ν=(i/4)H (1) 0(γ ± R)eiβz,

with k=ωμε,β=ω/ν,γ±=k±2β2,R=ρ2+ρ22ρρcos(φφ), and H() denotes the Hankel functionU is the three-dimensional unit dyadic.

Substituting (9) with ρ′=0 into (8), then after a lengthy mathematical manipulation, can be expressed as the following:

E˜=Ωeiβz[(βγ+k+eiπ2ρ̂+γ+φ̂)H1(1)(τ+)+γ+2k+H0(1)(τ+)ẑ
+(βγkeiπ2ρ̂γφ̂)H1(1)(τ)+γ2kH0(1)(τ)ẑ],
(10)

where Ω=qμε16π,τ ±=γ ± ρ ρ,̂ and φ^ are the unit vectors along ρ and φ directions, respectively. After a similar manipulation, the solution for Eq. (6) (the corresponding magnetic field ) can be obtained:

H˜=q16π±eiβz+iπ2[(βγ+k+eiπ2ρ̂+γ+φ̂)H1(1)(τ+)+γ+2k+H0(1)(τ+)ẑ
+(βγkeiπ2ρ̂+γφ̂)H1(1)(τ)γ2kH0(1)(τ)ẑ],
(11)

3. Discussions

In strong chiral media, there are two different characteristic phase velocities of two different eigenwaves for their state of polarization which are given in Eq. (3). Therefore, it appears that in such media the CR condition, where the particle’s velocity must be greater than the medium’s phase velocity, can be met for two different values of particle’s velocity. We can identify three regimes of the charge velocity with three different CR patterns in strong chiral media.

1)|ν±|<ν.

In this case, each frequency component of the radiated field in the CR consists of two cylindrical waves + (for the wave number k +) and - (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.

Using the large-argument (|τ ±|≫1) asymptotic expression for the Hankel functions, H0(1)(τ±)(2πτ±)eiτ±iπ4, H1(1)(τ±)(2πτ±)eiτ±i3π4, Eq. (10) and Eq. (11) can be modified as follows:

E˜=E˜++E˜=η{γ+[(βk+ρ̂+γ+k+ẑ)iφ̂]ei(βz+τ+π4)
+γ[(βkρ̂+γkẑ)iφ̂]ei(βz+τπ4)},
(12)
H˜=H˜++H˜=ηεμ{iγ+[(βk+ρ̂+γ+k+ẑ)iφ̂]ei(βz+τ+π4)
iγ[(βkρ̂+γkẑ)+iφ̂]ei(βz+τπ4)},
(13)

where η=(q16π)2μπρε. We can rewrite ±, ± in the following way:

E˜±=ηγ±[(cosθ±ρ̂+sinθ±ẑ)iφ̂]ei(βz+τ±π4),
(14)
H˜±=(±iεμ)ηγ±[(cosθ±ρ̂+sinθ±ẑ)iφ̂]ei(βz+τ±π4),
(15)
cosθ±=βk±
(15.1).

For the far electric and magnetic fields, the corresponding Poynting vectors are written as:

S˜=12Re(E˜×H˜*)=S˜++S˜
(16)

And the Poynting vectors for the left- and right- polarized waves are written as:

S˜±=12Re(E˜±×H˜±*)=12(q8π)2(γ±πρ)με(γ±k±ρ̂+βk±ẑ)=S±ρρ̂+S±zẑ
(17)

And the total energies per unit area radiated out in ρ^ and direction for the left- and right-polarized waves are written as:

W˜±z=W±zẑ=20S˜±zdω=0{(q8π)2(βγ±k±πρ)με}dω
(18)

For

W˜±ρ=W±ρρ̂=20S˜±ρdω=0{(q8π)2(γ±2k±πρ)με}dω
(19)

strong chiral media κ>με,W +z>0,W >0,W -z<0,W >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]

] and left-handed medium with weakly loss [12

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]

]. CR in this case is shown in Fig. 1(a). The constant phase fronts of + and - 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 ( + and -) 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 -|.

In this case, we see that γ + is real and γ - is imaginary. Then +, +, +, + are the same with that in case 1, and the right-polarized waves are evanescent in the ρ^ direction. We can find a forward-propagating CR. CR in this case is shown in Fig. 1(b). The constant phase front of + forms a cone around the zẑ direction. The direction θ + that k⃑ + makes with zẑ is determined from cosθ +=β/k +. Thus there is a single cone of radiation for the left-polarized waves ( +) 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 +.

Fig. 1. CR in strong chiral mediaκ>με. (a): case 1(|ν±|<ν); (b): case 2(ν+<ν<ν-|).

3)ν<|ν±|.

In this case, both γ + and γ - are imaginary. The fields decrease exponentially in the ρ^ direction, and there is no CR field in strong chiral media.

In the former discussions the materials are considered being lossless. If losses exist in strong chiral media, we have to consider the constitutive parameters 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]

]. Then the wavenumbers k ± are complex. Considering the analysis in Ref. 20, the condition for Cerenkov radiation is now ν2>(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 ±. If the losses are small, there is almost no difference between the direction of k⃑ + and the direction of +, and the direction of k⃑ - is almost opposite to that of -.

4. Conclusions

In summary, we have analyzed the problem of CR emitted from a charged particle traveling with a constant velocity in strong isotropic lossless chiral media. The analytical expressions for the electromagnetic field components have be given. We showed that there are a variety of CR patterns in strong chiral medium under different particle-velocity regimes. In each case, the characteristics of the CR have been discussed. And in one 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. CR has offered many applications in a variety of fields such as particle physics, high-energy physics, and cosmic-ray physics [21

21. J. V. Jelly, Cerenkov radiation and Its Applications (Pergamon, London, 1958).

]. The characteristics of CR in strong chiral medium can find application in velocity-sensitive particle detection and radiation generation.

References and links

1.

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

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. 47, 2075–2084 (1999). [CrossRef]

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. 84, 4184–4187 (2000) [CrossRef] [PubMed]

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]

6.

T. G. Mackay and A. Lakhtakia, “Plane waves with negative phase velocity in Faraday chiral mediums,” Phys. Rev. E. 69, 026602–026610 (2004). [CrossRef]

7.

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

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]

11.

L. D. Landau, E. M. Liftshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media (Pergamon, New York, ed. 2, 1984).

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]

16.

I. V. Lindell, A. H. Sihvola, S. A. Tretyakov, and A. J. Viitanen, Electromagnetic Waves in Chiral and BiIsotropic Media (Artech House, Boston, 1994)

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]

18.

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

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 3D, 589–622 (1984). [CrossRef]

21.

J. V. Jelly, Cerenkov radiation and Its Applications (Pergamon, London, 1958).

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

  1. V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of ε and μ," Sov. Phys. Usp. 10, 509-514 (1968). [CrossRef]
  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. 47, 2075-2084 (1999). [CrossRef]
  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. 84, 4184-4187 (2000) [CrossRef] [PubMed]
  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]
  6. T. G. Mackay and A. Lakhtakia, "Plane waves with negative phase velocity in Faraday chiral mediums," Phys. Rev. E. 69,026602-026610 (2004). [CrossRef]
  7. T. G. Mackay, "Plane waves with negative phase velocity in isotropic chiral mediums," Microwave Opt. Technol. Lett. 45, 120-121 (2005). [CrossRef]
  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]
  11. L. D. Landau, E. M. Liftshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media (Pergamon, New York, ed. 2, 1984).
  12. 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]
  13. S. J. Smith, 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 Èerenkov 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]
  16. I. V. Lindell, A. H. Sihvola, S. A. Tretyakov, and A. J. Viitanen, Electromagnetic Waves in Chiral and BiIsotropic Media (Artech House, Boston, 1994)
  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]
  18. C. Zhang and T. J. Cui, "Spatial dispersion and energy in strong chiral medium," Opt. Express 15, 5114-5119 (2007). [CrossRef] [PubMed]
  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 3D, 589-622 (1984). [CrossRef]
  21. J. V. Jelly, Cerenkov radiation and Its Applications (Pergamon, London, 1958).

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