Exact revision of the elliptically bent mirror theory

Chengwen Mao, Xiaohan Yu,* Tiqiao Xiao, Aiguo Li,Ke Yang, Hua Wang, Fen Yan, and Biao Deng

Shanghai Synchrotron Radiation Facility, Shanghai Institute of Applied Physics,Chinese Academy of Sciences, Shanghai 201800, China

*Corresponding author: yuxiaohan@sinap.ac.cn

Received 11 November 2010; revised 30 March 2011; accepted 30 March 2011;posted 31 March 2011 (Doc. ID 137934); published 25 May 2011

One of the main hurdles for nanometer focusing by a bending mirror lies in the theoretical surface errorsby its approximations used for the traditional theory. The impacts of approximations and analyticalcorrections have been discussed, and the elliptically bent mirror theory has been described duringexact mathematical analysis without any approximations. These approximations are harmful for thefocusing system with bigger grazing angle, bigger mirror length, and bigger numerical aperture. Theproperties of equal-moment and single-moment bentmirrors have been described and discussed. Becauseof its obvious advantages, a single-moment bending mirror has high potential ability for nanometerfocusing. © 2011 Optical Society of AmericaOCIS codes: 080.4228, 340.6720, 340.7470.

1. Introduction

Synchrotron radiation facilities produce high-qualitylight with wavelengths ranging from the infrared tohard x-ray regions. Hard x rays have exceptionalproperties that are useful in the chemical, elemental,and structural analysis of matter with such analysismethods as x-ray diffraction, x-ray scattering, x-rayfluorescence, x-ray absorption, and x-ray photoelec-tron spectroscopy. For these analytical methods,the spatial resolution is one of the key parametersfor the investigation of the structure, elementaldistribution, and chemical bonding state of specialsamples. There are a variety of hard x-ray focusingoptical systems, such as mirrors [1], Fresnel zoneplates [2], compound refractive lenses [3], multilayerLaue lenses [4], waveguides [5], and kinoform lenses[6]. Since the hard x-ray wavelength is typicallyon atomic scales and smaller, hard x-ray analyticaltechniques have the potential for single-nanometerspatial resolution. The imperfections of the focusingelements are the main reasons for the uncomfortablefocusing spot size.

Dynamical bending devices for x-ray focusing areconventional equipment in synchrotron radiationbeamlines. Key features of the bent mirror are theirhigh flexibility and variability requirement of opticalparameters, such as source distance p, focal distanceq and grazing angle θ [7]. The mirror is usually a flatconstant-thickness substrate bent by unequal endcouples [8]. Since the flat surface can be made withatom-size-level precision by machining technologynowadays, this focusing system also has the poten-tial for single-nanometer spatial resolution. Thedevelopments in bent mirror optics have produceda 90nm focus spot size for hard x rays (20keV) ona very long beamline (140m) [9]. To simplify calcula-tion, there are some approximations for the tradi-tional classical elliptically bent mirror theory [8],such as the ellipse equation approximation (seeSubsection 2.A), coordinate approximation, thick-ness approximation, and curvature approximation(see Subsection 2.B). The aberrations caused by allapproximations are divergent along the length ofthe mirror. In this regard, the mirror’s length andthe analytical spatial resolution are limited by theimperfections of these elements.

In this paper, the intrinsic errors caused bythese approximations are calculated, and the exact

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expression and the relationship among optical para-meters are deduced. The ideal mirror surface forpoint-to-point focusing is an ellipse, and for plane-wave-to-point focusing, it is a parabola [10]. Sincethe latter is one of the special cases of the former withthe source distance being infinity, the theory of para-bolic bent mirrors is broadly similar with that ofelliptically bent mirrors.

2. Theory

A. Geometrical Considerations and the Bending Theory

An elliptical cylinder mirror is defined by the opticalparameters p, q, and θ [7] and has semimajor axis aand semiminor axis b (see Fig. 1). It is represented inthe X–Z coordinate system by

X2

a2 þ Z2

b2¼ 1: ð1Þ

Ideally, the same ellipse can also be represented byan exact analytical expression at the x–z coordinatesin Fig. 1 shown as follows:

zðxÞ ¼ sin θðpþ qÞ4pqþ ðp − qÞ2cos2θ× f2pq − 2½ðpqÞ2 − pqx2 − xpqðp − qÞ cos θ�1=2− x cos θðp − qÞg; ð2aÞ

z00ðxÞ ¼ sin θðpþ qÞ2pq

×�1 −

x2

pq−

xðp − qÞ cos θpq

�−3=2

;

ð2bÞor by a high-order polynomial with approximation intraditional theory [8,11],

zðxÞ ¼X∞i¼2

aixi: ð3Þ

The ai coefficients for the ellipse are

a2 ¼ sin θðpþ qÞ4pq

;a3 ¼ a2 cos θðp − qÞ2pq

;

a4 ¼ a2½4pqþ 5ðp − qÞ2cos2θ�16ðpqÞ2 ;… ð4Þ

Because Eq. (3) is a common practice to limit theTaylor set by several first orders that results in theerrors of approximation, Eq. (2) is used for accuratecalculation in this paper.

Approximating the mirror as a beam, the mirror isbeing bent by the action of two end couples M1 andM2, as defined in Fig. 2. There are two different co-ordinates, x and s, where x is appropriate to describethe exact ellipse expression and s is appropriate fordescribe the moment distribution MðsÞ, the radiusρðsÞ, and width distribution bðsÞ of the bent mirror.R and ρ are the radii of the working plane on coordi-nates x and s, Rc and ρc are the radii of the centralplane on coordinates x and s, t represents thethickness of the mirror, L is the length of mirror,and l is the projection length of the bent mirror oncoordinates x.

The calculated equation for the shape of thebent beam is the classical beam theory [12], shownas follows:

EIρcðsÞ

¼ MðsÞ; ð5Þ

where E is Young’s modulus, I is the moment of in-ertia of the beam cross section (I ¼ bt3=12, b and t arethe width and thickness of the mirror), and MðsÞ ¼ðM1 þM2Þ=2þ ðM2 −M1Þs=L is the moment distri-bution on the mirror. To construct a nominally exactelliptical shape, it is a better solution to do this bymodifying the width of the mirror [8], so that I inEq. (5) becomes IðsÞ. It is also necessary to obtain theexact radius of curvature at each value of s as speci-fied in Eq. (6).

The approximations for simplified calculation andtheir unfavorable impacts, according to the tradi-tional elliptically bent mirror theory, are given inSubsection 2.B with exact correction.

B. Approximation of Coordinates, Thickness, andCurvature

Since the incidence angle θ is small, the differencebetween x and s is ignored when the traditional the-ory is applied. Because of the ellipse-shaped mirror,the exact relationship between x and s is shown as

Fig. 1. Working condition and coordinate definition of the ellip-tical focusing mirror.

Fig. 2. Schematic of a bent mirror with two different couples anddefinitions of coordinates, mirror parameters, etc.

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sðxÞ ¼Z

x

0½z0ðnÞ2 þ 1�1=2dn; ð6Þ

where z0 is the first derivative of Eq. (2). From Eq. (6)and Fig. 2, we may infer that L ¼ sðl=2Þ − sð−l=2Þ,ρðsÞ ¼ RðxÞ, ρcðsÞ ¼ RcðxÞ. The slope-error distri-bution caused by the approximation of coordinatesis plotted in Fig. 3(a), with parameter configurationp ¼ 5m, q¼15cm, and θ¼5mrad, 10mrad, 20mrad.

In Fig. 2, the mirror surface is the working planewith radius ρðsÞ on coordinates s or RðxÞ on coordi-nates x for focusing the rays. According to the classi-cal beam theory, the profile of the central plane istypically described by radius ρcðsÞ or RcðxÞ. The dif-ference of two radii is the half-thickness t=2:

RcðxÞ ¼ RðxÞ þ t=2 or ρcðsÞ ¼ ρðsÞ þ t=2: ð7Þ

According to the traditional bent mirror theory, thethickness is also ignored for t ≪ ρ. For qualitativecalculating the slope-error distributions caused byapproximation of thickness, the approximation ofcurvature is introduced. Since the mirror slope issmall when compared with unity, the ellipse curva-ture is approximately shown as follows:

1=RaðxÞ ≈ z00ðxÞ: ð8Þ

And the shape of the central plane can be deducedfrom Eq. (7):

z00c ðxÞ ¼z00ðxÞ

1þ 1=2tz00ðxÞ : ð9Þ

So the slope difference between the working planeand the central plane is represented as

ΔT ¼ z0 − z0c ¼t2

Zz00 · z00cdx: ð10Þ

The slope errorΔT is plotted in Fig. 3(b), with param-eter configuration p ¼ 5m, q ¼ 15 cm, θ ¼ 5mrad,10mrad, and thickness t ¼ 1 cm, 2 cm.

According to the traditional theory, the curvatureis usually represented as Eq. (8) with approximation.Mathematically, the exact curvature of any smoothcurve (such as an ellipse) is represented as [13]

1=R ¼ ðz02 þ 1Þ−3=2z00: ð11Þ

The difference between R and Ra is

δ ¼ R − Ra: ð12Þ

The curve with radius distributions Ra is defined asza with a similar expression like Eq. (11). The slopeerror based on the curvature approximation can beexpressed as

Δa ¼Z

z00½ð1þ z02a Þ3=2 − 1�dx: ð13ÞFigure 3(c) shows the slope error caused by the ap-proximation of curvature with parameter configura-tion p ¼ 5m, q ¼ 15 cm, and θ ¼ 10mrad.

It obviously shows that, without any exceptions,the slope errors of the mirror downstream causedby all approximations increased sharply. The unfa-vorable imperfections of these approximations arefatal for hard x-ray nanometer-level focusing.

Fig. 3. (Color online) Slope-error distributions: (a) approximationof coordinates, p ¼ 5m, q ¼ 15 cm, and θ ¼ 5mrad, 10mrad,20mrad; (b) approximation of thickness, p ¼ 5m, q ¼ 15 cm, andθ ¼ 5mrad, 10mrad, and t ¼ 1 cm, 2 cm; (c) approximation of cur-vature, p ¼ 5m, q ¼ 15 cm, and θ ¼ 10mrad.

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C. Width Variability for Bending an Elliptical Mirror

To construct a nominally exact elliptical shape, theexpression for the width needed to produce the de-sired elliptical shape is

bðsÞ ¼ b0ρcðsÞMðsÞEI0

; ð14Þ

where b0 and I0 are the width and the moment of in-ertia at the center of the mirror, respectively. Sincethe mirror surface is an exact ellipse, the expressionof the central plane on coordinates x can be easilydeduced from Eqs. (7) and (11):

RcðxÞ ¼ðz02 þ 1Þ3=2

z00þ t2: ð15Þ

Although it is hard to deduce ρcðsÞ with the exactanalytical expression on coordinates s, we can obtainthe corresponding relationship between ρc and s fromEqs. (6) and (15).

Equation (14) shows that the width of the mirror isstrictly related with the moment distributions ateach value of s calculated according to Eq. (6). Therelationship between the moments M1 and M2 canbe defined as [11]

M1 ¼ kM2: ð16Þ

The expression of M2 can be deduced according toEqs. (14) and (15):

M2 ¼ 4a2EI0ð1þ kÞð1þ a2tÞ

: ð17Þ

Figure 4 shows the relative width distribution on co-ordinates s for different bending models. When k ¼ 0,the moment M1 ¼ 0 and the mirror is bent by onlyM2; when k ¼ 1, M1 ¼ M2 and the mirror is bentby two equal moments.

Since k ¼ 0, the mirror is bent only by a single mo-ment, and the upstream ends of the mirror are sup-ported without extra moment. The single-moment

bent mirror has several significant advantages.First, although this type of mirror is bent by any un-designed moment, the surface shape is also a perfectellipse with nanoradian-level slope error, for the hardx-ray bent mirror has full variability of the grazingangle at the effective range [14]. Second, the mechan-ism system and the testing process can be simplifiedfor the mirror bent only by a single moment. Finally,the focusing resolution is determined by the preci-sion of width distributions and the zero-moment sup-port position. The only problem is that the width ofthe zero-moment support position is also zero as cal-culated according to Eq. (14) and can be solved by arevised design with the active mirror length beingreduced, as shown in Fig. 5(a). It should be noted thatthe zero-moment position must be supported withhigh precision.

Since k ¼ 1, the mirror is bent by two equal mo-ments. We know that the width of the upstream endsis the largest and the width of downstream ends viceversa in Fig. 4. Themirror bent by equal moments, asshown in Fig. 5(b), also has such advantages as sim-plifying the mechanism and testing process like themirror bent by a single moment. The main problemfor a mirror bent by equal moments is that the lengthof the mirror is limited by a slope of width distribu-tions that is not small.

For engineering considerations, it is appropriatethat the mirror ends are designed with equal widths.We consider the moment distribution MðsÞ is propor-tional to the line between the ends of the curvaturedistribution. The straight line between the ends ofthe curvature can be expressed as

yðsÞ ¼ Ksþ R−1c ðl=2Þ − Ksðl=2Þ; ð18Þ

Fig. 4. (Color online) Relative width distributions for differentmodels of moment distribution with parameter configurationp ¼ 5m, q ¼ 15 cm, θ ¼ 5mrad.

Fig. 5. Width-variable mirror: (a) bent by single moment, k ¼ 0;(b) bent by equal moments, k ¼ 1.

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with coefficient K , the slope of the straight line

K ¼ ½R−1c ðl=2Þ −R−1

c ð−l=2Þ�=L: ð19ÞThe moment distribution for an equal end-width

mirror is

MðsÞ ¼ 2a2EI0R−1

c ðl=2Þ − Ksðl=2Þ × yðsÞ: ð20Þ

Figure 6 shows the relative width distributions forequal end-width mirrors.

It should be noted in Fig. 6 that b0, the width ofthe mirror center, is not the maximum.

3. Discussions

Equation (14) is the exact expression that describesthe width distributions for elliptically bent mirrors.The moment distributions by two end couples can beexpressed by a linear function MðsÞ such as Eq. (20).In practical applications, the moment distributionsare also modulated by other factors [8]. For exam-ple, under the influence of gravity along the mirrorlength, the upward-surface mirror takes extra mo-ment distributions Mg, which have correspondingrelationship with width distribution bðsÞ. Thermalload on the mirror [15] and the different bendingmechanical systems maybe also take correspondingmoment distributions Mt and Mb. The momentdistributions with compensation should be repre-sented as

Mtotal ¼ M þMg þMt þMb þ � � � ð21Þ

Finally, the practical width distribution is ex-pressed by inserting Eq. (21) into Eq. (14). Obviously,the spatial resolution is also determined by the accu-racy of Eq. (21).

4. Conclusion

We have discussed all the approximations used forthe traditional elliptically bent mirror theory and de-duced the exact expression. The theoretical obstaclesfor bending a mirror to an ideal ellipse are removed.We have theoretically designed mirrors bent by a

single moment and equal moments. Because of itsadvantages, the mirror bent by a single moment isthe dynamic bending device with high potentialability for nanometer focusing.

We are grateful to Dr. Wanqian Zhu for signifi-cant discussions. This work was supported by theShanghai Postdoctoral Fund (10R21418200), theNational Natural Science Foundation of China(NSFC) (10805071, 11075200), and the NationalBasic Research Program of China (2010CB934501,2007CB936003, 2010CB834301).

References

1. H. Mimura, S. Handa, T. Kimura, H. Yumoto, D. Yamakawa,H. Yokoyama, S. Matsuyama, K. Inagaki, K. Yamamura,Y. Sano, K. Tamasaku, Y. Nishino, M. Yabashi, T. Ishikawa,and K. Yamauchi, “Breaking the 10nm barrier in hard-x-ray focusing,” Nat. Phys. 6, 122–125 (2010).

2. C. G. Schroer, “Focusing hard x rays to nanometer dimensionsusing Fresnel zone plates,” Phys. Rev. B 74, 033405 (2006).

3. A. Snigirev, V. Kohn, I. Snigireva, and B. Lengeler, “A com-pound refractive lens for focusing high-energy x-rays,”Nature384, 49–51 (1996).

4. H. C. Kang, J. Maser, G. B. Stephenson, C. Liu, R. Conley,A. T. Macrander, and S. Vogt, “Nanometer linear focusing ofhard x rays by a multilayer Laue lens,” Phys. Rev. Lett. 96,127401 (2006).

5. A. Jarre, C. Fuhse, C. Ollinger, J. Seeger, R. Tucoulou, andT. Salditt, “Two-dimensional hard x-ray beam compressionby combined focusing and waveguide optics,” Phys. Rev. Lett.94, 074801 (2005).

6. K. Evans-Lutterodt, A. Stein, J. M. Ablett, N. Bozovic,A. Taylor, and D. M. Tennant, “Using compound kinoformhard-x-ray lenses to exceed the critical angle limit,” Phys.Rev. Lett. 99, 134801 (2007).

7. L. Zhang, R. Hustache, O. Hignette, E. Ziegler, and A. Freund,“Design optimization of a flexural hinge-based bender forx-ray optics,” J. Synchrotron Radiat. 5, 804–807 (1998).

8. M. R. Howells, D. Cambie, R. M. Duarte, S. Irick,A. A. MacDowell, H. A. Padmore, T. R. Renner, S. Rah, andR. Sandler, “Theory and practice of elliptically bent x-raymirrors,” Opt. Eng. 39, 2748–2762 (2000).

9. F. Adams, L. Van Vaeck, and R. Barrett, “Advanced analyticaltechniques: platform for nano materials science,” Spectro-chim. Acta B 60, 13–26 (2005).

10. P. Kirkpatrick and A. V. Baez, “Formation of optical images byx-rays,” J. Opt. Soc. Am. 38, 766–773 (1948).

11. B. X. Yang, M. Rivers, W. Schildkamp, and P. J. Eng,“GeoCARS microfocusing Kirkpatrick–Baez mirror benderdevelopment,” Rev. Sci. Instrum. 66, 2278–2280 (1995).

12. A. C. Ugural and S. K. Fenster, Advanced Strength andApplied Elasticity (Prentice-Hall, 1995).

13. A. Pressley, Elementary Differential Geometry, SpringerUndergraduate Mathematics Series (Springer-Verlag, 2001).

14. C. Mao and X. Yu, “A study on the grazing angle variability ofbent elliptical microfocusing mirror,” AIP Conf. Proc. 879,686–689 (2007).

15. S. Yuan, M. Church, V. V. Yashchuk, K. A. Goldberg,R. S. Celestre, W. R. McKinney, J. Kirschman, G. Morrison,T. Noll, T. Warwick, and H. A. Padmore, “Elliptically bentx-ray mirrors with active temperature stabilization,” X-RayOpt. Instrum. 2010, 784732 (2010).

Fig. 6. Relative width distributions of equal end-width mirrorswith parameter configuration p ¼ 5m, q ¼ 15 cm, θ ¼ 5mrad.

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