599
Internat. J. Math. & Math. Sci.
Vol. 8 No. 3 (1985) 599-614
STUDY OF NONLINEAR WAVE PROCESSES IN PLASMAS
USING THE FORMALISM OF A SPECIAL LORENTZ
TRANSFORMATION FOR A SPACE-INDEPENDENT FRAME
S. N. PAUL* and B. CHAKRABORTY
Department of Mathematics
Jadavpur University
Calcutta- 700032, India
and
L. DEBNATH
Department of Idathematics
University of Central Florida
Orlando, Florida 32816
(Received ’ay 11, 1984 and in revised form December 7, 1984)
A study is made of pot;linear waves in plasmas using the formalism of a
special Lorentz transformation for a space-independent frame, S’. This special
transformation is used to transform the space-time dependent equations in a cold,
ABSTRACT.
relativistic, magnetized plasma to the S’ frame. Then the transformed equations are
employed to derive the expressions for the Lagrangian and the Hamiltonian in the S’
frame. The Lagrengian and the llamiltonian for a strong circularly polarized laser
beam have also been obtained in the S’ frame. The exact form of the nonlinear
dispersion ’elation is derived for circularly polarized waves. Then the results for
the frequency and the wave number shifts of these waves in a cold, magnetized
relativistic plasma are obtained with some Discussion on the nature of the frequency
shifts. Finally, numerical results are presented for the radiation of Nd-glass laser
in dense plasmas.
KEY WORDS AND PHRASES.
Nonlinear waves in plasmas,
relativistic plasma,
Lorentz
trans format ion
1980 MATHEMATICS SUBJECT CLASSIFICATION CODE.
76 X 65.
INTR0UCTION.
Using a special Lorentz transformation (SLT), the space-time dependent variables
of at, electromagnetic wave in a plasma in the laboratory frame, S become space
independent in a moving frame, S’ when the latter frame moves with a relative
V is the group velocity of the wave. Winkles and Eldridge [I] first
velocity
emplo)ed the SLT to obtain self-consistent solutions of the relativistic Vlasov-
1.
c2/V,
600
S. N. PAUL, B. CHAKRABORTY AND L. DEBNATH
Maxwell equations, and have shown that a pure transverse wave cannot exist, but the
coupled longitudinal field does necessarily appear. They also obtained a nonllnear
dispersion relation correct up to the squares of the field amplitude. Thus the
Lorentz transformation (LT) is found to be very useful because of the fact tF.at it
can transform a system of nonlinear partial differential equations for a plasma into
a set of ordilary differential equations.
everal eLthors including Clemmow [2-3], Chiap. and Clemmow [4], Kennal and
Pellat [5], Shih [6], Decoster [7], Lee and Lerche [8-11], Clemmow and Harding [12]
halve used the special Lorentz transformation to study nonlinear problems in plasma
dynamics. On the other hand, Akhiezer and Polovin [13], Wong [14], Wong and Lojko
[15] have employed sir.ilar transformation relations for the investigation of nonRecently, Paul and Chakraborty
linear wave prepagation in relativistic plasmas.
L16-17 developed the theory of transformations of nonlinear plasma equations, and
xtended it tc yield the nonlinear precessional rotation of an elliptlcally
polarized, electromagnetic wave in an unmagnetized, cold, collisiunless plasma in
addition to the nonlinear shifts of wave parameters.
The main purpose of the present work is to develop the work of Winkles and
Eldridge [i] for the transformation of field variables from a laboratory inertial
rame, S to the space independent frame, S’. Such a study is very useful for the
investigation of several nor, linear effects in plasmas. In particular, the study of
self-action effects including self-focusing, self-steepening, self-phase modulation,
self-precession is expected to be simpler by using the transformation relations for
the S’ frame for several reasons. First, some of the field variables become either
constant or zero. For instance, the number density of electrons and the scalar
potential become constant, and the oscillation of the magnetic field vanishes.
Second, some nonlinedr terms, which appear in the S frame, vanish in the S’ frame.
Consequently, tt, nonlinear terms are fewer in number in the S’ frame than in the S
frame.
Notivated by the above discussions, we shall use the special Lorentz transformation to transform the space-time dependent equations in a cold, relativistic, magnetized plasma to the S’ frame. These transformed equations are then employed to derive
the expressions for the Lagrangian and the Hamiltonian for a strong circularly
polarized laser beam in the S’ frame. The exact expression for the nonlinear
dispersion relation is derived for circularly polarized waves. Then the results for
the frequency and the wave number shifts of the waves in a cold, magnetized
relativistic plasma are obtained. Some attention is given tu the nature of the
frequency shift for different intensities of the wave and the static magnetic field.
Finally, some numerical results for the radiation of Nd-glass laser in dense plasmas
are presented.
BASIC EQUATIONS AND ASSUMPTIONS.
We make the following assumptions"
(i) The plasma is cold, homogeneous and stationary, and is subject to a strong
2
watts/cm resulting in electron velocity
radiation with intensity less than
2.
3x1022
NONLINEAR WAVE PROCESSES IN PLASMAS USING LORENTZ TRANSFORMATION
601
becoming relativistic. The icn motion is negligibly small in comparison with the
electron motion.
(ii) The forces due to other sources including gravitational and ponderomotive
forces are also negligible.
With these assumptions, the basic equations of plasma in the S-frame are given
by
e
[D +
-eE_
_v.(Nv_)
(v_.v_)j
DN +
v_
x
!xH_:
,
2
moY_V
(I
y
(2.1)
(2.2)
-D H_
(2.3)
4e(Nv)
-D _E-
(2.4)
E_
4e
v.H
D
H_)
0
_v.E_
where
E (v_
(N-Ni)
(2.5)
0
(2.6)
_=
(2.7abc)
c
d,d N are the umber densities of ions and electrons, m
and N
and -e are
the rest ,ss and charge of an electron, and other parameters have their usual
meanings.
3.
SPACE-INDEPENDENT FRAME AND TRANSFORMATION OF GUANTITIES lO IT.
The Lorentz transfornmtion from the S-frame to the S’-frame moving with a
V
relative velocity
(t’ +
t
where
y
(-B 2 1/2’
o)
parallel to the z-axis is given by
-
VoZ’/C2)o
(z’ +
z
Vot’)Yo,
x
Vo
6o
x’
y
y’
(3.1)
(3.2ab)
We follow Decoster [7] to obtain
and find from the relations
Bo
(3.1) that
Yo
t’
z
z’
o
oo sinh Vo
z
cosh o + -sinh Vo
cosh o + ct’ sinh o
cosh
t
tanh
(3.3)
Vo’
(3.4abc)
x
’, y y’
It is obvious that v o is the hyperbolic angle for the S’-system relative to the
S-system. The reverse transformation from S’ to S is obtained by changing the
sign of V o or Vo’ that is,
o
h o
t’
t co sh
z’
z
x’
x,y =y
cos,
z
s inh
o
ct s inh
’o
(3.5abc)
S.N. PAUL, B. CHAKRABORTY AND L. DEBNATH
602
These relations show that the wave phase
( cosh
o
o)t
-(k cosh
/k
c2/Vo
o
(3 6)
c sinh o)Z’
where k and
are the constant wave number and wave frequency respectively of an
electromagnetic wave.
Followir, g Winkles and Eldridge [1] we consider the transformation from S to
S’ at the velocity V o kc2/ to obtain the phase velocity
t-kz
kc sinh
V
(3.7)
Then (3.6) reduces to
t
(1-Bo)2
kz
t’
(3.8)
This transformation enables us to change the field variables from the spac-time
aependent S frame to the space-independe,t S’-frame of primed variables. It is to
be noted here that for transverse waves, the phase velocity V(
Ik) > c and so
V 0 < c Therefore the velocity of S’ relative io S is not unphysical.
Feplacing t’ by T we can write
T
t
Vz
z
kz
c2
(1-o)1/2
2
Xo
1/(1-
(3.9ab)
v ._T_T
Yo @t
v
(1-V-E){
IYo
2
c2
-2)
1/( I-
Vo
T)
c
(3.10ab)
2
t
c
(1--)
V"
kz
8
at- o
T
(3.1lab)
Yo 8
-F
’
[18] considered a linear transfomtion rule
(3.12ab)
T t kz, V u/k,
for
some
effect
collision
to solve the plasma equations (2.1) to (2.6) neglecting
c
V
nonlinear problems. Boyd and Sandersen [19] considered the special value,
Akhiezer et l.
for some investigations.
The transformation of some field variables from the S-frame to the S’-frame is
9iven by
z
cosh v o + v sinh
cosh + v z sinh
vx
Vx/o(l’ + BoV/C) (cv’)/(Cx
Vy
Vy/Yo(l+BoV/C) (cv)/(c
o
o
o
z’ BoC)/(l+^v/c)u c(V’z cosh o + c sinh o)/(c cosh o + Vz
Ex Yo (E’x + BoH)) E’x cosh o + H’y sinh o
sinh o
[y o ([’y go ’)x [’y cosh o
v
z
Hx
H
y
Hz
(v +
goEy’) (H’x cosh o E’y snh o
Yo(H’y + BoE ) H’y cosh o + E’x sinh o
Yo (H’x
H’z
(3.13abc)
sinh
o
(3 14abc)
(3.15abc)
NONLINEAR WAVE PROCESSES IN PLASMAS USING LORENTZ TRANSFORMATION
603
The transformation of ,ass is
m’Yo(1 + oVz/C)
m
(3.16)
as may be seen from Hughes and Young [20, pi8-19] where m
o (but not
rest mass. The momentum colr, ponents are transformed into the form
Px mVx m’o (I+BoV/C)
v’z sinh
(c cosh $ o
is the
P’
m"’
Vx
o
m)
(3.17abc)
Pz mVz
m
Yo(1BoV’z/C)
c(v’z cosh
(:. Eosh
sinh
o+ + c sinh
Vo
o v
p,
+
z Yo m
0
Yo (P +
Defining now the momentum like quantities q and q’ as
c2 +
q
q’
c2 +
moVoY’)
CBoY
(m
where
have
V’
p2)1/2,
(m
p,2)1/2
mc
(l-V’2/c 2)
’2
mV
moC22 (l-V’2/c 2)
q,2
40
(3.18ab)
2
+
m’
2c 2
(3.19)
is the velocity of the S’-/rame relative to the rest system
Pz P’z cosh o + q’
q
sinh
o
SO
So we
(3.20)
m’YoC(l+BoVz/C) m Yo(C + BoV z)
+
sinh v
yo(q + P’B
o
z o) q’ cosh o P
mc
If t, N’, N o are the symbols for the number density in the three systems
respectively, then again following Hughes and Young [20] we get
N’
No/(l-V’/c)
S, S’, S O
(3.22)
so we can write
moN’ moNo/(1-V’C/c)1/2
n,’ N
(3.23)
o
Similarly, we can also write
moN
I’
N
m’
m’c
m
mc
A_
and
Yo(Az+o
A’z
’
o
’
Vo A’z
q’
q
is vector potential and
form a four vector (_A
we
can write
and
potential),
/
cosh
sinh
Az
sinh
cosh +
yo(BoA’z +
Since
(3.24)
m NO
o’
A_+/-
(3.25)
is the scalar
+/-
o
(3.26ab)
o
-o
by
The reverse transformation is obtained by replacing
Equation (3.21) gives
(3.27)
constant
N o N’ sech V’
as
and (3.25) cdn be written
(3.28)
sinh (T’-v)}
N O q cosh ’/{q cosh
N N’q/q’
Here ? is the hyperbolic angle for the S-system relative to the S O frame and v’
is the same for the S’ system. Therefore, in the S-system, the number density is
not cor, stant, and the electron ar.d ion densities are not necessarily equal.
(v’-v)-Pz
S.N. PAUL, B. CHAKRABORTY AND L. DEBNATH
604
TRANSFORMATION CF FIELD EQUATIONS TO THE SPACE INDEPENDENT FRAME
Using the transformation relations of section 3, we obtain from (2.1) for the
x-component
aP’
v’ aP’x
4.
Yo (- v1 Tt-
YoV-Vo + (o-)’ (rt)/v(+o’/)
-eYo(E+BoH) + (e/c) (V’z+Vo )Yo(H+oE’)/(l+oV’z/C)x -v’H’yz/YoC( l+BoV z’/c)
}’
e V’y
BYo-l)v
l)V’}z eH>,{Vo(Yo-I)+(
Bo c E x’ YoV-Vo+(Yo.
H’z
+
B’o
oC +oV z"/ C)
c
c(I+B o v’/C)z
v.’/’c
This equation can be simplified to the form
Px
aT
Y
eE’x
_
V o (Y-l)
c-YoV-Vo
Yo {YoV Vo+(Yo -l)v}
eV v’ H’
z
,}
YoC{ynV_ -Vo+(y
o 1)v z
DT
x
eE + YoV-Vo’)
(4.2)
P’y becomes
Similarly from (2 1) the equation for
_Oi
(4.1)
Vo(Yo_l)
Yo {YoV-Vo+(Yo 1)v}
eV v’x H’z
+
(4.3)
YoC{YoV_Vo+(Yo_ l)V,z}
Now the equation for P’Z obtained from (2 I) is
aP mV@Y’
@--+
eV{Ez + Bo(B"E’) + (.B’xH’)z
-,}
Yo{Yo V Vo + (Yo 1)z
where
v_’ y’3(B_’--v’2 3/2 (_ )(v_,. aB-T
1/2(1 __)
c
c
p/(p2_ + m2oC2)1/2, d p2 + m2oc2 mc2y2, we obtain
)Y’
Since
B
(4.4)
(4.5)
ar,
aB_
-T
aP. P_(_P.aP_/aT)
moC, T
(4.6)
m-3oC2,3
Therefore,
P’(P’.aP_’/aT)
(P’-aP’/T)
aB’
(’"
@--)-
(moC )4
(moC Y, )2
1
moc2 2y,’4
(P_- ---)
(4.7)
Hence
aP’
V
aP’
+
Multiplying (4 2) by
after simplification
moC2Y’
P’x’
(p’
3-)
+ (B’xH’) z}
Ez + V o("E’)
+
Yo{Yo Vo (o l)Vz}
eV{
(4.8)
(4 3) by P’y and (4.8) by P’z and then adding, we get
NONLINEAR WAVE PROCESSES IN PLASMAS USING LORENTZ TRANSFORMATION
eV(PE + P’E’yY)(Yo(Yo v Vo) + Vz’(Yo2 Yo
Yo (v + Vz’l{YoV Vo + (Yo "llVz}
eVP’E’
z z
V o + (Yo
Xo{Yo V
eV
+
605
1)}
)v}
2
+ 4 V) +
2(-P’xH- )z ’v’(V
Yo
VYo3(yoV_ Vo)}
YoVo
z
3
oc(v+ v’z)(Yov-vo ){Yo v-vo+ (Yo- )V’z}
eV(P’xv’_ )zHz
Vo +
CXo{oV
Therefore, the equation for P’z
eVE’z
aP’z
v
@T
yo{Yo V + (YG
(Yo
(4.9)
1)Vz}
becomes
I)
0
V
z}
+ Vz(V(Yo2
+ F"E’){V(yVy
Vo)
YoVo
e(P’E’x.Lx
Y
+
V
moYoY Vo(’V Vz){YoV
eVP’E’
z z
3
moYoY’Vo{YoV V o +
Yo +
_)v’}
+-(Yo
o
1)
1) v z’}
(Yo
)zV2{Vz(V + Vy o6 oYo5
5
e(P_’xH_
+
4
yo(YoVo
Vo)2
moYoY ,Vo(V ’)(Yo v Vo){YoV Vo (Yo 1)v}
eVVo(P’xv’) H’
moC3Y oY’{Yov Vo (Yo 1)Vz}
c
Vo )}
+
+ v
z
(4.zo)
The equation of continuity (2.2) assumes the form
Yo
YoF(E’)
c(
,N’y
V-- B--y
y
+
o
60B
z’’)}
(I +
0
It can be written as
N’ (I
B
Yo
]
So
(4.12)
O.
(4.13)
N o, where N O is a constant.
And so, N’
obtained from (2 3) is given by
Equation for H’
X
1@
iB
V
B-T(Ey oHx
-T(Hx So E’y)
(4.14)
It can he written as
So So)H’x
So,
And the equation for
H,
oV
___)
r
Hence,
The x-component of equation
(2.4)
H’
Y
is
(4.16)
constant
obtained from equation
@-rL E,X(
(4.15)
0
+ H,(B
y
constant
(2.3), is given by
V
-)]
=0
(4.17)
(4.18)
S. N. PAUL, B. CHAKRABORTY AND L. DEBNATH
606
2
2
YO
V’-
)
YO
B-r(Hy BoEr)
+
)
4eN o v’x
@E’x
Or
4eNov,
-T(Ex + BoHy) + cYo(1 + XoB)
(4.19)
(4.20)
yo (I’ + oBz
Similarly, the y- and z-components of equation (2.4) become
4eN
ov
(4.21)
’o (1 + BoB z
aT
and
BE’
vz
4eNo(V’ V o)
Yo + ,oz
go z) in powers
/
B-T-
(4.22)
0 when
Since v z
of gz
We shall now expand I/(I +
-V o +
to avoid a physical impossibility, lherefore,
-v o, we put v z
6v
Yo(l
+
2 6Vz
2
yo(-VYo[l
.BoBz
46v2_
+ yo--/
...
(4 23)
So, the veioc! ty components become
v2
yo4(T
2
Vyyo[1 Yo2(T + y(T
2;v’
VxYo[l Yo(Tz
vx
Vy
+
...]
(4.24)
...]
(4.25)
6v,
2 -) + 4
yO(-V
yo(--V-) 2
Z
Vz YO2-v,[1
...]
(4.26)
where v’x and v’y are the first order velocity components in the directions paral6 v’ is the second order velocity component along
lel to OX and OY respectively
z
-OZ in the S’-frame.
The continuity equation (2.2) gives
N
6N’
and so
where
No/( Yo6V
z2’/V)
oYoVz/V
N O + 6N’
(4.27)
(4.28)
v
are neglected.
Therefore, equations (4.20)-(4.22) for the electric field components become
N o is a constant, and higher powers of
BEx
-T
4e
Yo
.
(No
+
(I +
aT
2
N’ )v x
BoB z)
2
Yom op
e
Yomoe
e
(4.29)
vx
(4 30)
v’y
BE
YmP
6
Vz
e
@r
(4Noe2/mo)1/2,
(4 31)
where
being the electron plasma frequency. Again from the
transfornwtion relations (3.14abc) and (3.15abc), neglecting the transverse magnetic
field components in the S’-frame, we obtain from the transfermation relations
=p
Ex
Hx
=p
YoE, Ey yo E’y’ E z E’z
-YoBoE, Hy -YoBoE, H z
(4.32abc)
Hz
Ho
(4.33abc)
607
NONLINEAR WAVE PROCESSES IN PLASMAS USING LORENTZ TRNASFORMATION
THE LAC-RANGIAN AND HAMILTONIAN IN THE SPACE-INDEPENDENT FRAME
Following Lcndau and Lifshitz [21], the expressions for the Lagrangian and
5.
Hamiltonian in plasmas in the laboratory frame are given by
-No[moC2(i .,,,1/2
c
I:
aF, d
(A_ v_) eel
+
NoV-[ moV-2
’
where th(. ector
+ e
e
H.
8
(5.1)
(5.2)
A_]
-)
(
scalar potentials A and
H curl A
E
respectively are defined by
(5.3)
v_
(5.4)
-t"
(5.5)
+
c at
div.
p
E2
+
A_
The Lorentz gauge condition (5.5), when transformed to the space-independent frame,
ives
2
2
YOv sT (A’z
YOc aT (’
o ’)
+
oAz’)
+
(5.6)
$I
z cancel from both sides, and since a constant potenti is neglected here, it turns out that
Si,ce the ter,ts contai,ing
’
-- -
(5.7)
O.
lherefore, in the c’-frame., the Lagrangitn and the Hamiltonian are found to be
2
4
2
Z +
Yo 5v’ + Yo sv’ 2
2
z
z
:-No[moC2{1- c(V+v-+ :’Vz,2)(1
2
-Y +VYo 5v’z
+eYo(1
6v
z
2
+eYoSoAz]’
+
Yc
(I
Yo
No[moC2{! c--(v+v_
2
){
+ E ,2
2)(E+ E
B0
z
Yo
oVz’2)(I
2
+
where
av
(i
5v’z 2
2
o)(E+E_)
A.. A’x iAy’ E+_ E iE’y, v V’x iV’y.
For a circularly polarized laser beam, we put
[x a cos e, E’y a sin o, E’z
+/-
+
2
YoAz5Vz
+
(5 8)
4
Yo
+
2
o
-
1(A+v_+ A_v+)
2
2
H’
V
4
2
E z,2
0
+
eYoBo A’]z
(5 9)
(5.10abc)
where a is the constant amplitude of the wave and o
+’T the subscripts + and
signs indicate tle left and right circular polarized components respectively. In
ae-+io and the solution of the plasma field equations (3.1abcd)this case,
(3.9ab) is exact.
For a circularly polarized wave (5.8) and (5.9) reduce to
2
2
2
+
(5.11)
(I
Z’
T
2
4
pmoC
p
E+
-No[moC2(1 e2a2
e2a
+----]
mo
ya2
c
S.N. PAUL, B. CHAKRABORTY AND L. DEBNATH
608
2 a2
2
"1/2
e2a2
No[moC2(1 mpmoC
4 2 ") ]
N’
Yo
(I
’8
c2
(5.12)
V--)
lhe results for the Lagrangian and Hamiltonian in the laboratory inertial frame are
in (5.11) and (5.12), where
to
obtained simply by changing
2
2
2
2)
+/-/Yo
o
I(+/- k2+/-c
6.
NONLINEAR DISPERSION RELATION
For a purely transverse circularly polarized wave,
E.+_ a exp(+/-iB) and E z O.
Following the method of averaged Lagrangian developed by Whitham [22-25] and
exterlded by Dysthe [26], Das and Sihi [27-29] and others, the Lagrangian derived in
this section can be used for finding the nonlinear effects including the shifts of
wave parameter in the space-independent frame and then in the laboratory frame with
the help of the transformation relations in Section 3.
(6.1)
So, we have
Pz’
O,
V’z _Vo + 6Vz
(6.2)
-V
From (4.1, (4.3), (2.7abc), (3.13abc), and (3.17abc), we obtain
P -eE
ieVv+/-H o
-+
X3oC(V
2
Yomomp
v
e
E
P_ moV_/[1
+
c
v’x
v+
(6.3)
V O)
(6.4)
(v;-
v’)]
(6.5)
c2
iv’y’ and a dot denotes the (eriative
P’x iP’y’ E’ E’x + iE’,
where P+/Y
with respect to time.
Using (6.1) in (6.3) to (6.5), the nonlinear dispersion relation for the left
and right circular polarization components in the space-independent frame can be
obta ned
V o =’+/+_,2
+/(6.6)
0
[I
o
2p
1/24
e2a22]
pmoC
Y(V
V)
2 2
where
Replacing
laboratory frame becomes
eHo/moC.
’+
by
+/yo,
the dispersion relation (6.6) in the
o.
c6.7)
2
m4m
p oc
In the absence of magnetic field, (6.7) becomes
where
o
is the
m2 k2c2 m2(1
+ 2)o
p
dimensionless amplitude parameter ea/momC.
(6 8)
It is to be mentioned
NONLINEAR WAVE PROCESSES IN PLASMAS USING LORENTZ TRANSFORMATION
609
that (6.8) is identical to the results of Paul and Chakraborty [16-17]. Expanding
the right-hand side of (6 8) in power of
and retaining only +he first two terms
0
of binomial expansion we obtain the results of Arons and Max [30] if their symbols
are adopted.
2
FREQUENCY AND WAVE NUMBER SHIFT OF AN ELECTROMAGNETIC WAVE
Nonlirearity in plasmas produces many interesting results in the medium and on
the waves. Electromagnetic wave would have shifts in the wave parameters (wave
number and frequency) as a result of nonlinear interactions in plasmas [31-41]. We
shall derive here the expressions for the shifts of frequency and wave number of a
circularly polarized wave propagating through a cold, magnetized relativistic plasma.
(a) Frequency Shifts:
For the temporal evolution problem, we can write
7.
k+
+ o
and
k
(7.1)
+ 6
are the increments of frequencies of the left and right circular polarTherefore, neglecting higher powers of 6m+, we
find that frequency shifts of the two polarization components are
where
6m
ization components of the wave.
o
I
2
m- i
m11+
P
+
j
(7.2)
P
where
m+_02-
k2c 2.
(7.3)
(b) Wave number shifts:
In the spatial evolution problem, we write
+
m,
o +
k+
k+
(7.4)
6k+
ak+ are the increments of wave numbers of the left and right circular polarization components of the wave. Therefore, neglecting higher powers of 6k/; the
where
expressions for the wave number shift of the two polarization components can be
derived as
2
m2
k+
+
o
m m-m2
P
a
2
2]
o
(7.5)
P
where
2
o2
2
k _* c
(7.6)
DISCUSSIONS AND CONCLUSIONS
From (7.2) and(7.5) it is observed that the intensity of the wave and static
magnetic field have important roles to create the shifts of the wave parameters. It
8.
610
S.N. PAUL, B. CHAKRABORTY AND L. DEBNATH
is seen that
{i) when
kc and m
o the LCP components of the wave do have
a frequency shift when its intensity a f
but the fFequency shifts of
the RCP components exist only when a is greater or less than
m
and (ii) when m > kc and m
p the frequency shift of LCP
W
e
components becoane nonzero when
buz the RCP components always have
f (o
frequency shifts for all possible values of
The nature of the frequency shifts
o
RCP and LCF components of the electromagnetic wave can be well understood from
Table and Figures I-3.
m >
v (n2/m2),
/(.p/)x
(/2- 2)1/2
/)1/2
Frequency shift of the LCP
component V.S. Intensity of the wav
when > kc and
o)"
Fig.
.
component
Frequency shift of the
V.S. Intensity ef the
(when
kc
Fig. 2
>
and
RCP
Yve
o).
Frequency shift of the LCP
Fig. 3
component V.S. Intensity of the wave
(whenm > kc and
9.
NUMERICA[
ESTIMATION
For the radiations of Nd-glass laser having wave length
1.06 um, frequency
1.78 x
c/s, Power
watts/cm 2 (which is less than the threshold power
generate self-action effects) passing through a dense plasma (N
the
o
frequepcy shift of LCP components
1.01 x
c/s and that of RCP components
-8.6 x
when
i.e
Ho
gauss
Under the situation m
the frequency shift of the LCP and RCP compoP’
nents become 4.42 x
c/s and 4.31 x 10 12 c/s respectively when Ho
gauss.
For magnetic field
gauss, the frequency shift of the LCP and RCP components
become 3.48 x
c/s and 5.22 x
c/s respectively.
1015
1016
1015
I014c/s
!020/c.c.),
108
1012
106
1012
105
!012
NONLINEAR WAVE PROCESSES IN PLASMAS USING LORENTZ TRANSFORMATION
611
O
0
3
CLOSING REMARKS
Transformation of field variables from the laboratory frame to the spaceindependent frame is applicable only for a single electromagnetic wave because the
condition required for this transformation, i.e., V
kc/ c
is not satiso
fied for two or more waves interacting in plasma.
To derive the results for the shifts of wave parameters with the help of special
Lorentz transfor.ation, field variables should be transformed to the space10.
2IV
S.N. PAUL, B. CHAKRABORTY AND L. DEBNATH
612
independent frdme at the beginning, and then one should proceed with the calculatior,.
If equations are solved in the laboratory frame before transforming them to the space
independent frame and transformations are used only in the subsequent steps, correct
results would not appear in the space-independent frame.
11.
ACKNOWLEDGEMENT
This work was partil!y supported by the University of Central Florida.
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*Permanent Address of S. N. Paul’.
13, T. C. Goswami Street
Serampore, Hoogh ly
Pin 7!2 201
West engal,
Ip.dia.