�?erenkov radiation in materials with negative permittivity and permeability

Cerenkov radiation in materials with

negative permittivity and permeability

Jie Lu, Tomasz M. Grzegorczyk, Yan Zhang, Joe Pacheco Jr,Bae-Ian Wu, and Jin A. Kong

Center for Electromagnetic Theory and Applications, RLE, MIT,77 Mass. Ave. Cambridge, 02139, USA

[email protected], [email protected]

Min Chen

Laboratory of Nuclear Science, Department of Physics, MIT,77 Mass. Ave. Cambridge, 02139, USA

[email protected]

Abstract: The mathematical solution for Cerenkov radiation in anovel medium, left-handed medium (LH medium), which has both neg-ative permittivity and permeability, is introduced in this paper. It isshown that the particle motion in the LH medium generates power thatpropagates backward. In this paper, both dispersion and dissipation areconsidered for the LH medium. The results show that in such a ma-terial, both forward power and backward power exist. In addition, weshow that the losses will affect the Cerenkov angle. The idea of build-ing a Cerenkov detector using LH medium is introduced, which couldbe useful in particle physics to identify charged particles of variousvelocities.c© 2003 Optical Society of America

OCIS codes: (160.0160) Materials; (160.1890) Detector materials

References and links1. 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).

2. R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index ofrefraction,” Science 292, 77–79 (2001).

3. V. G. Vesalago, “The electrodynamics of substances with simultaneously negative values of ε andµ,” Soviet Physics USPEKHI 10, 509–514 (1968).

4. P. A. Cerenkov, “Visible radiation produced by electrons moving in a medium with velocitiesexceeding that of light,” Phys. Rev. 52, 378–379 (1937).

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, Cambridge, 2000).7. Jie Lu, T. M. Grzegorczyk, Y. Zhang, J. Pacheco Jr, B. I. Wu, and J. A. Kong, “Cerenkov

radiation in left handed material,” in Proc. Progress in Electromagnetics Research Symposium(Cambridge, MA, 2002), 917.

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

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

(C) 2003 OSA 7 April 2003 / Vol. 11, No. 7 / OPTICS EXPRESS 723#2103 - $15.00 US Received February 03, 2003; Revised February 26, 2003

mailto:[email protected]



1. Introduction

In recent years, materials exhibiting simultaneously negative permittivity (ε) and per-meability (µ) over a frequency band (which we shall call Left-Handed-LH-media, byopposition to Right-Handed-RH-to denote standard media), have received much atten-tion. In particular, the propagation of electromagnetic waves in LH media has beenthoroughly investigated by many researchers [1, 2], and various new phenomena havebeen discovered. Probably the most fundamental one is the discovery that LH mediacan be characterized by a negative index of refraction.

In his pioneering 1968 paper, Vesalago [3] mentioned, in addition to many otherunique properties of LH media, the fact that Cerenkov radiation will be reversed, butno mathematical consideration was given. Normally, Cerenkov radiation occurs whena charged particle moves in a material with a speed faster than that of light in thatmaterial. Cerenkov radiation in normal media was first experimentally obsevered byP. A. Cerenkov in 1934 [4], and later theoretically explained by I. M. Frank and I. G.Tamm [5].

While many researchers have referred to Vesalago’s statement about the reversal ofCerenkov radiation, none have addressed the physical importance of this phenomenon indetail. It is the purpose of the first part of this paper to derive the mathematical solutionfor Cerenkov radiation in LH media in order to demonstrate existence of backwardradiation.

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

2. Mathematical formulation of Cerenkov radiation inside isotropic LH me-dia

2.1 Formulation of the problem

As stated above, Cerenkov radiation occurs when a charged particle travels througha material at a velocity higher than that of light in that material. The principle ofCerenkov radiation is depicted in Fig. 1[6]: A charged particle located at point O that

A

BO

z

θ

θ

v

k

E

Fig. 1. Cerenkov radiation in a normal material (RH media).

is moving along z with a velocity v which satisfies

|v| > c

|n| , (1)


where n is the refractive index of the medium, and c ≈ 3× 108m/s is the speed of lightin free space. The line connecting points A to B forms the phase front of the radiation,which is propagating with the wave vector k = ρkρ + z ω

v , where ω = 2πf is the angularfrequency of the radiation, and kρ is the transverse component of the wave vector k. Theemitted radiation has an electric field vector polarized parallel to the plane determinedby the direction of the particle speed and the direction of the radiation. Note that in realsituations, many charged particles form a beam, so that the radiation has cylindricalsymmetry and forms the well-known Cerenkov cone. The angle of the cone is given byθ (see Fig. 1) and is determined by

cos θ =1βn, (2)

where β = vc < 1.

These characteristics have been well predicted in RH media by I. M. Frank and I.G. Tamm [5] by using classical electromagnetic theory. Following the formulation in [5],we shall rederive the mathematics for Cerenkov radiation in LH media.

2.2 Mathematical solution

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

J(r, t) = zqvδ(z − vt)δ(x)δ(y) , (3)

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

∇2A+ω2

c2n2A = −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[

1ρ

∂

∂ρ

(ρ∂

∂ρ

)+ k2ρ

]g(ρ) = −δ(ρ)

2πρ, (5)

where kρ = ωv

√β2n2 − 1, 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) = i4 H

(1)0 (kρρ) which corresponds to an outgoing wave, for which

k = kρρ+ kz z,

• case 2: g(ρ) = − i4 H

(2)0 (kρρ) which corresponds to an ingoing wave, for which

k = −kρρ+ kz z,

where kz = ωv > 0.

Before choosing any solution, we calculate the electric and magnetic fields for bothcases, from which the total energy per unit area radiated out in ρ and z directions infar field is obtained [7]:

• Case 1:

Wz(ρ) =∫ ∞

−∞Sz(r, t) dt =

q2

8π2ρv

∫ ∞

0

kρεdω (6a)

Wρ(ρ) =∫ ∞

−∞Sρ(r, t) dt =

q2

8π2ρ

∫ ∞

0

k2ρωεdω (6b)


• Case 2:

Wz(ρ) =∫ ∞

−∞Sz(r, t) dt =

q2

8π2ρv

∫ ∞

0

kρεdω (7a)

Wρ(ρ) =∫ ∞

−∞Sρ(r, t) dt = − q2

8π2ρ

∫ ∞

0

k2ρωεdω (7b)

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

For the sake of illustration, we first consider a normal material, with ε > 0 and µ > 0.From Eqs. (6a) to (7b), we see that Wz(ρ) > 0 for both cases, but Wρ(ρ) > 0 for case1 and Wρ(ρ) < 0 for case 2. These two cases correspond to forward (same direction asvelocity of the particle) outgoing energy, and forward ingoing energy, respectively. FromSommerfeld’s radiation condition (no energy can come from infinity, since radiationmust be emitted from a source), we choose case 1 as the correct solution for Cerenkovradiation in normal materials with both ε and µ positive [8].

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

• case 1: Wz(ρ) < 0, Wρ(ρ) < 0 which corresponds to a backward and ingoingradiated energy.

• case 2: Wz(ρ) < 0, Wρ(ρ) > 0 which corresponds to a backward and outgoingradiated energy.

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

Wz

Wρ

v

k

Fig. 2. Directions of energy flow and wave vector for a charged particle moving in

an LH medium for case 1 [g(ρ) = i4

H(1)0 (kρρ) ].

Wz

Wρ

v

k

Fig. 3. Directions of energy flow and wave vector for a charged particle moving in

an LH medium for case 2 [g(ρ) = − i4

H(2)0 (kρρ) ].

If we again suppose that there are no sources at infinity, the solution that needsto be chosen is the one corresponding to case 2. In addition, both the permittivityand the permeability need be negative to assure a real k that can support propagatingwaves. Finally, in the far field for isotropic lossless materials (so that the directionsof the Poynting vector S is opposite to that of the wave vector k), the angle betweenthe 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 thereforedemonstrated that the energy is propagating backward as predicted in [3].

Yet, we still need to consider how the momentum is conserved, which relates to thedefinition of momentum in LH media. The standard definition of the momentum of anelectromagnetic wave is D(r, t) × B(r, t) = εµS(r, t) [6]. Upon using this definition, wesee that the momentum is

D(r, t) ×B(r, t) = εµE(r, t) ×H(r, t) = εµS(r, t) . (8)

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

The solution to this paradox is to be found in the quantum theory of Cerenkovradiation [8], in which the momentum of a photon is defined as p = hk, where p is themomentum, and h is the Plank constant divided by 2π. For case 2, kz > 0 which impliesa 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 inthe opposite direction of its momentum. When the wave crosses the boundary froman LH medium into a RH medium, the component of wave vector kz (thus also themomentum direction) will change sign (from +z to −z direction in our case), but thePoynting vector E(r, t)×H(r, t), which defines the energy flow, remains backward (−zdirection). Therefore once inside the RH medium, both energy flow and momentum ofthe wave will again be in the same (backward) direction.

3. Cerenkov radiation in dispersive LH media

It is already known from [3] that LH media must be frequency dispersive in order tosatisfy positive energy constraints.

A common model to represent the permittivity ε(ω) and permeability µ(ω) has beengiven in [2], which we shall use here. For the sake of simplicity, we shall first consider alossless case, for which the model becomes:

µr(ω) = 1 − ω2mp − ω2

mo

ω2 − ω2mo

(9a)

εr(ω) = 1 − ω2ep − ω2

eo

ω2 − ω2eo

(9b)

The following critical points can be identified

ωmc =

√ω2

mp + ω2mo

2for which µr(ωmc) = −1 (10a)

ωec =

√ω2

ep + ω2eo

2for which εr(ωec) = −1 (10b)

ωc =

√ω2

epω2mp − ω2

eoω2mo

ω2ep + ω2

mp − ω2eo − ω2

mo

for which µr(ωc)εr(ωc) = 1 (10c)

A summary of the various frequency bands generated and their properties is shown inFig. 4. The lower dark region corresponds to n2 > 1, for which Cerenkov radiation canhappen (supposing that β = 1).


RHM LHM

µr>1 µr<−1 −1<µr<0 0<µr<1εr>1 εr<−1 −1<εr<0 0<εr<1

n2>1 0<n2<1

εr>1µr<−1

εr<−1 εr<−10<µr<1 0<µr<1

RHM

n2>1 0<n2<1n2<0 n2<0n2<0

0 ωmo ωeo ωmc ωc ωecωmp ωep ω

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

The solution to Cerenkov radiation depends on frequency, as already shown inEqs. (6a) to (7b). The band in which the region of n2(ω) > 1 overlaps with theRH media region, ω ∈ [0, ωmo], corresponds to positive εr(ω) and positive µr(ω), andn2(ω) = εr(ω)µr(ω) > 1. From the conclusion of the previous section, the solution inthis frequency band is the one of case 1. Similarly, for the LH media band and withn2(ω) > 1, ω ∈ [ωeo, ωc], the solution is the one of case 2. The nonvanish field compo-nents produced by the propagating waves (in far field) are:

Ez(r, t) =q

4π

√2πρ

[∫ ωmo

0

(−)kρ

√kρ

ωε(ω)cos(φ+) dω +

∫ ωc

ωeo

kρ

√kρ

ωε(ω)cos(φ−) dω

](11)

Eρ(r, t) =q

4πv

√2πρ

[∫ ωmo

0

√kρ

ε(ω)cos(φ+) dω +

∫ ωc

ωeo

√kρ

ε(ω)cos(φ−) dω

](12)

Hφ(r, t) =q

4π

√2πρ

[∫ ωmo

0

√kρ cos(φ+) dω +

∫ ωc

ωeo

√kρ cos(φ−) dω

], (13)

where φ± = ωt ∓ kρρ − ωzv ± π

4 , with the upper sign corresponding to case 1, and thelower sign corresponding to case 2. The Poynting vector S(r, t) = zSz(r, t)+ ρSρ(r, t) =E(r, t) ×H(r, t) is given by

Sz(r, t) = Eρ(r, t)Hφ(r, t) =q2

8π3ρv

×[∫ ωmo

0

dω

∫ ωmo

0

dω′√kρk′ρε(ω)

cos(φ+) cos(φ′+)

+∫ ωc

ωeo

dω

∫ ωc

ωeo


cos(φ−) cos(φ′−)

+∫ ωmo

0

dω

∫ ωc

ωeo


cos(φ+) cos(φ′−)

+∫ ωc

ωeo

dω

∫ ωmo

0


cos(φ−) cos(φ′+)

], (14)


Sρ(r, t) = −Ez(r, t)Hφ(r, t) =q2

8π3ρ

×[∫ ωmo

0

dω

∫ ωmo

0

dω′kρ

√kρk′ρ

ωε(ω)cos(φ+) cos(φ′+)

−∫ ωc

ωeo

dω

∫ ωc

ωeo

dω′kρ

√kρk′ρ

ωε(ω)cos(φ−) cos(φ′−)

+∫ ωmo

0

dω

∫ ωc

ωeo

dω′kρ

√kρk′ρ

ωε(ω)cos(φ+) cos(φ′−)

−∫ ωc

ωeo

dω

∫ ωmo

0

dω′kρ

√kρk′ρ

ωε(ω)cos(φ−) cos(φ′+)

]. (15)

By using the identity [6]∫ ∞

−∞cos(ωt+ α) cos(ω′t+ α′)dt = πδ(ω − ω′) cos(α− α′) , (16)

we can get the total energy per unit area radiated out in the z direction, Wz(ρ), and ρdirection, Wρ(ρ):

Wz(ρ) =∫ ∞

−∞Sz(r, t)dt =

q2

8π2ρv

[∫ ωmo

0

dωkρε(ω)

+∫ ωc

ωeo

dωkρε(ω)

](17)

Wρ(ρ) =∫ ∞

−∞Sρ(r, t)dt =

q2

8π2ρ

[∫ ωmo

0

dωk2ρωε(ω)

−∫ ωc

ωeo

dωk2ρωε(ω)

](18)

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

• In the z direction:From Eq. (17), we see that the first integral is in a RH medium band (ε(ω) > 0and µ(ω) > 0), and the energy flows along the positive z direction, which is thesame as the direction of the particle motion. However, the second integral is in anLH medium band (ε(ω) < 0 and µ(ω) < 0), the energy flows along the negativez direction, which corresponds to backward power. The total energy crossing thex− y plane is determined by two frequency bands, and the net result will dependon which one is stronger. If we look at a single frequency, the energy will go indifferent directions.

• In the ρ direction:From Eq. (18), the first integral is in an RH medium band, so that the energyflows out of the ρ direction. The second integral is in an LH medium band, inwhich ε(ω) < 0, but there is a negative sign before the integral, which makes thewhole second term being positive. Therefore the energy in this LH medium bandalso goes out in the ρ direction.

4. Cerenkov radiation in lossy LH media

From Kramers-Kronig’s relations, we know that ε(ω) and µ(ω) have to be complex tosatisfy causality. Therefore, in order to predict the behavior of Cerenkov radiation in


real LH media, we have to consider the situation when both the permittivity and thepermeability are complex.

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

ε(−ω) = ε(ω)∗ with εI(ω) > 0 (19a)µ(−ω) = µ(ω)∗ with µI(ω) > 0 . (19b)

For lossy media, the condition for Cerenkov radiation is [9]

�{n2(ω)} > 1β2, (20)

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

• For RH media: g(ρ) = i4 H

(1)0 (kρρ),

kρ =√

ω2

c2 µrεr − ω2

v2 = kR + ikI , where kI > 0, kR > 0.

• For LH media: g(ρ) = − i4 H

(2)0 (kρρ),

kρ =√

ω2

c2 µrεr − ω2

v2 = kR + ikI , where kI < 0, kR > 0.

For an RH medium band, we obtain a result identical to [9]. However, for an LHmedium band, the nonvanishing fields are

Eρ(r, t) =q

4πv

√2πρ

∫LH

|kρ|1/2

|ε(ω)| cos(ωt+ kRρ− ωvz − π

4+θ

2− θε) ekIρ dω (21a)

Ez(r, t) =q

4π

√2πρ

∫LH

|kρ|3/2

ω|ε(ω)| cos(ωt+ kRρ− ωvz − π

4+

3θ2

− θε) ekIρ dω (21b)

Hφ(r, t) =q

4π

√2πρ

∫LH

|kρ|1/2 cos(ωt+ kRρ− ωvz − π

4+θ

2) ekIρ dω (21c)

where θ is the angle of kρ by letting kρ = ωv ηe

iθ, and θε is the angle of ε(ω) in the complexplane by letting ε(ω) = |ε(ω)|eiθε . By using Eqs. (21a) to (21c), we can calculate theenergy per unit area radiated out in the ρ and z directions for LH media and comparethose with the corresponding components in RH media as obtained in [9]

• For RH media:

Wρ(ρ) =∫ ∞


q2

8π2ρ

∫RH

|kρ|2ω|ε(ω)|e

−2kIρ cos(θ − θε) dω (22a)

Wz(ρ) =∫ ∞

−∞Sz(r, t) dt =

q2

8π2ρv

∫RH

|kρ||ε(ω)|e

−2kIρ cos(θε) dω (22b)

• For LH medium:

Wρ(ρ) =∫ ∞


q2

8π2ρ

∫LH

−|kρ|2ω|ε(ω)|e

2kIρ cos(θ − θε) dω (23a)

Wz(ρ) =∫ ∞

−∞Sz(r, t) dt =

q2

8π2ρv

∫LH

|kρ||ε(ω)|e

2kIρ cos(θε) dω (23b)


We can see that the direction of power radiation is determined by the angles of ε(ω)and kρ.

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

µr(ω) = 1 − ω2mp − ω2

mo

ω2 − ω2mo + iγmω

(24a)

εr(ω) = 1 − ω2ep − ω2

eo

ω2 − ω2eo + iγeω

(24b)

1 1.01 1.02 1.03 1.04 1.05 1.06 1.07-10

-5

0

5

10

15

20

ℜ {n}ℑ {n}

ℜ {n }2

x 10 Hz12

frequency

Fig. 5. �{n}, �{n}, �{n2} at a range near the resonant frequency.

The real and imaginary parts of the complex refractive index n as well as the real partof n2 are plotted in Fig. 5. Note that for such a model, we always have �{ε(ω)} > 0 and�{µ(ω)} > 0, where �{·} indicate the imaginary part operator. These considerationsare summarized in table 1.

Table 1. The range of angle for ε-θε-and kρ-θ.

Properties RH medium band LH medium band�{ε(ω)} > 0 < 0θε [0, π

2 ] [π2 , π]

kI > 0 < 0θ [0, π

2 ] [3π2 , 2π]

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

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


θs)

S(ω) =ρη cos(θ − θε) + z cos(θε)√η2 cos(θ − θε)2 + cos(θε)2

for RH media (25)

S(ω) =−ρη cos(θ − θε) + z cos(θε)√η2 cos(θ − θε)2 + cos(θε)2

for LH media (26)

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

k(ω) =ρη cos(θ) + z√η2 cos(θ)2 + 1

for RH media (27)

k(ω) =−ρη cos(θ) + z√η2 cos(θ)2 + 1

for LH media (28)

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

0 2 4 6 8 10 12-0.5

0

0.5

1

1.5

2

2.5x 10

-5

WρWz

x 10 Hz11frequency

Ene

rgy

per

unit

area

(eV

/mm

)2

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

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

Figure 7 shows the radiation pattern of Cerenkov radiation at different γ. We can seefrom Fig. 7(a) that, when the losses are high, there is mainly forward power (there is ac-tually a peak near the resonant frequency corresponding to backward power). Since thelosses 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 wealso find that the angle of forward power changes. The reason is that the radiation isdominated by the frequency in the region near the resonant frequency, where lossessare so small and the decay term is not strong enough to suppress the amplitude. Ifthe distance ρ increases, the decay term will become dominant, therefore the lobe forbackward 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 powerare both close to 90◦ as the losses decrease. This is due to the value of the refractiveindex which becomes extremely large, see Eq. (2).

30

210

60

240

90

270

120

300

150

330

80 0

(a) γ = 1 × 1010 rad/s, normalizing

constant 4.6 × 10−6.

30

210

60

240

90

270

120

300

150

330

80 0

(b) γ = 1 × 108 rad/s, normalizing


30

210

60

240

90

270

120

300

150

330

180 0

(c) γ = 1 × 107 rad/s, normalizing


30

210

60

240

90

270

120

300

150

330

180 0

(d) γ = 1 × 106 rad/s, normalizing


Fig. 7. Radiation pattern of Cerenkov radiation for a material characterized byEqs.(24).

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


0.9 0.94 0.98 1.02 1.06 1.1

x 1012

-100

-50

0

50

100

150

angl

e (

)

θc

θs

Hzfrequency

◦

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 Cerenkov radiation in aleft-handed material, for both lossless and lossy situations. We have found, consistentlywith the prediction in [3], that Cerenkov radiation in LH media exhibits a backwardpower, yet maintaining a forward k vector.

With a simple model for the permittivity and the permeability, we have observedthat the radiation pattern of the Cerenkov radiation presents lobes at very large angles,close to 90◦ with respect to the particle motion, which is in constrast with the angleobtained in classical gas environment. With such a large angle, we expect more photonsto be generated, lying the fundamental idea of improved Cerenkov detectors based onthe use of LH media.

Acknowledgements

This work was supported in part by the MIT Lincoln Laboratory under Contract BX-8133 and the Office of Naval Research under Contract Nos. N00014-10-1-0713, andN00014-99-1-0175.


Date post:	05-Oct-2016
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�?erenkov radiation in materials with negative permittivity and permeability

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