vlasov s.n. iap ras e-mail: [email protected]
DESCRIPTION
Influence of peripheral field on structure of nonlinear focus arising at propagation of a wave beam in cubic nonlinear media. Vlasov S.N. IAP RAS e-mail: [email protected] 603950 Russia , N-Novgorod , Uljanov street, 46, e - mail : vlasov @ hydro . appl . sci - nnov . ru. - PowerPoint PPT PresentationTRANSCRIPT
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Influence of peripheral field on structure of nonlinear focus arising at propagation of a wave beam in cubic nonlinear media
Vlasov S.N.
IAP RAS
e-mail: [email protected] Russia, N-Novgorod, Uljanov street,46,
e-mail: [email protected]
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Contents
1. Jntroduction. Motivation.
2. Construction of solution. The first order approximation.
3. The second order approximation. Influence of periphery of beam.
4. Numerical modelling of influence of "wings" on field in nonlinear focus.
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1. Intoduction
2
2, 2 0r zi
0 ,pr k r
The initial equation
' ,E 2,r
- transverse Laplacian,2 2 ,r x y
Self-focusing part of beam
“Wings" of beam or nonself-focusing to a part of a beam
( )r
r
Amplitude structure of a beam at self-focusing
sfz - Point of a collapse
(1)
sfz
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Ray structure of self-focusing an axially symmetric beam
r
sfz
z
ln( )(0, ) ~ sf
sf
z zz
z z
sfz z
ln[ ln( )](0, ) ~ sf
sf
z zz
z z
sfz z
(3)
(2)
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Self-simular solution of V.I. Talanov (1966)
1.2
2, 0,r
,r ~ exp[ ],r
r
sfz
z
,crP P
1(0, )
sf
zz z
Ray structure of self-focusing an axially symmetric beamself-simular solution of first type
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Ray structure of self-focusing an axially symmetric beamself-simular solution of second type
2.22
2, 2 1 0,r C C r
,r r
sfz
z
1sin( )~
C r dr
r
~ ln ,P r
1(0, )
( )sf
zz z
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0 5 10 15 20 25
-5
-4
-3
-2
-1
0
1
Cross-section structures of a beam, showing the dependences of growth rate of a field at nonlinear focus from cross-section structures
ln
ln4, (0, ) ~
zz
z
( ln )3, (0, ) ~
ln zz
z
12, (0, ) ~z
z1
1, (0, ) ~zz
r
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2. Construction of solution. The first order approximation[L,P,S,S;K,Sh,Z]
(4)
2
2
1,
D
r r r
2
2, 2 0r zi iD
, , ;r z
or
2, ;
( ) ( )
dr dz
2
2, 2 2 2 0i i i iD
( )C
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exp( ),A i 2 3 A-( ) A-2 A-2C( ) A +A =0,
2 2 2 2 2 2( ) 0,A A d CA D C A d
( ) 0,C
1 20 2
( )( ) ( )( ) .... ....,
( ) ( ) ( )nn
AA AA A
1 2 1 20 02 2
( )( ) ( )ln ... ... ( ) .... ....,
( ) ( ) ( )n nn n
HH HH
1 20 2
( ) .... .....,( ) ( ) ( )
nn
CC CC C
1 20 2
[ ( )] .... ...,( ) ( ) ( )
nn
DD DD C D
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2
2 30 00 0 0 0 0 0 02
1' 2 ( ' ) ,
A AA A C H A
20 0
0 0 20
' ;D A d
CA
0 4 8 12
0
0.4
0.8
1.2
1.6
2
0 4 8 12
0
0.4
0.8
1.2
1.6
2
0 4 8 12
0
0.4
0.8
1.2
1.6
2
0 4 8 12
0
0.4
0.8
1.2
1.6
20 4 8 12
-6
-4
-2
0
0 4 8 12
-6
-4
-2
0
0 4 8 12
-6
-4
-2
0
0 4 8 12
-6
-4
-2
0
0( )A
0( ) 0 0C
0 0,3C 0 0,2C
0 0,1C
0 0C
0 0,3C
0 0,2C
0 0,1C
Comparison of amplitudes of homogeneous beams
Comparison of phases of homogeneous beams
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20exp[ ],2
iC 22 22, 0 0 0 0[2 ] 0,C H C iD
2, 0,U 22 2
0 0 0 02 ,U C H C iD
0 4 8 12
0
1
2
3
0 4 8 12
0
1
2
3
0 4 8 12
0
1
2
3
0 4 8 12
0
1
2
3 20 0
0
( )W A d
0 0,3C
0 0,2C
0 0,1C
0 0C
Dependences of power of homogeneous beams from cross-section coordinate
ReU 1 2
The real part of potential U
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и
0 0( ),H C 0 0( ),D C0 0(0), (0) (0),A 0,
0 1
1~ ,A
0 1~ ,
, 00
~ exp( ),T
DC
0
02
D
C 0 0,C
0 4 8 12 16 20
-25
-20
-15
-10
-5
0
2 20 0 0
00 0 (1 )
exp[ ln ] exp[ ]2 2 2~
C D Ci iCa a
10 0 0 0
0 0
1(0)exp{ [ ]} (0)exp{ },
2
Ta A H A
C C
0 0 1ln , ln , lnD a C
0
1
C
0lnD
1lnC0ln a
Dependence 0ln ,D 0ln a 1lnC
on value0
1
C
0 0 0 ,H C H
0a
r
1 0.65î öT
00
1.25~ exp[ ],D
C 1 0.59T
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The explanatory to a way of a choice of a principle of growth rate of a field on an axis
Self-focusing part of beam
“Wings" of beam or nonself-focusing to a part of a beam
sfz
( )r
( )r
,2( )r
,2( )r
r rThe first way The second way
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First way
0 0 1
21/ 400
(0)exp[ ] ( ) ,
2
A C Tr Const
CH
0exp ,C d 0 0,C 10 ,
2
TC
1~ exp[ 2 ],T
ln( )(0, ) ~ ,sf
sf
z zz
z z
,
,2
2 2 2 2 22 1 , ,2 1 ,( )
,2 2
kr
k kw
r
Q r r Q rP r dr
2
0
0
kr
W r dr
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Second way
0 0 0 01 1 1 1 1
( ),
a a a C Q
r r r
10
0
~ exp[ ],T
aC
00
~ exp[ ] / 2 ,T
CC
0 ,
ln ln ln
TC
,
exp{ },ln ln ln
T
ln[ ln( )] ln ln[ ln( )](1 ),
2 ( ) 2 ln[ ln( )]f f
f f
z z z z
T z z z z
, ,
,2 ,2
,2 2 21 2
,2
lnk kr r
kw
r r
rdr drP Q Q Q
r r r
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3. The second order approximation.22
1 1 1 0 00 12
21 0 0 1 0 0 0 0 1 1 1 0 1
12 ( ) ( )
2[( )( ' ) ( ' )] 3 ,
A AA A
AC A C h A C h A A
2
0 0 1 1 0 0 1 01 0 11 0 1 2 2
0 0 0 0
2 [ 2 ]2' '
C A Ad D A D A A dA C AC
A A A A
0 (0) 2,A 0 (0) 0,dA
d 0 (0) 0, 0 0.2,C 0 2.317122448,H
0 0.0042702864,D 1(0) 0,A 1 (0) 0,dA
d 1(0) 0, 1 0,H
1 0.1; 0.05,C 1 12 ,D C
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0 10 20 30
0
0.5
1
1.5
2
2.5
0 10 20 30
0
0.5
1
1.5
2
2.5
1( )A
1 0.1C 1 0.05C
Dependence of amplitude 1( )A
on cross-section coordinate at
0 0.2C и 1 0H
0 10 20 30
-100
-80
-60
-40
-20
0
20
0 10 20 30
-80
-40
0
1( )
1 0.1C 1 0.05C
and phase
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0 10 20 30
0
1
2
3
0 10 20 30
0
1
2
3
0 10 20 30
0
1
2
3
1( )A
0 1.2, .05C C
0 1.3, .2C C
0 1.1, .003C C
Dependence of amplitude on cross-section coordinate
at various parameters iC
1( )A and 1 0H
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10 ,
CC
1
0 ,2
TC
1 1 10 1
2exp[ ] exp[ ],
BC C C B
C T
1 10
( ) 20 exp[ ],B
A CC
1
1
2exp[ ] exp[ ]
2 2
T Ba b
T
1
21
1
2,
2( )
2
Ca
TB
T
1
1
1 1
2,
2( )
2
Cb
TB
T
2
,( )
ddz
1.68
2.181
ln( ){1 0.34 ( ) ln( ) }
( ) ln( )sf sf
sfsf sf
z z z zA z z
z z z z
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4. Results of numerical calculations
.
0 0.4 0.8 1.2 1.6
0
10
20
30
40
50
z
(0, )z 2.26cr
P
P 1.81
cr
P
P 1.3
cr
P
P
2 2( 2 ) 0
N
Niz
2 250 ,N
N 2
0( ,0) exp[ ],2
rr
2
2
1,
r r r
0
10
2.2; 2.6; 2.9
1.3;1.81; 2.26cr
N
P
P
0 1.93,cr
Dependence of the amplitude of a field on axes, the equation
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.
z
(0, )z 2.26cr
P
P 1.81
cr
P
P 1.3
cr
P
P Dependence of the
amplitude of a field on axes, the equation
2 2
( 2 ) 0N
Niz
2 250 ,N
N 2
0( ,0) exp[ ],2
rr
2
2
1,
r r r
0
5
2.2; 2.6; 2.9
1.3;1.81; 2.26cr
N
P
P
0 1.93,cr
0 0.4 0.8 1.2 1.6
0
10
20
30
40
50
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2 2.2 2.4 2.6 2.8 3
47.2
47.6
48
48.4
48.8
49.2
0
max 1.11f 1.66f
,f
f
Dependence of the maximal field on size of an initial field for a various degree of focusing
2 2( 2 ) 0,
N
Niz
2 250 ,NN
2
0
1( ,0) exp[ (1 )],
2
rr i
f
2
2
1
r r r
10,N
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0 0.5 1 1.5 2 2.5
0
10
20
30
40
50
(0, )t0 2.6
2.4
2.2
t
Dependence of a field in the center of a cavity from time
2 2
3( 2 ) 0,N
Nit
2 250 ,NN
2
0( ,0) exp[ ],2
rr
2
3 2
2,
r r r
10,N
0 2.1cr
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(0, )z
0 2.6
2.2
z0 0.5 1 1.5 2 2.5
0
10
20
30
40
50
60
Dependence of a field on an axis in system with the combined nonlinearity
2 4 2
4( 2 ) 0,N
Niz
2 250 ,NN 2
4 50 ,
2
0( ,0) exp[ ],2
rr
2
2
1,
r r r
10N
Литература
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11. Fraiman G.M., Smirnov A.I., The interaction representation in the self-focusing theory , Physica D, 1991, v.52, p.16-35
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Использование явления СФ для исследования пробоя при сверхкоротком взаимодействии света с веществом, в сб. Квантовая электроника, Наукова Думка , Киев , 33, с.89(1987)
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23. С.Н.Власов, Л.В.Пискунова, В.И.Таланов, Трехмерный волновой коллапс в модели нелинейного уравнения Шредингера, ЖЭТФ, 1989, т.95, n.6, с.1945
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