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!!