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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 15 — Jul. 18, 2011
  • pp: 13825–13830
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Enhanced reversed Cherenkov radiation in a waveguide with double-negative metamaterials

Zhaoyun Duan, Chen Guo, and Min Chen  »View Author Affiliations


Optics Express, Vol. 19, Issue 15, pp. 13825-13830 (2011)
http://dx.doi.org/10.1364/OE.19.013825


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Abstract

In order to enhance the radiation energy of reversed Cherenkov radiation (RCR), on the basis of the single charged particle model, we developed a theoretical method using a charged particle beam bunch to enhance RCR in a circular waveguide partially filled with anisotropic double-negative metamaterials (DNMs). In this case, the reversed radiation mechanism is further illustrated. Numerical example shows that the radiated energy can be effectively enhanced by increasing the charged particle number in a short bunch and thus be readily detectable. The method reported here offers a theoretical basis for directly observing RCR using a charged particle beam bunch.

© 2011 OSA

1. Introduction

Artificially structured metamaterials, firstly introduced by Veselago [1

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

], exhibit some fascinating electromagnetic properties, which are not found in naturally occurring materials and composites, such as the reversal of Snell’s law, Doppler effect, and Cherenkov radiation. Since Smith and Shelby et al. demonstrated the realization of the first double-negative metamaterials (DNMs) [2

2. 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(18), 4184–4187 (2000). [CrossRef] [PubMed]

,3

3. 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]

] based on the pioneering work of Pendry et al. [4

4. J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. 76(25), 4773–4776 (1996). [CrossRef] [PubMed]

,5

5. J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech. 47(11), 2075–2084 (1999). [CrossRef]

], there has been a lot of interest to understand the exotic behavior of different metamaterials and to explore their potential applications to components, devices, antennas, and so on. Typical examples are the advanced lenses and optics [6

6. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000). [CrossRef] [PubMed]

,7

7. T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, “Terahertz magnetic response from artificial materials,” Science 303(5663), 1494–1496 (2004). [CrossRef] [PubMed]

] as well as invisible cloaks [8

8. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006). [CrossRef] [PubMed]

,9

9. D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006). [CrossRef] [PubMed]

].

Cherenkov radiation (CR) [10

10. P. A. Čerenkov, “Visible radiation produced by electrons moving in a medium with velocities exceeding that of light,” Phys. Rev. 52(4), 378–379 (1937). [CrossRef]

,11

11. I. M. Frank and I. E. Tamm, “Coherent visible radiation of fast electrons passing through matter,” Dokl. Akad. Nauk SSSR 14, 109–114 (1937).

] is extensively employed in high-energy particle physics, cosmic-ray physics, and high-power radiation sources [12

12. V. P. Zrelov, Cherenkov Radiation in High Energy Physics (Israel Program for Scientiðc Translations, Jerusalem, 1970).

]. The research interest has arisen in reversed Cherenkov radiation (RCR) in DNMs due to the “reversed” behavior and easy control of effective electromagnetic parameters as well as potential applications to particle detectors and microwave/millimeter wave generation [13

13. J. Lu, T. Grzegorczyk, Y. Zhang, J. Pacheco Jr, B.-I. Wu, J. Kong, and M. Chen, “Cerenkov radiation in materials with negative permittivity and permeability,” Opt. Express 11(7), 723–734 (2003). [CrossRef] [PubMed]

19

19. S. N. Galyamin and A. V. Tyukhtin, “Electromagnetic field of a moving charge in the presence of a left-handed medium,” Phys. Rev. B 81(23), 235134 (2010). [CrossRef]

]. Although these publications [20

20. S. Antipov, L. Spentzouris, W. Gai, M. Conde, F. Franchini, R. Konecny, W. Liu, J. G. Power, Z. Yusof, and C. Jing, “Observation of wakefield generation in left-handed band of metamaterial-loaded waveguide,” J. Appl. Phys. 104(1), 014901 (2008). [CrossRef]

,21

21. S. Xi, H. S. Chen, T. Jiang, L. X. Ran, J. T. 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] [PubMed]

] have claimed indirect observation of RCR by use of either free electrons or a phased electromagnetic dipole array to model moving charged particles, RCR has not been really observed using a charged particle beam. We have theoretically investigated the RCR in a circular waveguide either fully [22

22. Z. Y. Duan, B.-I. Wu, J. Lu, J. A. Kong, and M. Chen, “Reversed Cherenkov radiation in a waveguide filled with anisotropic double-negative metamaterials,” J. Appl. Phys. 104(6), 063303 (2008). [CrossRef]

] or partially [23

23. Z. Y. Duan, B.-I. Wu, J. Lu, J. A. Kong, and M. Chen, “Cherenkov radiation in anisotropic double-negative metamaterials,” Opt. Express 16(22), 18479–18484 (2008). [CrossRef] [PubMed]

] loaded with the anisotropic DNMs using a single charged particle model and found that the total radiated energy even at high frequencies, such as terahertz band, is too small due to the large loss of DNMs to be easily detected. In order to greatly enhance the total radiated energy, in this paper, we propose a charged particle beam bunch method for the direct observation of RCR.

2. Theoretical analysis of RCR

We use the electromagnetic theory to analyze the field components generated by a charged particle beam bunch with a total charge qmoving through the channel in a circular waveguide partially loaded with DNMs (Fig. 1
Fig. 1 A schematic diagram of a charged particle beam bunch moving uniformly with a constant speed υ¯ in a circular waveguide partially filled with anisotropic DNMs. ρ0 and 2z0 are the radius and the axial length of the charged particle bunch, respectively.
). There are two sets of cylindrical coordinates, one (ρ,θ,z) for the source and one (ρ,θ,z)for the point where fields are being calculated. The anisotropic DNMs can be characterized by the permittivity and permeability tensors [22

22. Z. Y. Duan, B.-I. Wu, J. Lu, J. A. Kong, and M. Chen, “Reversed Cherenkov radiation in a waveguide filled with anisotropic double-negative metamaterials,” J. Appl. Phys. 104(6), 063303 (2008). [CrossRef]

], the elements of which are described by the formulae (1) and (2) in [3

3. 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]

], respectively. With the assumption that the space charge effect is negligible and the charged particles are uniformly distributed, the charged particle beam bunch with the radius ρ0 and the axial length 2z0 travels uniformly along the z-axis with a constant speed υ¯. Note that the beam bunch length should satisfy the condition 2z0<λcp, with λcp=υ/f being the charged particle wavelength and f being the operating frequency. The charge density ρP and the current density J¯ can be written in the following forms:

ρp=q0N2πρ02z0T(t,z)  and  J¯(z,t)=z^υq0N2πρ02z0T(t,z),
(1)

whereT(t,z)=1,(z(υtz0,υt+z0)), q0is the electric charge of a single charged particle, and N is the charged particle number.

The Fourier transform of the current density in the frequency domain is given by

J¯(z,ω)=12πdtJ¯(z,t)eiωt=z^υq0N2π2ρ02z0eiωz/υ1ωsinωz0υ.
(2)

Here, the cross section of the circular waveguide is divided into three layers, the first is filled with charged particle beam bunch (0<ρρ0), the second with vacuum (ρ0<ρa), and the third with DNMs (a<ρb), as shown in Fig. 1.

For the first layer, we use the Maxwell’s curl equations in phasor form, adopt the method of the separation of variables, follow the electromagnetic potentials approach [22

22. Z. Y. Duan, B.-I. Wu, J. Lu, J. A. Kong, and M. Chen, “Reversed Cherenkov radiation in a waveguide filled with anisotropic double-negative metamaterials,” J. Appl. Phys. 104(6), 063303 (2008). [CrossRef]

,23

23. Z. Y. Duan, B.-I. Wu, J. Lu, J. A. Kong, and M. Chen, “Cherenkov radiation in anisotropic double-negative metamaterials,” Opt. Express 16(22), 18479–18484 (2008). [CrossRef] [PubMed]

], and carry out a series of tedious manipulations. Finally, we arrive at the scalar wave equation

[1ρρ(ρρ)+kρ2]Az(ρ,z,ω)=μ0υq0N2π2ρ02z01ωsinωz0υeiωυz=p(z,ω),
(3)

where the radial wave number kρ has this form kρ2=ω2ε0μ0ω2/υ2, Az is the component of the vector potential along the z-axis, and ε0 and μ0 denote the permittivity and permeability respectively in vacuum.

The analytical solution to the above wave equation is [24

24. T. Shiozawa, Classical Relativistic Electrodynamics (Springer, 2004).

]

Az(ρ,z,ω)=VG(ρ,ρ,z,z,ω)p(z,ω)dV,
(4)

where G(ρ,ρ,z,z,ω) is the Green’s function for the inhomogeneous Helmholtz Eq. (3), namely, the solution corresponding to a unit point source satisfies the following equation [22

22. Z. Y. Duan, B.-I. Wu, J. Lu, J. A. Kong, and M. Chen, “Reversed Cherenkov radiation in a waveguide filled with anisotropic double-negative metamaterials,” J. Appl. Phys. 104(6), 063303 (2008). [CrossRef]

,23

23. Z. Y. Duan, B.-I. Wu, J. Lu, J. A. Kong, and M. Chen, “Cherenkov radiation in anisotropic double-negative metamaterials,” Opt. Express 16(22), 18479–18484 (2008). [CrossRef] [PubMed]

]:

[1ρρ(ρρ)+kρ2]G(ρ,ρ,z,z,ω)=12πρδ(ρρ)δ(zz),
(5)

and the integration is carried out in the volume V in which the source p(z,ω) is distributed.

Once obtaining the solution to (3), we derive the analytical expressions of field components based on the electromagnetic potentials approach.

Similarly, we also derive the field components for the second and the third layers by solving the homogenous equation corresponding to (3), respectively. Here, we just concentrate on the RCR in the third layer when the CR condition is satisfied. Note that the CR condition describes that the charged particle speed exceeds the phase velocity of radiated electromagnetic wave in the DNM [12

12. V. P. Zrelov, Cherenkov Radiation in High Energy Physics (Israel Program for Scientiðc Translations, Jerusalem, 1970).

]. Thus, matching the boundary conditions atρ=ρ0, ρ=a, and ρ=b, we derive the unknown coefficients for the different layers. As a result, all the field components for the three layers are entirely determined. Once knowing the field components, we can depict the reversed radiation behavior in the third layer according to the time-averaged Poynting vector <S¯>. In addition, the total radiated energy per unit length can be derived. From the analytical expressions, we find that for the case with the beam length 2z 0 much smaller than the RCR wavelength of interest, the total radiated energy is proportional to the square of charged particle number N. This result means that the total radiated energy of a short charged-particle beam bunch can be greatly enhanced to make the radiation readily observable. Here, we note that the aforementioned conclusion is made with the assumption that the charged particles are uniformly distributed in the first layer, and for simplicity, the space charge effect has not been taken into account but is smaller under the condition that the charged particle number is smaller.

3. Numerical results and discussions

The contribution to RCR is mainly from the peaks from the poles of the spectral density function in the spectral densityfi, as shown in Fig. 3
Fig. 3 Radiated energy spectral density over the radiation frequency band.
. The peak contribution increases with decreasing material loss. In other words, as the loss increases, the peak number decreases and the spectral density become smoother over the radiation frequency band. Thus, we obtain the maximum coherent radiation at a certain frequency corresponding to a peak when the electrons radiate in phase. This may provide a new method to produce a novel microwave or terahertz wave. In addition, numerical results show that the spectral density, as described in the unbounded DNMs [14

14. Z. Y. Duan, B.-I. Wu, J. Lu, J. A. Kong, and M. Chen, “Reversed Cherenkov radiation in unbounded anisotropic double-negative metamaterials,” J. Phys. D Appl. Phys. 42(18), 185102 (2009). [CrossRef]

], is not sensitive to the degree of anisotropy for the charged particle beam bunch model.

The effect of the charged particle velocity and the material loss on the total radiated energy per unit length is illustrated in Fig. 4
Fig. 4 Total radiated energy as a function of the loss for different charged particle velocities.
. The total radiated energy is an increasing function of the charged particle velocity due to the fact that increasing the charged particle velocity results in expanding the operating frequency band in which the RCR occurs. As stated before in the spectral density, the total radiated energy naturally decreases with increasing loss.

Based on the present analysis reported here, the RCR for a short electron beam bunch is greatly enhanced in comparison with that for a single electron case [23

23. Z. Y. Duan, B.-I. Wu, J. Lu, J. A. Kong, and M. Chen, “Cherenkov radiation in anisotropic double-negative metamaterials,” Opt. Express 16(22), 18479–18484 (2008). [CrossRef] [PubMed]

]. For example, when N and γ are 5×107and 5×108 rad/s, respectively, the RCR for such an electron beam bunch with z0= 2.5 mm and ρ0=0.1 mm is enhanced by a factor of ~2.3×1015(close toN2) times relative to the radiation intensity of a single electron. The radiation is so strong that it should be readily detectable by use of a spectrum analyzer. In addition, in order to further improve the total radiated energy, the DNMs can be scaled to higher frequencies such as the terahertz band, as described in [23

23. Z. Y. Duan, B.-I. Wu, J. Lu, J. A. Kong, and M. Chen, “Cherenkov radiation in anisotropic double-negative metamaterials,” Opt. Express 16(22), 18479–18484 (2008). [CrossRef] [PubMed]

].

4. Conclusion

Acknowledgments

This research was partially supported by National Natural Science Foundation of China (Grant No. 60971031), Sichuan Youth Foundation (Grant No. 2010JQ0005), Foundation of Returned Visiting Scholar managed by Ministry of Education, China, and the Fundamental Research Funds for the Central Universities (Grant No. ZYGX2010X010).

References and links

1.

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

2.

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(18), 4184–4187 (2000). [CrossRef] [PubMed]

3.

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.

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. 76(25), 4773–4776 (1996). [CrossRef] [PubMed]

5.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech. 47(11), 2075–2084 (1999). [CrossRef]

6.

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000). [CrossRef] [PubMed]

7.

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, “Terahertz magnetic response from artificial materials,” Science 303(5663), 1494–1496 (2004). [CrossRef] [PubMed]

8.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006). [CrossRef] [PubMed]

9.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006). [CrossRef] [PubMed]

10.

P. A. Čerenkov, “Visible radiation produced by electrons moving in a medium with velocities exceeding that of light,” Phys. Rev. 52(4), 378–379 (1937). [CrossRef]

11.

I. M. Frank and I. E. Tamm, “Coherent visible radiation of fast electrons passing through matter,” Dokl. Akad. Nauk SSSR 14, 109–114 (1937).

12.

V. P. Zrelov, Cherenkov Radiation in High Energy Physics (Israel Program for Scientiðc Translations, Jerusalem, 1970).

13.

J. Lu, T. Grzegorczyk, Y. Zhang, J. Pacheco Jr, B.-I. Wu, J. Kong, and M. Chen, “Cerenkov radiation in materials with negative permittivity and permeability,” Opt. Express 11(7), 723–734 (2003). [CrossRef] [PubMed]

14.

Z. Y. Duan, B.-I. Wu, J. Lu, J. A. Kong, and M. Chen, “Reversed Cherenkov radiation in unbounded anisotropic double-negative metamaterials,” J. Phys. D Appl. Phys. 42(18), 185102 (2009). [CrossRef]

15.

Y. P. Bliokh, S. Savel’ev, and F. Nori, “Electron-beam instability in left-handed media,” Phys. Rev. Lett. 100(24), 244803 (2008). [CrossRef] [PubMed]

16.

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] [PubMed]

17.

S. N. Galyamin and A. V. Tyukhtin, “Electromagnetic field of a moving charge in the presence of a left-handed medium,” Phys. Rev. B 81(23), 235134 (2010). [CrossRef]

18.

Z. Y. Duan, B.-I. Wu, S. Xi, H. S. Chen, and M. Chen, “Research progress in reversed Cherenkov radiation in double-negative metamaterials,” Progress In Electromagnetics Research 90, 75–87 (2009). [CrossRef]

19.

S. N. Galyamin and A. V. Tyukhtin, “Electromagnetic field of a moving charge in the presence of a left-handed medium,” Phys. Rev. B 81(23), 235134 (2010). [CrossRef]

20.

S. Antipov, L. Spentzouris, W. Gai, M. Conde, F. Franchini, R. Konecny, W. Liu, J. G. Power, Z. Yusof, and C. Jing, “Observation of wakefield generation in left-handed band of metamaterial-loaded waveguide,” J. Appl. Phys. 104(1), 014901 (2008). [CrossRef]

21.

S. Xi, H. S. Chen, T. Jiang, L. X. Ran, J. T. 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] [PubMed]

22.

Z. Y. Duan, B.-I. Wu, J. Lu, J. A. Kong, and M. Chen, “Reversed Cherenkov radiation in a waveguide filled with anisotropic double-negative metamaterials,” J. Appl. Phys. 104(6), 063303 (2008). [CrossRef]

23.

Z. Y. Duan, B.-I. Wu, J. Lu, J. A. Kong, and M. Chen, “Cherenkov radiation in anisotropic double-negative metamaterials,” Opt. Express 16(22), 18479–18484 (2008). [CrossRef] [PubMed]

24.

T. Shiozawa, Classical Relativistic Electrodynamics (Springer, 2004).

OCIS Codes
(130.2790) Integrated optics : Guided waves
(230.7370) Optical devices : Waveguides
(350.3618) Other areas of optics : Left-handed materials
(160.3918) Materials : Metamaterials

ToC Category:
Physical Optics

History
Original Manuscript: May 11, 2011
Revised Manuscript: June 16, 2011
Manuscript Accepted: June 17, 2011
Published: July 5, 2011

Citation
Zhaoyun Duan, Chen Guo, and Min Chen, "Enhanced reversed Cherenkov radiation in a waveguide with double-negative metamaterials," Opt. Express 19, 13825-13830 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-15-13825


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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(4), 509–514 (1968). [CrossRef]
  2. 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(18), 4184–4187 (2000). [CrossRef] [PubMed]
  3. 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. J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. 76(25), 4773–4776 (1996). [CrossRef] [PubMed]
  5. J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech. 47(11), 2075–2084 (1999). [CrossRef]
  6. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000). [CrossRef] [PubMed]
  7. T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, “Terahertz magnetic response from artificial materials,” Science 303(5663), 1494–1496 (2004). [CrossRef] [PubMed]
  8. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006). [CrossRef] [PubMed]
  9. D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006). [CrossRef] [PubMed]
  10. P. A. Čerenkov, “Visible radiation produced by electrons moving in a medium with velocities exceeding that of light,” Phys. Rev. 52(4), 378–379 (1937). [CrossRef]
  11. I. M. Frank and I. E. Tamm, “Coherent visible radiation of fast electrons passing through matter,” Dokl. Akad. Nauk SSSR 14, 109–114 (1937).
  12. V. P. Zrelov, Cherenkov Radiation in High Energy Physics (Israel Program for Scientiðc Translations, Jerusalem, 1970).
  13. J. Lu, T. Grzegorczyk, Y. Zhang, J. Pacheco, B.-I. Wu, J. Kong, and M. Chen, “Cerenkov radiation in materials with negative permittivity and permeability,” Opt. Express 11(7), 723–734 (2003). [CrossRef] [PubMed]
  14. Z. Y. Duan, B.-I. Wu, J. Lu, J. A. Kong, and M. Chen, “Reversed Cherenkov radiation in unbounded anisotropic double-negative metamaterials,” J. Phys. D Appl. Phys. 42(18), 185102 (2009). [CrossRef]
  15. Y. P. Bliokh, S. Savel’ev, and F. Nori, “Electron-beam instability in left-handed media,” Phys. Rev. Lett. 100(24), 244803 (2008). [CrossRef] [PubMed]
  16. 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] [PubMed]
  17. S. N. Galyamin and A. V. Tyukhtin, “Electromagnetic field of a moving charge in the presence of a left-handed medium,” Phys. Rev. B 81(23), 235134 (2010). [CrossRef]
  18. Z. Y. Duan, B.-I. Wu, S. Xi, H. S. Chen, and M. Chen, “Research progress in reversed Cherenkov radiation in double-negative metamaterials,” Progress In Electromagnetics Research 90, 75–87 (2009). [CrossRef]
  19. S. N. Galyamin and A. V. Tyukhtin, “Electromagnetic field of a moving charge in the presence of a left-handed medium,” Phys. Rev. B 81(23), 235134 (2010). [CrossRef]
  20. S. Antipov, L. Spentzouris, W. Gai, M. Conde, F. Franchini, R. Konecny, W. Liu, J. G. Power, Z. Yusof, and C. Jing, “Observation of wakefield generation in left-handed band of metamaterial-loaded waveguide,” J. Appl. Phys. 104(1), 014901 (2008). [CrossRef]
  21. S. Xi, H. S. Chen, T. Jiang, L. X. Ran, J. T. 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] [PubMed]
  22. Z. Y. Duan, B.-I. Wu, J. Lu, J. A. Kong, and M. Chen, “Reversed Cherenkov radiation in a waveguide filled with anisotropic double-negative metamaterials,” J. Appl. Phys. 104(6), 063303 (2008). [CrossRef]
  23. Z. Y. Duan, B.-I. Wu, J. Lu, J. A. Kong, and M. Chen, “Cherenkov radiation in anisotropic double-negative metamaterials,” Opt. Express 16(22), 18479–18484 (2008). [CrossRef] [PubMed]
  24. T. Shiozawa, Classical Relativistic Electrodynamics (Springer, 2004).

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