Zoran M. Trifković
Assistant Professor
University of Belgrade
Faculty of Mechanical Engineering
Božidar V. Stanić
Professor Emeritus
University of Belgrade
Faculty of Electrical Engineering
Second Harmonic Generation in
Nonlinear Transformation of
Electromagnetic Waves in Suddenly
Created Cold Magnetized Plasma.
Transversal Propagation
The nonlinear transformation of linearly polarized source electromagnetic
wave in electron and electromagnetic plasma oscillations, stationary
(rectification and space-varying) modes and travelling electron and
electromagnetic plasma waves, due to a weak nonlinearity, has been
analyzed by using the second order perturbation theory in radio
approximation. The efficiency of excitation of the transversal secondharmonic electric wave modes with double wave number with respect to
the wave number modes in the linear transformation has been studied for
different values of source wave frequency, taking the electron cyclotron
frequency as a parameter).
Keywords: rapidly created plasma, stationary modes, second harmonic
generation, nonlinear transformation, Laplace and Fourier
transformation.
1. INTRODUCTION
2. PROBLEM FORMULATION AND SOLUTION
Rapidly created plasmas appear practically in all
pulse gas discharges, laser created plasmas, lightning,
and plasmas created by nuclear explosions. The
transformation of electromagnetic (EM) waves in such
time varying linear media has been the subject of
interest in many recently published papers. The basic
results of these contributions are summarized in [1].
In this paper we have assumed that for t < 0 the
linearly polarized (LP) source electromagnetic wave
(EMW) with angular frequency ω0 and the wave
number k0 is propagating in free space in the positive
z direction. The static magnetic field is assumed to be
along positive y direction, B0 = yB0 , where y is unit
vector in positive y -direction. At t = 0 the entire free
space is ionized with an electron plasma density
increase from zero to some constant value N 0 . The
transformation of the source EM plane wave, due to a
weak nonlinearity, has been analyzed by using the
second order perturbation theory. The efficiency of
excitation of newly created modes in anisotropic plasma
medium has been obtained in a closed form and studied
for different values of source wave frequency and for
electron cyclotron angular frequency equal to the
angular electron plasma angular frequency.
Electric and magnetic fields of the linearly polarized
source EMW for t < 0 are given by
Received: December 2004, Accepted: Mart 2005.
Correspondence to: Zoran Trifković
Faculty of Mechanical Engineering,
Kraljice Marije 16, 11120 Belgrade 35, Serbia and Montenegro
E-mail: [email protected]
© Faculty of Mechanical Engineering, Belgrade. All rights reserved
e0 ( z , t ) = E0 cos(ω0t − k0 z ) ⋅ x ,
(1)
ho ( z , t ) = H o cos(ωo t − ko z ) ⋅ y ,
(2)
where x , y and z are the unit vectors in positive
direction of x , y and z axis, respectively,
H 0 = ε 0 / µ0 E0 is magnitude of magnetic field, and
Eo is magnitude of electric field. ( µo and ε o are
permeability and permittivity of free space,
respectively).
The EM e ( z , t ) , h( z , t ) , electron velocity u( z , t ) and
electron density fields n( z , t ) in magneto-plasma
medium have to satisfy the following equations:
∂n( z, t )
+ ∇ ( n ( z , t ) u( z , t ) ) = 0 ,
∂t
∇ × e ( z , t ) = − µ0
∂h( z, t )
,
∂t
∇ × h( z, t ) = − N 0 qu( z, t ) + ε 0
(3)
(4)
∂e ( z, t )
,
∂t
(5)
du( z, t ) ∂u( z, t )
=
+ ( u( z , t )∇ ) u( z, t ) =
dt
∂t
=−
q
q
q
e ( z, t ) − u( z, t ) × B0 ( z, t ) − u( z, t ) × h( z , t ) , (6)
m
m
m
FME Transactions (2005) 33, 41-46
41
The equation (6) is nonlinear due to the second term
on its left-hand side and last term on its right-hand side.
EM, electron velocity and electron density fields for
weakly nonlinear plasma can be further expressed in the
following form:
e ( z , t ) = e1 ( z , t ) + e2 ( z , t ) ,
(7)
h( z , t ) = h1 ( z , t ) + h2 ( z , t ) ,
(8)
u( z , t ) = u1 ( z , t ) + u2 ( z , t ) ,
(9)
n ( z , t ) = n1 ( z , t ) + n2 ( z , t ) ,
(10)
where the subscript “1” refers to the linear and the
subscript “2” to the weakly nonlinear field. Substituting
(7-9) into (3-6) one obtains the following system of
equations for linear and weakly nonlinear fields:
∂h ( z , t )
∇ × e1 ( z , t ) + µ0 1
=0,
(4a)
∂t
∂e1 ( z , t )
=0,
∂t
∂u1 ( z , t ) q
q
+ e1 ( z , t ) + u1 × B0 = 0
∂t
m
m
∇ × h1 ( z , t ) + N 0 qu1 ( z , t ) − ε 0
(5a)
b = ωp4 + (ωp2 + ωb2 ) ⋅ ω02 ,
where ω0 , ωp = ( qN 0 / ε 0 m )
1/ 2
∇ × e2 ( z , t ) + µ 0
(
)
(
)
ω 4 − 2ωp2 + ωb2 + k 2 c 2 ω 2 + k 2 c 2 ωp2 + ωb2 + ωp4 = 0 ,
(12a)
ε = 1−
ωp2
(
)
ω 2 1 + ωb2 ωp2 − ω 2 

.
Term n1 ( z , t ) is given in [4].
Ω 1, Ω 2
Ωb = 1
(3b)
Ω 2b
2
(4b)
∂e2 ( z, t )
=
∂t
−qn1 ( z , t )u1 ( z , t ) ,
+1 +
Ωb
2
Ω1
Ω 2b + 1
Ω 2b
2
+1 −
(5b)
Ωb
Ω2
2
45°
∂u2 ( z, t ) q
q
+ e2 ( z, t ) + u2 ( z , t ) × B0 =
∂t
m
m
−
Ω0
Figure 1. Normalized angular frequencies of newly created
linear wave modes Ω1,2 = ω1,2 / ωp versus normalized


n ( z, t )
q
u1 ( z , t ) ×  µ0 h1 ( z , t ) + 1
B0  −
m
N
0


− ( u1 ( z , t )∇ ) u1 ( z , t )-
frequency of source wave
2.2 Nonlinear transformation of EM source wave
(6b)
where q and m are electron charge and mass,
respectively.
2.1 Linear transformation of source EMW
Linear plasma theory analysis, analyzed by using the
first order perturbation theory, gives that two
transmitted and two reflected wave modes are generated
due to interaction between LP source EMW and
suddenly created magneto-plasma medium, with the
following frequencies [2] (see also Fig.1 in [3]):
42 ▪ VOL. 33, No 1, 2005
a = ω 2p + (ω 02 + ω 2b) / 2 ;
Ω 0 = ω0 / ωp for value of
normalized cyclotron angular frequency Ω b = ωb / ωp = 1 .
n1 ( z , t ) ∂u1 ( z , t )
⋅
−
∂t
N0
qn ( z , t )
− 1
e1 ( z , t ) ,
mN 0
ω1,2 = a ± a 2 - b ;
(12b)
(6a)
∂h2 ( z , t )
=0,
∂t
∇ × h2 ( z, t ) + qN 0 u2 ( z, t ) − ε 0
and ωp = qB0 /(ε 0 m)
are the source wave, electron plasma and electron
cyclotron angular frequencies, respectively. Transmitted
waves have the angular frequencies ω1 and ω2
( ω1 , ω2 > 0 ), whereas the reflected waves have angular
frequencies - ω1 , - ω2 . Dispersion relation and
dielectric constant have the following forms,
respectively:
and
∂n2 ( z, t )
+ ∇ ( N 0 u2 ( z , t ) ) = −∇ ( n1 ( z , t )u1 ( z , t ) ) ,
∂t
(11)
The weakly nonlinear fields are defined by Eqs.
(3b)- (6b). The corresponding initial conditions are
evaluated from the continuity of EM and velocity fields
at t = 0 and assumption that newly created electrons in
plasma are at rest at t = 0 , i.e.:
e2 ( z , t = 0+ ) = h2 ( z , t = 0+ ) = u2 ( z , t = 0+ ) = 0
(13)
In order to solve the system of partial differential
equations (3b)-(6b), with the prescribed initial
conditions given by Eq. (12), we shall apply the Laplace
transform in time
L( f ( z , t )) =
∞
∫ f ( z, t ) exp(-st )dt = F ( z , s) ,
(14a)
0
FME Transaction
and, as the plasma is unbounded, Fourier transform in
space
F( f ( z , t )) =
2
6
i =1
i =1
t ,r
t ,r
+ ∑ E22xi
cos(ωiβ t ∓ 2k0 z ) + ∑ E24xi
cos(ϕi t ∓ 2k0 z ) ,
(16a)
∞
∫ f ( z, t ) exp(-jk )dz = F (k , s) . (14b)
0
In the domain of complex frequency s = jω and
wave number k , the EM and velocity fields are defined
by the following system of linear algebraic equations,
obtained from (3b)-(6b):
j k E2 x (k , s ) + µ 0sH 2 y (k , s ) = 0 ,
2
h2 y (z, t) = H20y cos(2k0 z) + ∑ H23tyi,r cos(ωiβ t ± 2k0 z) +
i=1
6
+ ∑ H 24tyi,r cos(ϕi t ± 2k0 z ) ,
2
e2 z ( z , t ) = E20z sin(2k0 z ) + ∑ E12 zi sin(ωiα t ) +
i =1
(3bx)
(4by)
where
F (k , s ) = FL {-qn1 ( z , t )u1x ( z , t )} ,
(15a)
G (k , s ) = FL {-qn1 ( z , t )u1z ( z , t )} ,
(15b)
2
i =1
6
i =1
t ,r
+ ∑ E24zi
sin(ϕi t ∓ 2k0 z )
(4bx)
q
sU 2 x (k , s )-ωbU 2 z (k , s ) + E2 x (k , s ) = H (k , s ) , (5bx)
m
q
sU 2 z (k , s ) + ω bV2 x (k , s ) + E2 z (k , s ) = I (k , s ) , (5by)
m
q
sU 2 z (k , s) + E2 z (k , s) = J (k , s) ,
(5bz)
m
6
+ ∑ E22zi sin(ϕi t ) + ∑ E23tzi,r sin(ωi β t ∓ 2k0 z ) +
-j k H 2 y (k , s ) + N 0 qU 2 x (k , s) − ε 0 s E2 x (k , s ) = F (k , s ) ,
N 0 qU 2 z (k , s )-ε 0 sE2 z (k , s ) = G (k , s ) ,
where
ϕ 1 = ω 1 ; ϕ 2 = ω 2 ; ϕ 3 = ω 1-ω 2 ; ϕ 4 = ω 1 + ω 2 ;
ϕ 5 = 2ω 1 ; ϕ 6 = 2ω 2 ; ω iα = ω i (ω 0 = 0) ;
(18)
ω i β = ω i (ω 0 → 2ω 0) i = 1,2 ,
(see Figs.2a and 2b). In Eqs. (16a)-(17) the t, r
superscripts refer to transmitted and reflected wave
modes, respectively. The upper sign in ∓ refers to the
transmitted wave mode and the lower to the reflected.
Φ3
Ωb =1
Φ4


∂u ( z , t )
− q µ 0u1x ( z, t )h1 y ( z, t ) 
−u1z ( z , t ) 1z
∂t


q

,
I (k , s) = FL −
n1 ( z , t )u1z ( z , t )

 mN 0

 n ( z , t ) ∂u ( z , t ) ω b

⋅ 1z
−
n1 ( z , t )u1x ( z , t ) 
− 1
∂t
N0
N0


(15d)
 qµ

J (k , s) = FL - 0 ( u1x ( z , t )h1 y ( z, t )-u1 y ( z, t )h1x( z, t ) )  .
m


(15e)
2
6
e2 x ( z , t ) = E20x + ∑ E21 xi cos(ωiα t ) + ∑ E22xi cos(ϕi t )
i =1
FME Transaction
i =1
(17)
i =1


∂u ( z , t )
+ q µ 0u1z ( z , t )h1 y ( z , t ) 
−u1z ( z , t ) 1x
∂t




q
H (k , s ) = FL −
n1 ( z , t )e1x ( z , t )
,
 mN0

 n ( z , t ) ∂u1x ( z , t ) ω b

1
n1 ( z , t )u1z ( z , t ) 
⋅
+
−
N0
N0
∂t


(15c)
Operator FL in the above equations has the meaning of
the Fourier-Laplace transformation.
In the domain of complex frequency s=jω and wave
number k one should solve the obtained system of linear
algebraic equations, (3bx)-(5bz), and perform the
inverse Laplace and Fourier transforms. After these
steps, nonlinear EM and electron fields in weakly
nonlinear plasma are obtained in the following form:
(16b)
i =1
Ω0
(a)
Ωb =1
Ω 1β
Ω 2β
Ω0
(b)
Figure 2. Normalized angular frequencies of newly created
(a)
and
nonlinear wave modes Φ 3,4 = ϕ3,4 / ωp
Ω1,2 β = ω1,2 β / ωp (b) versus normalized frequency of
source
wave
Ω 0 = ω0 / ωp
for
value
of
normalized
cyclotron angular frequency Ω b = ωb / ωp = 1 .
VOL. 33, No 1, 2005 ▪ 43
The effect of switching nonlinear magneto-plasma
medium is creation of EM field with the following
components: rectification electric mode E20x , stationary
space-varying magnetic mode with wave number 2k0 ,
eight oscillating electric wave modes with angular
frequencies ωiα , i = 1, 2 and ϕi , i ∈ [1, 6] , eight
electric and magnetic transmitted and reflected wave
modes with angular frequencies ωiβ , i = 1, 2 and ϕi ,
i ∈ [1, 6] ,with wave number 2k0 , and creation of
electric electron plasma field with the following
components: stationary space-varying mode with wave
number 2k0 , eight oscillating modes with angular
frequencies ωiα , i = 1, 2 and ϕi , i ∈ [1, 6] , eight
transmitted and reflected wave modes with angular
frequencies ωiβ , i = 1, 2 and ϕi , i ∈ [1, 6] ,with wave
number 2k0 .
In this paper the efficiency of excitation of electric
second harmonics ( ±ω1,2 , 2k0 ) of EM field
is
analyzed. Distribution of amplitudes of these
newly
created
wave
modes
in
plasma,
normalized
E20 =
to
qE02
/ mcωp = 586 E02
/ ωp V/m ,
(for E0 = 100 V/m and ωp = 106 Hz ), versus the
angular frequency of the source wave, normalized on
electron plasma angular frequency ωp , are presented in
Figs.3a and 3b, respectively, thereon the electron
cyclotron angular frequency is taken to be equal to
angular plasma frequency.
3. NUMERICAL RESULTS
Inspecting carefully the Figs.1, 2a and 2b one concludes
that resonant excitation is possible only for
oscillating and wave
modes
having
the
angular frequency ω2 β . For ωb = ωp , we get
ω2 β rezonant ≈ 1, 25ωp , i.e. the
value of angular
frequency frequency of the source wave which
enables resonant excitation is approximately
ω0 rezonant ≈ 0, 63ωp .
3.1 Transmitted and reflected electric components
( ± 2ω1,2, 2k0) of EMW in plasma
Amplitudes of these components have the following
form:
E 20
4t , r
⋅ X t , r Ω1 , Ω 0 , Ω b ⋅
E 2 x5 ( ±2ω1 , 2k0 ) = ±
32
(
2
(
)
)
⋅ ∑ Yiβ Ω1β , Ω 0 , Ω b ⋅
i =1
(
) (
( )(
)(
)
44 ▪ VOL. 33, No 1, 2005
)
4t , r
2
(
)
E 20
⋅ X t ,r Ω1 , Ω 0 , Ω b ⋅
32
(
)
⋅ ∑ Yiβ Ω1β , Ω 0 , Ω b ⋅
i =1
(
) (
( )(
)(
)
Ω + Ω Ω 2 − 1  2 Ω 2 − 1 ± Ω Ω  
2
2 b 
iβ
iβ
 0




2 2
2
2
2
⋅ + Ω 2 Ω b + Ω b Ω 2 − 1 Ω2 − Ω b − 1
 , (20)


+ Ω iβ Ω 22 − 1 Ω i2β − Ω b2 − 1



where the superscripts t, r refer to transmitted and
reflected components, respectively. The upper sign in
± refers to transmitted and lower to reflected
component, and
(
(
)
X t ,r Ω1,2 , Ω 0 , Ω b =
(
)
Yiβ =  Ω iβ Ω i2β − A 


)
)
(
Ω 0 Ω b Ω1,2 ± Ω 0
2
Ω1,2
−1
,
(
2
−a
Ω1,2
A = 1+
)
2
),
4Ω 02 + Ω b2
2
(21)
, (22)
where Ω 0 , Ω b , Ω1,2 and Ω iβ are normalized angular
frequencies, on electron plasma angular frequency ωp .
The relative power content of these wave modes is
given by:
S24xt ,5r = E24ty,5r ⋅ H 24ty,5r and S24xt ,6r = E24ty,6r ⋅ H 24ty,6r ,
(23)
where
H 42ty,5r = ±
ε 0 ⋅ Ω 0 ⋅ 4t , r ,
E
µ 0 Ω 1 2 x5
H 24ty,6r = ±
ε 0 Ω 0 4t , r
⋅
⋅E
.
µ0 Ω1 2 x 6
(24)
Figs.3a and 3b show that the amplitudes of transmitted
waves are much greater than the amplitudes of reflected
waves.
The peak value of the amplitude of transmitted wave
E24xt 5 (2ω1 , 2k0 ) ≈ 0,37 E20 ≈ 2, 20 V/m is obtained for
value Ω 0 = 1,7 . Its relative power content, for that
value of angular frequency of the source wave, is about
15% ( ( S24xt 5 / S0 ) / E20 = 0, 025 , i.e. ( S24xt 5 / S0 ) ≈ 0,15 ,
see Fig.4a). The relative power content of the
corresponding reflected wave is negligible (see Fig.4b).
The peak value of the amplitude of transmitted wave
E24xt 5 (2ω1 , 2k0 ) ≈ 0,34 E20 ≈ 1,99 V/m is obtained for
Ω 0 = 1,9 . Its relative power content, for this value of
angular frequency of the source wave, is about 9%
2
2
Ω + Ω


1β Ω1β − 1  2 Ω1 − 1 ± Ω1Ω b 
 b




⋅  + Ω12 Ω b2 + Ω b Ω b2 − 1 Ω12 − Ω b2 − 1
 , (19)


 + Ω1β Ω12 − 1 Ω12β − Ω b2 − 1



(
E 2 x 6 ( ±2ω2 , 2k0 ) = ±
)
( ( S24xt 6 / S0 ) / E20 = 0, 016 , i.e. ( S24xt 6 / S0 ) ≈ 0, 09 , see
Fig.4a). The relative power content of corresponding
reflected wave is negligible (see Fig.4(b)).
FME Transaction
E 42tx5
S 42tx5 S 0
E 20
E 20
t
E4
2 x6
E 20
S 42tx6 S 0
E 20
Ωo
(a)
Ωo
(a)
E 42rx5
E 20
r
S4
2 x5 S 0
E 42rx6
E 20
E 20
S 42rx6 S 0
E 20
(b)
Ωo
Figure 3. Normalized amplitudes of excited secondharmonic (transmitted (a) and reflected (b)) wave modes
±ω1,2 versus
( k = 2k0 ) with angular frequencies
(b)
Ω o0
Ω
Figure 4. Normalized relative power content of the secondharmonic, transmitted (a) and reflected (b), wave modes
±ω1,2 versus
( k = 2k0 ) with angular frequencies
normalized frequency of source wave Ω 0 = ω0 / ωp for
value of normalized cyclotron angular frequency
Ω b = ωb / ωp = 1 .
normalized frequency of source wave Ω 0 = ω0 / ωp
4. CONCLUSION
ωiα = ωi (ω0 = 0) , ( i = 1, 2 ); sixteen transversal EM
and sixteen longitudinal electron wave modes (eight
transmitted plus eight reflected) ( k = 2k0 ) with angular
The initial value problem of interaction of LP EM
source wave with suddenly created weakly nonlinear
magneto-plasma medium, in particular case of
transverse propagation, is solved in the closed form in
the paper. The nonlinearities caused by the interaction
of newly created linearly magnetic and velocity modes
in plasma and perturbation of electron density on
direction of waves propagation are taken into account in
the equation of continuity, Maxwell equation for space
variation of magnetic field and equation of electron
fluid motion. LP EM source wave in suddenly created
linear cold magneto-plasma splits in two transmitted
(with angular frequencies ω1 , ω2 ) and two reflected
(with angular frequencies −ω1 , −ω2 ) traveling waves.
In the case of weakly nonlinear cold magneto-plasma
eight transversal EM and eight longitudinal electron
oscillations ( k = 0 ) with the following angular
ϕ1 = ω1 ,
ϕ 2 = ω2 ,
ϕ3 = ω1 − ω2 ,
frequencies:
ϕ4 = ω1 + ω2 ,
ϕ5 = 2ω1 ,
ϕ6 = 2ω2
and
FME Transaction
for
value of the normalized cyclotron angular frequency
Ω b = ωb / ωp = 1 .
frequencies ϕi , i ∈ [1, 6] and ωiβ = ωi (ω0 → 2ω0 ) ,
( i = 1, 2 ), are excited. Additional stationary spacevarying electron longitudinal and EM transversal wave
mode ( k = 2k0 ) with zero value of angular frequency
( ω = 0 ) and stationary EM rectification mode
( ω = 0 , k = 0 ) with angular frequencies are excited.
The efficiency of creation of all of these modes can be
controlled by the source wave angular frequency and
the magnitude of external static magnetic field. The
resonant transformation of LP source wave is noticed in
the case when the angular frequency of newly created
traveling and oscillating modes has value equal to ω2 β .
REFERENCES
[1] Kalluri, D.K.,Electromagnetics of Complex Media,
CRC, Boca Raton, 1999.
VOL. 33, No 1, 2005 ▪ 45
[2] Goteti, V.R. and Kalluri, D.K, Wave propagation in
a switched magnetoplasma medium: Tranverse
propagation, Radio Sci., 26, 61, 1990.
[3] Trifković, Z., Stanić, B., Frequency shift of a
source EMW due to sudden changes of magnetized
plasma. Transverse propagation, ETRAN XLVII,
Contributed papers, Vol. 2, p. 213-215, 8-13 jun
2003, Herceg Novi.
[4] Trifković, Z., Influnce analysis of time discontinuity
on EMW transformation in cold weakly nonlinear
magnetized plasma, PhD Thesis, Faculty of
Electrical Engineering, Belgrade, 2002.
46 ▪ VOL. 33, No 1, 2005
ГЕНЕРИСАЊЕ ДРУГОГ ХАРМОНИКА ПРИ
НЕЛИНЕАРНОЈ ТРАНСФОРМАЦИЈИ
ЕЛЕКТРОМАГНЕТСКОГ ТАЛАСА У НАГЛО
СТВОРЕНОЈ МАГНЕТИЗОВАНОЈ ХЛАДНОЈ
ПЛАЗМИ. ТРАНСВЕРЗАЛНО ПРОСТИРАЊЕ
Зоран Трифковић, Божидар Станић
У раду је анализирана трансформација изворног
линеарно поларизованог електромагнетског таласа,
услед слабих нелинеарности, у електронске и
електромагнетске
осцилације,
стационарне
(просторно променљиве и једносмерне) и таласне
електромагнетске и електронске модове у плазми,
применом пертурбационе теорије другог реда.
Ефикасност екситације трансверзалних електричних
модова, двоструко веће фреквенције и двоструко
већег таласног броја од модова добијених у
линеарној теорији, анализирана је за различите
вредности угаоне фреквенције изворног таласа и за
вредност угаоне електронске жиро-фреквенције која
одговара вредности угаоне електронске пазмене
фреквенције.
FME Transaction