NEW GRAZING INCIDENCE SPECTROMETER FOR HOT PLASMA DIAGNOSTICS

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NEW GRAZING INCIDENCE SPECTROMETER FOR HOT PLASMA DIAGNOSTICS

M. Pouey

To cite this version:

M. Pouey. NEW GRAZING INCIDENCE SPECTROMETER FOR HOT PLASMA DIAGNOSTICS.

Journal de Physique Colloques, 1983, 44 (C8), pp.C8-201-C8-212. �10.1051/jphyscol:1983814�. �jpa-

00223322�

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JOURNAL DE P H Y S I Q U E

Colloque C8, suppl6ment au

n O 1 l ,

Tome

44,

novernbre

1983

page C8-201

NEW G R A Z I N G I N C I D E N C E SPECTROMETER FOR HOT PLASMA D I A G N O S T I C S

M . Pouey

Laboratoire de Physique des Gaz e t des P~asmas*, UniversitG Paris X I , 91405 Orsay Cedex, France

RBsum6

:

Les diagnostics des plasmas chauds cr6Bs par impact laser ou en confinement inertiel requihrent , en particulier, I'emploi de dispositifs dispersifs de hautes performances travaillant dans l'ultraviolet extreme.

Or, en dessous de

100

nm, les dispositifs dispersifs font necessairement appel

Q

des rkseaux concaves. Dans toutes les configurations actuelles

(montages

Q

champ plan), la position de la fente d'entree et le lieu des images diffractkes sont d6duits des solutions des Bquations des focales tangentielles. L'emploi de r6seaux holographiques,

a

distribution de traits non uniforme

, enregistres s u r des supports sph6riques

, permet l'obtention, pour plusieurs longueurs d'ondes, d'images diffractees dont la qualit6 n'est limitee que par les termes d'aberrations sphdriques, si pour des points objet

A

( r , a ) , image

B

(r9,f3) et ceux dlenregistrement de l'holo- gramme C(pC,n) et

D ( p D , 6 ) ,

on satisfait

B

la condition

:

sin cr sin R

=

e e

^{p c}

sinq

= p D

sin6

=

a.

L'objet et Ifimage doivent donc &re situks s u r un cercle de rayon R/a orthogonal au cercle de Rowland.

Dans ce montage, avec un seul r6seau , en faisant varier

a e t , corr6lativement,

R on peut obtenir plusieurs images diffractees dont la qualitB ne sera altBr6e que par des termes aberrants du quatrikme ordre.

Abstract

:

In laser induced confinement fusion or in ion driven fusion the understanding of the physical process taking place in the plasma

* ~ s s o c i G au C.N.R.S.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1983814

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CB-202 JOURNAL DE PHYSIQUE

atmosphere surrounding the target is extremely important. Plasma diagnostics require then line ratios and line broadening at very short wavelengths. This paper deals with the design of a stigmatic grazing incidence device fitted with a spherical holographic grating.

I

-

INTRODUCTION

In laser-induced inertial confinement fusion the

understanding of the physical process taking place in the plasma

atmosphere surrounding the target is extremely important since it is

the locus where the energy is absorbed and transported to the core

region

^;

similar situation occurs in the case of ion driven fusions

devices C21 . Various optical probing could be used [31 including

the spectroscopial one at very short wavelengths. The electronic density

ne could be deduced from Stark broadening of H-like ions 141 and He-

like ions line ratios, which exploit the temperature dependence of the

electron excitation rates of various n

= 2

levels, together with the

density dependence of collisional mixing of metastable levels, give ne and

the electronic temperature Te

L5 -

61 . For a berylium targent the working

wavelengths are

7.59

nm in the first case and, for example,

6

and

10

nm

in the second one which means that grazing incidence concave grating

mounts are to be used. To reduce the large amount of astigmatism various

proposals have- been made

:

in spectrographic system working on the

Rowland Circle, following the first study of Rense and Violett [71

a toroidal mirror in front of the entrace slit is used to balance the

astigmatism at one wavelength 181 . Due to the development of the

holographic technique it is now possible to investigate the focussing

properties of toroidal grating having unequally spaced grooves over all

the pupil. Indeed, by a proper choice of the recording point sources we

can generate over the grating surface a phase variation that might

balance aberrations C9

-

161 .

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Following the concept first developped by Rowland, in all these devices the horizontal focus equation determines the image locus. Design rules have been recently 117-191 obtained without paying a priori any attention to this latter equation since if the device is to be stigmatic the horizontal and the vertical focussing conditions are identical.

In this paper, we restrict ourselved to the design of a spectroscopic device fitted with spherical holographic grating working at grazing incidence. For line profile measurements the device shoul' be at least astigmatism and comas free since only these aberrations are generating dyssimetrical broadening of the Line Spread Function.

I I - THE EIKONAL FUNCTION AND THE WAVE FUNCTION

The first step in image evaluation is to consider the

"Eikonal Function", i.e., the "Characteristic Functionll connected with the Optical Length of each ray. The derived aberration theory enables Us, through the Fermatts principle, to determine the aberrations in the image plane. Indeed a s the optical path length must be stationary, the geometrical optical aberrations are obtained by a simple derivation of this optical path, expressed in terms of the pupil coordinates. The best image plane is then determined by considerating the conditions in which the main first terms of the derived series vanish.

In the following we adopt classical notation

L171

and we consider spherical or toroidal holographic grating, the pole of which being centered at the origin of a Cartesian coordinate system. The X

axis is normal to the grating, the y axis lies in the principal plane, and the z axis is parallel to the ruling plane. For a point object A ( r , a , z ) and an image point B ( r l , B+A

B

,zt + A z l ) the light path function is given by :

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JOURNAL DE PHYSIQUE

F

=

<AP> + <PB>

+

mnh

, (1)

where P

( W

,l) is any point on the grating surface, n the number of grooves per mm, X the wavelength and m the order. After expansion in function of the pupil coordinates

W

and 1, F can be expressed by

[ l

71

:

i

j

,,

^,,W ⁺

^cl1wi

⁺

^{r.. cc..}

⁺

^m--"

^{D..) I}^W ^,

F = F 0 + C 1 + C

11 11

X

11 ( 2 )

where

C..

and

D..

(i

+ j

_> 2) are respectively, the aberration coefficients

11 11

of the equivalent type

I

grating and those associated with the two recording point sources. The Fermat's principle yields the following equations

:

where A(x,l)

=

Fo-F is the Wave Aberration Function.

As generally it is impossible to minimize simultaneously the dependence of

A B

and Azl on

W

and 1, the horizontal focus is usually chosen for spectroscopic work. The first order focussing conditions are then the grating equation, the magnification and the horizontal focussing condition

:

The vertical focussing condition is given by

:

where

:

cos

T)

K 2 = p c - ^l

t

^-

cos 6

K ; = ~ D - -

t

tR being the vertical radius of the toroidal blank. At normal incidence

by a proper choice of t and of the recording parameters, the amount of

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astigmatism could be balanced at two wavelengths. At grazing incidence this condition could be fulfilled over the spectral range of typical mono- chromators with straight grooves grating since cos

a +

cos

B

varies very little. It is also the case of flat field spectrograph the image plane being approximatively perpendicular to the central ray. At this step, it remains only one recording parameters to balance two comas terms. If, at normal incidence, it is possible to balance the

image broadening generated b y this latter C171

such a correction is not allowed at grazing incidence due to the adtual low values of

P.

Improvements could be then achieved by the use of a prefocussing toroidal mirror working at magni- fication

113

1161 .

1 1 1 - CONDITIONS FOR STIGMATISM

As first pointed out by Cordelle

et al. [ l 8 1

stigmatic imaging with

spherical holographic grating

having a non uniform grooves distribution is allowed at one wavelength if the recording point sources and the object and image are harmonic conjugate with respect to a circle of radius

R .

Absolute stigmatism is then given by

:

CAP> + <PB> + P [<CP> - <DP> 1

=

Cte (6) The wavelength at which the stigmatism could be obtained is strongly dependent of X. and the location of the stigmatic image at one wave- length is determined for s given position of the object. For example, at normal incidence the object and the recording point C must be located at the curvature centre of the blank, i.e., at the object tangential focus.

However,

if the device is stigmatic, i t is not necessary to r e s t r i c t the location of the object and the image to the positions deduced from the tangential focussing equation ; indeed the only s t r i c t l y required condition is t o satisfy to the Fermat's principle.

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JOURNAL

DE

PHYSIQUE

For such a device the wave aberration function A (w,l) = F -F 0 must be proportionnal to an unique function $I ( e , e t ,

cc, B,n, 6 ,

p c , pD)

,

the coefficients being function only of the pupil coordinates. As all the aberration coefficients are, in fact, only function of

T,

and of sin

a

-- and pc sinn in the objects space and

$-@

^and

^%

^sin6 ^{in the}

image space, by assuming that :

the wave aberration function takes the form :

T S 2 T t S i 2 2

+

L

- + PHK ^Wl

e pc + 7 - PHtKh p D 1

-

4~

Moreover, by taking into account of E q .

8 ,

can be expressed in function of

B

by ^:

F = i -

sin _e2 a

sttB

²

^{+ P}

^(P,, ^{sin 2}

⁶

^-^pc ^{sin 2}^II)

= - S - a [sin a + sin

B

- P (sin6 - sinq )

I

Since the coefficient of a in the Eq. 9 is simpiy the grating equation, the focussing condition will be unique for both the horizontal and the vertical plane. But the corresponding $I function does not allow the absolute stigmatism due to the fourth order remaining aberrations terms.

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I V

-

S T I G M A T I C COMA FREE DEVICES

Assuming that a is a constant, the object and the image are to be located on a circle of radius R/a

orthogonol to the Rowland Circle

(Fig. 1). The function Q , which caracterizes the focussing condition, is given for a spherical grating by

:

a cos2 a

-

sin a cos a

+

a cos

2

B

-

sin 6 cos B

sin a sin B

where now

C

is a coefficient only function of the recording parameters and then constant for a given grating. In such a case,

E q .

10 being satisfied, the wave aberration function

( E q . 8)

reduces to the fourth order terms

;

the device will be the free of defocussing, astigmatism and comas,

the Line Spread Function being then symmetrical.

- 1

For 6

= - q ,

i.e. for the maximum value of n

= 2

sin

6

X. the coefficient

C

is given by

:

C = -

a cotg 6

2

(11)

Particular solution of

E q . 1 0

occurs for 6

=

6,

a being deduced from

the relation

a cos'

a -

sin a cos

a

,

^Csin =

cos B

=

cos ,

sin

a ⁽¹²⁾

Table

I

summarizes the corresponding solutions obtained for

a = - 80°

and various a values for a device working in the m

= - 1

order.

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JOURNAL DE PHYSIQUE

Fig.

:

1 S t i g m a t i c coma f r e e imaging c i r c l e s .

For one g r a t i n g ( a fixed

;

o b j e c t d i s t a n c e r

=

Ra s i n a

;

image d i s t a n c e r '

=

Ra sinB) t o each angle of incidence a

( a v i r t u a l o b j e c t on t h i s schema) corresponds one

6

value

(one wavelength) f o r which t h e image q u a l i t y i s only

l i m i t e d by s p h e r i c a l a b e r r a t i o n s

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TABLE I a

= -

^80°

^{@ = 6}

In the more general case, if we consider the grating efficiency as the key parameter, for a given angle of incidence (i.e. a = -

SO0)

and a given angle diffraction (i.e.

B

= 75O), the C value, for each a value, is deduced from Eq. 10 and Eq.

11

and gives the

6

values. Typical examples are given on Table 11. In both cases we note that by increasing a , we decreases the value of the wavelength for which stigmatism occurs.

a

1 2 3 4 6

TABLE I t a

= -

^80° f3

=

75O 6

46O384 58O095 63O977 67O442 71 O284

Now assuming that C is fixed, i.e. for a given grating and then a given focussing circle, to each

a

values correspond, at least, an

B value (Table 111) for which the image quality is only limited by the following spherical aberrations-terms

ntrlpm 2,9672 3,4793 3,6830 3,7849 3,881 7

'

^nm

21,54 14,86 11.82

9,96 7.66 6,20 5.13

'nm 87, 91 39,05 23,40 16,20 9,71

ntrlpm 0,8767 1,2708 1,5973 1,8956 2,4643 3,0449 3,681 8 a

1 2 3 4 6 8 10

6 1 Z0352 18O064 22'938 27O549 36O961 47O983 63O941

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JOURNAL DE PHYSIQUE

T A B L E I I I a = l O 6

=

63O94

If A (w,l) is smalIer or of the same order a s the wavelength, the ratio of the corresponding light intensity at the maximum of the diffraction pattern obtained with a line object and a rectangular aperture, to that of the same instrument without phase errors can be derived from the Fresnel-Kirchhoff equation. The Strehl criterion [ 13

1

which gives the lowest tolerated value of the latter ratio, assuming that the theoretical

diffracted profile remains unchanged, enables us to determine the optimum tolerated value W of the width of the grating. If p

*

= L/W*,

this latter is given by :

the practical limiting resolving power being given by :

For larger aperture, i.e. for wavefront errors larger than the wavelength, we consider a quality factor [ l j ] that is the radius of gyration of the image pattern described by the geometrical irradiance distribution. By using the equations for the ray intercepts in the image, the quality factor is more easily evaluated by averaging over the exit pupil ; actually the

r . m .

S. value of the width of the diffraction pattern,

in

the dispersion plane, is given by

w 3

mn < A A > =

m,

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where W

>

W

*

and

the corresponding practical resolving power is then given by

X/ <

A

X >.

V -

CONCLUSIONS

Actual design rules for holographic concave grating devices are derived from solutions of the tangential focus equation. For grazing incidence, stigmatic coma free images could be obtained at various wavelengths for the same spherical holographic grating providing that the object and the image are located on a circle orthogonal to the classical Rowland circle.

This new focussing process has many practical applications not restricted to the above described field (CNRS-ANVAR patents pending). Amony them are new high resolution monochromators for line profile measurements which will be described later [ l 9 1

.

REFERENCES

[ l ] NUCKOLLS ( J . ) , WOOD ( L . ) , THIESSEN (A.), ZIMMERMAN ( G . ).

Nature, 1972, 239, 139.

C21 DEUTSCH (C.) - Bull. Soc. Fr. Phys., 1980, 38, 14.

DEUTSCH (C . ) , GOMBERT (M.M.), MINOO (H.) - Comments Plas.

Phys. and Cont. Fusion, 1978, 4, 1.

[ 3 1 BALDIS (H.A.)

-

Opt. Eng., 1982, 21, 751.

[ 4 ] HELD ( B . ) - These dlEtat, Orsay, Nov. 1981.

C51

GABRIEL (A.H.), JORDAN

( C . )

- Case Studies in Atomic Physics, N o r t h Holland, Edit., Amsterdam, 1972, p. 209.

[61 PRADHAN (A.K.), SHULL (J.M.) - Astrophys. J . , 1981, 249, 821.

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C8-212 JOURNAL

DE

PHYSIQUE

RENSE (W . A . ) , VIOLETT ( T . ) - J. O p t . Soc. Am. , 1959, 49, 139.

GARIFO (L. ), WALVEZZI (A.M.), TONDELLO ( G . ) - A p p l . O p t .

1979, 18, 1900.

FLAMAND ( J . )

,

PASSEREAU ( G . ), THEVENON (A. ) , LERNER (J.M. )- Proc. I n t . Conf. o n Vac. U l t r . Rad.

Phys.,1980,

111,

21.

C H R I S P (M.P.) -

1982

O p t . Soc. o f Am. Meeting. T o b e p u b l i s h e d in A p p . O p t i c s .

S y n c h r o t o n Radiation I n s t r u m e n t a t i o n , N u c l e a r I n s t r . a n d M e t h o d s ,

1980,

vol.

172

;

1982,

vol.

195.

R e f l e c t i n g O p t i c s f o r S y n c h r o t r o n R a d i a t i o n , S P I E ,

1981,

vol.

315.

POUEY

( M - )

- J. O p t . Soc. Am.,

1964, 64, 1616.

POUEY ( M - ) - J. P h y s . ,

1978,

39-C4,

188.

P o u E Y (M. )

,

HOWELLS (M.R. )

,

TAKACS (P. Z. ) - i b i d

11, 1982, 195,

223.

POUEY (M.), HOWELLS (M.R.), TAKACS ( P . Z . ) - i b i d

12, 1981, 315,

37.

POUEY (M.) - S u p . J. S p e c . Soc. J a p a n ,

1974,

p .

67.

CORDELLE ( J . ) , FLAMAND ( J . ) , PIEUCHARD (G. - O p t . I n s t r . a n d T e c h n . O r i e l , E d i t . , New C a s t l e ,

1970,

p . 177.

POUEY (M.) - T o b e p u b l i s h e d in J. o f O p t i c s .