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Microwave-Plasma Interaction Simulations
Mandeep Singh 2009PH10718,Rupinder Singh 2009PH10740
Supervisor: Dr. Malik H.K
Abstract: The perturbed electron density in a non-magnetized plasma medium caused due to the interaction of opp
travelling microwaves within the medium is profiled using numerical simulations Runga-Kutta, 4th
order method
effect of the interaction is studied for three different initial electron density profiles viz. uniform distribution,
distribution, & Gaussian distribution. The study includes the density perturbation caused due to opposite travelling w
with variable phase difference. The density steepening effect was simulated for different microwave frequenc
intensities, and different electron temperatures. The change in the electron density due to the propagation of a EM
through an unbounded plasma with various initial Electric field profiles is studied. The behavior of the TE10 mod
rectangular waveguide filled with plasma is studied and the corresponding perturbation in the plasma density is st
against the EM- parameters and electron temperature.
Email: [email protected].
INTRODUCTION :
The interaction of microwaves with plasma has been studied in great detail till date. Stu
concerned with high Intensity microwaves in plasma have shown interesting results like elecbunching, wavelength elongation, etc.
The movement of an electromagnetic wave in under-dense plasma generates a force on
constituent charge particles known as the ponderomotive force. The ponderomotive force chan
electron density distribution and the dielectric permittivity of the plasma. As a result, the
modification in the profiles of the electric and magnetic fields of the microwave into the plasma.
The nonlinear Lorentz force on an electron in an electric field can be equated to
ponderomotive force on electron. This effect is proportional to the gradient of the microw
intensity.
For the ponderomotive force to be effective its magnitude should be comparable to or gre
than the pressure gradient force of the plasma medium.
We consider the plasma effect through the permittivity () and assume a balance betw
the effects of the ponderomotive force and the electron pressure. Using Runga-Kutta- 4th Order
study the ensuing electron density profiles, specifically the modification in plasma density whstanding wave pattern is formed by the propagation of two opposite travelling waves of e
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intensity and frequency (assuming equal change in wavelength for both the waves in the medi
The ions on the other hand are assumed to be fixed due to their heavy mass.
The Second order differential equation used for Runga-kutta simulations is achieved by u
the Maxwell and fluid equations. MATLAB was used as the platform for the coding of the Ru
Kutta simulations.
The output of the study shows the dependency of the microwaves Electric field profile
respect to wave Intensity, frequency, and profile of the density of the plasma. Also the results s
the upshift in the frequency. The density bunching effect is also very profound as shown in
graphs.
THEORY:
A- Microwave in an unbounded plasma
We have plasma with electric permittivity (z > 0) and Microwave field is incident on it alo
direction. We have Maxwells equations in the absence of any charges or current as:
E= - 1c
B
t
(1)
B =1
c
D
t(2)
D = 0 (3)
B = 0 (4)
E, B Are the electric field and magnetic field strength respectively. The time independent w
equation for the electromagnetic wave travelling in the plasma can be derived from the ab
Maxwells equations:
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2E- E( )+ w 2
c2
e E= 0 (5)Where and c is the electromagnetic frequency and speed of light respectively.
The electromagnetic wave travelling through plasma will modify the charge de
distribution and electric permittivity in the plasma due to theponderomotive force acting on cha
particles..
Ponderomotive force is basically the nonlinear Lorentzs force. The force acting on ion
neglected due to high mass of as compare to the electrons. Its average value force per unit vol
is given by:
Fpe =
14p ne ene E
2 (6)
There are two forces acting on the plasma i.e plasma pressure gradient force and
Ponderomotive force. Equating these forces:
1
4p nee
neE
2= T
ene (7)
Where Te is the electron temperature and is the electric permittivity.
Rearranging equation (7)
22
2
1 e x
e e
dn dE e
n dz m T dz (8)
We can get the electron plasma density by integrating the above (8) equation.
So the electron plasma density is given as:
ne(z) = n
e0e- e
2Ex (z)/mw 2Te
(9)
Where ne(z) is the electron density in the presence of electromagnetic wave field and ne0 is
maximum electrons density without electric field. The electric permittivity is given by:
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2 2
2 2/0
2 2
41 1 x e
pe e E m T ee n em
(10)
Where w pe = 4p e2ne /m( )1/2 is the electron plasma frequency.For obtaining the effect of Microwave on the plasma we plug-in equation (10) to obtain
perturbed Electric Field
2 22 22
/0
2 2 2
41 0x e
e E m T x ex
d E e ne E
dz c m
(11)
Hence superimposing the numerical simulations for both the directions, we can obtain a new Electric
distribution.
B- Stationary waves in a Rectangular waveguide formed by reflection: TE10 mode
We have studied the propagation of microwave in a rectangular waveguide and then we h
considered the interaction of microwave with the plasma, which is filled, in the rectang
waveguide.
For the interaction we have considered fundamental TE10 mode of the microwave, w
propagates in the empty rectangular waveguide and then encounters plasma. We have u
Maxwells equations for evaluating the field components of the fundamental mode in evacu
waveguide and then obtained the coupled differential equations for the fundamental mod
plasma filled waveguide.
To solve these equations, for the amplitude of the electric field of the microwave and
wavelength under the effect waveguide width, plasma density and microwave frequency, fo
order Runge-kutta method has been used. Firstly we have simulated for the wave travelling in
direction only and then we have obtained solution for the two oppositely travelling waves i.e. fo
standing waves formed in the rectangular waveguide filled with plasma.
We are considering two rectangular waveguides of width b and height h of which one is
evacuated and another one is filled with unmagnetized homogenous isothermal plasma and bot
these are joined coaxially. Here we assume that the fundamental TE10 mode excited by the
microwave after travelling a distance in evacuated waveguide encounters plasma in the second
waveguide. So the wave governing Maxwell equations are (Stated earlier.)
Here B and E are the magnetic and electric field for the microwave respectively and o for the
evacuated waveguide. The field components for the fundamental TE10 mode for the microwave can be
obtained from the Maxwells equation:
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sini kz t
y o
xE E e
b
(12)
sini kz t
x o
k xB E e
b
(13)
sini kz t
z o
i xB E e
b b
(14)
So the above equation gives us the relation between the density and the electric field of the TE10 mode a
shows that density will modify itself with the field. The electric permittivity of the plasma will also change
accordance with the electric field given as
2
21p
o
, where
22
pe o
nem
. Now clubbing this
electric permittivity with the Maxwells equations and the field components of the fundamental TE10 mode
obtained. We obtain:
22 2
2 2 20
yxr y
EEE
x z c
(15)
Where
2
21p
r
. So we substitute the electric field and get two coupled differential equations.
sin expyx
E A z ikzb
as we consider the variation the electric field same along the x-direction in
interest of the conducting waveguide for obtaining the pattern of the standing field in the plasma filled
waveguide. Here we have to separate out the real and imaginary parts of the amplitude of the wave as
r iA z A z iA z and we obtain the coupled equations as :
2 2 2
2 2 2 22 2 22 sin / 0rir r r i r AA k k A A A x b Az z b c
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C- Microwave with different initial Electric Field profiles
Consider the incident EM wave on the unbounded plasma in case 1 of the profile:
2
22
sin
p
o
y
a ikx
oE E e kx e
(16)
Where p an integer
We have considered the values of p = 2,3,4
Using maxwells equations and substituting the perturbed amplitude of the Electric Field we get
differential equation for which the solution needs to be determined. Next we superimpose these
results to form a standing wave in the plasma and then calculate the perturbed density of electro
Using the expression for the perturbed electron density caused due the pondermotive force
generated by the Electric field of the radiation we studied the effect of frequency of microwave,
Intensity of the radiation and the electron temperature on the density profile of the electrons.
The Second order differential equation used for Runga-kutta simulations is achieved by u
the Maxwell and fluid equations. MATLAB was used as the platform for the coding of the Runga
Kutta simulations.
ne(z) = n
e0e- e
2Ex (z)/mw 2Te
(18)
The average Ponderomotive force per unit volume acting on the electrons (ions are take n fixed to their heavy mass) in the plasma is the same as the non-linear Lorentz force
RESULTS AND DISCUSSION:
A- Stationary waves in an unbounded plasma
CHANGE IN WAVELENGTH OF THE RADIATION:
2 2 2
2 2 2 2
2 2 22 sin / 0r
irr r i r
AAk k A A A x b A
z z b c
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The numerical simulations show that the wavelength in the plasma changes and the prof
the Electric Field distorts from the true sinusoidal shape.
Figure 1. Change in wavelength of a microwave radiation during propogation in a plasma.
PROFILES OF ELECTRON DISTRIBUTION:
1- UNIFORM DISTRIBUTION
2- LINEARLY DEPENDENT DISTRIBUTION3- GAUSSIAN DISTRIBUTION
UNIFORM DISTRIBUTION:
Taking initial electron density to be uniform over the complete plasma medium,
simulations were carried out to study the behavior of the perturbed electron density profile
respect to frequency, Intensity and electron temperature.
i.e Taking ne0 = CONSTANT;
and solving equation (11);
Temperature dependence:
The solution was carried out for different levels of electron temperature. The output/re
are shown in Fig. 1 below.
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Figure2. Density perturbation for different electron temperatures using radiation with Intensity (I) = 108
V/m, frequency (w) = 10
ne0 = 1.24*1018
m-3
.
Frequency dependence:
Figure3. Density perturbation for different electron temperatures using radiation with Intensity (I) = 108
V/m, Temperature (Te) =
ne0 = 1.24*1018 m-3.
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Intensity dependence:
Figure4. Density perturbation for different electron temperatures using radiation with Temperature (Te) = 2 , frequency (w) = 10
ne0 = 1.24*1018
m-3
.
LINEARLY DEPENDENT DISTRIBUTION:
Taking initial electron density to be linearly changing with distance over the complete pla
medium, the simulations were carried out to study the behavior of the perturbed electron de
profile with respect to frequency, Intensity and electron temperature.
i.e ( ) *e eon z n b z
where b is the slope of the distribution line;
eon is the value of electron density at z=0;
Temperature dependence:
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Figure5. Density perturbation for different electron temperatures using radiation with Intensity (I) = 108
V/m, frequency (w) = 10
ne0(z=0)= 1.24*1018
m-3
, slope = 8*1019
.
Frequency dependence:
** Figure6. Den
perturbation for different electron temperatures using radiation with Intensity (I) = 108
V/m, Temperature (Te) = 200000 eV, ne0 (
1.24*1018
m-3
, slope = 8*1019
.
Intensity dependence:
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Figure7. Density perturbation for different electron temperatures using radiation with Temperature (Te) = 20000 eV, frequency
1010
s-1
, ne0 (z=0)= 1.24*1018
m-3
, slope = 8*1019
.
GAUSSIAN DISTRIBUTION:
Taking the distribution to be Gaussian about a point zo we carried out the simulations.
2
2
( )
2( )oz z
e eon z n e
Again the simulations were carried out against microwave parameters and electron temperature
Temperature dependence:
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Figure9. Density perturbation for different electron temperatures using radiation with Intensity (I) = 108
W/cm2, frequency (w) = 1
1, ne0 (z=0)= 1.24*10
22m
-3, standard deviation = 0.2.
Frequency dependence:
Figure10. Density perturbation for different electron temperatures using radiation with Temperature (T e) = 200000 eV , , Inten
108
W/cm2, ne0 (z=0) = 1.24*10
22m
-3, standard deviation = 0.2
Intensity dependence:
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Figure11. Density perturbation for different electron temperatures using radiation with Temperature (Te) = 200000 eV, frequen
= 1010
s-1
, ne0 (z=0)= 1.24*1022
m-3
, standard deviation = 0.2
Results: Stationary waves in a rectangular waveguide filled with plasma
Change in Density with change in the frequency of the incident radiation
Figure12. p=2, Density perturbation for different frequencies using radiation with Intensity (I) = 10
6
V/m, Temperature (Te) = 3eV, ne0 = 2*10
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Figure13. p= 2, Density perturbation for different Intensitiess using radiation with Temperature (T e) = 3 eV, frequency (w) = 1
ne0 = 2*1016
m-3
.
gure14. p=2,Density perturbation for different electron temperatures using radiation with Intensity (I) = 106
V/m, frequency (w) = 109
s-1
2*1016
m-3
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esults for Stationary waves in an unbounded plasma with varying initial Electric Field Profile.
Change in Density with change in the Intensity of the incident radiation, p=2
(a) I = 106V/m (b) I=10
7V/m (c) I = 10
8V/m
Figure15. p= 2, Density perturbation for different Intensitiess using radiation with Temperature (Te) = 3 eV, frequency (w) =109
s
= 2*1016
m-3
.
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Change in Density with change in the electron temperature of the plasma
Figure16. p=2,Density perturbation for different electron temperatures using radiation with Intensity (I) = 106
V/m, frequency (w) = 109
s-1
2*1016
m-3
.
Change in Density with change in the frequency of the incident radiation
ure17. p=2, Density perturbation for different frequencies using radiation with Intensity (I) = 106
V/m, Temperature (Te) = 3eV, ne0 = 2*1016
m
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Change in Density with change in the Intensity of the incident radiation, p=3
(a) I = 106
V/m (b) I = 107
V/m (c) I = 108
V/m
gure18. p= 2, Density perturbation for different electron temperatures using radiation with Temperature (Te) = 3 eV, frequency (w) = 109
s
2*1016
m-3
.
(a) T = 3eV (b) T = 10eV (c) T = 100eV
gure19. p=2,Density perturbation for different electron temperatures using radiation with Intensity (I) = 106
V/m, frequency (w) = 109
s-1
2*1016
m-3
(a) f = 1 Ghz (b) f = 1.5 Ghz (c) f = 2 Ghz
gure20. p=2, Density perturbation for different frequencies using radiation with Intensity (I) = 106
V/m, Temperature (Te) = 3eV, ne0 = 2*1016
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P=2 P=3 P=4
Intensi ty =
107V/m
T= 3eV
Frequency
= 109s-
1
Temperat
ure = 100
eV
Intensi ty =
106V/m
Frequency
= 109s-
1
Frequenc
y = 2Ghz
Intensi ty =
106V/m
T = 3eV
Figure 21.Comparative density profiles for different values of p
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DISCUSSION A: Stationary waves in unbounded plasma
1- It is evident from the above graphs that for each of the initial density profiles the pertu
density distribution becomes more and more broader i.e. effect of the ponderomotive force
electron bunching is reduced as we increase the temperature.
2- Also as we increase the frequency of the incident radiation, the bunching gets more and m
closely spaced.
3- As we increase the intensity of the radiation, the electron bunching increases and we get m
and more sharp peaks.
4- The graphs below (for p =2) depict the change in the perturbed densities when observed fro
up, Blue color indicating the maximum change (decrease) in the electron density in that
regions whereas the red color indicating almost no change in the density.
Discussion B: Stationary waves in a rectangular waveguide
5- In our simulations we have seen that the electron density of the plasma changes with certai
parameters like intensity of the microwave, frequency of the microwave, temperature.
6- As shown in the graphs it is evident that the spacing between the electron bunches formed
the plasma come close to each other (in the figure blue regions in the graph come closer to
each other).
7- If we consider the effect of temperature the electron bunching decreases with the increase i
the temperature of the plasma (in the figure the blue region decreases and the red region
increases) because the pressure gradient which fights the ponderomotive force will increase
and result will be the decrease in bunching.
8- With the increase in the intensity the electron bunching increases i.e. blue region in the figu
increases because as we will increase intensity the ponderomotive force acting on the elect
will increase there will be more bunching.
Dicsussion C: Stationary waves for different Initial Electric Field profiles
9- As shown in the graphs, as you increase the temperature the blue area decreases. This is
accordance with the theory of electron bunching as higher temperature increases the avera
energy of electrons and hence it is not possible to bunch many electrons with the same
Intensity.
10- Consider the parameter of Intensity: As the Intensity of radiation is increased the bunching
effect is increased and the blue colored region increases in size.
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11- As the frequency is increased the bunching spacing is reduced and the areas where the de
decreases come more and more closer.
12- Plots show similar behavior for p=4.
FUTURE SCOPE:
In our project we have studied microwave-plasma interaction. Firstly we studied the e
caused by microwave propagating in one direction through plasma. Secondly we have studied
effects of two-opposite travelling waves in the plasma. Then rectangular waveguide was introdu
to study confined propagation of the EM wave through the plasma. For the future such statio
waves can be created in a cylindrical waveguide and then the change in the electron density ca
studied.
REFERENCES:
[1] Zhi-zhan Xu, Jian Yu, Yong-hong Tang,Density-profile steepening by laser radiation in a magnetized inhomogeneous pla
Volume 33, Number 6 June1986
[2] P. Vyas and M. P. Srivastava , Density profile steepening due to selfgenerated magnetic fields in plasmas produced by
irradiation of spherical targets, Phys. Plasmas 2, 2835 (1995)
[3] Y. Sentoku, T. E. Cowan, A. Kemp and H. Ruhl, Phys. Plasmas 10, 2009 (2003).
[4] V. K. Tripathi and C. S. Liu, IEEE Trans. Plasma Sci. 17, 583 (1989).
[5] Z. M. Sheng, K. Mima, Y. Sentoku, K. Nishihara and J. Zhang, Phys. Plasmas 9, 3147 (2002).