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Optics Express

  • Editor: Michael Duncan
  • Vol. 11, Iss. 7 — Apr. 7, 2003
  • pp: 723–734
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Čerenkov radiation in materials with negative permittivity and permeability

Jie Lu, Tomasz M. Grzegorczyk, Yan Zhang, Joe Pacheco Jr, Bae-Ian Wu, Jin A. Kong, and Min Chen  »View Author Affiliations


Optics Express, Vol. 11, Issue 7, pp. 723-734 (2003)
http://dx.doi.org/10.1364/OE.11.000723


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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

1. Introduction

In recent years, materials exhibiting simultaneously negative permittivity (∊) and permeability (µ) over a frequency band (which we shall call Left-Handed-LH-media, by opposition to Right-Handed-RH-to denote standard media), have received much attention. In particular, the propagation of electromagnetic waves in LH media has been thoroughly investigated by many researchers [1

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

, 2

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]

], and various new phenomena have been discovered. Probably the most fundamental one is the discovery that LH media can be characterized by a negative index of refraction.

In his pioneering 1968 paper, Vesalago [3

3. V. G. Vesalago, “The electrodynamics of substances with simultaneously negative values of ∊ and μ,” Soviet Physics USPEKHI 10, 509–514 (1968). [CrossRef]

] mentioned, in addition to many other unique properties of LH media, the fact that Čerenkov radiation will be reversed, but no mathematical consideration was given. Normally, Čerenkov radiation occurs when a charged particle moves in a material with a speed faster than that of light in that material. Čerenkov radiation in normal media was first experimentally obsevered by P.A. Čerenkov in 1934 [4

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

], and later theoretically explained by I. M. Frank and I.G. Tamm [5

5. I. M. Frank and I. G. Tamm, “Coher ent visible radiation of fast electrons passing through matter,” Compt. Rend. (Dokl.) 14, 109–114 (1937).

].

While many researchers have referred to Vesalago’s statement about the reversal of Čerenkov radiation, none have addressed the physical importance of this phenomenon in detail. It is the purpose of the first part of this paper to derive the mathematical solution for Čerenkov radiation in LH media in order to demonstrate existence of backward radiation.

The remainder of this paper is organized as follows. In Section 2, we derive the solution for Čerenkov radiation in isotropic LH media. Section 3 considers the effects of dispersion, which are inherent to all LH media. Section 4 discusses the effect of the existence of loss.

2. Mathematical formulation of Čerenkov radiation inside isotropic LH media

2.1 Formulation of the problem

As stated above, Čerenkov radiation occurs when a charged particle travels through a material at a velocity higher than that of light in that material. The principle of Čerenkov radiation is depicted in Fig.1[6

6. J. A. Kong, Electromagnetic Wave Theory (EMW, Cambridge, 2000).

]: A charged particle located at point O that is moving along with a velocity v¯ which satisfies

Fig. 1. Čerenkov radiation in a normal material (RH media).
v¯>cn,
(1)

where n is the refractive index of the medium, and c≈3×108 m/s is the speed of light in free space. The line connecting points A to B forms the phase front of the radiation, which is propagating with the wave vector k¯=ρ̂kρ+ẑωv , where ω=2πf is the angular frequency of the radiation, and k ρ is the transverse component of the wave vector . The emitted radiation has an electric field vector polarized parallel to the plane determined by the direction of the particle speed and the direction of the radiation. Note that in real situations, many charged particles form a beam, so that the radiation has cylindrical symmetry and forms the well-known Čerenkov cone. The angle of the cone is given by θ (see Fig.1) and is determined by

cosθ=1βn,
(2)

where β=vc<1 .

These characteristics have been well predicted in RH media by I.M. Frank and I. G. Tamm [5

5. I. M. Frank and I. G. Tamm, “Coher ent visible radiation of fast electrons passing through matter,” Compt. Rend. (Dokl.) 14, 109–114 (1937).

] by using classical electromagnetic theory. Following the formulation in [5

5. I. M. Frank and I. G. Tamm, “Coher ent visible radiation of fast electrons passing through matter,” Compt. Rend. (Dokl.) 14, 109–114 (1937).

], we shall rederive the mathematics for Čerenkov radiation in LH media.

2.2 Mathematical solution

The flow of charged particles can be described as a current of speed v¯ =ẑv, written as

J¯(r¯,t)=ẑqvδ(zvt)δ(x)δ(y),
(3)

where δ is the standard Dirac function. Under the Lorenz gauge condition, the wave equation for vector potential is given as

2A¯+ω2c2n2A¯=μJ¯.
(4)

In the frequency domain, and upon performing a cylindrical coordinate transformation, Eq.(4) can be reduced to a standard Poisson’s equation of the following form

[1ρρ(ρρ)+kρ2]g(ρ)=δ(ρ)2πρ,
(5)

where kρ=ωvβ2n21 , and g(ρ) is the two dimensional scalar Green’s function.

It can be seen that Eq. (5) has two independent solutions,

  • case 1: g(ρ)=i4H0(1)(kρρ) which corresponds to an outgoing wave, for which =K ρ ρ^ +Kz,
  • case 2: g(ρ)=i4H0(2)(kρρ) which corresponds to an ingoing wave, for which =K ρ ρ^ +Kz,

where kz=ωv>0 .

Before choosing any solution, we calculate the electric and magnetic fields for both cases, from which the total energy per unit area radiated out in ρ^ and directions in far field is obtained [7

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

]:

  • Case 1:
    Wz(ρ¯)=Sz(r¯,t)dt=q28π2ρv0kρdω
    (6a)
    Wρ(ρ¯)=Sρ(r¯,t)dt=q28π2ρ0kρ2ωdω
    (6b)
  • Case 2:
    Wz(ρ¯)=Sz(r¯,t)dt=q28π2ρv0kρdω
    (7a)
    Wρ(ρ¯)=Sρ(r¯,t)dt=q28π2ρ0kρ2ωdω
    (7b)

Even though the integration limits are from 0 to ∞, the above results are only valid for those frequencies that satisfy Eq.(1).

For the sake of illustration, we first consider a normal material, with ∊>0 and μ>0. From Eqs.(6a) to (7b), we see that Wz(ρ¯ )>0 for both cases, but W ρ(ρ¯ )>0 for case 1 and W ρ(ρ¯ )<0 for case 2. These two cases correspond to forward (same direction as velocity of the particle) outgoing energy, and forward ingoing energy, respectively. From Sommerfeld’s radiation condition (no energy can come from infinity, since radiation must be emitted from a source), we choose case 1 as the correct solution for Čerenkov radiation in normal materials with both ∊ and μ positive [8

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

].

However, for LH media where ∊<0 and μ<0, the results are reversed. From Eqs.(6a) to (7b), we isolate the following two cases:

  • case 1: Wz(ρ¯ )<0, W ρ(ρ¯ )<0 which corresponds to a backward and ingoing radiated energy.
  • case 2: Wz(ρ¯ )<0, W ρ(ρ¯ )>0 which corresponds to a backward and outgoing radiated energy.

The different cases are illustrated in Figs.2 and 3, where the energy flow in LH media is shown for both cases.

Fig. 2. Directions of energy flow and wave vector for a charged particle moving in an LH medium for case 1 [g(ρ)=i4H0(1)(kρρ)] .
Fig. 3. Directions of energy flow and wave vector for a charged particle moving in an LH medium for case 2 [g(ρ)=i4H0(2)(kρρ)] .

If we again suppose that there are no sources at infinity, the solution that needs to be chosen is the one corresponding to case 2. In addition, both the permittivity and the permeability need be negative to assure a real k that can support propagating waves. Finally, in the far field for isotropic lossless materials (so that the directions of the Poynting vector is opposite to that of the wave vector ), the angle between the direction of the Poynting vector and that of the velocity of the charged particle is again given by Eq.(2), but with the refractive index being negative. We have therefore demonstrated that the energy is propagating backward as predicted in [3

3. V. G. Vesalago, “The electrodynamics of substances with simultaneously negative values of ∊ and μ,” Soviet Physics USPEKHI 10, 509–514 (1968). [CrossRef]

].

Yet, we still need to consider how the momentum is conserved, which relates to the definition of momentum in LH media. The standard definition of the momentum of an electromagnetic wave is (, t(, t)=∊μ(, t) [6

6. J. A. Kong, Electromagnetic Wave Theory (EMW, Cambridge, 2000).

]. Upon using this definition, we see that the momentum is

D¯(r¯,t)×B¯(r¯,t)=μE¯(r¯,t)×H¯(r¯,t)=μS¯(r¯,t).
(8)

When both ∊ and μ are negative, (, t(, t) and (, t) are in the same direction, which implies a momentum pointing backward. By momentum conservation, this implies that the momentum of the charged particle increases, which results in an energy increase. This is in contradiction with the third fundamental law of thermodynamics, which stipulates that charged particles radiate energy out and therefore lose energy.

The solution to this paradox is to be found in the quantum theory of Čerenkov radiation [8

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

], in which the momentum of a photon is defined as =ħK̅, where is the momentum, and ħ is the Plank constant divided by 2π. For case 2, kz>0 which implies a forward propagation, while the component in the ρ^ direction is cancelled. Therefore, momentum and energy are conserved. Inside LH media, energy flow of the wave is in the opposite direction of its momentum. When the wave crosses the boundary from an LH medium into a RH medium, the component of wave vector kz (thus also the momentum direction) will change sign (from +z ̂ to - direction in our case), but the Poynting vector (, t(, t), which defines the energy flow, remains backward (- direction). Ther efore once inside the RH medium, both energy flow and momentum of the wave will again be in the same (backward) direction.

3. Čerenkov radiation in dispersive LH media

It is already known from [3

3. V. G. Vesalago, “The electrodynamics of substances with simultaneously negative values of ∊ and μ,” Soviet Physics USPEKHI 10, 509–514 (1968). [CrossRef]

] that LH media must be frequency dispersive in order to satisfy positive energy constraints.

A common model to represent the permittivity ∊(ω) and permeability μ(ω) has been given in [2

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]

], which we shall use here. For the sake of simplicity, we shall first consider a lossless case, for which the model becomes:

μτ(ω)=1ωmp2ωmo2ω2ωmo2
(9a)
τ(ω)=1ωep2ωeo2ω2ωeo2
(9b)

The following critical points can be identified

ωmc=ωmp2+ωmo22for whichr(ωmc)=1
(10a)
ωec=ωep2+ωeo22for whichr(ωec)=1
(10b)
ωc=ωep2ωmp2ωeo2ωmo2ωep2+ωmp2ωeo2ωmo2for whichμr(ωc)r(ωc)=1
(10c)

A summary of the various frequency bands generated and their properties is shown in Fig.4. The lower dark region corresponds to n 2>1, for which Čerenkov radiation can happen (supposing that β=1).

Fig. 4. Frequency bands for RH and LH media obtained from the model of Eqs. (9a) and (9b).

Ez(r¯,t)=q4π2πρ[0ωmo()kρkρω(ω)cos(ϕ+)dω+ωeoωckρkρω(ω)cos(ϕ)dω]
(11)
Eρ(r¯,t)=q4πv2πρ[0ωmokρ(ω)cos(ϕ+)dω+ωeoωckρ(ω)cos(ϕ)dω]
(12)
Hϕ(r¯,t)=q4π2πρ[0ωmokρcos(ϕ+)dω+ωeoωckρcos(ϕ)dω],
(13)

where ϕ±=ωtkρρωzv±π4 , with the upper sign corresponding to case 1, and the lower sign corresponding to case 2. The Poynting vector (r̅, t)=ẑSz(r̅, t)+ρ^ S ρ(r̅, t)=(r̅, t(r̅, t) is given by

Sz(r¯,t)=Eρ(r¯,t)Hϕ(r¯,t)=q28π3ρυ
×[0ωmodω0ωmodωkρkρ(ω)cos(ϕ+)cos(ϕ+)
+ωeoωcdωωeoωcdωkρkρ(ω)cos(ϕ)cos(ϕ)
+0ωmodωωeoωcdωkρkρ(ω)cos(ϕ+)cos(ϕ)
+ωeoωcdω0ωmodωkρkρ(ω)cos(ϕ)cos(ϕ+)],
(14)
Sρ(r¯,t)=Ez(r¯,t)Hϕ(r¯,t)=q28π3ρ
×[0ωmodω0ωmodωkρkρkρω(ω)cos(ϕ+)cos(ϕ+)
ωeoωcdωωeoωcdωkρkρkρω(ω)cos(ϕ)cos(ϕ)
+0ωmodωωeoωcdωkρkρkρω(ω)cos(ϕ+)cos(ϕ)
ωeoωcdω0ωmodωkρkρkρω(ω)cos(ϕ)cos(ϕ+)].
(15)

By using the identity [6

6. J. A. Kong, Electromagnetic Wave Theory (EMW, Cambridge, 2000).

]

cos(ωt+α)cos(ωt+α)dt=πδ(ωω)cos(αα),
(16)

we can get the total energy per unit area radiated out in the direction, Wz(ρ¯ ), and ρ^ direction, W ρ(ρ¯):

Wz(ρ¯)=Sz(r¯,t)dt=q28π2ρv[0ωmodωkρ(ω)+ωeoωcdωkρ(ω)]
(17)
Wρ(ρ¯)=Sρ(r¯,t)dt=q28π2ρ[0ωmodωkρ2ω(ω)ωeoωcdωkρ2ω(ω)]
(18)

Because the speed of the high energy charged particle is very close to c, we can take the limit β→1. The interference between the components of the RH medium band and the LH medium band vanish due to time averaging.

  • In the direction:

    From Eq.(17), we see that the first integral is in a RH medium band (∊(ω)>0 and μ(ω)>0), and the energy flows along the positive direction, which is the same as the direction of the particle motion. However, the second integral is in an LH medium band (∊(ω)<0 and μ(ω)<0), the energy flows along the negative direction, which corresponds to backward power. The total energy crossing the x-y plane is determined by two frequency bands, and the net result will depend on which one is stronger. If we look at a single frequency, the energy will go in different directions.

  • In the ρ^ direction:

    From Eq.(18), the first integral is in an RH medium band, so that the energy flows out of the ρ^ direction. The second integral is in an LH medium band, in which ∊(ω)<0, but there is a negative sign before the integral, which makes the whole second term being positive. Theref ore the energy in this LH medium band also goes out in the ρ^ direction.

4. Čerenkov radiation in lossy LH media

From Kramers-Krönig’s relations, we know that ∊(ω) and μ(ω) have to be complex to satisfy causality. Theref ore, in order to predict the behavior of Čerenkov radiation in real LH media, we have to consider the situation when both the permittivity and the permeability are complex.

The complex permittivity ∊(ω) and permeability μ(ω) must satisfy

(ω)=(ω)*withI(ω)>0
(19a)
μ(ω)=μ(ω)*withμI(ω)>0.
(19b)

For lossy media, the condition for Čerenkov radiation is [9

9. M. H. Saffouri, “Treatment of Čerenkov radiation from electric and magnetic charges in dispersive and dissipative media,” Nuovo Cimento 3D, 589–622 (1984). [CrossRef]

]

{n2(ω)}>1β2,
(20)

where ℜ{·} is the real part operator. The argument of the Hankel functions is now complex. However, the solutions of Eq. (5) are unchanged. In order to ensure finite electric and magnetic fields at ρ→+∞, we write

  • For RH media: g(ρ)=i4H0(1)(kρρ) , kρ=ω2c2μrrω2v2=kR+ikI,wherekI>0,kR>0.
  • For LH media: g(ρ)=i4H0(2)(kρρ) , kρ=ω2c2μrrω2v2=kR+ikI,wherekI<0,kR>0 .

For an RH medium band, we obtain a result identical to [9

9. M. H. Saffouri, “Treatment of Čerenkov radiation from electric and magnetic charges in dispersive and dissipative media,” Nuovo Cimento 3D, 589–622 (1984). [CrossRef]

]. However, for an LH medium band, the nonvanishing fields are

Eρ(r¯,t)=q4πv2πρLHkρ12(ω)cos(ωt+kRρωvzπ4+θ2θ)ekIρdω
(21a)
Ez(r¯,t)=q4π2πρLHkρ32ω(ω)cos(ωt+kRρωvzπ4+3θ2θ)ekIρdω
(21b)
Hϕ(r¯,t)=q4π2πρLHkρ12cos(ωt+kRρωvzπ4+θ2)ekIρdω
(21c)

where θ is the angle of k ρ by letting kρ=ωvηeiθ , and θ is the angle of ∊(ω) in the complex plane by letting (ω)=(ω)eiθ . By using Eqs. (21a) to (21c), we can calculate the energy per unit area radiated out in the ρ^ and directions for LH media and compare those with the corresponding components in RH media as obtained in [9

9. M. H. Saffouri, “Treatment of Čerenkov radiation from electric and magnetic charges in dispersive and dissipative media,” Nuovo Cimento 3D, 589–622 (1984). [CrossRef]

]

  • For RH media:
    Wρ(ρ¯)=Sρ(r¯,t)dt=q28π2ρRHkρ2ω(ω)e2kIρcos(θθ)dω
    (22a)
    Wz(ρ¯)=Sz(r¯,t)dt=q28π2ρvRHkρ(ω)e2kIρcos(θ)dω
    (22b)
  • For LH medium:
    Wρ(ρ¯)=Sρ(r¯,t)dt=q28π2ρLHkρ2ω(ω)e2kIρcos(θθ)dω
    (23a)
    Wz(ρ¯)=Sz(r¯,t)dt=q28π2ρvLHkρ(ω)e2kIρcos(θ)dω
    (23b)

We can see that the direction of power radiation is determined by the angles of ∊(ω) and k ρ.

For a real physical model of permittivity and permeability, we should add an imaginary part to Eqs.(9a) and (9b), which now become

μr(ω)=1ωmp2ωmo2ω2ωmo2+mω
(24a)
r(ω)=1ωep2ωeo2ω2ωeo2+eω
(24b)
Fig. 5. ℜ{n}, ℑ{n}, ℜ{n 2} at a range near the resonant frequency.

The real and imaginary parts of the complex refractive index n as well as the real part of n 2 are plotted in Fig. 5. Note that for such a model, we always have ℑ{∊(ω)}>0 and ℑ{μ(ω)}>0, where ℑ{·} indicate the imaginary part operator. These considerations are summarized in table 1.

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

table-icon
View This Table

We see that we still have backward power in LH media, and the angles θ and θ determine the direction of the power. The lossless limit implies that θ=0 for both LH and RH media, whereas θ=0 for RH media but θ=π for LH media. The expressions for the energy will reduce to Eqs. (17) and (18).

When losses exist, the directions of power propagation Ŝ(ω) (denoted by the angle θs)

Ŝ(ω)=ρ̂ηcos(θθ)+ẑcos(θ)η2cos(θθ)2+cos(θ)2forRHmedia
(25)
Ŝ(ω)=ρ̂ηcos(θθ)+ẑcos(θ)η2cos(θθ)2+cos(θ)2forLHmedia
(26)

are different from that of phase propagation (denoted by the angle θc)

k̂(ω)=ρ̂ηcos(θ)+ẑη2cos(θ)2+1forRHmedia
(27)
k̂(ω)=ρ̂ηcos(θ)+ẑη2cos(θ)2+1forLHmedia
(28)

For the purpose of illustration, we plot the energy distribution as computed from Eqs. (22a) to (23b) by taking the model of Eqs. (24a) and (24b), and taking the values ωmpep=2π×1.09×1012 rad/s, ωmoeo=2π×1.05×1012 rad/s, and γme=γ. All values are calculated at the same distance ρ for all frequencies.

Fig. 6. Energy distribution over frequency for γ=1×108 rad/s.

Figure 6 shows the energy distributions Wz and W ρ over frequency at γ=1×108 rad/s. The high peak is in the RH medium regime for which Wz>0, corresponding to a forward outgoing power. The small peak at f≈1.07×1012 Hz corresponds to W ρ>0 and Wz<0.

Figure 7 shows the radiation pattern of Čerenkov radiation at different γ. We can see from Fig.7(a) that, when the losses are high, there is mainly forward power (there is actually a peak near the resonant frequency corresponding to backward power). Since the losses are so high, the peak value is very small compared to the main lobe in Fig. 7(a). As the losses decrease, the backward power becomes more and more evident, and we also find that the angle of forward power changes. The reason is that the radiation is dominated by the frequency in the region near the resonant frequency, where lossess are so small and the decay term is not strong enough to suppress the amplitude. If the distance ρ increases, the decay term will become dominant, therefore the lobe for backward power will be suppressed, and the pattern will become like the one of Fig. 7(a).

Another noticeable phenomenon is that the angles of forward and backward power are both close to 90° as the losses decrease. This is due to the value of the refractive index which becomes extremely large, see Eq. (2).

Fig. 7. Radiation pattern of Čerenkov radiation for a material characterized by Eqs.(24).

In addition we see that the direction of phase propagating is different from that of power propagating. This difference is due to the losses. We find from Fig. 8, that for an RH medium band, there is almost no difference between these two angles, which is due to the imaginary part is very small at this frequency band, therefore the angles θ and θ are very small. For an LH medium band however, the direction of phase propagation is almost opposite to that of the power.

Fig. 8. The distributions of angle over frequency at γ=1×108 rad/s

5. Conclusion

In this paper, we have given the mathematical solution for Čerenkov radiation in a left-handed material, for both lossless and lossy situations. We have found, consistently with the prediction in [3

3. V. G. Vesalago, “The electrodynamics of substances with simultaneously negative values of ∊ and μ,” Soviet Physics USPEKHI 10, 509–514 (1968). [CrossRef]

], that Čerenkov radiation in LH media exhibits a backward power, yet maintaining a forward vector.

With a simple model for the permittivity and the permeability, we have observed that the radiation pattern of the Čerenkov radiation presents lobes at very large angles, close to 90° with respect to the particle motion, which is in constrast with the angle obtained in classical gas environment. With such a large angle, we expect more photons to be generated, lying the fundamental idea of improved Čerenkov detectors based on the use of LH media.

Acknowledgements

References and links

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, 4184–4187 (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 ∊ and μ,” Soviet Physics USPEKHI 10, 509–514 (1968). [CrossRef]

4.

P. A. Čerenkov, “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, “Coher ent visible radiation of fast electrons passing through matter,” Compt. Rend. (Dokl.) 14, 109–114 (1937).

6.

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

7.

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

8.

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

9.

M. H. Saffouri, “Treatment of Čerenkov radiation from electric and magnetic charges in dispersive and dissipative media,” Nuovo Cimento 3D, 589–622 (1984). [CrossRef]

OCIS Codes
(160.0160) Materials : Materials
(160.1890) Materials : Detector materials

ToC Category:
Focus Issue: Negative refraction and metamaterials

History
Original Manuscript: February 3, 2003
Revised Manuscript: February 25, 2003
Published: April 7, 2003

Citation
Jie Lu, Tomasz Grzegorczyk, Yan Zhang, Joe Pacheco Jr., Bae-Ian Wu, Jin Kong, and Min Chen, "�?erenkov radiation in materials with negative permittivity and permeability," Opt. Express 11, 723-734 (2003)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-7-723


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References

  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).
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