Bin Qiao
Office: Room 544 (South), Physics Building
Tel: 62745005
2014 Autumn Semester, course for graduate student Lecture notes: Physics of Laser-Plasma Interaction
III.1 Wave equations
III.1 wave equations
3.1


ZtZpe n0i(x) |E2|
)exp()( tixEE Z GGG
4.1 Equation for EM waves4.1 Equation for EM waves

We consider linear propagation of an electromagnetic wave in nonuniform plasma without external or self-generated DC magnetic fields. Its electric field component:
)exp()( tixEE Z &&&
ZtZpe n0i(x)Neglect high order terms when are small
)p()(
&&&& u 0i( ) g g
quantities |E2|, the equation of motion for the electron fluid is given by eee
)exp()( tixE eue Z
GGG
EE i
J GGG
u
,

EE i
J GGG
III.1 wave equations 2( ) ( )E E Eu u
G G
E
, )()( E c iB
uu u HHH
o
uu 0)(1 2
c ε E)
III.1 wave equations EB
,0,0 E G

Zpe -1,Z<Zpe
ZZpe
(critical density)H=0
9
9
EEjB )(14 22 ZZ Z
S
21 21.1 10 / . m crn O O P u
2 2 2 2 pe c kZ Z
III.1 wave equations z
).exp()()( ),,( ),(00 tizExEzznn ZZHH GGG
E
.0 , 0 1 ,
E
, )()( E c iB
uu u HHH
o
uu 0)(1 2
E
, )()( E c iB
uu u HHH
o
uu 0)(1 2

0 0
c Z \ ³
( , ) ( ) zH Z H Z
z
).exp()()( ),,( ),(00 tizExEzznn ZZHH GGG
E
.0 , 0 1 ,
“ ’ ”z
0
0
' ' 2
H

0 0
c Z \ ³
( , ) ( ) zH Z H Z E(z)
, 0''2'' 02
“ ’ ”z
0
0
' ' 2
“ ’ ”z
0
0
' ' 2

Solutionsfortheelectricandmagneticfieldcomponents
E
E c

WKB Ho0Oo f
0 0
0 0
“ ’ ”z
0
0
' ' 2
III.2 WKB approximation
III.3 Analytical Solution in nonuniform plasma with a constant linear density profile
()
3.3

,0)1
0),(
2
2
2
2
2
2
2
2
o
equationStokesE d Ed

,
).()()( ii BAE
III.3 Analytical solution in nonuniform plasma… Ai([ ) Bi([ ) [ o fBi([ ) of ,E0
[0WKB 1 z
L H

3/2
6/1
. 43


.)] 23

.)] 43
)()( [D[ iAE 2 1 2 2 cos[ ( ) ] exp[ ( )] exp[ ( )]
¯ ¿


.)] 23

.)] 43
)()( [D[ iAE 2 1 2 2 cos[ ( ) ] exp[ ( )] exp[ ( )]
¯ ¿
E24

.)(6.3|| 3/12max
c L
E E
III.3 Analytical solution in nonuniform plasma… WKB
H

min
Z Z H S
0


min
Z Z H S
0


min
Z Z H S
0
E ( z) =
III.3 Analytical solution in nonuniform plasma… Exz
Faradayy
)(')(2)( 6/1 [ Z
S[ M i
i FS AeE
[
u
AiryWKB

WKB|[|>1.03WKB 1.5 [ 1.045 1.4
, 0),(2
2
2
2
)()( 3/1 2
L znn cr
BEWKB[ AiryWKB

WKB|[|>1.03WKB 1.5 [ 1.045 1.4
, 0),(2
2
2
2
)()( 3/1 2
L znn cr
III.3 Analytical solution in nonuniform plasma…



Landau