ON THE THEORY OF EXCITATION OF RIPPLED PLASMA RESONATORS BY A RELATIVISTIC ELECTRON BEAM

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Submitted on 1 Jan 1979

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ON THE THEORY OF EXCITATION OF RIPPLED PLASMA RESONATORS BY A RELATIVISTIC

ELECTRON BEAM

L. Bogdankevich, M. Kuzelev, A. Ruchadze

To cite this version:

L. Bogdankevich, M. Kuzelev, A. Ruchadze. ON THE THEORY OF EXCITATION OF RIPPLED PLASMA RESONATORS BY A RELATIVISTIC ELECTRON BEAM. Journal de Physique Collo- ques, 1979, 40 (C7), pp.C7-783-C7-784. �10.1051/jphyscol:19797378�. �jpa-00219375�

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JOURNAL DE PHYSIQUE CoZZoque C7, suppZ6ment au n07, Tome 40, JuiZZet 1979, page C7- 783

ON THE THEORY OF EXCITATION OF RIPPLED PLASMA RESONATORS BY A RELAl-IVISnC ELECTRON BEAM

L.S. Bogdankevich, M.V. Kuzelev and A.A. Ruchadze.

Lebedev PhysicaZ I n s t i t u t e , Moscow, U.S.S.R.

1. I n $his paper the problem of ex- c i t a t i o n of r i p p l e d plasma resonators by a r e l a t i v i s t i c e l e c t r o n beam is discussed.

I n t h e classical nonrelat i v i s t i c e l e c t ~ o - n i c s onlg numerical methods f o r solving t h i s problem may be u s e f u l l . But i n t h e case of r e l a t i v i s t i c e l e c t r o n beams i t may be eolved a n a l i t i c a l l y .

The system c o n s i a t s 09 a hollow re- l a t i v i s t i c e l e c t r o n beam and rippled plasma resonator (see fig.1). The length of thg resonator 1, i a more then its radius

Ke *

which is of the same order r i p - pled.

,!;7Etnd

more- then

'

I

t h e depth ^I I

p i plg

I:.::....:;;.

^J'::::)

.. .

^.<.

-

^{q - .}

O L , t E a t

. . . .. . ::. .- ., ^...:

^::':+:.,: ^;^:

^; ^',.

^'

I

is I

k

s>%"tr/~~ >>

A .

The thiokneas of beam is l e s s then t h e middle r a d i u i

5 .

The resonator i s f u l l f i l e d by cold c o l l i - s i o n l e s s plasma with density 4'

,

which is more then the beam electrododenaity 48.

A l l t h e eystem is put i n atrong longitu- d i n a l magnetic f i e l d go and t h e following

l o n g i t u d i n a l and t r a n v e r s a l com onents r e s p e c t i v e l y of beam v e l o s i t y

8

In t h e system under t h e condztione ( 1 ) i t can be excited by t h e e l e c t r o n beama onlg electromagnetic m v e e of E-tg- pea Therefore below r e i n v e e t i g a t e t h e symmetrical mqdera of E-waves, f or which t h e f i e l d equations can be w r i t t e n i n t h e form

/

2

3 E ^A ^A

T 1 Z 2 3+-

""€E,-0,

where

4

~ : ~ ~ 4 - ~ 4 - '

E = : ~ - ~ _

-

^0".

(0-4u,)*

assume, t h a t from t h e l e f t edge of t h e resonator (2s o ) e l e c t r omagnetio waves completely r e f l e c t , but from t h e r i g h t edge ( 21& ) they r e f l e c t ,on19 p a r $ i a l l y and r e f l e c t f o n c o e f f i c i e n t i ~ 2 , ~ .

2. IPhe problem folmulated above can simply be solved if (A.

/&%, 424z~4

.Be- low we give t h e s o l u t i o n of t h a problem

k

UP t o t h e order of L?!'?/&~, b Z 4 .For t h e f i e l d &omponent

L;; (9

^-wecan-write down

=r ^& ^x(;e ^pgxp(I'st ^triw.9

^'5'

where-. is a l o n g i t u d i n a l wave number,

and pfl'is Beasel-root,

^- 2 pos)z

⁰

.

The Broulion harmonics k: s a t i s f y some requerent r e l a t i o n , from %ich i t f ollows d i s p e r s i o n equation f o r e l e c t r o - magnetic waves i n a i p l d aveguide(up t o t h e order ofdy4<t%$

~ r g ^J

^,-

I n t h i s equation is w r i t e down non- c o r r e c t l y and onla f o r & = o

.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19797378

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When the e l e c t r o n beam is absent

( lcfg 4 0 ) then from t h i s equation we

reeeave t h e s p e c t r a of electromagnetic E-waves i n a r i p p l e d waveguide ⁽

A +-

^).

Theee apectra a r e shown i n fig.2 i n t h e region of p e r i o d i s i t y ^,-KO< Ua <tKo.From t h t h i a f i g u r e it f ollowa , t h a t i n t h e plasma waveguide t h e r e two types of E-waveer: h f with W and If with W

<

up . m e I f waves i n o r i g i n a r e @asma waves and when

plasma

is va-

. I 1" ¹

¹

^,/

niahed

(trii-9

_waves

'L I A

f o r e t h e e x c i t a t i o n of If waves is iden- t i c a l t t h i r e x c i t a t i o n i n smooth waveguide.. %,3f

Below we investiga t e o n 4 e x c i t a t i o n

It must be emphasized, then i n t h e r i p p l - ed waveguides can be excited by e l e c t r o n beam both not onla forward, but a l s o back- ward waves. The group v e l o s i t y of back- ward wavea i s a n t i p a r a l l e l t o t h e i r phase velosity.

3. Hon we can i n v e s t i g a t e t h e r i p p l - ed r e s o n a t o r s shown i n fig.1 and t h e i r e x c i t a t i o n by a r e l a t i v i j t i c e l e c t r o n beam. m e frequencies of foward and back- ward E-waves e x c i t e d by a n e l e c t r o n beam fn t h e resonator a s i n t h e waveguide a r e given by (9). But f o r t h e growt r a t e i n t h e case of reaortator r e have

81 Im

w

= Y m 6 4 f -Zmd'4--

-4

&

^'4%;^'

f

a&+ -

^axs- ⁽¹⁰⁾

Here &- &'% t h e %%enumbers of E-waves i n t h e r'fppled waveguides, ca r r i n g energy i n p o e i t i etand negative P - d i r e c t i o n and

Im 1%

a r e t h e i r a m p l i f i e r coeffi- c i e n t s respectiveI.3.

From t h e d i s p e r s i o n equation ( 7 ) i t can be ahown t h a t i n t h e caae of e x c i t a - t i o n of forward wavee i n t h e r i p p l e d reso- 7 -

n a t o r

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velocity of wavee i n t h e absence of beam Bow from t h e condition ( W g + 6 )

.

>0 we r e a i v e t h e threshold c u r r e n t of an

e l e c t r o n beam f o r t h e e x c i t a t i o n of r i ~ ~ -

. -

X (13)

pled wave generators t h e s e l e c t i o n of r a - d i a l modes can be reaczhed bg choosing t h e indectinn r a d i u s of beam 2,

,

which must c o b c i d e with maximum of the f u n c t i o n

L:

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f o r s u i t a b l e r a d i a l wave

nu&er

p.

Ref erences e

1. Kovaliov 3I.F. Sov.J6rn.Electronic The- chnic8,Shf e l e c t r o n i c s 2;102 (1978).

2. Bogdankevich L.S., Ruchadae A.A. Sov.

dourn. fTP 192 (1977).

. ^,

^uzelev^M.V., ^Rnchad-

3. Bogdankevic&.S K

ze A.A. Sov. Journ. Plasma Phyaica

2:

90 (1979).

ON THE THEORY OF EXCITATION OF RIPPLED PLASMA RESONATORS BY A RELATIVISTIC ELECTRON BEAM

HAL Id: jpa-00219375

https://hal.archives-ouvertes.fr/jpa-00219375