DETERMINATION OF THE RAY PHASE FOR PLASMA CYLINDER USING MODULUS OF THE PROBING RADIATION TRANSMITTIVITY

HAL Id: jpa-00219417

https://hal.archives-ouvertes.fr/jpa-00219417

Submitted on 1 Jan 1979

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

DETERMINATION OF THE RAY PHASE FOR PLASMA CYLINDER USING MODULUS OF THE

PROBING RADIATION TRANSMITTIVITY

V. Zatsepin, A. Golubev

To cite this version:

V. Zatsepin, A. Golubev. DETERMINATION OF THE RAY PHASE FOR PLASMA CYLINDER USING MODULUS OF THE PROBING RADIATION TRANSMITTIVITY. Journal de Physique Colloques, 1979, 40 (C7), pp.C7-861-C7-862. �10.1051/jphyscol:19797416�. �jpa-00219417�

JOURNAL DE PHYSIQUE ColZoque C7, suppZ6ment au n07, Tome 40, JuiZZet 1979, Page C7- 861

DETERMINATION OF THE RAY PHASE FOR PLASMA CYLJNOER USING MOOULUS

(X THE PROBING RADIA-TION TRANSMITT'MTY

V.G. Zatsepin and A.V. Golubev.

I n s t i t u t e for Physical *okZems, Moscow U.S.S.R.

Abstract : Method f o r determination of t h e phase s h i f t along t;he ray t r a j e c t o r i e s i n side-on probing of plasma c y l i n d e r i s pre- sented. The unknown phase f u n c t i o n

+(#)

i s r e s t o r e d by c a l c u l a t i o n of t h e f i r s t d e r i v a t i v e

g'(y)

which i s proportional t o t n e probing beam d e f l e c t i o n angle and de- f i n e s t h e r e f r a c t i v e decrease of plasma t r a n s m i t t i v i t y . This method i s usable under strong absorption and r e f r a c t i o n

conditions, a s w e l l a s i n t h e case when phase of t h e probing beam i s s u b s t a n t i a l l y

d i s t o r t e d .

I n t r o d u c t i o n : Modulus

~ ( p )

and phaseY(p) of t h e probing be am, t r a n s m i t t e d perpendi- c u l a r l y t o t h e plasma axis, a r e connected with apparatus f u n c t i o n

x($)

of the dia- g n o s t i c i n s t a l l a t i o n and plasma phase fun- -

c t i o n

@($)

by convolu+ion equation I11 : i Y(y)-

7 ~ ( 3 - 9 e-'

^"3)

j2

2Pe- ---

⁽¹⁾

Experiments computer s i w l a t i o n s show t h a t f u n c t i o n

y(9)

i s strongly d i s t o r t e d due t o f i n i t e width of t h e probing beam when phase gradient

d*/dy

i s too l a r g e . The measured f u n c t i o n

y

becomes discon- tinuous and d i f f e r s from

9

a t t h e value multiple t o 2% if t h e phase increment

&

!k

a t t h e beam width exceeds

2

'3C

.

^The

r e s u l t i n g phase d i s t o r t i o n prevents t h e use of i t e r a t i v e methods f o r determination of t h e function

$ ((Y

[3]

.

Nevertheless, phase information may be r e s t o r e d by t h e experimental modulus of t r a n s m i t t i v i t y

$9)

which i s a f u n c t i o n a l of t h e unknown fun- c t i o n

$ (3) .

T r a n s m i t t i v i t y c o e f f i c i e n t in eqn. (I) i s defined primarily by t h e f i r s t d e r i v a t i v e

d@/dy=~6 ,

where

~=2'z/A

an8 @ i s t h e d e f l e c t i o n angle of t h e ray.

Determination of t h e Phase Function : Restoration of t h e ray phase was performed

by means of t b e i t e r a t i v e procedure. A t t h e n-th s t e p of i t e r a t i o n s we receive c e r t a i n approximate s o l u t i o n @a($) which provides t h e corresponding value of the modulus f n f ~ ) by eqn. (1). The f u n c t i o n a l

9 (9) = 9i@ q3),Q *7yi.,.]

depends p r i n c i p a l l y upon t h e f i r s t d e r i v a t i v e

Gr($) ,

^{so one}

can w r i t e t h e f i n i t e increrllent of

s(9)

a s follows :

-*.A@f+

^{. , a}

68 - a+*

(2)

It i s possible t o improve the function using eqn.(2) a t t h e next s t e p of i t e r a t i o n s . S u b s t i t u t i o n UP t h e exact - and approximate values of

9

and+' i n (2)

- 9 -

(3)

gives t h e next approximation

'Phis f u n c t i o n should be c l o s e r t o m o w n

d e r i v a t i v e

gf(9)

because eqn. (4) takes i n t o account t h e discrepancy between ex- perimental and approximate modulus

~ ( 9 )

and

Sn(p) .

^{I n} order t o f i n d t h e i n i t i a l approximation

$2

one can use the c a l i b r a t i n g f u n c t i o n :

;+'y

0 , ( 6 7 = 1 r 3 ~ x ~ 1 ~ - ts)

-0

which defines t h e dependence of transmit- t i v i t y modulus on t h e angle

@ = a / / ~

^{of the}

beam d e f l e c t i o n . Comparison of t h e expe- rimental and c a l i b r a t i n g f u n c t i o n s p e r m i t s t o obtain

$:P;(y)

d i c h i s usually r a t h e r close t o t h e ualmom function+?])

.

Influence of Absorption : I n the case oP strong damping the accuracy of t h e phase f u n c t i o n r e s t o r a t i o n may be s u b s t a n t i a l l y diminished. The influence of absorption i s defined by t h e complex ray phase @$)=

~ e P ( ~ ) t i l ~ $ ( ~ ) ) .

If the r a d i a t i o n l o s s i s vanishingly small then I m * = G and t h e damping f a c t o r

$(y)rerpE~m@y))J

i s equal

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19797416

t o u n i t y . Otherwise, t h e modulus i n (4) References :

be with re@;ard to absor

I?]

. V a i n s t e i n , 2 . A ,Yishchenko, Zh.Tech p t i o n . It i s a p p r o p r i a t e t o n o t i c e , t h a t F i z , 46, 11, 2271 (1976)

r a d i a t i o n damping produces t h e maximum d i s -

t o r t i o n of

3,(9)

^when

^QzO

( p r o b i n g a l o n g [23 E.A.Tishchenko, A.V.Go1ubev and V,B.La- t h e plasma d i a m e t e r ) , meanwhile t h e grea- z a r e v , paper at- t h i s conference

t e s t r e f r a c t i v e l o s s e s a r i s e a t t h e ex- [3] V.G.Zatsepin, E.A.Tishchenko, A.V.Golu- tremums of

@'($I .

T h i s p e c u l i a r i t y per- bev, p a p e r a t t h i s conference

m i t s t o f i n d an approximate v a l u e of t h e a b s o r p t i o n f a c t o r

f (3)

f o r f4-=0 :

S

tq(0) =i

~ ( Q I /pfi(~),

(6) ^{PQ.! I I}

where

9

i s t h e e x p e r i m e n t a l f u n c t i o n and

$Y i s t h e o r e t i c a l modulus of t r a n s m i t t i - v i t y without a b s o r p t i o n . I f one knows t h e side-on d i s t r i b u t i o n of t h e plasma emissi-

v i t y

Q(y)

a t t h e wavelength of p r o b i n g

, ^-

t h e n p r o f i l e of may be e a s i l y d e t e r - -

mined, because f u n c t i o n

Q (2)

i s p r o p o r t i - o n a l t o t h e product of i n t e g r a l a b s o r p t i o n

c o e f f i c i e n t a l o n g t h e r a y p a t h and plasma Fig.?

t e m p e r a t u r e , which i s assumed t o be c o n s t a n t

to

i n t h e r e g i o n of s t r o n g a b s o r p t i o n , Fun- c t i o n

f

t h e n becomes :

I \

^{4'2 a5}

P

- g 2 ( 0 )

Ja(i)={d- &li-m), ^-

⁽⁷⁾

R e s u l t s : The appl.ication of t h i s procedure

-47 -5 f f 5 @

Fig.2

i s i l l u s t r a t e d i n Figs.? and 2 showing t h e f * # ( ~ d

r e s u l ' t s of t h e r a y phase r e s t o r a t i o n f o r

Fig., Restored ( +, ) and measured (

Y.

⁾

plasma of A r a r c d i s c h a r g e

121 .

^Fig.1 _{phase of} A r a r c discharge and appa- p r e s e n t s t h e modulus and phase of t h e ap-

r a t u s f u n c t i o n of t h e i n t e r f e r o m e t e r p a r a t u s f u n c t i o n

X(#)

and measured phase

Y(,O)

of t h e t r a n s m i t t i v i t y c o e f f i c i e n t Fig.2 C a l c u l a t e d ( s o l i d l i n e ) and e x p e r i - w i t h a c h a r a c t e r i s t i c kinks accompanying mental ( c i r c l e s ) modulus of t r a n s - t h e l o s s of

2%

i n t h e r e g i o n s where @?J) m i t t i v i t y and a b s o r p t i o n f a c t o r

f .

h a s ertremums. G c t i o n %2&~) was o b t a i n e d a f t e r seven i t e r a t i o n s made i n accordance with Eqns. (4) and ( 7 ) . Experimental fun- c t i o n

g(f)

^{i s} shown i n Fig.2 ( c i r c l e s ) . It a g r e e s w e l l w i t h a t h e o r e t i c a l modulus of t r a n s m i t t i v i t y ( s o l i d l i n e ) , c a l c u l a t e d f o r r e s t o r e d f u n c t i o n

@

( t j ) and a b s o r p t i o n f a c t o r

f

(IJ) from Eqn. (7).

The accuracy of t h i s method i s determined by t h e e r r o r s of t h e measured f u n c t i o n $ @ ) and depends on t h e f u n c t i o n a l c o n n e c t i o n of

g (9)

^and

@ ( y ) .

Computer s i m u l a t i o n s have shown t h a t f o r c o n c r e t e f u n c t i o n s

~ ( y )

w i t h e r r o r s AY-J 5 O.O? t h e a c c u r a c y of t h e r e s t o r e d f u n c t i o n

(1)

i s b e t t e r t h a n 5%.