Relativistic effects on the TM-modes of one boundary corrugated parallel plate waveguide filled with uniaxial warm drifting plasma

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Relativistic effects on the TM-modes of one boundary

corrugated parallel plate waveguide filled with uniaxial

warm drifting plasma

Rajendra Prasad, Ramjee Prasad, A.N. Mantri

To cite this version:

Relativistic effects

on

the

TM-modes

of

one

boundary

corrugated parallel plate

waveguide

filled

with uniaxial

warm

drifting plasma

Rajendra

Prasad

_(1),

_Ramjee

Prasad

₍₂₎

and A. N. Mantri

₍₃₎

(1)

Department

of Natural Sciences, Mutah

University,

P.O. Box _{7, Mutah, Karak,} Jordan

(2)

Faculty

of Electrical

_Engineering,

Delft

University

of

Technology,

P.O. Box 5031, 2600 GA

Delft,

The Netherlands

(3)

Department

of

_Physics,

Banaras Hindu

University,

Varanasi-221005, India

(Reçu

le 24 _janvier 1990, révisé le 23 mars _1990,

_accepté

le 3 avril

₁₉₉₀₎

Abstract. - The relativistic effects

on the TM-modes of one

_{boundary corrugated parallel plate}

waveguide

filled with uniaxial warm

_{drifting plasma}

have been

_{investigated.}

The

dispersion

relation for the TM-modes has been derived and the numerical

computations

have been carried

out to

_study

the effects of increased

(relativistic)

drift

_velocity

and the _temperature under this condition. The effects of drift

velocity

are

_quite

distinct and

significant

over the non-relativistic case. The effect of _temperature in the relativistic case is

_{remarkably insignificant}

unlike the

non-relativistic case where it was

_{quite pronounced.}

Classification

Physics

Abstracts

52.40D

1. Introduction.

In a recent _paper

_{[1] we}

studied the characteristics of the

electromagnetic

wave

_propagation

through

one

_{boundary corrugated parallel plate waveguide}

filled with warm uniaxial

_drifting

plasma, using purely

non-relativistic treatment.

However,

to simulate the actual behavior of the

_plasma

and to obtain

_greater

_insight

of the

_problem,

the more realistic

_approach

would

obviously

be the relativistic one. The relativistic effect has now been included in this

_sequel

to

reference

_[1].

In the

_present

_paper, the drift

_velocity

of the bounded

_{plasma along}

the axis of the

waveguide

has been assumed

_{large enough}

to be treated as a relativistic

_parameter.

The effects of the relativistic drift

_velocity

and

_temperature

of the

_plasma,

under this

condition,

on the TM-modes of the

_waveguide

have been studied in detail. These results are of

great

importance

due to their wide

_application

in

_{plasma diagnostics}

[7].

2. Basic theoretical

_concept.

Referring

to the

_geometry

of the

_problem

_(one

_{boundary corrugated parallel plate}

waveguide)

described in reference

_{[1], consider,}

two frames of

_reference,

S and S’.

Let,

S be attached

_(stationary)

to the

waveguide

and be described

_by

the

_space-time

coordinates

_(X,

Y,

1510

Z,

t).

S’ is fixed to the

_plasma

medium

_moving

with a constant

_{velocity, Vo}

with

respect

to

_S,

and is described

_by

the coordinates

_(X’,

Y’,

Z’,

t’).

The frame S’ moves

_{only along}

the

longitudinal

Z-axis with

_{velocity, Vo}

=

ho à,.

The field vectors _vary

_harmonically

as,

-where,

w’ is the

_angular

wave

_frequency

and

_y’

is the

_{complex propagation}

constant.

2.1 LORENTZ TRANSFORMATION _{EQUATIONS. - The Lorentz transformation for the}

space-time coordinates is well known and is

_{given [2] by,}

and

where,

r =

( 1 -

a 2)- 1/2

with a =

VOIC

and C is the

velocity

of

light

in

free-space.

The Lorentz transformation

_{equations (without rotation)}

for the electric

(E)

and the

magnetic

(H)

field vectors in

_free-space

are

given

[3, 4]

by,

and

where,

J.Lo and Eo are

_permeability

and the

_permittivity

of the

_{free-space, respectively.}

The Lorentz transformations for the wave _vector,

_frequency

and

_plasma

_parameters

are

given [4, 5] by,

where,

p,

py,

pz are the wave numbers

along

X, Y,

Z

_directions,

_Pz

_{= - j y,}

a is the electron thermal

_velocity

indicative of the average

temperature

of the

plasma,

_{cv p}

is the

_angular

plasma frequency,

defined as

_{co p}

=

(Ne2/mëo)1/2, N

is the electron

density,

and m and e are the mass and

_charge

of the

electron,

respectively.

In view of

_{equation (2.1.3),}

we can

_write,

2.2

_EQUATIONS

IN UNBOUNDED PLASMA MEDIUM. - Maxwell

_{equations [6] describing}

the time harmonic fields in an unbounded warm

_plasma

medium

(in

the frame

_S’)

can be written

as,

where,

s’ is the dielectric tensor of the

_plasma

medium and is

_{given [1]}

_by,

where,

It is

_imperative

to note that the

_expression

for

_e;z

does not contain the

_velocity

term

_[1]

because the

_plasma

is attached to the S’-frame.

Now,

using equations (2.2.1 )

to

_(2.2.6),

the wave

_equations

and the field

_expressions

can be derived and are obtained as,

where,

kô

= cv’

/C

and the

subscript

T denotes the _transverse

_components

of the field vectors.

3. TM-modes

_(11;

_{= 0 ) dispersion}

relation.

The wave

_{equation (2.2.7),}

in this case, takes the

form,

The

_boundary

conditions to be

_applied

are as referred in reference

[1]

and are

_{given by,}

where,

X = 0 is the lower

boundary

surface,

X

= e

(Z’ )

is the upper

boundary

surface, b

and

L are the

_amplitude

and the

_wavelength

of the

_periodic

function,

respectively,

and d is the average distance between the two

_plates.

Thus,

in view of

_{equation (3.2),}

the solution of

_{equation (3.1 )}

is obtained as,

where,

h =

n -ff / e (Z),

with n =

0, 1,

2,

3,

... and

Eo

is constant.

1512

where,

(ù c is the cut-off

frequency

of the

_parallel

_{plate waveguide}

filled with

_free-space

and is

expressed

as,

Now,

transforming equation (3.4)

from S’ to S frame

_by

_using

Lorentz transformation

equations (2.1.1), (2.1.3)

and

_(2.1.4)

and

_normalizing

the variables as,

after a rather

_{lengthy algebra,}

we obtain the

_dispersion

relation as,

where,

Since,

we are

_dealing

with the lossless case, we _assume, G

_{= j B}

in

_{equation (3.7)}

and rewrite

it as,

It can

_easily

be seen that

_for

_{1 a}

₁

«

_1,

that

_is,

_{1 il a 1 z}

_{1 BI,}

equation (3.9)

reduces to the

equation (29)

of reference

_[1].

This confirms the exactness of the derivation of

_{equation (3.9)}

for the relativistic case.

Further,

using equations (2.1.3), (2.1.4), (2.1.5), (2.2.9), (2.2.10)

and

_(3.4)

the

_following

field

_components

of TM-modes in S frame are obtained _as,

where,

4. Numerical results and discussion.

In order to

_study

the relativistic effects on the

_phase

characteristics of the TM-modes of one

boundary corrugated waveguide

[b/d = 0.01

and

_{(Z-Vot)/L=O.5],}

the

_dispersion

Fig.

1. - Phase characteristics of the

_corrugated

waveguide

filled with uniaxial

_{plasma, il c}

= 2.0,

a = 0.4, 8 = 0.03,

b/d

= 0.01 and

(Z - Vo

t)/L

= 0.5.

Fig.

2. - Phase characteristics of the fast and the slow

waves

_showing

the effect of relativistic drift

velocity.

5 = 0.03,

ne

= 2.0,

b /d

= 0.01 and

1514

Fig.

3. - Phase characteristics of the fast and the slow waves

_showing

the effect of temperature.

a = 0.5,

ne

= 2.0,

b/d

= 0.01 and

(Z - Vo

t)/L

= 0.5.

Figure

1

_represents

the f2 - B characteristics for warm

_{(8 = 0.03 ),}

uniaxial

_{(Bo =}

ooâz)

and

_{drifting ( a}

=

0.4) guided

wave

_(f2c

=

2.0) propagation.

As before

[1],

_one

may

identify

in

_figure

_1,

HGI as the

_{« waveguide}

wave _», ABOC as the « fast wave », DE as the

« slow wave » and BG as the

_{non-propagating}

mode. The

portion

AB of the « fast wave » is termed as the « backward wave »

_{(the slope}

_{dn IdB}

_{being negative).}

On

_comparing

the non-relativistic results of reference

_[1]

with those of

_figures

1 to 3 for the relativistic case, we notice the

_following

remarkable distinction between the two results.

In

_figure

_1,

the

_{« waveguide}

wave » becomes wider and the vertex remains at almost the same

_point.

The « slow wave » becomes almost linear

_(straight).

The effects of increased drift

velocity

is shown in

_figure

2. The « slow wave »

_approaches

the

_origin

as a increases and at a = 0.5 it

originates

at the

_origin.

The « backward wave »

_disappears

at a = 0.5. For a

> 0.6,

both the « fast » and the « slow » waves

_{join together}

at the start and

_originate

at the

same

_{point (origin).}

At a = 0.8

(dashed curve),

the « slow _{wave »} becomes

exactly

linear

(straight line).

The effect of the drift

_velocity

on the OC

portion

of the « fast wave » is the same as before.

Figure

3 exhibits the effect of

temperature

(under

relativistic drift

_{velocity condition)}

on the

« fast » and the « slow » waves. It can be noticed in the

_figure

3 that

_practically

there is no effect of

_temperature.

For 8 =

0.0,

0.01 and

0.02,

the curves are

_exactly

the same. For 8 =

0.03,

there is a

_slight

deviation

but,

within statistical error, the curves

_(fast

and slow

waves)

occupy almost the same

_respective

_positions

as those

_{corresponding}

to the other values

References

[1]

PRASAD R. and PRASAD R., Nuovo Cimento D 9

₍₁₉₈₇₎

132.

[2]

NEY E. P.,

Electromagnetism

and

_{Relativity (Harper}

and Row, New

_York)

1962.

[3]

PAPAS C. H.,

Theory

of

_{electromagnetic}

wave

propagation (McGraw-Hill,

New

York)

1965.

[4]

MOLLER C., The

_theory

of

_{relativity (Oxford university}

press,

Oxford)

1952.

[5]

CHAWLA B. R. and UNZ H.,

Electromagnetic

waves in

_{moving magneto-plasma (Kansas}

_university

press,

Lawrence)

1971.

[6]

KRALL N. A. and TRIVELPIECE A. W.,

Principles

of

_{plasma physics (McGraw-Hill,}

New

_York)

1973.