Pram~.na - J. Phys., Vol. 35, No. 2, August 1990, pp. 187-191. © Printed in India.
A note on grating based soft X-ray monochromators
V C SAHNI
Nuclear Physics Division, Bhabha Atomic Research Centre, Bombay 400085, India MS received 2 March 1990
Abstract. A few facts related to the aberrations for toroidal and plane gratings are presented. Some design aspects of the plane grating-based monochromator devised by Petersen (1982) are discussed. We also point out that for a cylindrical grating astigmatic correction can be reduced and suggest that following Petersen's scheme some improvements in the monochromator performance could be achieved with such a grating.
Keywords. Soft X-ray monochromator; cylindrical grating; aberration corrections.
PACS Nos 07.85; 33.20
1. Introduction
The development of soft X-ray grating monochromators for use with synchrotron radiation sources has seen consistent advances over the last two decades. A comprehensive list of references and an updated status of most of these developments has recently been provided by Padmore (1989). Principally, the thrust of most of the developments, whether using plane, spherical or toroidal gratings has been to devise configurations so as to reduce the aberration corrections (and thus increase the resolving power). Concurrently, technological improvements in the surface quality of optical elements and designs of the drive mechanisms used for tracking, have upgraded m o n o c h r o m a t o r performance remarkably over the years. In this note we propose to recapitulate some of the optimization schemes used for reducing aberration corrections and then turn to some design aspects of the Petersen's plane grating monochromator. We also point out a new way, based on the use of a
cylindrical
grating, to achieve an improved m o n o c h r o m a t o r performance. A brief summary of the geometrico.l optic approach to grating aberrations is presented in § 2.Some practical aspects of Petersen's m o n o c h r o m a t o r are examined in § 3. In § 4 we discuss the case of cylindrical grating and a summary of our conclusions is given in § 5.
2. Aberrations for different grating shapes
We begin by recalling basic facts of geometrical-optic-grating-abberation theory and in particular some facts relating to toroidal and plane gratings.
Assume O is the centre of a grating of some specific shape (toroidal, spherical, plane, elliptical etc.) whose surface is given by an equation of the form
z = f ( x , y). (1)
187
The coordinate system is chosen such that the tangent plane at O coincides with the X Y plane and the origin of the coordinate system is at O. Also grating groves are taken to be parallel to Y axis, equidistant on the tangent plane, with a spacing d.
Let us suppose that a polychromatic light source A, with coordinates (X,t, YA, ZA), is located at a distance r from O and the grazing incidence angle is denoted by ~. In the geometrical optic approach one takes an arbitrary point P(x,y,z) on the grating surface and follows the course of a ray AP, diffracted towards a point B, at a distance r' from O, with coordinates (XB, YB, ZB) (see figure 1). One computes the optical path length L = (AP + PB) as a power series in x,y and then using Fermat's principle sets up the equations for a k-th order diffracted image at B corresponding to wavelength 2
c~L k2 dL 0
c~-=--d- and ~ - y = . (2)
If fl denotes the grazing angle of diffraction corresponding to the incident ray AO, it emerges from the above analysis that the usual grating equation
k2 = d(cos ~ - cos fl) (3)
is obtained only when all terms except the linear ones are ignored (Johnson 1983). In general, inclusion of higher order terms leads to "aberration corrections" and there exists a nomenclature identifying some lower order terms with corresponding corrections. For instance the x 2 term is identified with the 'focusing correction', y2 term with 'astigmatism', x a term with 'coma', x y 2 t e r m with astigmatic coma etc.
Further, it turns out that, in general, the extent of these corrections depends upon the grating shape, and for optimized performance a careful study is required. For instance although toroidal gratings are primarily deployed for reducing astigmatism, Howells (1980) had shown that through a suitable choice of the object (r) and image (r') distances and the minor radius (p) of the torus, it is possible to satisfy the focusing
/ I
¥
\,,B
Figure 1. The coordinate axes and other parameters are depicted.
Grating based soft X-ray monochromators 189 condition for two wavelengths 21 , 22, and at the same time correct astigmatism to a sufficient accuracy for a sizeable wavelength range. In fact for the astigmatic term to be zero we require
p = (sin ct + sin fl)/(1/r + 1/r') (4)
while vanishing of focusing term demands that
sin20t/r + sin2B/r'= (sins + sinfl)/R (5)
where R is the major radius of the torus, In addition Howells examined the astigmatic coma term and studied the conditions under which the reduction of this term could be optimized at the expense of focusing requirement.
Later, Petersen (1982) working with plane gratings, proposed an optimally focused and kinematically simple two element optics for soft X-ray monochromators. Over the years this concept has been put into practice with increasingly better quality optics and the performance of the latest version (Arvanitis et al 1989) is remarkably impressive. In the "fixed focus" version of Petersen's scheme, the grazing angle of incidence ~t and the grazing angle of diffraction fl are constrained to follow the condition
sin/Y/sin ct = c( # 1). (6)
Petersen showed that with a suitable choice of c, his arrangement allows one to construct an instrument that can perform extremely well over a wide spectral range.
3. Some practical aspects of Petersen's monochromator
In a practical realization of Petersen's scheme Reimer and Torge (1983), essentially deployed a large plane mirror which is rotated about an axis located outside its surface so that it deflects the horizontal incident beam onto the plane grating which is rotated about an axis passing through the grating surface (see figure 2). The plane mirror rotation axis is so located that reflected beam impinges on the grating in such a way so that the emerging (horizontal) diffracted beam practically does not shift vertically when the monochromator scans differer't wavelengths. The horizontally diffracted beam is intercepted by an ellipsoidal mirror such that the monochromatized beam is focused at one of the focii of the ellipsoidal mirror and a sample could be positioned there. The principal virtue of this scheme lies in that using simple kinematic
PLANE T~
MIRROR
INCIDENT
i , ~ / / / / / / / / . # , ~ , ~ = = - ~ . . V , P ~ N E .IRROR GRATING
Figure 2. Diagram illustrates the practical layout of optical components in Petersen's monochromator.
movements of the plane grating and of the plane mirror preceding it, one is effectively able to construct a (virtual) monochromatic source at a fixed point in space. Then the eUipsoidal focusing mirror positioned so that the virtual source is at its one focus, images it at the second focus. A monochromator of this type is going to be utilized for soft X-ray absorption studies on the storage ring INDUS-1 coming up at Indore and we deem it appropriate at this point to discuss one aspect of this design explicitly. The point relates to locating the rotation axes of the plane mirror and the plane grating.
To identify their locations and the precision necessary in their positioning we use the following procedure (Sahni 1990).
By our choice of coordinate system these rotation axes are parallel to Y axis and let M and G denote the location of these axes (in the X Z plane) corresponding to the plane mirror and grating respectively. First using elementary coordinate geometry we find the coordinates of the point (S) of intersection of the (horizontal) principal incident ray with the plane mirror. As S also lies on the reflected ray, the equation of the latter can be found out and the coordinates of the point (T) of intersection of the reflected ray with the grating can be obtained (see figure 2). If z denotes the grazing angle of incidence on the plane mirror, then for the horizontally emerging diffracted ray 0t + fl = 2~. It follows from eqs (3) and (6) and that the angle z decides the wavelength setting of the monochromator. Now, in general, as the plane mirror and the gratings are rotated, corresponding to different wavelength settings, the coordinates of point T change, leading to a shift in the vertical position of the emerging diffracted beam. However, it is a straightforward matter to examine the extent of this vertical shift for various locations of points M and G. An optimized choice for their positions can be made by using an iterative scheme on a computer (Sahni 1990). For a grating with 1200 grooves per mm, one can easily identify locations which lead to a vertical spread of less than 3 or so microns over a spectral range from ~ 100eV to ,,~ 1000eV of photon energy, over which photoabsorption experiments with INDUS-1 could be performed. One can also examine with this program the precision requirements in positioning these axes and we find that a few micron accuracy suffices.
4. Cylindrical grating based monochromator
Although in the above mentioned mode the focusing condition is satisfied for all wavelengths, there are significant astigmatic distortions in the image as revealed by the ray tracing studies (Petersen 1986). However, if in the Petersen's arrangement, in place of a plane grating, one employs a cylindrical grating one can not only continue to satisfy the focusing requirement for all wavelengths but also minimize the astigmatic distortions for certain wavelengths. In order to see how this comes about let us adapt the expressions for toroidal gratings (Howells 1980) to the case of cylindrical grating. It can be easily shown that focusing condition for a cylindrical grating is given by
r' = - rsin 2 p/sin 2 ~, (7)
and once again an optical arrangement similar to Petersen's can be deployed to satisfy this condition. Moreover the condition for astigmatic distortions to be absent implies
Gratin9 based soft X-ray monochromators 191 that the radius p of the cylindrical grating should satisfy (4). Using (4) and (7) we find
p = r sin 2///(sin fl - sin ~).
Using (6) we obtain
p = rc2sina/(c- 1).
(8)
(9) Taking the values r = 15m, c = 2.25, and grating with 1200 grooves per mm used by Petersen (1982) one can estimate the value of radius of the cylindrical grating that will make astigmatic distortions for a chosen wavelength actually zero and improve monochromator performance around that range. For example a radius of 2"1 m corresponds to • = 2"0 ° (i.e. photon energy of ~ 600eV). This being a reasonable value we feel it should be feasible to make such a monochromator. We should mention that even though the spectral range over which astigmatic correction can be minimized in this way is somewhat limited, it can be quite valuable in certain situations.
5. S u m m a r y
We have briefly discussed aberrations for different grating shapes and their implications on soft X-ray monochromators. Some practical aspects of the Petersen's plane grating monochromator have been examined. We have also shown that a cylindrical grating based monochromator with an arrangement similar to Petersen's can give an improvement in performance.
R e f e r e n c e s
Arvanitis D, Rabus H, Domke M, Puschmann A, ComeUi G, Petersen H, Troger L, Lederer T, Kaindl G and Baberschke K 1989 Appl. Phys. A49 393
Howells M R 1980 Nucl. Instrum. Methods 172 123
Johnson R L 1983 in Handbook on synchrotron radiation (ed.) E E Koch (Amsterdam: North Holland) Vol. 1A ch 3
Padmore H A 1989 Rev. Sci. lnstrum. 60 1608 Petersen H 1982 Opt. Commun. 411 402
Petersen H 1986 SPIE Soft X-ray optics and technolooy 733 262 Reimer F and Torge R 1983 Nucl. lnstrum. Methods 208 313 Sahni V C 1990 BARC Report (under preparation)
