RECENT PROGRESS IN V.U.V. INSTRUMENTATION

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RECENT PROGRESS IN V.U.V.

INSTRUMENTATION

M. Pouey

To cite this version:

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RECENT PROGRESS IN V.U.V. INSTRUMENTATION

M. POUEY

Laboratoire de Physique des Plasmas UniversitC Paris-Sud, B2t. 212,91405 Orsay, France

Rtsumt. - Les differents concepts, utilises dans le calcul des dispositifs spectroscopiques pour

l'ultraviolet lointain, sont examines en detail ainsi que leurs performances actuelles. Les differents types d'interferometres susceptibles d'&tre utilises sont ensuite passks en revue. Enfin, les methodes holographiques sont envisagees pour etendre, a ce domaine spectral, la spectroscopie par transformee de Fourier et la restitution de surfaces d'onde.

Abstract. - In this survey paper we consider first the fundamental aspects of the various designing

processes and we point out the recent advances in V.U.V. spectroscopic devices. In the second part we discuss the design and operation of interferometers and, finally, we investigate the possibility of extensions of Fourier Transform Spectroscopy and Optical Reconstruction.

1. Introduction. - Few instruments in the history of mankind have contributed more to the extention of human capabilities in fundamental and applied research than diffraction grating devices. The develop- ment of atomic and solid state physics, the spectral determination of the composition of stellar and planetary atmosphere are just two among the major examples, which range from metrology to metallurgy and biology. Fundamental to .the current improvements in higher resolutions and efficiencies are interferometric servo-control and wavefront testing, holographic technique and electromagnetic theory of groove-determined grating efficiency. Moreover, both the theoretical and experimental results [l] that have emerged since the last Conference on V.U.V. Radiation Physics three years ago, were surveyed during the appropriate panel' chaired by W. R. Hunter [2].

In this survey paper we point out the recent advances in V.U.V. spectroscopic devices; but as detailed theoretical and experimental results of most of the focussing properties of gratings are described in the Proceedings [3], we shall restrict ourselves to consider only the fundamental aspects of the various designing processes. To conclude this first part the now available performances of various devices will be reported.

In the second part we discuss the design and operation of interferometers with division of amplitude and those with division of wavefront.

In the last part we investigate the possibility of recording, together with a suitable background, the amplitude and the phase of a scattered or diffracted electromagnetic wavefront in order to reproduce,

in the V.U.V. spectral range, the electromagnetic field distribution of the original wavefront. Extensions of Fourier Transform Spectroscopy and of Optical Reconstruction of three-dimensional macroscopic scenes as well as high resolution lensless X-ray microscopy are, of course, the goal of theese investi- gations.

2. Image evaluations used for the design of V.U.V. grating device.- 2.1 THE EIKONAL FUNCTION. -

The first historical as well as practical step in image evaluation is to consider the Eikonal Function i.e., the Characteristic Function connected with the Optical Length of each ray. The derived aberration theory, which is one of the most important theoretical backgrounds of geometrical optics, enables us, through the Fermat's principle, to determine the aberrations in the image plane. Indeed as 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 in then determined by considering the conditions in which the main first terms of the derived series vanish. Following this process the first and now classical devices, such as the Row- land Circle, Wadsworth Mount and Seya-Namioka monochromators were designed earlier [4, 51. The object and image plane are those corresponding to the paraxial rays, and the image quality is characte- rized by the amount of geometrical aberrations, the only remaining parameter to reduce these latters being the aperture.

Recently however great improvements on the classical devices have been obtained, namely in

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grazing incidence, by the use of aspherical holographic gratings with straight grooves, a balance of the astigmatism being obtained through the asphericity of the blank [6, 71. Other improvements of spectrographs are more correlated either with refinements of the mechanical design (modular concept, optical or automatic adjustements) [S, 9, 101 or with the use of the ultra high vacuum technique for synchrotron devices [l l].,

Spherical holographic gratings with unequally spaced grooves have been used to improve Seya- Namioka monochromators [l21 or to design similar devices [13]. Aberrations are then expressed by the typical relation

where Ao is the laser wavelength and

1

the working wavelength in the mth order. Cij characterizes the

aberrations of the mounting, Dij being the optical

correction arising from the hologram. Since in order to form grating lines holographically, four recording parameters are to be determined, four simultaneous equations are to be solved. The first one determines the nominal number of lines per mm and the second oqe is needed to assume the focalization in the horizontal plane. This then leaves only two degrees of freedom which can be used to reduce aberrations, generally the astigmatism and one coma term [14].

Howewer, as every aberration is a complex function of the recording parameters, minimization of one set of particular terms may cause the other terms, which have not been taken into consideration (namely those associated with the height of the entrance slit), to become enlarged in one case or at least to remain unchanged. Because of this ray-tracing evaluations have been extensively carried out (see for example refs [15, 16, 171). Although this type of design method has been found to be useful in practice, namely for grazing incidence devices fitted with toroidal gratings and prefocussing toroidal miror [7, 8, 101, it still has some drawback particularly for normal or quasi normal incidence mounts. The diver- gence between the computed and the observed diffracted patterns arise from the fact that spot diagrams are derived from a geometrical theory of the grating neglecting the intensity law of geometrical optics (See

5

2.3).

The geometrical rays being at each point its normal, the wave surface emitted by the source is spherical, and in general, the point at which the normal to the diffracted wave surface intersects with the image plane deviates from the Gaussian image (Fig. 1).

FIG. 1. - 1) Stigmatic optical system. 2) Wave aberration function A . 3) Role played by a defect of focus in the case of spherical aberration : H, marginal image plane, H, paraxial image plane.

H, best plane of focus.

Introducing the so called wave aberration function as the difference between the wave surface of the optical system and a reference asphere having its centre at the Gaussian image point and passing throught the centre of the exit pupil, Nibjoer was the first to connect the Geometrical Optical aberra- tions with the wave aberration function A . Through simple geometrical consideration the wave surface equation can be expressed in function of r' the radius of the reference sphere, of A and of the pupil coordi- nates (W

<

W = 2 a ; I < L = 2 b).

The equation of the normal and the deviations (y', 2') from the Gaussian point of the intersection

in the image plane at X' = r', are given by (Fig. 2) :

2.2 THE WAVE FUNCTION. - From a theoretical

point of view the only way to improve this situation _{y r}₌ a A

_.

_{z r =}_r'-.a A

is to give a physical meaning to the characteristic a w dl (1

'1

function, connecting the ray with a.physica1 quantity,

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contribution of each of them determines the local deviation of the diflracted wavefront.

Following the same concept, the wavefront of the diffracted waves could be constructed on the basis of the local elementary gratings, the recording parameters being, according to Fresnel's principle, determined by matching these waves to spherical waves.

FIG. 2.

-

Schematic diagram of the optical system. Point A(r, a, z)

is a self luminous point in the entrance slit, B(rr, 8, 2') a spectral image point. P(u, W , l) a point on the nth groove counted from the origin 0, and C(r,, v ) and D(r,, 6) are the point l~ght sources used for

recording interference fringes on the grating blank.

These two methods, which in fact incorporate ray tracing formula into the design process through the concept of the wave function, have been recently investigate and preliminar results have been reported at this meeting [18, 191. Considering many object points on the entrance slit, bundle of rays representing the same amount of light energy and diffracted by elementary gratings, Lepere et al. have

also obtained computed line profiles of diffracted images. Despite the fact that such methods neglect the diffraction or interference-edge effects, they incorporate the main design concept, which implicites that any tolerances placed on an optical system must be imposed on the complete wavefront over the exit pupil taken as a whole, and not on each aberration

term separately. In fact this idea can be fully turned into account only through the more precise physical- optics estimate of the light distribution.

Nevertheless stigmatic imaging at one wavelength may be obtained by using the Weierstrass's point of the grating in fact; assuming positive-working photoresist, the light path function F is, for the ray

APB (Fig. 2) mA F = ( A P )

+

( P B )

+

mnl = FG

+

z F H

,

(1") with n l o = [ ( C P ) - ( D P ) ] - [ ( C O ) - ( D O ) ] , (l " l )

F, being the optical path length associated with the

working conditions and

t;,

the holographic recording one. In the recording geometry the sources C and D can occupy several special positions : they may coincide with each other, or with the center of curvature of the grating or they may be harmonically conjugated [20, 21, 22, 231. Recently aplanetic devices at one wavelength (stigmatism and spatial resolution along the slit height) have been described for normal [24] and grazing incidence [25]. Generally the stigmatism balancing effect is very important but the wavelength range of maximum resolution remains rather narrow. AIthrough solutions have been found for spectrographs [25], use of such a type of process is not very well suitable for monochromators. For example a recording geometry that has two diverging sources situated symmetrically with regard to the 2, X plane has no astigmatism and no fourth-order aberration and can be used to correct one coma term only of Seya Namioka monochromators (astigmation of the device being uncor- rected) [17, 231.

A normal incidence monochromator using a mecha-

nically ruled stigmatic concave grating has been also designed by Harada [26] following the same prin- ciple. One of the advantages of the mechanically ruled gratings is that $hey offer the freedom to choose the value of l o . The device is found to reduce the

amount of astigmation to 1/50 and the spectral band width to 115 of those of the conventional Seya- Namioka monochromator. In fact this gain arises not only from the use of an ellipovidal groove distribution but also from the fact that the device works at quasi normal incidence (2 0 = 120). Moreover to avoid too great an amount of defect of focus a translation of the grating is needed as in the classical symmetrical monochromators [27, 281. In fact, if they are designed from the second order theory

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2.3) better performances may be obtained simply with a classical toroidal grating (uniform groove distribution).

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from the reference sphere is then directly connected noted previously. As a result of the introduction of

to the broadening of the difracted images. The Strehl the ray-density method to measure the image quality,

criterion we consider a quality factor that is the radius of

gyration of the image pattern obtained from this latter method, the ,geometrical irradiance distribution being used as a weighting factor. For this averaging

4 ab _{in the image plane, we get}

(yf2

+

z12) I ( y l , z') dy' dz'

.

- m

which gives the lowest tolerated value of the latter ratio, assuming that the theoretical diffracted profile remains unchanged, enables us [31] via the phase balancing method to determine the best plane of focus. In other words, using this physical process, an optimal reference sphere whose center is located at the image point of maximum irradiance can be determined (see Fig. 1) as well as tolerances for each aberration [31]. Moreover the whole contribution .of

the aberrations to the image broadening can be mini- mized [32].

For large aperture normal-incidence or for grazing incidence mounts where the wavefront error remains larger than the wavelength, the phase variation is large in the domain of integration, and the exponen- tial factor of the Kirchhoffs integrant changes its sign many times : accordingly the main contribution to the integral comes on& from the neighbourhood of critical points satisfying the conditions dA/dw = 0

and dA/i?E = 0. This fact explains the disagreements observed many times between computed and experimental profiles, and the necessity to introduce a weighting factor.

To introduce this factor for evaluating an image in the geometrical limit we use the method first described by Herzberger [33]. This method (( yields

a pattern of points formed in the image plane by a system of rays emanating from a single object point and wziformly distributed over the entrance pupil.

This uniform distribution is necessary i f each ray is to represent the same amount of light energy B. The approximation made in the Herzberger formalism can be justified [34] by starting from Maxwell equations solved in the geometrical case limit and by using the time averaging of the Poynting vector. This leads to an energy-conservation law associated with each ray. Or we can investigate Liouville's theorem, as was done by Arakengy [35]. Using the Geometrical

law of intensity or the brightness theorem, we obtain the irradiance distribution

which is exactly the result obtained by Miyamoto [36] and is in agreement with the introduction of critical points in the solution of the Kirchhoff integral, as

As in the case of small phase errors, ( 0') can be more-easily evaluated by averaging over the exit pupil. By using the equations for the ray intercepts in the image, we obtain

( m 2 ) =

' l a

4 ab

lb

(El2

₊

(!$l2

_{d w d l}

- a - b

A(w, 1 ) being a even function of E. It is obvious that this method of averaging by use of the exit-pupil radius as a weighting factor is simpler than that defined by Eq. (4). Moreover, as pointed out by De Velis [34], this average over the exit pupil takes into account simultaneously the Herzberger criterion and the fact that there are more rays at the outer edges of the exit pupil than near the center. Here again a correlation between the aberration function

and the r.m.s. value of the dzffractedpattern is obtained and tolerances can be derived [31, 321.

2.4 THE RESPONSE FUNCTION.

-

It is also possible

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p

being the angle of diffraction (and a the angle of incidence). Applied to the case of simple defect of focus we found a condition which is equivalent [3 l] to the tolerance derived from the Strehl criterion and which can be applied only to very low aperture device (less than f/100 or f/300 depending of the incidence).

Now in the case where the extension of the spot diagramm is about 30 times or more larger than width of the sinc function and if finer details than the length of about 2.5 times the stigmatic diffracted pattern (no phase errors) are smoothed out, the geometric and wave optical intensity distribution are almost equal to each other. Indeed in this case the effect of a considerable amount of aberrations predominates that of diffraction.

Then in the intermediate case, which is the most frequent in the V.U.V. spectral range, and more generally if detailed structure of the diffracted patterns are required, image performance must be evaluated by computing directly the intensity distribution from the Fresnel-Kirchhoff equation. The surface of the grating is then divided into many strips of width Aw. In each strip, the phase

is replaced by an equation of straight line tangent to q(w) and the integration is performed by a tra- pezoidal approximation [40]. Finally we note that fairly good agreement has been obtained by Katayama et al. [37] between the response function calculated as above and calculated by the Fourier transform of such computed intensity distribution of a point image.

3. Grating devices : recent improvements. - 3 . 1 OPTICAL DESIGN AND CHARACTERISTIC FUNCTIONS. - According to the diffraction theory

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2.3), the first order conditions are ignored and phase-balancing of the aberrations together with image quality critaria are affected. The purpose is to locate the point image B in the image space such that the deformation of the wavefront aberration from the corresponding reference sphere is a minimum (Fig. 1). Shifts, off the Gaussian image plane, as a function of the amount of the aberrations, are then introduced in order to minimize the aberrations over all the exit pupil. In evaluating tolerances for the residual aberrations, the optimum focus is assumed to be determined by maximum irradiance in the image plane at the best plane of focus. Despite the fact that such principles are valid for any type of devices [41, 42, 431 including the spectrographs, we shall restrict ourself to consider in the following only the case of monochromators involving a simple rotation of the grating. They are indeed of the most common and practical use.

Application of Fermat's principle yields then : (for mathematical details see Refs. [32] and [44]).

- The grating equation

lo(sin a

+

sin p) = ml(sin B - sin r]) , ₍₇₎ with the nominal number of lines per mm defined by nAo = sin d - sin > 0 , (8) the dispersion relation di = AllAx being given in the image spacL by

cos

p

d j =

-

r' mn '

- the horizontal focusing condition cos2 a

₊

cos2

p

- cos a - cos

e'

m l

+

-[p,cos2 y

-

p,cos2 6 - COSY

+

COS^] =

l 0

where R is the radius of curvature in the horizontal plane and

the right hand side of Eq. (10) characterizing the defect of focus, balancing the four order terms (co&ficients of

X

and v are those determined in the geometrical optical case) which is generally omitted in classical theories.

For a given grating (R, n, W and L being fixed) and a given 8 value, the object r and image r' distances are first derived from Eq. (10) which must he satisfied at least for two wavelengths. Using the preceding image-evaluation criteria and an assumption that an aberration-free image is formed at the distance rf(l) and observed at the fixed r' slit distance we obtain the tolerated value of the instrumental defocusing 1321. The third fundamental condition is related to astigmatism correction. The height of astigmatism Za(l) is for a monochromator given by :

z44

=

L

$- 1) (Y,, Yh) (1 1) with

rl/(Yt, Y,) = cos ~ ( ~ 0 s Y - cos Yt)

+

sin y(sin y

-

sin y,)

.

(12) For each particular

r

value of the magnification and each 8 value, the right hand side of Eq. (ll), where

2 e' cos 8

.

2 pe' cos 8

COS y, =

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must be minimized over all the interval 0-y, which represents the angular rotation of the toroidal grating (vertical radius t R ) employed to scan the spectral range of interest. In Eq. (l l), p = - 1 for mountings working in negative orders and p = 1 in the opposite case, dz0 characterizing the amount of astigmatism generated by the non uniform grooves distribution. y, is then the angle at which astigmatic correction is effected as a function of the toroidal characteristic of the grating and y, the angle at which astigmatic correction is effected as a function of the recording parameters employed in the holographic formation of the grating grooves. To illustrate this, figure 3

n, Q, e and e'. In figure 4, three solutions are graphed for the function $(y, ; y,), the last one corresponding to the optimized condition (y, = 140 2'; y, = 20° 6') for y, = 400. From the optimized condition according to figure 4, it is sufficient for practical purposes [44] to relate y,, y, and y, as given by :

nlo

d,, = -

-

sin (ym/2) cos-' ( ~ ~ 1 2 . 8 5 )

.

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And to further illustrate practical examples for the following conditions : R = 1.5 m, n = 3 600 lins/mm, 6 = 330, r' = 1 286.8 mm, r = 1 236.4 mm, reference is made to the graph shown in figure 5 representing, the height of astigmatism in function of l for various gratings, the last one 4 representing a toroidal grating having a non-uniform groove distribution.

FIG. 5. - Height of astigmatism in function of 2 : 1) refers to a

FIG. 3. - Optimum values of y, and y, in function of 7,". spherical grat~ng with straight grooves, 2) to a spherical grating w ~ t h a non-uniform groove distribution, 3) to a toroidal grating

with straight grooves and 4) to a toroidal grating with a non- uniform grooves distribution according to Fig. 4.

Eqs. (7), (10), and (14) have only one degree of freedom for the location of the two recording point sources C and D. This degree of freedom can be used to minimize the image broadening generated by the coma terms. Indeed, from image-evaluation criterion we know that each aberration term does not have

the same importance, because from Eq. (5) :

where

m l m l

i = c ~ 3 + z D 0 3 ; < = C Z l z D Z 1 , FIG. 4. - Normalized astigmatism function ij(y,, yh) in function

graphs the variations of y, and y, which minimize

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be achieved by the use of the holographic technique cannot be established without considering one particular mounting.

The performances of the devices will be characte- rized by their resolution and their luminosity which

. -

are two related quantities. From the Rayleigh criterion (unlimited incident wavefront) the theoretical resolving power is given by :

(sin a

+

sin p) R , = m n W = W

1 9 (16)

Q ' = W cos ,!3/rf being the image aperture. Now for a point object and diffraction limited pupil having a width W smaller or at least equal to WO the optimum value deduced from the Strehl criterion [42, 431, the limiting resolving power is given by :

RI,, = 0.8 mn WO = 0.8 R,

.

(1 7) For larger values of W we have from Eq. (5) :

Finally the practical resolution d1, which takes into account the width of the entrance slit and the broadening due to image curvature (function of the height h' of the slit) determines the width

S' = d1,ldi of the exit slit [32]. The luminosity resolution product is then given by

h' L, R, = KWL mnA ,

r (1 9)

where R, is the practical resolving power and K is a factor characterizing the energetic losses arising from astigmatism for a given h' value. Eq. (19) implies that the practical resolution d1,, is given by the half width of the triangular profile, which corresponds to the convolution product of the rectangular

image profile of the entrance slit and the rectangular profile of the exit slit. To illustrate, figure 6 graphs the R, values in function of the dispersion for two values (S' = 10 mp and S' = 50 mp) of the exit slit width ( W = L = 54 mm).

3.2 PROPERTIES OF THE ASYMMETRICAL MOUNTINGS.

- These monochromators involving a simple rotation of the grating can be fitted either with spherical or toroidal grating having or not a non-uniform grooves distribution (l). Computations made for

various available gratings lead to the following conclusions : there exists one particular 8, value of 8 close to 350 for which

r

= 1 (see also

5

3.4) but this 8, value changes with the grating area; whatever R, n, W , L may be, we obtain

r

> 1 for

8 < 8, with r < R cos 8 and

r

1 for 8 > 8, with r'

<

R cos 8 if m = - 1 (r being equal to r' and vice versa for m = f l)..The limiting resolution and then the practical resoldtion decreases with 1 for low 8 values, but increases with 1 generally for

8 values larger than 200. In thelstrehl case the limiting resolving power varies between l l 520 and 14 400 for n = 1 200 lines/mm if R = 50 cm (L = 25 mm) and between 16 320 and 26 880 if R = 100 cm (L = 40 mm) if 100

<

8 < 400, the maximum value being close to 350. In fact the 8 range is alsq limited by the spectral range to be corrected ; for e x a ~ p l e the corresponding resolution is obtained only fofi 29 d 8 G 40° in the

wavelength range 30-300 nm. Now in the geometrical- optical case, for a given wavelength, the Rli, value is maximum for 200 8 G 300 whatever R, n, W

and L may be (curves l , 2 , 3 on figure 6 gives the R,,,,, value at 100 nm for 10° G 8 G 850). The optimum value of q (Eq. 18) for various gratings is given in the following table I for W = L = 54 mm

If we consider the case R = 50 cm, n = 1 200 l/mm we see that a increase of L, R, qroduct by a factor 7.8 introduces a decrease of the limiting resolution by a factor 10 (the net gain in flux being about 78).

In other respects figure 6 shows that for a 50 cm radius of curvature grating the average resolution is about 50 mp at 100 nm, whatever the number of lines per mm may be. As for a lm radius of curvature grating, the resolution is about 30 mp the performances of the most commonly used devices being summarized by the following relations

di being given in &mm.

FIG. 6. -Practical resolving power R, in function of the dispersion di (A (mm) for S' = 10 p and S' = 50 p). R,,,,, values at 100 nm for various simple rotating grating monochromators (loo < 0 < 85")

in function of the dispersion. The last curve (-.-) characterizes a 90° device fitted with a toroidal grating havlng 600 ]/mm.

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As the luminosity resolution product [32, 451 reaches a maximum a low 6 values (100 < 6 < 200), and taking into account the preceding results, the most performant normal incidence monochromators are those for which n = 1 200 l/mm and I3 200. Then, since the last meeting, th'e most important results are more correlated with an increase in luminosity [l41 arising from the stigmatism correction than with an increase in resolution. For example the use of a toroidal grating in the earlier designed ASMSO type monochromators [32, 451 (R = 50 cm,

n = 1 200 l/mm, 0 = 140) working in all'the spectral range at the maximum value of the luminosity resolution product with jixed entrance and exit slit

widths, leads to an increase in luminosity by a factor 3 for a h' = 10 mm exit slit height (gain of a factor 9 with respect to a classical Seya Namioka monochromator), this factor increasing, of course, rapidly with lower h' value [46] (factor 10 for h' = l mm and 70 for h' = 0.1 mm). Practical solutions have also been found [44, 471 for asymmetrical mountings with multiple entrance [48] or exit beams [32, 451. In grazing incidence however, due to the high value of the astigmatism, and of the strong correlation between the cross'ed coma term %nd the resolution, gains in resolution have been also achieved. With very careful mechanical design and optical balance of the spherical aberration [6, 71, resolution of 5 mp over a 200 mp X 200 mp field has been obtained for the Rowland circle spectrograph (Q'

--

f/300, q

"

3).

Stigmatic properties of aspherical holographic gratings have been also tested for spectrographs [6, 71 and monochromators. In the last case the flux, available at the exit slit, is measured in function of the exit slit height h'. Comparisons with theoretical

values [49] have been made (in collaboration with H. Damany) for a stigmatic monochromator working at 2 Q = 40° and fitted with a 40 cm radius of curvature 1 200 l/mm holographic grating having a non-uniform groove distribution [32]. A good agreement between theoretical and experimental values has been obtained for various wavelengths (Fig. 7) when the r' and h' values used in the calculations are those of the diaphragms (and not of the slit which is of course not exactly in the same plane).

FIG. 7. - Normalized output for various exit slit heights h' (entrance slit height h = 2 mm). Experimental results are in agreement with the computed one if the considered r and r' values are those of the

diaphragrams. instead of that of the slits (broken curve).

3.3 MONOCHROMATORS FOR LINE PROFILE MEA- SUREMENTS. - One way to improve the actual situation is to find new configurations which essentially are coma free, since this aberration generates the greater contribution to the line broadening, which is, in addition, dyssimetrical. Typical examples [50] of this effect are shown on figure 8 with some profiles obtained at 2 378

A

(low pressure Hg lamps) and at 1 609

A

(NI emitted by a Damany type source) with an ASMSO type monochromator, fitted succes- sively with a spherical and a toroidal holographic grating (n = 1 200 l/mm). The contribution of the toroidal blank to the dyssimetrical broadening, through the crossed coma term in Czl L2 W is clearly shown, as well as the usefulness of a diamond-shaped pupil which suppress wholly or partly the rays coming from each corner of the grating.

To design stigmatic coma free devices the following process has been used [44, 471. Basically the second order focusing condition as given by Eq. (10) is always used, but the object and image distances are no more determined, as in the last applications, in function of two wavelengths for which the foresaid equation -must be solved. The variations of the aberrations in function of

T

and I3 are first considered in order to select particular configuration having specific optical properties [44, 471.

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FIG. 8. - Experimental profiles obtained with an asymmetrical dev~ce fitted w ~ t h a spherical grating (1-2). a toroidal grating (1-3)

and a toroidal grating with a diamond shaped pupil (1-4).

coma term is strictly proportionnal to the wavelength and then if the recording parameters satisfy to

l o n l o

DZ1 = - - Czl =

-

mL 2 R 2 er2 ' (21)

the monochromator will be free of the coma term in L 2 W at any wavelength. Practically it is sufficient, to assume that

n Ao D,, = q*

----

2 R 2 er''

if the resulting aberration

where Q, = WIR, remains lower than the tolerated

value deduced from the optical criteria.

In the same circumstances, the second coma term takes the form :

As for

r

= 1, Eq. (10) is satisfied only for rotations of grating less than 5O, for large aperture devices covering a 300-2 200

A

spectral range we have :

n < 600 lineslmm ;

e = e l =

(42

COS y O ) - l N 0.708 077

.

₍₂₄₎ Now if t = 112 and D,, = 2 D,,, solution of Eq. (2 1) occurs for

n l o

a = - v

, sin d =

-

2 '

the height of astigmatism being given by

Za(A) = 2 L ( l - cos y cos y; l )

.

₍₂₆₎

Let RC be the limiting resolving power, associated only with the coma broadening (assuming that the other aberrations have a negligible contribution), we have from Eq. ( 5 )

where T ( p ) = 18.48 if p = 1 and T ( p ) = 24.64 if p = 0.644. This last equation is very useful in practice thince it is possible to predict the performances of a mounting having a given aperture Q, whatever the value of R, n may be since the others aberrations are also proportionnal to Q:. For example Fig. 9 shows the R,, Rli,, and R, (S' = 10 m p ; h' = 10 mm)

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FIG. 9. - R,, values in function of 1 for two YOO devices. The R, theorc~~cal value is also shown as well as the corresponding limiting resolving power Rllm = R,, which is the same for the two gratings.

the size of the instrument by a tactor 4 the practical resolution is degraded only by a factor 1.3 to 1.5. And experimental proofs obtained with a prototype (Fig. 10) (') of 75 cm radius of curvature toroidal grating, were presented at the coi~lerence LJUJ (Fig. 11).

Use of specific grating with non uniform groove distribution will enhance these performances (for

R = 50 cm and C2 = 0.1, q is about 16.2 instead of the value 32.4 given in table I).

Applications of such devices are evident in V.U.V. plasma diagnostics [51] for doppler and stark broadening measurements of line widths as a function of time which lead to ion temperature and ion number density determination. Also, as for lower resolution the working spectral range can be extend to 5 000

W,

determination of electronic temperature and relative species number densities can be performed using line ratio techniques. Finally due to the low diver- gence of the synchrotron beams high resolution stigmatic monochromator can be designed. Moreover by combining a spatial beam splitter (composed of a plane reflecting mesh and a plane reflecting miror [52]) with two toroidal gratings working at 900, two beams of equal bandwiths can be obtained in order to build either high luminosity spectro- photometers or der'ivative spectrometers [44].

Finally using the same procedure as discussed above, stigmatic grazing incidence monochromators using a 'toroidal mirror and a cylindrical grating have been also designed [47]. The source is first focused by the mirror behind the grating which works like a plane mirror in the horizontal plane

(r

= - l), the cylindrical blank of the grating assuming the focus of the diffracted beams in the vertical plane. 4. V.U.V. interferometers. - Since the wavefronts can now be in a great part monitored extensions of ihe classical interferometric methods can be investigated at present. Then in the following, we describe some devices which have been or could be used in the V.U.V. spectral range.

Among the two general methods of obtaining interference, the first one, called division of amplitude, uses beam-splitters; the beam is then divided into one or more partially reflecting surfaces at each of which the light is partly reflected and partly trans- mitted. As extended sources can be used high luminosity devices may be obtained, althought conventional beam-splitters introduce a severe limitation in wavelength. The second method, which is called division of wavefront, is useful only with sufficiently small sources, the beam being divided by passage through apertures placed side by side. Among them are the three apertures or the grating interferometers which may be considered as an extension of the more familiar young double slit interferometers.

Despite the fact that detailed analysis of the holographic interferometry is outside, the scope of this paper, some comments on the basic schemes and applications of the techniques in the V.U.V. field will be also reviewed.

FIG. 10. - A.S.M. 90° monochromator. 4.1 INTERFEROMETERS WITH DIVISION OF AMPLI- TUDE. - TO our knowledge only few attempts have

(2) This instrument has been worked out with the assistance of been made to inter'eromerers

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FIG. I l . - Experimental profiles obtained with an 900 device fitted with a toroidal grating and for an entrance slit height of 1 mm (4-1) of 10 mm (4-2).

interferograms for U.V. wavelength measurements was reported in 1971 by G. M. C. Freeman [53]. The instrument comprises a concave grating, which disperses the radiation from the entrance aperture and collimates it, the Michelson itself, and a concave focusing mirror. With high quality reflecting optics, the component limiting the wavelength range is the beam splitter. A fused silica substrate is used for

il

down to 170 nm and lithium fluoride or magnesium fluoride can be used down to 110 nm, the semi- reflecting surface film being metallic, usually alu- minium. To avoid reflection at the second surface of the beam splitter or of the compensator, these components are to be set at Brewster angle. An analysis of the noise on observed interferograms at

il

= 546 nm, ;l = 253 nm and 2 = 185 nm has shown that wavelength measurements of the U.V. lines in terms of the visible line can be performed with a precision of a few parts in 1 9 ~ provided that the line width is small enough.

Regarding the multiple beams interferometers only SiO, and MgF, interferometers have been sic&&-

fully tested [54]. With silica spectrosil B substrates (flat to 21200 in the green) and A1

+

MgF, layer the hyperfine structure of the transition 2S,12-2P,l, of Hg I1 at 1.94.2 nm has been recorded (reflective finesse 27, peak transmission 26

%,

theoretical resolvance

-

500 000). Use of pure MgF, substrates (flat to 2/80 in the green) allows an extension of the spectral range down to 150 nm (reflective finesse 3, peak transmission 4

X).

The main limiting factor remains probably the roughness defects and the lack of accurate refractive index measurements.

Fi-om the point of view of interference spectroscopy, use of .simple two-beam interferometers, having a useless profile with' a half-width equal to one half of the inter-order spacing, or use of Fabry Perot interferometers give a result approximating the desired spectrum. Like in the S.I.S.A:M. [55] the two mirors of Michelson can be replaced by two gratings, rotating so that, as seen from the source, their images rotate in opposite direction. The signal received - ~ is then a

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result in convolved with a sin x / x profile and it is desirable to reduce the secondary maxima by suitable diaphragms placed at the gratings.

In Fourier spectroscopy following initial work by Jacquinot [56] and Fellgelt [57l a record, called the interferogram, is taken of the variation of the flux through a two-beam interferometer as the time- difference is changed. The simple theory shows that, ideally, this is the Fourier cosine transform of the spectral distribution of the light and this distribution is found by the same transform applied to the measured interferogram. Among the principal advantages are a simultaneous recording of all spectral elements (recording-time independent of the spectral width) and luminosity. However the 'method requiries very accurate (ruling-engine quality) moving mirror motion (or scanning) and both digital compu- tation and analogue methods of Fourier analysis. Then it is not so evident that' such a method could achieve high resolution (better than 0.1 cm-') in the vacuum ultraviolet [58] even if they are used in conjunc- tion with synchrotron which are quasi noise-free powerful sources. However, the response of such interferometers to a light pulse of very short duration may not be given by the conventional formula, which generally corresponds to a steady-state situation. Indeed as fringes are produced by interference of a large number of wavefronts of regularly increasing phase difference, this latter implies a temporal delay between the wavefronts [59].

4.2 INTERFEROMETERS WITH DIVISION OF WAVE- FRONT. - In interferometers with division of wavefront, the two coherent sources may be generated by a simple mirror used at grazing incidence, the first one being the primary source and the second one its initial image in the mirror. In the so called Lloyd's mirror arrangement, the fringe which lies in the plane of the mirror surface is an intensity minimum. Experi- mental proofs were first obtained in the X rays at 8.33

A

by Kellstrom [60] and a higher-order Lloyd interferometer, using a transmission grating acting as a beam splitter, has been used to test, in the violet, optical surface at grazing incidence, the most suitable device for the V.U.V. spectral range having been described by Day [61]. The device makes use of the reflexions from the surface of two plane parallel flat mirrors placed at the exit slit of a monochromator. The fringes, obtained by the superposition of the direct beam and of the various reflected olie (twice or more), are sharper than those of a Frabry Perot. And intensity distribution is reminiscent of Fresnel's zone plate, in that only one wavelength is focused but all wavelengths produce some intensity a t the resonant position.

A schematic diagram corresponding to Young's experiment could be, in the V.U.V. spectral range, the following : the source is the exit slit of a stigmatic monochromator having a R, resolution, the

corresponding coherence length being AR,/n. The light is then focused on the image plane (image distance d) by a stigmatic toroidal mirror, in front of which is placed an aperture mask with two parallel slits S, and S,. If o is the separation of the slits, the separation i of adjacent bright fringes will be given by dA/o (i = 20 mp for 1 = 100 nm, d = 2 m and o = 10 mm). ~ e s ~ i t e the fact that such conditions are not so attractive, this type of device (Ray- leigh interferometer) is very interesting for refractive index n, measurements. And to avoid the technical and physical problems associated with media of differing dispersion or with the lack of transparent materials, only the refractive index of flowing gases (like in molecular beams) in front of S, may be investigated. For a difference of optical path AL = (n, - 1) I, (l, being the width of the jet), the displacement of the central fringe will be Am times the separation of an adjacent bright fringes where

From measurement of Am, ib and A, the n - l values can be determined providing that AL remains less than the length of coherence. This is the case even with a 50 cm radius of curvature grating having 600 l/mm (R,

-

1 000) since for l, = 50 mm the fringe visibility remains unaffected for n - 1 values less than 1.110-3 at A = 180 nm (indeed for Ar n - 1 is about 3.3 X 10-7). In fact the sensitivity of the device must be increased by using an aperture mask with three colinear slits. By appropriate use of this interferometer [62], the optical path difference between the optical path of interest passing through the center slit and the reference wavefront passing through each outer slit, is manifested as a change in the relative intensity of the primary secondary fringes rather than a shift of the fringes. The sensitivity (111 000 ?)

of such a three-aperture interferometer could be about twenty-five times that of the classical Ray- leigh one.

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as coherent light sources, one may use a source larger than what is permissible in a conventional slit-interference set-up. And it is now possible to take advantage of the recent advances in the design of free-standing transmission gratings [64] to develop diffraction grating interferometers. For example use of a transparent grating as a coherent beam splitter located on the common path of a two 900 monochromator as described before

(g

3.3) generates two coherent and quasi monochromatic sources ;

the fringes are then located in the plane of the image, of the transparent grating given by a toroidal mirror for example. As the transparent grating is used in divergent light the corresponding phase errors must be balanced by a proper choice of the groove distribution following classical holographic techniques (the holographic interference pattern must be obtained with non parallel beams). Indeed the method of fabri- cating free-standing transmission gratings [65] involves interferographic manufacturing of dispersion gratings. At grazing incidence the preceeding trnasmission grating could be associated with a Wolter-Schwarzs- child type I1 telescope [66]. Nevertheless it is clear that such an interferometer would be perhaps too sophisticated for new attempts in the present situation. (They must be, however, more versatile than the Laue or Bragg X-ray interferometer [67]). Essentially the main difficulties arise from the fact that information encoding and decoding are simultaneously realized. A two step process, as described in the following paragraph, would be a more realistic approach.

4 . 3 HOLOGRAPHIC INTERFEROMETRY. - In the wavefront-reconstruction imaging method, first described by Gabor in 1948, the amplitude and the phase of a scattered electromagnetic wavefront is recorded (usually photographically) together with a suitable coherent background in such a way that it is possible to reproduce the electromagnetic field distribution of the original wavefront [29]. In other spectral range lasers have generated a great upsurge of expe- riments and applications ; some of them could be easily extended by using V.U.V. lasers (3). For example let us consider only Fourier-transform spectroscopy, assuming that the two diffracted beams obtained from the transparent grating are now simply reflected by plane mirrors in such a way that the two wavefronts form a small angle

Z

with each other. The interference-fringe system photographi- cally recorded in a plane parallel to the bisector of the wavefront is proportional to a non-coherent superposition of the monochromatic fringe systems corresponding to each wavelength. The equation of the fringe system in the plane of the photographic plate is given by

where I(A) is the spectral intensity distribution of the source. The fringes in the recorded grating are modulated in position and in intensity, according to the distribution of the electromagnetic field in the diffracted or scattered wave near the photographic plate. And from Eq. (29) it is clear that Z(x) is the spectral Fourier-transform hologram of Z(i). Conse- quently, when the modulated interference-fringe grating is illuminated by a plane wave, it will reproduce, in the focal plane of the lens, in two distinct sets of diffracted waves, precisely the recorded phase and the amplitude modulations. The spectrum (one pair of spectral lines for each grating recorded) is then symmetrically displayed on the two sides of the optical axis. G. W. Stroke et al. [68] who were the first to suggest such a method, have use compensate Michelson-Twyman-Green interferometer with tilted mirror (wedge fringes). But in this case, as in the J. Ch. Vienot device [68] the luminosity is limited by the small value of the tolerated angular diameter of the sources. With grating orders as coherent light sources the fringe visibility remains equal to l whatever the size of the source may be [63, 691. An other great advantage in the V.U.V. spectral range is the quite easy generation of the reference beam acting as a coherent background. Indeed investigation of other methods using spherical reference wavefront implies the use of small holes of micron size [70]. 5. Conclusions. - Regarding the classical spectroscopic devices, high resolution required namely in molecular spectroscopy, will be achieved by an increase of the size of the instruments. Otherwise, new devices using zone plates and transmission gratings will be developed for the soft X-ray spectral range. Moreover, extension of the classical holographic techniques to vacuum ultraviolet seems to be one of the most exciting new trends in the field of lensless imaging devices. Indeed, if only possible extensions of Fourier Transform spectroscopy have been examin- ed in the last paragraph, it is clear that the same experimental set up could be used for optical reconstruction of three dimensional macroscopic scenes, or for X-ray crystal structure determination. Indeed the anticipated availability of X-ray lasers at 1 A has raised the hope of atomic scale imaging of organic molecules by holography [71]. This implies

' that even grainless recording media are unspitable

for imagery on an atomic scale as their resolu'tion is limited by molecular dimensions. Another limitation would arise from the intensity requirements [72]. Despite these limitations, the most spectacular developments of V.U.V. instrumentation will be, in the

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