radiation of a charge flying from vacuum into anisotropic dispersive medium
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RADIATION OF A CHARGE RADIATION OF A CHARGE FLYING FROM VACUUM FLYING FROM VACUUM INTO ANISOTROPIC INTO ANISOTROPIC DISPERSIVE DISPERSIVE MEDIUMMEDIUM
Sergey N. Galyamin, Andrey V. TyukhtinRadiophysics Department, Physical faculty, Saint Petersburg State University
0
R
||
0 0
0 0
0 0
vacuum
2
,i2
1d
2
2p
|| d2
2||p
||i2
1
1
Formulation of the problem Vacuum – аnisotropic plasma-like medium interface
z
tctVz
q V
Taking into account the specific frequency dispersion and losses
Detailed investigation of the structure of the electromagnetic field
Goals:
Left-handed frequency band
2c
1rm sm pe
2n
3
Motivation Vacuum – “left-handed medium” (LHM) interface
Isotropic LHM: ,i2
1)(de
2
2pe
2
dm2rm
2pm
i21)(
E
H
k
S
q V
z0
вакуум
R
)(),( vacuum Reversed Cherenkov-transition radiation (RCTR) occurs in both vacuum and
medium
pe rm
reversed VCR:
4
Spatial distribution of the Fourier harmonic
– lines parallel to the Poyting vector of VCR – lines parallel to the Poyting vector of RCTR in medium – lines parallel toi Poyting vector of RCTR in vacuum
TIRCR
Motivation Vacuum – “left-handed medium” (LHM) interface
Z k
5
Field spectrum in vacuum 19pe2pm2 c10102
14R сm
5-
5
0
910Re H
5-
5
0
5-
5
0
5-
5
0
pe/
1
8.0
6.0
4.0
6.0 7.0
pe/6.0 7.0
pe/6.0 7.0
453015
T
IR
CR
T
IR
CR
T
IR
CR
(Vm–1s)
1q nC
,0rm2 ,10 e2p3
m2de2d ,10 pe22
pe1 pe26
de1 10 TIRCR
Motivation
6
b(1,2) b(1,2) exp i d2
qH H t
c
)iexp()(][d )2,1()1(1
1)2,1(2)2,1(2)b(1, zkkHkkBkH zz
(1) 2 2 2zk c k )( 2
||221
||)2(
kckz 0Im )2,1( zk
1 2(1) (2) 1
||(1)(2) (1) 2 2 2 2 2 1 2 2
1 || 21 1 ( )
z z
z z
ck k cB
k k k s c k s
Transition radiation (TR) Reversed Cherenkov – transition radiation (RCTR)
Solution of the problem Vacuum – аnisotropic plasma-like medium interface
(1,2) q(1,2) b(1,2)H H H V.L. Ginzburg, V.N. Tsytovich. “Transition radiation and transition scattering»
q(1,2) q(1,2) 1exp( i )d ,2
qH H V
c
, , 0zE E H
q(1,2) (1)1,2 1,21iH s H s 2 2 1 2
2 || 1s V
1,2Im 0s 2 2 2
1 i 1s V
Vavilov-Cherenkov radiation (VCR):
s Real ),max(),min( p||pp||p
z V t Self-field of the charge
Scattered field
7
||p
p
1
||
0
1
0 ||pp
||
Self-field of the charge Anisotropic plasma-like medium
– branch points
q(2) q(2) q(2)WCH H H
guasistatic (“quasiqulomb”) field
wave field (VCR)
0
Re
Im
– cuts 2Im 0s
c
q(2)2 1 2C (i ) (i ) exp d
qH s J s
c V
p
p | |
q(2)W 2 1 2
2( ) ( ) sin d ( )
qH s J s
c V
8
1||p
cin units of ,
| | p3
d| | d 10
0120- 80- 40-
05.0
0H
05.0- q
10 9.0qH
0120- 80- 40-
q
10 9.01.0
0H
1.0-
qH
p||p 5.1 ||pp 5.1
Self-field of the charge Anisotropic plasma-like medium
2 2 2 2|| 0x zk k c
c
c
gV
k
V
zk
xk
p|| p
V
gV
k
||c
||c
xk
zk
p|| p
forward radiation backward radiation
9
Analytical approach Numerical approach
Fourier harmonics of the scattered field
)iexp()(d )2,1()1(1)2,1(
2)2,1(2)b(1, zkkHk
kBkH z
z
“Half-shadow” regions estimation
Impact of losses in medium on the radiation field
Analytical investigation of the integrands behavior, choosing the appropriate
integration step and integration interval
Numerical investigation of the integrands behavior, choosing the appropriate
integration step and interval
Total scattered field
tH
cq
H iexpd
2b(1,2)b(1,2)
Asymptotic representation in the far field zone
(with respect to the incident point) 2/11 ~,~ R
Scattered field Vacuum – anisotropic plasma-like medium interface
Fourier harmonics of the scattered field
10
Field in vacuum, the case of backward VCR: p||p
sin1kk
2
2
b2
SDP
Re
Im
II I IV
IIVIII122
1 ck
1,2Im 0k
Analytical approach
Scattered field Vacuum – anisotropic plasma-like medium interface
2s
1k
2k
2kk
k
1k
||22
2 ck
RCTR ||||2tg
11
2 2 4 2 2p|| p||2 2 2 2
p|| p
(1 ) (1 )
2 4
2 2 2 2 2 2 1RCTR p || p|| p( ) ( ) ( )
1101 2||10
11sin
11
2Rc
2
2||
||101
1
1tg8
c
R
10 1 1R R 1 11 12 (1 / )c R R R
Conditions of the RCTR existence in vacuum
“half-shadow” regions
)(||p )(RCTR
Results Vacuum – anisotropic plasma-like medium interface
1b(1)S exp( i )~
k RH
R
b(1)Pb(1)Sb(1) HHH Field asymptotic in vacuum, 1~ R 2/1~
Spherical wave of TR:
Cylindrical wave of RCTR:2 (1)b(1)P (1)
2 2 1123
i( ) exp( i ( ) | |) ( ) ,
( )z
sqH H s k s z
c g
(1)3 2( ) ( )zg c k s
1 1k R
12
)(||p
Spatial distribution of the Fourier harmonic
– lines parallel to the Poyting vector of VCR – lines parallel to the Poyting vector of RCTR in medium – lines parallel toi Poyting vector of RCTR in vacuum
Z101
2
220
||2tg
Results
13
1
10
2
S
20
Z
1R
tS
)(S2 R
2
1
2R
2
10
)(||p
Spatial distribution of the Fourier harmonic Results
14
910Re H
2-
2
0
1
8.0
6.0
4.0
||p/1 2.1
756045
2-
2
0
2-
2
0
2-
2
0
||p/1 2.1 ||p/1 2.1
R
CT
R
RC
TR
RC
TR
,c102 110||p
14R cm 1q nC
,10 ||p3
d||d ,10 ||p
2pe1 ||p
6de1 10
||pp 5.1
)(||p
Field spectrum in vacuumResults (Am–1s)
15
19pe2pm2 c10102
14R сm
5-
5
0
910Re H
5-
5
0
5-
5
0
5-
5
0
pe/
1
8.0
6.0
4.0
6.0 7.0
pe/6.0 7.0
pe/6.0 7.0
453015
T
IR
CR
T
IR
CR
T
IR
CR
(Am–1s)
1q nC
,0rm2 ,10 e2p3
m2de2d ,10 pe22
pe1 pe26
de1 10 TIRCR
Vacuum – “left-handed medium” (LHM) interface
16
0 2 10 10 4 10 10 6 10 10 8 10 10
200
0
200
0 2 10 10 4 10 10 6 10 10 8 10 10
200
0
200
0 2 10 10 4 10 10 6 10 10 8 10 10
200
0
200
cm5.1cm15.0z
cm6.0z
1Am, H
s,t
cm3.0z
vacuum
zcm03.0
cm2.3
observer
medium
1q nC z
Time evolution of the total fieldResults
17
Investigation of the electromagnetic field generated at a charge flight from vacuum into anisotropic plasma-like medium
Conclusion
Analytical approach:•Rigorous condition of the RCTR presence•Spatial structure of the far-field Fourier harmonics•“Half-shadow” areas •Impact of losses
Numerical approach:•Field spectrum results •Time evolution of the total field•Possibility of the RCTR dominance in the total field
Reversed Cherenkov-Transition Radiation (RCTR)
18
Thank you for your attentionThank you for your attention!!
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