Design optimization of “straight groove” toroidal grating monochromators for synchrotron radiation

Nuclear Instruments and Methods 172 (1980) 149-156 © North-Holland Publishing Company

DESIGN OPTIMIZATION OF "STRAIGHT GROOVE" TOROIDAL GRATING MONOCHROMATORS FOR SYNCHROTRON RADIATION * W.R. McKINNEY and M.R. HOWELLS Brookhaven National Laboratory, Upton, New York 11973, U.S.A.

The use of approximate focussing to achieve a moderate resolution monochromator was pioneerd by the workers at the National Bureau of Standards. More recently the advantages of high flux, mechanical simplicity and low cost have been exploited by other groups, and monochromators of this class have been built employing aberration correction. We consider briefly the theory of the straight groove (uncorrected) case and advance some ideas aimed at achieving optimal designs for various purposes. Some particular solutions proposed for use at the National Synchrotron Light Source are discussed, and detailed ray traces are shown indicating the expected performance.

because the aberrations worsen with increasing angle o f incidence. We assume that the grating frequency in grooves/mm has been chosen as a compromise between the desire for highest possible dispersion, and the requirement that/3 < 90 ° for the long wavelength end of the region ("horizon wavelength" condition). In addition, we assume that the choice o f spectral order, which involves several interrelated factors, has already been made.

1. Introduction The availability of storage rings giving intense broadband radiation in the spectral range 1 0 - 5 0 0 A has prompted renewed interest in the improvement of grazing incidence monochromators [1]. Of particular importance is the moderate resolution case designed for highest throughput. This report will outline the reasons behind the various choices leading to a toroidai grating m o n o c h r o m a t o r design, and indicate a method for finding optimum values for some o f the parameters in the straight-grooved case [2]. We have chosen to fully explore this case because, at present, all aberration corrected gratings contain photoresist on their surfaces, and are therefore unsuited for use with high intensity synchrotron radiation. We shall also assume fixed entrance and exit slit to grating distances and wavelength scanning by simple rotation of the grating about its pole.

3. Grating parameters We choose the coordinate system of Haber [3] and Namioka [4] (see fig. la). The toroidal grating pole is 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 planes. The toroidal surface is defined b y the following:

2. Choice o f incidence angle, grating frequency, and order

x 2 + y 2 + z 2 = 2Rx - 2R(R - p)

In this analysis, we assume that the angle between the entrance and exit directions has been previously chosen. This angle, 20, should be just large enough to provide reasonable reflectivity at the short wavelength end of the region o f interest; and it should not be made larger than necessary for good reflectivity,

This is the "bicycle tire" or " t y p e A " toroid. We now write the optical path function,

+ 2(R - p ) [ ( R

- x ) 2 +y211/2

F = AP + PB + n m X , where P is any point on the grating surface, A and B are the object and image points, respectively, and N is the groove number. At some labor, Haber [3] expanded the AP and PB terms as functions o f y and z. We adopt this expression, although we group the

* This research was supported by the U.S. Department of Energy: Contract No. EY-79-C-02-0016. 149

IV. XUV MONOCHROMATOR SYSTEMS

150

W.R. McKinney, M.R. Howells / "Straight groove" toroidal grating monochromators

which the contributions of each aberration can be assessed (when this is done by desk-top computer). The following points become clear. For the region o f 0 > 75 ° : 1) Just taking one corner of the grating Cv =Ymax, z = 2max) is not sufficient for finding real AX's since the maximum AX's and Az's are often found at other places on the grating. For example, at other corners or at z = 0 , y = Ymax. 2) To an excellent approximation (error K10%) AX can be computed by using only the defocus and astigmatic coma terms. 3) To the same approximation, Az can be computed by using only the astigmatism and astigmatic coma terms. In fact, except when p has been chosen for a vertical focus, (Co2 ~- 0) the astigmatism term alone is sufficient. These observations are summarized in the following model:

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Z 2

/

/j./"

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terms more in the manner o f Welford [5] -= defocus, C309 - coma, Co2z 2 = astigmatism, C1 zyz 2 ~- astigmatic coma, C4oY 4 ~ generalized spherical abberation; Co4z 4 - higher order astigmatism, Czay2z 2 - g e n e r a l i z e d spherical aberration. We ignore terms in the distance of the entrance point from the principal plane, i.e., we consider only a point source on the principal plane. Ray traces using out of plane entrance points confirm this is valid for slits on the order of a few m m high. These terms may now be used to construct exit focal plane spot diagrams b y the following, often incorrectly stated, formulae: C2oy 2

az ~

2r'z

R

CO2.

(2)

y and z define various points on the surface of the grating, K is the order, iV is the groove density (gr/)~), and r', y , z and R are in the same units. Factors of 1/R are pulled out o f Haber's coefficients, so that we have - cos ~ + ~/. -

cos/3 ,

(3)

.

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OF F z~=r ~, z~x=a3ay' ,

(r' = grating-to-exit-slit distance), d = grating spacing. A grid of points can be chosen on the grating surface and individual AX's and z~z's are found for each point generating a 'fourth order theory ray trace" in

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A

* indicates a dimensionless parameter, e.g., p*R = p,

r*R = r,

r'*R = r' .

First, we examine Co2. As was pointed out by Madden and Ederer [6], over the range of a typical grazing incidence instrument cos a + cos/3 varies very little. For example, consider the 0 = 87.5 °, 600 g/ram case. X, 10 A, 25 A, 50A,

cos a, 0.05049, 0.06079, 0.07795,

cos/3, 0.03675, 0.02644, 0.00923,

cos a + cos/3; 0.08724; 0.08722; 0.08719;

W.R. McKinney, M.R. Howells

/ "Straight groove"

At 0 = 75 °, the variation rises to 1 or 2%, depending on wavelength range. This fact implies immediately that the astigmatism will be very constant over the entire range of the instrument. After r*.and r'* have been set by other considerations, p* is then set by defining C02 = 0 in mid-range, although the precise wavelength is unimportant. Thus we see by eqs. (5) and (2) that astigmatism can be corrected by the toroidal shape alone. The choice of entrance slit to grating and exit slit to grating distances requires a careful inspection of eq. (1). First, we rewrite it using our definition r = r*R.

AX:K----~I2(~)r*C2o +(z)Zr*c12].

toroidal grating rnonochromators

151

Vertical collection angle = (2Ymax/r) COS~; horizontal collection angle = 2Zmax/r. We now pick a somewhat arbitrary rectangular shape for the grating. The full width at zero height of the exit spot for our point source object may now be written, A•FWZH = A~.+ -- A~t_ ,

AX+ and AX_ correspond to the rays with the Iargest sideways deflection from the central ray. The computer chooses the y and z values within our chosen grating shape which gives these maximum AX deflections. A 3 × 3 matrix of points on the grating is sufficient to include all of the y and z values that might be involved. It is important to realize at this point that AXFWZH can be calculated completely independently of the overall size of the monochromator. Only groove density, order, entrance acceptance angles, p*,

(6)

This permits us to recognize that y / r and z/r are proportional to the vertical and horizontal collection angles of the instrument. (We assume vertical dispersion.)

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(b) Fig. 2. The values of the defocus aberration term (C2o) and the astigmatic coma aberration term (C 12) for two cases; (a) a symmetrical T.G.M., (b) an asymmetrical T.G.M. optimized for resolution in the range 80-320 A, positive order, IV. XUV MONOCHROMATOR SYSTEMS

W.R. McKinney, M.R. Howells / "Straight groove" toroidal grating monochromators

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uration of fig. 2b was computed, using the Wisconsin R , p and grating size. Notice that this procedure optimizes the design for minimum AX over the working range. One could equally easily optimize for the best resolving power (minimize AX/X) or best energy resolution (minimize AX/X2). We retain their p* for the sake of the comparison, although it should be modified slightly for best astigmatism correction for our straight-grooved design. In practice, of course, the design would work with an existing grating blank whose p* was known by measurement. Curiously we find r and r',

N

r = r*R = 0.19424 (588.9 cm) = 114.4 cm, r' = r ' * R = 0.32872 (588.9 cm) = 193.6 cm,

-2

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A

X320 A

Fig. 3. Exit slit spot diagrams and intensity distributions for the case of AX optimization and the wavelengths of 80 A and 320 A.

r*, and r'* are necessary. In order to set r* and r'* we now consider the concrete example of the "Wisconsin T.G.M." which has an aberration corrected grating supplied by J o b i n - Y v o n . Our analysis will determine how close a straight-grooved grating can approach its performance. Its parameters are [7] R = 5.889 m, r = 1145 ram,

/9 = 0.367 m, r' = 1927 mm,

20 = 150 °, gr/mm = 1800 order = +1.

By examining eq. (3), we see that there will be only two positions of focus. This is shown in fig. 2a which plots Czo and C12 for an r* = r'* case. What is done by allowing r* @ r'* is to move one o f the foci from one order to the other order as in fig. 2b. This is the fundamental reason why T.G.M.'s are usually designed as asymmetric monochromators. It is clear from fig. 2b why it is desirable to do this in order to achieve approximate focusing over the best possible range. We now vary r* and r'* (or equivalently Xl and X2, the wavelengths of the two zeros o f C2o) until the AX~WZH is minimized over the range specified for the design. This is done by guessing an initial Xi and X2. The value o f AXFWZH is computed for several wavelengths in the design range. Xl and X2 (r* and r'*) are then varied iteratively by a grid search until the highest AXFWZH in the range cannot be lowered any farther without raising the AXFWZH o f some other wavelength to a greater value. This is how the config-

remarkably close to the values o f 114.5 cm and 192.7 cm given by J o b i n - Y v o n for their grating.

4. Ray traces To check our straight-grooved design, we proceed to an exact geometric ray trace based on the method of Noda et al. [8]. This program has been checked against another program based on solid analytic geometry which is similar to that of Kastner and Neupert [9]. Five point images were placed on an entrance slit 1000 #m tall. This accounted for our out of principal plane terms previously ignored in the analysis. A matrix of approximately 400 points is placed on the grating surface. The matrix is not square in order to have the rays more uniformly spaced in solid angle as they extend from source to I

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W.R. MeKinney, M.R. Howells

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(e.g., AE "" 0.7 eV at 80 A for the Wisconsin design). In order to address this problem, we continue with the basic Wisconsin parameters and investigate the effect o f optimizing the configuration for best energy resolution. We retain the following:

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Order = +1 , gr/mm = 1 8 0 0 , grating height 2.5 e.m, grating width 7.5 cm, wavelength range 8 0 - 3 2 0 A .

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X520A Fig. 5. Exit slit spot diagrams and intensity distributions for the case of ~ E optimization and the wavelengths of 8 0 320 A.

grating. In our opinion, this is often crucial to the validity of a grazing incidence ray trace. Fig. 3 shows the results o f these ray traces o f approximatley 2000 rays. The spot diagram shows the intersection points of the diffracted rays and a plane perpendicular to the central ray located at a distance r' from the grating pole. The intensity distribution below each diagram is found by discretely stepping a slit over the spot diagram and plotting the number of spots that fall within the slit at each step. Fig. 4. shows a plot of the full width at half maximum derived from these ray traces. This aberration limited performance is essentially similar to that achieved experimentally and predicted theoretically for the Wisconsin T.G.M. [7].

r'* = 0.30503 ,

~2 = 193.7 A ,

r' = 179.63 c m .

Checking our design by ray trace as before produces figs. 5 and 6. We see that b y moving the two foci tho shorter wavelengths, we can have a better overall for the same throughput (exposed grating area) as the best AX design. Note that this resolution can be

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5. Other criteria of optimization As we indicated above, the computer optimization may pick r* and r'* with other design targets than AX. The program may compute AX/X at the several wavelengths o f the optimization and iterate to the best overall resolving power. F o r the case of surface physics at synchrotron light sources, a constant AE is more appropriate, with AX/X 2 optimized. In fact, the requirement is really for a AE o f 0 . 1 - 0 . 2 eV; and the effect o f optimizing AX is that AE is much worse than this at the short wavelength end of the range

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IV. XUV MONOCHROMATOR SYSTEMS

156

-0.2

W.R. McKinney, M.R. Howells / "Straight groove" toroidal grating monochromators

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achieved purely by changing r and r ' and using the same grating as before. This would make p non-optimal and would give an image about 3 m m long. If we adjust p suitably then the images shown in fig. 5 a - g are obtained.

6. Choice o f mask geometry An alternative approach to the problem of poor energy resolution at the short wavelength end of the spectral region is to reduce the exposed area o f the grating. If the AX design were masked down with a rectangular mask to achieve the same aberration limited resolution (0.2 eV) as the AE design, then calculation shows that approximately 90% o f the grating area would be lost. This is shown b y fig. 7 where contours of constant energy deviation have been plotted for the AX case at 80 A. We have assumed AX~rWZH"~ 2 2X>,FWHM is drawing the dotted lines for a 0.2 eV

mask. It is seen that 0nly ~60% of the area would be lost if the mask exposed all of the area between the two contours of -+0.2 eV. Unfortunately, these contours are strong functions of wavelength. A more general problem remains to be solved; that of computing an optimum mask geometry for all wavelengths. It appears that control of the size and shape of the mask is an inferior way to approach the problem o f improving the short wavelength resolution as compared to manipulation of r and r'. However, adjustments of a mask would have the advantage of being feasible as operational functions, while changes in r and r ' would amount to a complete restructuring of the monochromator.

References [1] M. Pouey, J. de Phys., Co11. C4 (suppl. no. 7) 39 (1978) 188. [2] H.A. Rowland, Phil. Mag. 13 (1882) 469. [3] H. Haber. J. Opt. Soc. Am. 40 (1950) 153. [41 T. Namioka, M. Seya and H. Noda, Jap. J. Appl. Phys. 15 (1976) 1181. [5] W.T. Welford, in Progress in Optics, Vol. IV, ed. E. Wolf (North Holland, Amsterdam, 1965). [6] R.P. Madden and "D.L. Ederer, private communication; J. Opt. Soc. Am. 62 (1972) 722A. We gratefully acknowledge the receipt of an unpublished manuscript detailing the design of the original toroidal grating monochromator at the National Bureau of Standards. [7] B. Tonner, private communication. [8] H. Noda, T. Namioka and M. Seya, J. Opt. Soc. Am. 64 (1974) 1037. [9] S.O. Kastner and W.M. Neupert, J. Opt. Soc. Am. 53 (1963) 1180.