- Email: [email protected]
Radiation of multiple-scattered charged particle
ARTICLE IN PRESS
Nuclear Instruments and Methods in Physics Research A 563 (2006) 364–367 www.elsevier.com/locate/nima
Radiation of multiple-scattered charged particle V.M. Grichinea,b, a
P.N. Lebedev Physical Institute, 119991 Moscow, Russia b CERN/PH-SFT, Geneva 23 CH-1211, Switzerland Available online 13 March 2006
Abstract The aberration of Cherenkov radiation due to multiple scattering of radiating charged particle is considered in terms of the radiation energy loss. The energy-angular distribution of photons emitted by multiple-scattered charged particle is derived for the case of transparent medium. The cases energy-angular distribution of Cherenkov radiation (absorption) in left-handed materials are discussed. r 2006 Elsevier B.V. All rights reserved. PACS: 41.60.Bq; 29.40.Ka Keywords: Cherenkov radiation; Left-handed materials; Multiple scattering
In many large water Cherenkov detectors it is necessary to estimate the accuracy of rare event vertex reconstruction based on the analysis of Cherenkov light signals.The influence of multiple scattering on the Cherenkov radiation produced by the slow electrons in liquids affect the accuracy of the reconstruction of the event vertex. The problem has been already considered in a few publications (see, e.g., Ref. [1]). The analytical solution for the energyangular distribution of emitted photons, which is needed for simulation was not however found. Here we report on the radiation produced by multiplescattered charged particle in terms of the radiation energy loss, which is suitable for simulation in the spirit of simulation approach, when photons are tracked just starting from the trajectory of the radiating charged particle. The energy-angular distribution of emitted photons will be derived for the case of transparent medium. In addition, some features of recently realized left-handed materials will be considered for Cherenkov radiation (absorption). Let us consider a relativistic charged particle with the charge e moving along an arbitrary trajectory rðtÞ in a uniform, isotropic and absorbing medium with the comCorresponding author. CERN/PH-SFT, Geneva 23 CH-1211, Switzerland. Tel.: +41 22 76 75532; fax: +41 22 767 8130. E-mail address: [email protected].
0168-9002/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2006.02.152
plex dielectric permittivity ¼ 1 þ i2 and magnetic permeability m ¼ m1 þ im2 . Following the results of Ref. [2] and references therein, one can derive the relation for ¯ ? emitted into unit solid the mean number of photons N angle O, per unit energy _o, in unit time t: (Z 1 ¯ ? ðtÞ d3 N a mðoÞ dk ¼ Im 2 2 3 _ do dt dO 2p _c k ðoÞmðoÞ oc2 0 Z 1 dt ½k2 vðt þ tÞvðtÞ o2 1 ) exp iot ik½rðt þ tÞ rðtÞ , ð1Þ where a is the fine structure constant, _ is the Planck’s constant, vðtÞ ¼ r_ðtÞ is the charge velocity, and k is the modulus of the photon wave vector k. O is the solid angle defining the direction of k versus vðtÞ. For simplicity, we consider the frequency dispersion only. Multiple scattering can result in the radiation angular distribution broadening. Let us consider the relativistic charge moving in a medium and experiencing multiple scattering (Fig. 1). One can calculate the energy-angular distribution of the mean number of emitted photons averaged over all possible (in the sense of multiple scattering) trajectories. For simplicity we consider optical frequencies where magnetic permeability is negligible. The
ARTICLE IN PRESS V.M. Grichine / Nuclear Instruments and Methods in Physics Research A 563 (2006) 364–367
365
The equation for f l ðs; kÞ reads qf l ðs; kÞ þ ½sl þ ik cos yf l ðs; kÞ ¼ 0 qs where Z p sl ¼ 2pN sðwÞ½1 Pl ðcos wÞ sin w dw.
Radiation of multiple scattered charge
k
0
Then the normalized distribution function reads
n(s) ϑ(s)
f ðs; k; nÞ ¼
n
Due to the orthogonality of the Legendre polynomials only the first two terms of this sum will contribute to hIðk; oÞi. Integration of Eq. (3) with respect to s followed by the extraction of real part result in (k cos y ¼ ðk n0 Þ) yields 2po2 o 2k2 vs1 d k cos y hIðk; oÞi ¼ . þ 2 v v k cos y ov þ s21
Fig. 1. The notations for the multiple scattering radiation.
modified relation (1) reads (Z ) 1 ¯? d3 N a hIðk; oÞi dk 2 . ¼ 3 Im 2 2p _c _ do dt dO k ðoÞ oc2 0 Z 1 o 2 Re ds exp i s v v 0
Z dn f ðs; k; nÞ½k2 v2 ðn n0 Þ o2 .
hIðk; oÞi ¼
(2)
ð3Þ
4p
Here s is the trajectory path length, Z f ðs; k; nÞ ¼ d r f ðs; r; nÞ expðik rÞ R3
is the Fourier transform of the probability density distribution function, f ðs; r; nÞ, n is the trajectory direction unit vector at the path s, and n0 is the initial trajectory direction unit vector at the time t (s ¼ 0). The distribution function f ðs; k; nÞ satisfies the following kinetic equation [3]: qf ðs; k; nÞ þ iðk nÞf ðs; k; nÞ qs Z
½f ðs; k; n0 Þ f ðs; k; nÞsðjn0 njÞ dn0
¼N 4p
with the initial condition: f ð0; k; nÞ ¼ dðn n0 Þ. Here s is the elastic cross-section which is responsible for the multiple scattering and N is the atomic density. We suppose that the multiple scattering angles W are negligibly small compared to the radiation emission angle y, and put ðk nÞ ’ ðk n0 Þ. Then the distribution function can be expanded to in terms of the series of the Legendre polynomials Pl ðcos WÞ, cos W ¼ ðn n0 Þ: f ðs; k; nÞ ¼
1 X l¼0
f l ðs; kÞPl ðcos WÞ.
1 X 2l þ 1 Pl ðcos WÞ exp½ðsl þ ik cos yÞs. 4p l¼0
(4)
The first term reflects the contribution from straight (in average) trajectory and hence corresponds to usual Cherenkov radiation. The second term shows the averaged effect of multiple scattering on the angular distribution of radiation. We consider the case of transparent medium, when the calculation of Eq. (2) is less cumbersome. Following the method developed in Ref. [2], the contribution of multiple scattering reads: I
¯ ?msc d3 N a vs1 Im gðzÞ dz , ¼ _c p2 cos3 y _ do dt d cos2 y C
1 z2 1 1 þ gðzÞ ¼ , 2 z2 a2 ðz bÞðz b Þ ðz þ bÞðz þ b Þ o2 1 o þ is ; b ¼ . 1 cos y v c2 Here the integration with respect to z is performed along contour C (with radius R ! 1) shown in Fig. 2. The integral over the contour C is defined by the sum of residues in the singularities of gðzÞ, i.e. the points a, b and b . In the limit of transparent medium the integral is defined by the pole a only. The result reads (dx ¼ v dt): ¯ ?msc d3 N a bðn2 cos2 y þ b2 þ d2 Þ ¼ _c 2n cos y _ do dx d cos2 y Gmsc , ð5Þ p½ðcos2 y cos2 ymsc Þ2 þ G2msc a2 ¼
cos2 ymsc ¼
1 b2 d 2 ; b2 n2
Gmsc ¼
2d ; bn2
d¼
cs1 . o
The calculation for the straight line trajectory is trivial. One easily gets ¯ ?cr d3 N a 1 1 2 d cos y . ¼ _c n2 b2 _ do dx d cos2 y n2 b 2
ARTICLE IN PRESS V.M. Grichine / Nuclear Instruments and Methods in Physics Research A 563 (2006) 364–367
366
cos2 yx ¼
z
Gx ¼
x2 . b2 jxj2
Here x1 ¼ 1 m1 2 m2 , x2 ¼ 1 m2 þ 2 m1 , and jxj2 ¼ ð21 þ 22 Þ ðm21 þ m22 Þ. In the case of medium without magnetic permeability (m ¼ 1, for optical frequencies) we get for the mean number of emitted Cherenkov photons from the unit particle trajectory length:
R→∞
−b∗
x1 ; b2 jxj2
b
¯? d3 N a Gsin2 y , ¼ _ do dx d cos2 y _c p½ðcos2 y cos2 yo Þ2 þ G2
a
cos2 yo ¼ Fig. 2. The contour C of integration for the multiple scattering radiation.
In the case of negligible multiple scattering d ! 0 the energy-angular distribution will be reduced to the usual Cherenkov radiation: ¯? d3 N _ do dx d cos2 y ¯ ?msc ¯ ?cr d3 N d3 N ¼ þ _ do dx d cos2 y _ do dx d cos2 y a 1 1 2 1 ¼ y d cos . _c n2 b2 n2 b2
ð6Þ
The value of s1 is defined (for W51) by the mean squared angle of multiple scattering per unit length of the particle trajectory. For particles with moderate energy moving in heavy medium, the relative broadening of the radiation angular distribution can be estimated as Gmsc = cos ymsc 2d=nt0:1. It affects the angular distribution much substantially compared to the medium absorption. The multiple scattering correction however is about a few mrad in water for electrons with the energy 1210 MeV. For the case of movement with constant velocity v the integral with respect to t can be calculated and relation (1) is reduced to
¯? d3 N a m tan2 y ¼ Im _ do dx d cos2 y _c p½1 mb2 cos2 y where b ¼ v=c, c is the speed of light in vacuum, dx ¼ v dt is the element of the particle trajectory and y is the angle between k and v. Introducing a new complex variable, x ¼ m ¼ x1 þ ix2 , one gets the following general energyangular distribution of emitted Cherenkov photons: ¯? d3 N _ do dx d cos2 y
a m2 tan2 y 2 2 2 m sin y þ ½1 x1 b cos y ¼ _c 1 x2 b 2 Gx , p½ðcos2 y cos2 yx Þ2 þ G2x
1 ; b jj2 2
G¼
2 . b jj2
(8)
2
The integration of this relation with respect to cos2 y results in the energy spectrum of the radiation:
¯? d2 N a 1 1 Im 1 2 ln ¼ . _ do dx p_c b 1 b2 In the limit of a transparent medium, when 2 ! 0 and m2 ! 0, one can get the energy-angle distribution of the ¯ c in the transparent mean number of Cherenkov photons N medium with refractive index n (n2 ¼ 1 m1 ): ¯c d3 N am1 1 ¼ signðGx Þ 1 2 _ do dx d cos2 y _c b n2 1 d cos2 y 2 ð9Þ b n2 where d is the Dirac delta function and the introduction of sign ðGx Þ allows us to take into account the absorption of photons rather than their emission. This expression fixes
cos2c =
1 ∋112
Right-handed materials ∋2 > 0 2 > 0
∋1 > 0
∋2 > 0
1 > 0
2 > 0 c
∋2 > 0 2 > 0
∋1 > 0 1 > 0
e ∋2 > 0 2 > 0
Left-handed materials
Cases of Cherenkov radiation
ð7Þ
Fig. 3. The cases of Cherenkov radiation in right- and left-handed materials.
ARTICLE IN PRESS V.M. Grichine / Nuclear Instruments and Methods in Physics Research A 563 (2006) 364–367
explicitly the emission angle to be yc ¼ arccosð1=bnÞ. If bn41, integration of Eq. (9) with respect to cos2 y results in the well known formula of the Frank–Tamm theory (in the particular case, m1 ¼ 1 and n2 ¼ 1 ): ¯c d2 N a 1 1 2 ; bn41. (10) ¼ signðGÞ _c _ do dx b n2 One can consider another interesting case when 1 m1 40, but both 1 o0 and m1 o0. Here the radiation follows the group velocity qo=qk which is anti-parallel to the wave vector k. Therefore the radiation is emitted at the angle p yc 4p=2 (see review [4]). In addition, relation (9) shows that Cherenkov radiation energy loss can be negative (m1 ): ¯c d2 N a 1 ¼ signðGx Þ m1 1 _c _ do dx 1 m1 b2 and the radiation has the tendency to be absorbed along the direction p yc , if 2 o0, m2 o0 and hence Gx 40. Materials with both negative permittivity and permeability (left-handed materials) were recently experimentally investigated and anomalous behavior of refraction for was observed [5]. It would be interesting to investigate also the
367
angular distribution of the Cherenkov radiation in this case keeping in mind that the Cherenkov angle can be obtuse, and to check that Cherenkov photons are absorbed by relativistic charge in the case when 2 o0 and m2 o0. The cases of angular direction of Cherenkov radiation are shown in Fig. 3. A similar picture is valid for the cases of Doppler radiation in right- and left-handed materials. The author is thankful to the participants of I.E. Tamm Theory Division (P.N. Lebedev Institute) seminar for stimulating discussion. This investigation was supported partly by INTAS-2001-0323 grant and was done in the framework of GEANT4 Collaboration.
References [1] [2] [3] [4] [5]
M.G. Bowler, M.D. Lay, Nucl. Instr. and Meth. A 378 468. V.M. Grichine, S.S. Sadilov, Phys. Lett. B 559 26. H.W. Lewis, Phys. Rev. 78 (1950) 526. V.G. Veselago, Sov. Phys. Usp. 10 509. D.R. Smith, W.J. Padilla, D.C. Vier, S.C. Nemat-Nasser, S. Schultz, Phys. Rev. Lett. 84 4184.
Nuclear Instruments and Methods in Physics Research A 563 (2006) 364–367 www.elsevier.com/locate/nima
Radiation of multiple-scattered charged particle V.M. Grichinea,b, a
P.N. Lebedev Physical Institute, 119991 Moscow, Russia b CERN/PH-SFT, Geneva 23 CH-1211, Switzerland Available online 13 March 2006
Abstract The aberration of Cherenkov radiation due to multiple scattering of radiating charged particle is considered in terms of the radiation energy loss. The energy-angular distribution of photons emitted by multiple-scattered charged particle is derived for the case of transparent medium. The cases energy-angular distribution of Cherenkov radiation (absorption) in left-handed materials are discussed. r 2006 Elsevier B.V. All rights reserved. PACS: 41.60.Bq; 29.40.Ka Keywords: Cherenkov radiation; Left-handed materials; Multiple scattering
In many large water Cherenkov detectors it is necessary to estimate the accuracy of rare event vertex reconstruction based on the analysis of Cherenkov light signals.The influence of multiple scattering on the Cherenkov radiation produced by the slow electrons in liquids affect the accuracy of the reconstruction of the event vertex. The problem has been already considered in a few publications (see, e.g., Ref. [1]). The analytical solution for the energyangular distribution of emitted photons, which is needed for simulation was not however found. Here we report on the radiation produced by multiplescattered charged particle in terms of the radiation energy loss, which is suitable for simulation in the spirit of simulation approach, when photons are tracked just starting from the trajectory of the radiating charged particle. The energy-angular distribution of emitted photons will be derived for the case of transparent medium. In addition, some features of recently realized left-handed materials will be considered for Cherenkov radiation (absorption). Let us consider a relativistic charged particle with the charge e moving along an arbitrary trajectory rðtÞ in a uniform, isotropic and absorbing medium with the comCorresponding author. CERN/PH-SFT, Geneva 23 CH-1211, Switzerland. Tel.: +41 22 76 75532; fax: +41 22 767 8130. E-mail address: [email protected].
0168-9002/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2006.02.152
plex dielectric permittivity ¼ 1 þ i2 and magnetic permeability m ¼ m1 þ im2 . Following the results of Ref. [2] and references therein, one can derive the relation for ¯ ? emitted into unit solid the mean number of photons N angle O, per unit energy _o, in unit time t: (Z 1 ¯ ? ðtÞ d3 N a mðoÞ dk ¼ Im 2 2 3 _ do dt dO 2p _c k ðoÞmðoÞ oc2 0 Z 1 dt ½k2 vðt þ tÞvðtÞ o2 1 ) exp iot ik½rðt þ tÞ rðtÞ , ð1Þ where a is the fine structure constant, _ is the Planck’s constant, vðtÞ ¼ r_ðtÞ is the charge velocity, and k is the modulus of the photon wave vector k. O is the solid angle defining the direction of k versus vðtÞ. For simplicity, we consider the frequency dispersion only. Multiple scattering can result in the radiation angular distribution broadening. Let us consider the relativistic charge moving in a medium and experiencing multiple scattering (Fig. 1). One can calculate the energy-angular distribution of the mean number of emitted photons averaged over all possible (in the sense of multiple scattering) trajectories. For simplicity we consider optical frequencies where magnetic permeability is negligible. The
ARTICLE IN PRESS V.M. Grichine / Nuclear Instruments and Methods in Physics Research A 563 (2006) 364–367
365
The equation for f l ðs; kÞ reads qf l ðs; kÞ þ ½sl þ ik cos yf l ðs; kÞ ¼ 0 qs where Z p sl ¼ 2pN sðwÞ½1 Pl ðcos wÞ sin w dw.
Radiation of multiple scattered charge
k
0
Then the normalized distribution function reads
n(s) ϑ(s)
f ðs; k; nÞ ¼
n
Due to the orthogonality of the Legendre polynomials only the first two terms of this sum will contribute to hIðk; oÞi. Integration of Eq. (3) with respect to s followed by the extraction of real part result in (k cos y ¼ ðk n0 Þ) yields 2po2 o 2k2 vs1 d k cos y hIðk; oÞi ¼ . þ 2 v v k cos y ov þ s21
Fig. 1. The notations for the multiple scattering radiation.
modified relation (1) reads (Z ) 1 ¯? d3 N a hIðk; oÞi dk 2 . ¼ 3 Im 2 2p _c _ do dt dO k ðoÞ oc2 0 Z 1 o 2 Re ds exp i s v v 0
Z dn f ðs; k; nÞ½k2 v2 ðn n0 Þ o2 .
hIðk; oÞi ¼
(2)
ð3Þ
4p
Here s is the trajectory path length, Z f ðs; k; nÞ ¼ d r f ðs; r; nÞ expðik rÞ R3
is the Fourier transform of the probability density distribution function, f ðs; r; nÞ, n is the trajectory direction unit vector at the path s, and n0 is the initial trajectory direction unit vector at the time t (s ¼ 0). The distribution function f ðs; k; nÞ satisfies the following kinetic equation [3]: qf ðs; k; nÞ þ iðk nÞf ðs; k; nÞ qs Z
½f ðs; k; n0 Þ f ðs; k; nÞsðjn0 njÞ dn0
¼N 4p
with the initial condition: f ð0; k; nÞ ¼ dðn n0 Þ. Here s is the elastic cross-section which is responsible for the multiple scattering and N is the atomic density. We suppose that the multiple scattering angles W are negligibly small compared to the radiation emission angle y, and put ðk nÞ ’ ðk n0 Þ. Then the distribution function can be expanded to in terms of the series of the Legendre polynomials Pl ðcos WÞ, cos W ¼ ðn n0 Þ: f ðs; k; nÞ ¼
1 X l¼0
f l ðs; kÞPl ðcos WÞ.
1 X 2l þ 1 Pl ðcos WÞ exp½ðsl þ ik cos yÞs. 4p l¼0
(4)
The first term reflects the contribution from straight (in average) trajectory and hence corresponds to usual Cherenkov radiation. The second term shows the averaged effect of multiple scattering on the angular distribution of radiation. We consider the case of transparent medium, when the calculation of Eq. (2) is less cumbersome. Following the method developed in Ref. [2], the contribution of multiple scattering reads: I
¯ ?msc d3 N a vs1 Im gðzÞ dz , ¼ _c p2 cos3 y _ do dt d cos2 y C
1 z2 1 1 þ gðzÞ ¼ , 2 z2 a2 ðz bÞðz b Þ ðz þ bÞðz þ b Þ o2 1 o þ is ; b ¼ . 1 cos y v c2 Here the integration with respect to z is performed along contour C (with radius R ! 1) shown in Fig. 2. The integral over the contour C is defined by the sum of residues in the singularities of gðzÞ, i.e. the points a, b and b . In the limit of transparent medium the integral is defined by the pole a only. The result reads (dx ¼ v dt): ¯ ?msc d3 N a bðn2 cos2 y þ b2 þ d2 Þ ¼ _c 2n cos y _ do dx d cos2 y Gmsc , ð5Þ p½ðcos2 y cos2 ymsc Þ2 þ G2msc a2 ¼
cos2 ymsc ¼
1 b2 d 2 ; b2 n2
Gmsc ¼
2d ; bn2
d¼
cs1 . o
The calculation for the straight line trajectory is trivial. One easily gets ¯ ?cr d3 N a 1 1 2 d cos y . ¼ _c n2 b2 _ do dx d cos2 y n2 b 2
ARTICLE IN PRESS V.M. Grichine / Nuclear Instruments and Methods in Physics Research A 563 (2006) 364–367
366
cos2 yx ¼
z
Gx ¼
x2 . b2 jxj2
Here x1 ¼ 1 m1 2 m2 , x2 ¼ 1 m2 þ 2 m1 , and jxj2 ¼ ð21 þ 22 Þ ðm21 þ m22 Þ. In the case of medium without magnetic permeability (m ¼ 1, for optical frequencies) we get for the mean number of emitted Cherenkov photons from the unit particle trajectory length:
R→∞
−b∗
x1 ; b2 jxj2
b
¯? d3 N a Gsin2 y , ¼ _ do dx d cos2 y _c p½ðcos2 y cos2 yo Þ2 þ G2
a
cos2 yo ¼ Fig. 2. The contour C of integration for the multiple scattering radiation.
In the case of negligible multiple scattering d ! 0 the energy-angular distribution will be reduced to the usual Cherenkov radiation: ¯? d3 N _ do dx d cos2 y ¯ ?msc ¯ ?cr d3 N d3 N ¼ þ _ do dx d cos2 y _ do dx d cos2 y a 1 1 2 1 ¼ y d cos . _c n2 b2 n2 b2
ð6Þ
The value of s1 is defined (for W51) by the mean squared angle of multiple scattering per unit length of the particle trajectory. For particles with moderate energy moving in heavy medium, the relative broadening of the radiation angular distribution can be estimated as Gmsc = cos ymsc 2d=nt0:1. It affects the angular distribution much substantially compared to the medium absorption. The multiple scattering correction however is about a few mrad in water for electrons with the energy 1210 MeV. For the case of movement with constant velocity v the integral with respect to t can be calculated and relation (1) is reduced to
¯? d3 N a m tan2 y ¼ Im _ do dx d cos2 y _c p½1 mb2 cos2 y where b ¼ v=c, c is the speed of light in vacuum, dx ¼ v dt is the element of the particle trajectory and y is the angle between k and v. Introducing a new complex variable, x ¼ m ¼ x1 þ ix2 , one gets the following general energyangular distribution of emitted Cherenkov photons: ¯? d3 N _ do dx d cos2 y
a m2 tan2 y 2 2 2 m sin y þ ½1 x1 b cos y ¼ _c 1 x2 b 2 Gx , p½ðcos2 y cos2 yx Þ2 þ G2x
1 ; b jj2 2
G¼
2 . b jj2
(8)
2
The integration of this relation with respect to cos2 y results in the energy spectrum of the radiation:
¯? d2 N a 1 1 Im 1 2 ln ¼ . _ do dx p_c b 1 b2 In the limit of a transparent medium, when 2 ! 0 and m2 ! 0, one can get the energy-angle distribution of the ¯ c in the transparent mean number of Cherenkov photons N medium with refractive index n (n2 ¼ 1 m1 ): ¯c d3 N am1 1 ¼ signðGx Þ 1 2 _ do dx d cos2 y _c b n2 1 d cos2 y 2 ð9Þ b n2 where d is the Dirac delta function and the introduction of sign ðGx Þ allows us to take into account the absorption of photons rather than their emission. This expression fixes
cos2c =
1 ∋112
Right-handed materials ∋2 > 0 2 > 0
∋1 > 0
∋2 > 0
1 > 0
2 > 0 c
∋2 > 0 2 > 0
∋1 > 0 1 > 0
e ∋2 > 0 2 > 0
Left-handed materials
Cases of Cherenkov radiation
ð7Þ
Fig. 3. The cases of Cherenkov radiation in right- and left-handed materials.
ARTICLE IN PRESS V.M. Grichine / Nuclear Instruments and Methods in Physics Research A 563 (2006) 364–367
explicitly the emission angle to be yc ¼ arccosð1=bnÞ. If bn41, integration of Eq. (9) with respect to cos2 y results in the well known formula of the Frank–Tamm theory (in the particular case, m1 ¼ 1 and n2 ¼ 1 ): ¯c d2 N a 1 1 2 ; bn41. (10) ¼ signðGÞ _c _ do dx b n2 One can consider another interesting case when 1 m1 40, but both 1 o0 and m1 o0. Here the radiation follows the group velocity qo=qk which is anti-parallel to the wave vector k. Therefore the radiation is emitted at the angle p yc 4p=2 (see review [4]). In addition, relation (9) shows that Cherenkov radiation energy loss can be negative (m1 ): ¯c d2 N a 1 ¼ signðGx Þ m1 1 _c _ do dx 1 m1 b2 and the radiation has the tendency to be absorbed along the direction p yc , if 2 o0, m2 o0 and hence Gx 40. Materials with both negative permittivity and permeability (left-handed materials) were recently experimentally investigated and anomalous behavior of refraction for was observed [5]. It would be interesting to investigate also the
367
angular distribution of the Cherenkov radiation in this case keeping in mind that the Cherenkov angle can be obtuse, and to check that Cherenkov photons are absorbed by relativistic charge in the case when 2 o0 and m2 o0. The cases of angular direction of Cherenkov radiation are shown in Fig. 3. A similar picture is valid for the cases of Doppler radiation in right- and left-handed materials. The author is thankful to the participants of I.E. Tamm Theory Division (P.N. Lebedev Institute) seminar for stimulating discussion. This investigation was supported partly by INTAS-2001-0323 grant and was done in the framework of GEANT4 Collaboration.
References [1] [2] [3] [4] [5]
M.G. Bowler, M.D. Lay, Nucl. Instr. and Meth. A 378 468. V.M. Grichine, S.S. Sadilov, Phys. Lett. B 559 26. H.W. Lewis, Phys. Rev. 78 (1950) 526. V.G. Veselago, Sov. Phys. Usp. 10 509. D.R. Smith, W.J. Padilla, D.C. Vier, S.C. Nemat-Nasser, S. Schultz, Phys. Rev. Lett. 84 4184.