Optimization and theoretical performance of an adaptive xray mirrorJ. Susini, G. Förstner, L. Zhang, C. Boyer, and R. Ravelet Citation: Review of Scientific Instruments 63, 423 (1992); doi: 10.1063/1.1142720 View online: http://dx.doi.org/10.1063/1.1142720 View Table of Contents: http://scitation.aip.org/content/aip/journal/rsi/63/1?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Performance of highrate adaptive equalization on a shallow water acoustic channel J. Acoust. Soc. Am. 100, 2213 (1996); 10.1121/1.417930 Biologically motivated adaptive sonar system J. Acoust. Soc. Am. 100, 1849 (1996); 10.1121/1.416004 Adaptationinduced enhancement of vibrotactile amplitude discrimination: The role of adapting frequency J. Acoust. Soc. Am. 99, 508 (1996); 10.1121/1.414509 HLA/VLA broadband adaptive beamforming detection performance comparison in shallow water J. Acoust. Soc. Am. 98, 2932 (1995); 10.1121/1.414154 Conceptual design of an adaptive xray mirror prototype for the ESRF Rev. Sci. Instrum. 63, 489 (1992); 10.1063/1.1142740

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Optimization and theoretical performance of an adaptive x-ray mirror J. Susini, G. FOrstner, and L. Zhang European Synchrotron Radiation Facility, BP 220, F-38043 Grenoble Cedex, France

C. Boyer and R. Ravelet Laserdot/Groupe Aerospatiale, F-91460 Marcoussis, France

(Presented on 15 July 1991)

The performance of an adaptive mirror strongly depends on the quality of the overall mirror optimization and on the model on which the control algorithm is based. Fundamental parameters entering the mirror design are: mirror function, source and beam parameters, response time, actuator positions and forces. In a case of a wiggler source, we show that it is possible to decrease the thermal deformation generated by 4 kW absorbed power by a segmented mirror of one meter length from 600 prad to a few prad. The discussion is based on detailed mechanical analysis and on ray-tracing simulation.

I. INTRODUCTION

In order to take full advantage of the x-ray sources inserted in the 6 GeV storage ring of the ESRF, very im- proved, if not totally new, designs of the optical devices are necessary. They must be adapted to the specific properties of the radiation, particularly the small source size and beam divergence, and the high power of the beam emitted by the various insertion devices.’ Unfortunately, the finite- optics efficiency limits the beam quality:2 on one side, the emittance is increased by aberrations, microroughness, slope errors, misalignment, and other imperfections of op- tical devices, and on the other side, the maximum amount of brilliance that can be transmitted by the optics is limited by the high power and the subsequent heat-load effects associated with insertion devices. Thus, not only are the requirements very severe in terms of figure and finish of x-ray mirrors, but the design must also take the high heat load into account.3 The figure error is one of the relevant parameters. This parameter determines that the optical quality of mirrors is no longer static, but becomes dy- namic. In other words, it is important to keep the same high quality under severe boundary conditions which change with time. For many years these difficulties have been well-known in astronomy.4’5

The aim of this article is to answer to the following questions:

(i) By using an adaptive mirror (Fig. l), can the ther- mal deformation be decreased to an acceptable level.

(ii) Regarding the intrinsic properties of the adaptive technology, can further mirror functions be exploited that are to achieve focusing or defocusing of x-ray beams and even to correct for instabilities of various origins.

Whereas in another article of this conference6 we dis- cuss the practical aspects and the solutions of various tech- nical problems arising at the design of such an adaptive x-ray mirror, we here consider the theoretical aspects of such a mirror.

In order to assess the real possibilities of this system, the calculations parameters are based on the characteristics of the ESRF Beamline no2 “Multipole Wiggler/Material

Science,“’ which corresponds to the most severe require- ments in terms of power absorbed by the mirror. The in- sertion device is a multipole wiggler with a characteristic energy EC = 28.8 keV, covering an energy range from 5 to 60 keV. In a plane normal to the beam at 30 m from the source, the horizontal and vertical sizes of the beam is S, = 45 mm and S, = 3.5 mm, respectively. Regarding two cut-off energies of 10 and 40 keV, the absorbed power is 3.7 and 1.7 kW, and the glancing angle fl is 8.0 and 2.0 mrad, respectively. The mirror is composed of three segments: a central segment of 50 cm long and two side segments of 25 cm each.

II. THERMAL DISTORTION OF THE MIRROR

The total thermal slope error is composed of the two surface displacement components. The first one is the bump component which is associated with the temperature gradient along the mirror length, and the later one is the

-- - X-ray beam - - - ---

f---- I --

MIRROR

1

EXTERNAL SIGNAL

FIG. 1. General principle of an adaptive mirror.

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bending component due to temperature gradient between the upper warm surface and the lower cold surface of the mirror. Basic theoretical considerations show that if the mirror thickness e is much smaller than the half mirror length (e/L< 1 ), the bump component can be neglected with respect to the bending component.” Then, assuming a Gaussian distribution of the incident power, this “ther- mal slope error” & can be calculated from the following equations. ’ i

For the central segment (L, ) Ix 1 ) 0):

s

Xb-/U~ &h(X) = - e, e - s/2 &,

0 (1)

and for the side segments (L ) 1 x 1 ) L, ) XP/Q %l(X) = - e, s e-‘12dt 0

e. L

+L--Li L, SJ” -Q/o,

e - ‘I2 dt dx, (2) 0

where p is the glancing angle and o, is the half height of the beam along the vertical direction. The constant 0, de- pends on the total absorbed power and mirror parameters:

(3)

where a is the thermal expansion coefficient, k the thermal conductivity, P the total incident power, a the mean ab- sorption coefficient of the mirror and L, the half-mirror width.

For the case of the multipole wiggler and Sic/C mir- ror as described in Ref. 6, the value of 0, is equal to 363 prad at 10 keV, which should be the most critical situation in the interesting photon energy range ( 10-40 keV). Shadow simulations show that the intensity distribution of the reflected beam in the vertical direction is no more Gaussian and that the convex surface of the distorted mir- ror produces a blurred image. Then the vertical dimension of the image becomes 6.2 cm, which is nearly four times bigger than in the case of an ideal mirror (r = 3.9 where f is the ratio between the size of the blurred-image and the ideal-image size). The horizontal parameters are the same because the thermal deformation is only calculated in the vertical direction. In conclusion, the image size as well as the non-Gaussian intensity distribution of the beam re- flected by a thermally deformed mirror are not acceptable for high performance beamlines. We will now study the possibility of compensating for the thermal deformation of the mirror.

Ill. MECHANICAL CORRECTION OF THE THERMAL DEFORMATION

Because the mirror length is much bigger than the mirror width and its thickness, it can be considered as a mechanical beam. A set of actuators are placed along the mirror to pull or to push in order to compensate for the thermal bending. The action of actuator i is described as a

F* Fi F F,+! N FN M 1

FIG. 2. Definition of the coordinates system and relevant parameters for the mechanical analysis.

force Fi acting at a point Xi, normal to the mirror plane as shown in Fig. 2. By definition, Fi is positive when the actuators push onto the mirror. The bending moment M(x) at any point x can be easily obtained:

M(x)= - i, FAX - Xi> s (4)

where

if X(Xi

‘x-xi’=[ (XI).xi), if x>xi (5)

This moment M(x) bends the beam (mirror) into a curved shape, denoted by a radius of curvature R(x) or a slope e,,(x) relative to the axis of x. The governing equa- tion is given by

de,,(x~ i M --z-=-=- R(n) EI’ (6)

where E and I are the elastic moduli of the mirror material and the inertia of the cross section, respectively. Substitut- ing M(x) in Rq. (6) by Rq. (4) and after integration, the mirror shape can be calculated as follows for the central segment

e,,(x) = - iil & [(x-xi>2- (xc-Xi)2], (7)

where x, is the coordinate of the center spot and for the two side segments:

(x - d2 - $.y$) . (8)

Note that EL++ (7) and (8) show that the beam-slope function is a set of N - 1 parabolas, which are continuing at each point of action. Considering that any smooth curve can be approached by a parabola in a small enough range [x,-xi- t], the mirror can be theoretically bent into any smoothly curved surface by using a big enough number of actuators. Therefore, the thermal deformation can be com- pensated for by using a mechanical correction (Fig. 3). Whereas this was expected, the more difficult question is how to determine the best configuration (number and po- sition of actuators) for given boundary conditions.

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-60 ” ” ” ” ” ‘I”’ i , , ,-

-50 -50 -30 -30 -10 -10 10 10 30 30 50 50 MIRROR COORDINATES (cm) MIRROR COORDINATES (cm)

corrected shape

Mechanical deviation

FIG. 3. Minimization of the thermal deformation by a mechanical cor- rection by using a set of 11 actuators.

IV. OPTIMIZATION OF THE CONFIGURATION

The optimization of the actuator configuration consists of minimizing the number of actuators and determining their forces and positions in order to correct the thermal- slope error to within a requested criteria. Obviously, this optimization leads to both a minimization of the actuator number N and to a determination of the force and position, in agreement with the following condition:

I&(x) + O,,(x) I<&3 (xl, for any x, (9)

where X0(x) is the maximum acceptable slope error. We discuss here only the simplest yet realistic case

where the actuators are uniformly distributed. Thus, the position of the forces is given by the total number of actu- ators. The minimization calculation is performed by using a subroutine programme E02GCF in NAG FORTRAN library.i2

Figure 4(a) shows the residual deformation of the cor- rected mirror at EC = 10 keV, in terms of slope error and vertical displacement, by using five actuators on the central segment and 3 actuators on each side segment. The biggest slope error for the corrected mirror is 19 prad and the maximum surface deviation is 1.2 pm. These values have to be compared with the thermal slope error of a free mir- ror presented previously (363 ,urad and 55pm). The ray- tracing simulations show that the image size has the same dimensions as after reflection by an ideal-plane mirror (r z 1). Nevertheless, due to the action of the actuators, the shape becomes convex around two points of action and thus, each segment has, as expected, a nearly parabolic surface. Then, even the small amplitude ( < 0.5 pm) of the waveform deformation creates an heterogeneous-struc- tured image as shown in Fig. 4(b).

In order to improve these first results and considering the Gaussian-shape of the incoming beam, it seems incon- venient to minimize the deformation of the whole mirror. Moreover, at an energy cut-off EC = 10 keV, the footprint size a, is equal to 180 mm. This means that only the part - 180(x (mm) < 180 of mirror is irradiated. So obviously

in this case, only this part of the mirror needs to be cor-

s 10 0.5 :! 3

-2 30 0 s K

c K zo B 3 -0.5 g

s 2 10 -1 C $

P + A O -1.5 * 2 2 0 s -10 -2 z

K

* -20 ’ ’ ’ * ’ r,,,,,,.,,,.,.,,.,.~ 8 -7 9 -‘” -50 -30 -10 10 30 50

(a) MIRROR COORDINATES (cm)

..,~-.~.~:?~~ii..~~~~~~.‘~,~~~~:;.

E “.o~~~~~~~~~ ‘..., r:..\+ 4 --.<‘,:. <ye .<y.:.: -

“;‘.‘~-.“uk:~~:.~r,~.. . , * * .>-.L1’? -+.F~> -::..&&.&*-J 7:

(b) -’ 0 Y Horizontal [cml

FIG. 4. (a) Residual shape of a corrected mirror and corresponding slope errors. (b) Size and shape of the image after reflection on this mirror.

rected. The interesting length is defined as the interval - Ii,,/2 < x < lint/29 in which the slope error is considered

to be smaller than B&x). Figure 5(a) shows the residual slope error when the

previous optimization protocol is applied to only an opti- mized zone of 35 cm in length. Increasing the mirror per- formance for a given zone, increases the slope error at the mirror edges. Figure 5 (b) shows the rays reflected from the center segment. Now, the horizontal and vertical pho- ton distributions are Gaussian and homogeneous. Note that the slope error may be made significantly smaller by decreasing the interesting length for the central piece. It can also be shown that this improvement is more signifi- cant for the bigger actuators numbers.

The optimization model which takes the interesting length criteria into account must be applied for each en- ergy cut-off. Throughout these calculations only the value of the forces change, not their position. We will see in the next section the fastest way to relate the correction to be applied to the real shape of the mirror.

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5 2

60

z 40 B 2 20 v)

(a)

1.5

I z t

0.3 g

0 k 5

-0.5 g

-50 -30 -10 10 30 50 iMIRROR COORDINATES (cm)

F 3.5 4 z 2

[ $d&dh4*\~

r J

i LF----“-

i

i

ii j

It 7

z [ ‘. ..” :., t l,.l ,: t If i > +qF@gj~*p;~g; 1 p-F= * 0.0 :’ p+ &&&*r~J..~~;~~~~* L ^ ,-f,T% ‘A&-,.?* 94,:~&<* :: * -2s +.+d**$.e,&.p$; II. I/ ;zj :, . . .

(b)

-3.5 I It 1 (.,. . -8 0 8

Horizontal [cm]

FIG. 5. (a) Residual shape of a corrected mirror and corresponding slope errors. The actuators optimization is applied over 35 cm on the mirror center. (b) Size and shape of the image after reflection on this mirror.

V. SIMULATION OF THE STATIC-CONTROLLED ALGORITHM

The deformable mirror is now continuously inspected by an in situ interferometer measuring the local slope error across the mirror. By definition, the interaction matrix Mi describes the linear dependence between the slope mea- surements and the corrections to be applied to the mirror. For example, for the central segment of the mirror, we have to consider 10 actuators and 12 slope measurements (6 measurement points along the x and y axis), the ML size is 10X 12. The main step of the control algorithm is to construct &ii, which is the numerical determination of the matrix elements. It consists in the definition of the set of the slope error when only the ith actuator is driven. In practice, considering a given force applied to the mirror with the ith actuator, we calculate the corresponding slope errors for every measurement point onto the mirror by using a finite element model. This calculation is performed for each actuator.

In order to get the relevant relation between the cor- rections and the slope measurements, the matrix must be

426 Rev. Sci. Instrum., Vol. 63, No. 1, January 1992

inversed. But since the matrix is rectangular ( 12X lo), the usual inversion method cannot be applied, and we used the singular decomposition method based on a least square condition.13 The determination of the pseudo-inversed ma- trix Mi+ allows us to calculate the corrections command 1 C) from the slope measurements 1 P) by 1 C) = iWi* 1 P},

In an improved version of the algorithm, we are able to consider a reference slope IPr,f)#O. In this way, the min- imization of the thermal deformation is applied to a curved mirror. At 10 keV, with a curvature radius R, = 3.7 km, the ideal size being 0.13 mm in case of a magnification 1: 1, the thermal slope error produces an image size of 27.8 mm ( f = 215). After correction, the image size becomes 1.3 mm (r= lO).At40keVandforR, = 15km,theseparam- eters are Sideal = 0.11 mm, A’& = 3.4 mm (r = 33) and Se,, = 0.39mm (r= 3.5).

These results prove the capabilities of an adaptive mir- ror to compensate for thermal deformation under focusing conditions. As far as the correction for beam instabilities is concerned, the computer time associated with slope mea- surements should have a frequency of about 10 Hz. But the main problem is to find a relevant, accurate enough, and fast reference system for steering the reflected beam. This is still under study at ESRF.

VI. SUMMARY AND CONCLUSIONS

Among several alternatives, ESRF chose to develop an adaptive x-ray mirror because this technology permits it to meet several objectives at a time, namely, compensation of thermal deformation and variable curvature for focusing applications. An analysis combining thermomechanical models and ray-tracing calculations has been performed. This study shows that it could be possible to correct ther- mal bending by using mechanical actuators. The best con- figuration of actuators’ positions and forces to be applied for a given practical case has been calculated using an optimization model. SHADOW simulations show that the quality of the reflected beam (intensity, size, and shape of the image of the source) depends on the optimization cri- teria. Because of its inherent flexibility, adaptive optics should have useful and interesting applications in x-ray intrumentation that makes this technique very attractive. However, significant research and development is still needed for a reliable use in synchrotron radiation instru- mentation.

ACKNOWLEDGMENTS

The authors would like to thank C. Riekel, G. Marot, and J. P. Gaffard for useful and constructive discussions.

’ P. Elleaume, Nucl. Instrum. Methods A 266, 68 ( 1988). ‘A. K. Freund, X-ray Optics For Synchrotron Radiation, edited by R. M. Sweet and A. D. Woodhead (Plenum, London, 1989); Basic Life Sci. 51, 255 (1989).

‘A. K. Freund, F. de Bergevin, G. Marot, C. Riekel, J. Susini, L. Zhang, and E. Ziegler, Opt, Eng. 29, 928 (1990).

4See various articles in Proceedings of the International Society for Opti- cal Engineering, edited by F. J. Rodier, Orlando Conference, March 1989, Vol. 1114.

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‘J. P. Gaffard and C. Boyer, Appl. Opt. 26, 3772 (1987). ‘J. Susini, L. Zhang, G. Marot, P. Jagourel, and R. Ravelet, these pro- ceedings.

‘B. Lai and F. Cerrina, Nucl. Instrum. Methods A 246, 337 (1986). “B. Lai, K. Chapman, and F. Cerrina, Nucl. Instrum. Methods A 266,

544 (1988). ‘A. Kvick, Internal ESRF Report No. EXP/AK/90/02, 1990.

“L Zhang and G. Marot, Workshop on High Heat-Load X-Ray Optics, Argonne, August 1989, Report ANL/APSfTM-6.

“B. A. Boley and J. H. Weiner, Theory of Thermal Stresses (Krieger, Malabar, FL, 1985).

“The NAG FORTRAN Library Manual, 3, 2-3; and E02GCF Subrou- tine, 1-4.

t3G. H. Golub and C. Reinch, Numer. Math. 14, 403 (1970).

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