final btp thesis 2

7/28/2019 Final BTP Thesis 2

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2009PH10

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:


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;



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ne0(z=0)= 1.24*1018

m-3

, slope = 8*1019

.


** 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

.



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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



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2009PH107


W/cm2, frequency (w) = 1

1, ne0 (z=0)= 1.24*10

22m

-3, standard deviation = 0.2.


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



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2009PH107

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


s-1

2*1016

m-3


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2009PH107

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


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


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


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



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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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2009PH107

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).

final btp thesis 2

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