Abstract

The mathematical solution for Čerenkov radiation in a novel medium, left-handed medium (LH medium), which has both negative permittivity and permeability, is introduced in this paper. It is shown that the particle motion in the LH medium generates power that propagates backward. In this paper, both dispersion and dissipation are considered for the LH medium. The results show that in such a material, both forward power and backward power exist. In addition, we show that the losses will affect the Čerenkov angle. The idea of building a Čerenkov detector using LH medium is introduced, which could be useful in particle physics to identify charged particles of various velocities.

© 2003 Optical Society of America

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References

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  1. 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. 84,4 184�??4187 (2000).
    [CrossRef] [PubMed]
  2. R. A. Shelby, D. R. Smith, and S. Schultz, �??Experimental verification of a negative index of refraction,�?? Science 292, 77�??79 (2001).
    [CrossRef] [PubMed]
  3. V. G. Vesalago,�??The electrodynamics of substances with simultaneously negative values of ε and µ,�?? Soviet Physics USPEKHI 10, 509�??514 (1968).
    [CrossRef]
  4. P. A. Cerenkov,�??Visible radiation produced by electrons moving in a medium with velocities exceeding that of light,�?? Phys. Rev. 52, 378�??379 (1937).
    [CrossRef]
  5. I. M. Frank and I. G. Tamm, �??Coherent visible radiation of fast electrons passing through matter,�?? Compt. Rend. (Dokl.) 14, 109- 114 (1937).
  6. J. A. Kong, Electromagnetic Wave Theory (EMW,Cam bridge, 2000).
  7. Jie Lu, T. M. Grzegorczyk,Y . Zhang, J. Pacheco Jr, B. I. Wu, and J. A. Kong, �??�?erenkov radiation in left handed material,�?? in Proc. Progress in Electromagnetics Research Symposium (Cambridge,M A,2002), 917.
  8. V. P. Zrelov, �?erenkov Radiation in High-Energy Physics (Israel Program for Scientific Translations, Jerusalem,1970).
  9. M. H. Saffouri, "Treatment of �?erenkov radiation from electric and magnetic charges in dispersive and dissipative media,�?? Nuovo Cimento 3D, 589�??622 (1984).
    [CrossRef]

Compt. Rend. (1)

I. M. Frank and I. G. Tamm, �??Coherent visible radiation of fast electrons passing through matter,�?? Compt. Rend. (Dokl.) 14, 109- 114 (1937).

Nuovo Cimento (1)

M. H. Saffouri, "Treatment of �?erenkov radiation from electric and magnetic charges in dispersive and dissipative media,�?? Nuovo Cimento 3D, 589�??622 (1984).
[CrossRef]

Phys. Rev. (1)

P. A. Cerenkov,�??Visible radiation produced by electrons moving in a medium with velocities exceeding that of light,�?? Phys. Rev. 52, 378�??379 (1937).
[CrossRef]

Phys. Rev. Lett. (1)

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. 84,4 184�??4187 (2000).
[CrossRef] [PubMed]

Science (1)

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

Soviet Phys. USPEKHI (1)

V. G. Vesalago,�??The electrodynamics of substances with simultaneously negative values of ε and µ,�?? Soviet Physics USPEKHI 10, 509�??514 (1968).
[CrossRef]

Other (3)

J. A. Kong, Electromagnetic Wave Theory (EMW,Cam bridge, 2000).

Jie Lu, T. M. Grzegorczyk,Y . Zhang, J. Pacheco Jr, B. I. Wu, and J. A. Kong, �??�?erenkov radiation in left handed material,�?? in Proc. Progress in Electromagnetics Research Symposium (Cambridge,M A,2002), 917.

V. P. Zrelov, �?erenkov Radiation in High-Energy Physics (Israel Program for Scientific Translations, Jerusalem,1970).

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Figures (8)

Fig. 1.
Fig. 1.

Čerenkov radiation in a normal material (RH media).

Fig. 2.
Fig. 2.

Directions of energy flow and wave vector for a charged particle moving in an LH medium for case 1 [ g ( ρ ) = i 4 H 0 ( 1 ) ( k ρ ρ ) ] .

Fig. 3.
Fig. 3.

Directions of energy flow and wave vector for a charged particle moving in an LH medium for case 2 [ g ( ρ ) = i 4 H 0 ( 2 ) ( k ρ ρ ) ] .

Fig. 4.
Fig. 4.

Frequency bands for RH and LH media obtained from the model of Eqs. (9a) and (9b).

Fig. 5.
Fig. 5.

ℜ{n}, ℑ{n}, ℜ{n 2} at a range near the resonant frequency.

Fig. 6.
Fig. 6.

Energy distribution over frequency for γ=1×108 rad/s.

Fig. 7.
Fig. 7.

Radiation pattern of Čerenkov radiation for a material characterized by Eqs.(24).

Fig. 8.
Fig. 8.

The distributions of angle over frequency at γ=1×108 rad/s

Tables (1)

Tables Icon

Table 1. The range of angle for ∊-θ-and k ρ-θ.

Equations (47)

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v ¯ > c n ,
cos θ = 1 β n ,
J ¯ ( r ¯ , t ) = z ̂ q v δ ( z v t ) δ ( x ) δ ( y ) ,
2 A ¯ + ω 2 c 2 n 2 A ¯ = μ J ¯ .
[ 1 ρ ρ ( ρ ρ ) + k ρ 2 ] g ( ρ ) = δ ( ρ ) 2 π ρ ,
W z ( ρ ¯ ) = S z ( r ¯ , t ) d t = q 2 8 π 2 ρ v 0 k ρ d ω
W ρ ( ρ ¯ ) = S ρ ( r ¯ , t ) d t = q 2 8 π 2 ρ 0 k ρ 2 ω d ω
W z ( ρ ¯ ) = S z ( r ¯ , t ) d t = q 2 8 π 2 ρ v 0 k ρ d ω
W ρ ( ρ ¯ ) = S ρ ( r ¯ , t ) d t = q 2 8 π 2 ρ 0 k ρ 2 ω d ω
D ¯ ( r ¯ , t ) × B ¯ ( r ¯ , t ) = μ E ¯ ( r ¯ , t ) × H ¯ ( r ¯ , t ) = μ S ¯ ( r ¯ , t ) .
μ τ ( ω ) = 1 ω m p 2 ω m o 2 ω 2 ω m o 2
τ ( ω ) = 1 ω e p 2 ω e o 2 ω 2 ω e o 2
ω m c = ω m p 2 + ω m o 2 2 for which r ( ω m c ) = 1
ω e c = ω e p 2 + ω e o 2 2 for which r ( ω e c ) = 1
ω c = ω e p 2 ω m p 2 ω e o 2 ω m o 2 ω e p 2 + ω m p 2 ω e o 2 ω m o 2 for which μ r ( ω c ) r ( ω c ) = 1
E z ( r ¯ , t ) = q 4 π 2 π ρ [ 0 ω m o ( ) k ρ k ρ ω ( ω ) cos ( ϕ + ) d ω + ω e o ω c k ρ k ρ ω ( ω ) cos ( ϕ ) d ω ]
E ρ ( r ¯ , t ) = q 4 π v 2 π ρ [ 0 ω m o k ρ ( ω ) cos ( ϕ + ) d ω + ω e o ω c k ρ ( ω ) cos ( ϕ ) d ω ]
H ϕ ( r ¯ , t ) = q 4 π 2 π ρ [ 0 ω m o k ρ cos ( ϕ + ) d ω + ω e o ω c k ρ cos ( ϕ ) d ω ] ,
S z ( r ¯ , t ) = E ρ ( r ¯ , t ) H ϕ ( r ¯ , t ) = q 2 8 π 3 ρ υ
× [ 0 ω m o d ω 0 ω m o d ω k ρ k ρ ( ω ) cos ( ϕ + ) cos ( ϕ + )
+ ω e o ω c d ω ω e o ω c d ω k ρ k ρ ( ω ) cos ( ϕ ) cos ( ϕ )
+ 0 ω m o d ω ω e o ω c d ω k ρ k ρ ( ω ) cos ( ϕ + ) cos ( ϕ )
+ ω e o ω c d ω 0 ω m o d ω k ρ k ρ ( ω ) cos ( ϕ ) cos ( ϕ + ) ] ,
S ρ ( r ¯ , t ) = E z ( r ¯ , t ) H ϕ ( r ¯ , t ) = q 2 8 π 3 ρ
× [ 0 ω m o d ω 0 ω m o d ω k ρ k ρ k ρ ω ( ω ) cos ( ϕ + ) cos ( ϕ + )
ω e o ω c d ω ω e o ω c d ω k ρ k ρ k ρ ω ( ω ) cos ( ϕ ) cos ( ϕ )
+ 0 ω m o d ω ω e o ω c d ω k ρ k ρ k ρ ω ( ω ) cos ( ϕ + ) cos ( ϕ )
ω e o ω c d ω 0 ω m o d ω k ρ k ρ k ρ ω ( ω ) cos ( ϕ ) cos ( ϕ + ) ] .
cos ( ω t + α ) cos ( ω t + α ) d t = π δ ( ω ω ) cos ( α α ) ,
W z ( ρ ¯ ) = S z ( r ¯ , t ) d t = q 2 8 π 2 ρ v [ 0 ω m o d ω k ρ ( ω ) + ω e o ω c d ω k ρ ( ω ) ]
W ρ ( ρ ¯ ) = S ρ ( r ¯ , t ) d t = q 2 8 π 2 ρ [ 0 ω m o d ω k ρ 2 ω ( ω ) ω e o ω c d ω k ρ 2 ω ( ω ) ]
( ω ) = ( ω ) * with I ( ω ) > 0
μ ( ω ) = μ ( ω ) * with μ I ( ω ) > 0 .
{ n 2 ( ω ) } > 1 β 2 ,
E ρ ( r ¯ , t ) = q 4 π v 2 π ρ L H k ρ 1 2 ( ω ) cos ( ω t + k R ρ ω v z π 4 + θ 2 θ ) e k I ρ d ω
E z ( r ¯ , t ) = q 4 π 2 π ρ L H k ρ 3 2 ω ( ω ) cos ( ω t + k R ρ ω v z π 4 + 3 θ 2 θ ) e k I ρ d ω
H ϕ ( r ¯ , t ) = q 4 π 2 π ρ L H k ρ 1 2 cos ( ω t + k R ρ ω v z π 4 + θ 2 ) e k I ρ d ω
W ρ ( ρ ¯ ) = S ρ ( r ¯ , t ) d t = q 2 8 π 2 ρ R H k ρ 2 ω ( ω ) e 2 k I ρ cos ( θ θ ) d ω
W z ( ρ ¯ ) = S z ( r ¯ , t ) d t = q 2 8 π 2 ρ v R H k ρ ( ω ) e 2 k I ρ cos ( θ ) d ω
W ρ ( ρ ¯ ) = S ρ ( r ¯ , t ) d t = q 2 8 π 2 ρ L H k ρ 2 ω ( ω ) e 2 k I ρ cos ( θ θ ) d ω
W z ( ρ ¯ ) = S z ( r ¯ , t ) d t = q 2 8 π 2 ρ v L H k ρ ( ω ) e 2 k I ρ cos ( θ ) d ω
μ r ( ω ) = 1 ω m p 2 ω m o 2 ω 2 ω m o 2 + m ω
r ( ω ) = 1 ω e p 2 ω e o 2 ω 2 ω e o 2 + e ω
S ̂ ( ω ) = ρ ̂ η cos ( θ θ ) + z ̂ cos ( θ ) η 2 cos ( θ θ ) 2 + cos ( θ ) 2 for RH media
S ̂ ( ω ) = ρ ̂ η cos ( θ θ ) + z ̂ cos ( θ ) η 2 cos ( θ θ ) 2 + cos ( θ ) 2 for LH media
k ̂ ( ω ) = ρ ̂ η cos ( θ ) + z ̂ η 2 cos ( θ ) 2 + 1 for RH media
k ̂ ( ω ) = ρ ̂ η cos ( θ ) + z ̂ η 2 cos ( θ ) 2 + 1 for LH media

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