influence of fabrication errors on wölter mirror imaging performance

Influence of fabricationerrors on Wolter mirror imaging performance

Katsumi Sugisaki, Shin-ich Takahashi, Yasuji Yoshidomi, Kuninori Shinada,Osamu Mitomi, Eishiro Uchishiba, Ryuji Hamada, Tokio Kato, and Sadao Aoki

The resolution of the Wolter mirror, which is utilized as an objective in soft-x-ray microscopes, is limitedby fabrication errors. We studied the relation between fabrication errors and imaging performance ofthe Wolter mirror to determine how this performance could be improved. Figure errors, which arecharacterized by low spatial frequency, were analyzed by ray tracing, and surface roughness, character-ized by high spatial frequency, was analyzed by modified ray tracing. Modified ray tracing was basedon ray tracing but took scattering into account. The results of these analyses were compared withexperimental data. As a result, we obtained a simple and practical fabricating tolerance criterion thatmay be employed to obtain higher Wolter mirror resolution. Additionally, we discuss problems incurrent Wolter mirror fabrication techniques and the changes that might be made in both the design andthe fabrication process to improve imaging performance. © 1998 Optical Society of America

OCIS codes: 340.0340, 340.7470, 290.5880, 220.0220, 340.7460, 110.7440.

1. Introduction

The Wolter-type x-ray mirror1,2 offers many advan-tages, including high collecting efficiency and low ab-erration. For these reasons it has been widelyapplied as an objective in x-ray telescopes and has beenexperimentally applied in x-ray microscopes. In ad-dition, the Wolter mirror can potentially play an im-portant role in soft-x-ray imaging microscopes with

When this research was performed, K. Sugisaki and S. Takahashiwere with Tsukuba Research Laboratory, Nikon Corporation, 5-9-1Tokodai, Tsukuba, Ibaraki 300-2635, Japan. K. Sugisaki and S.Takahashi are currently with First R & D Department, R & DHeadquarters, Nikon Corporation, 1-6-3 Nishi-ohi, Shinagawa-ku,Tokyo 140-8601, Japan. Y. Yoshidomi, K. Shinada, O. Mitomi, E.Uchishiba, R. Hamada, and T. Kato were with Sagamihara R & DDepartment, Nikon Corporation, 1-10-1 Asamizodai, Sagamihara,Kanagawa 228-0828, Japan. Y. Yoshidomi, K. Shinada, E. Uchi-shiba, and R. Hamada are currently with Third R & D Department,R & D Headquarters, Nikon Corporation, 1-10-1 Asamizodai,Sagamihara, Kanagawa 228-0828, Japan. O. Mitomi is currentlywith Machinery R & D Department, Nikon Corporation, 276-6,Motoishikawa-Cho, Mito, Ibaraki 310-0843, Japan. T. Kato is cur-rently with System Integration Department, Nikon Corporation1-6-3 Nishi-ohi, Shinagawa-ku, 140-8601, Japan. S. Aoki is withthe Institute of Applied Physics, University of Tsukuba, 1-1-1 Ten-nodai, Tsukuba, Ibaraki 305-0006, Japan.

Received 5 February 1998; revised manuscript received 27 July1998.

0003-6935y98y348057-10$15.00y0© 1998 Optical Society of America

wavelengths from 2.3 to 4.4 nm for biological analysis,since such applications require a highly efficient objec-tive for prevention of excessive x-ray irradiation.

Many attempts to fabricate high-resolution Woltermirrors for x-ray microscope applications have beenmade.3–8 However, resolution has not yet been ad-equate. A Wolter mirror with a N.A. of 0.2 used withan x-ray wavelength of approximately 3 nm has therecognized potential to achieve better than 10-nmresolution at the theoretical diffraction limit, but ow-ing to residual errors caused by current fabricatingtechniques its imaging performance has been limited.

Investigating the relation between fabrication er-rors and imaging performance is necessary to im-prove this performance. Several such studies havebeen conducted.9–12 However, these studies wereconducted with respect to specific mirrors, and theirresults cannot be generalized to all Wolter mirrors.

In general, a fabrication error spectrum’s ampli-tude has a tendency to be larger for lower spatialfrequencies. Such low-frequency errors are referredto as figure errors. The figure error spectrum oftenhas several peaks in the low-spatial-frequency re-gion, which indicate periodic structures. Suchpeaks characterizing the figure error spectrum arecaused by machining procedures, as in the case ofperiodic marks left by tools.13,14 By contrast, thefabrication error spectrum’s amplitude tends to besmaller for higher spatial frequencies. Such high-frequency errors are referred to as surface roughness.

1 December 1998 y Vol. 37, No. 34 y APPLIED OPTICS 8057

wW

r

8

Surface roughness has a random structure and isdetermined by polishing.

There are two approaches for analyzing the influ-ence of fabrication errors on imaging performance.Geometrical optics is employed for figure errors; dif-fraction optics, for surface roughness. Geometricaloptics is based on geometrical reflection, ignoring dif-fraction phenomena. Geometrical optics is used forstudying errors of a size much larger than the x-raywavelength—large enough that diffraction need notbe considered. By contrast, diffraction optics takesinto account the effect of interference. Diffractionoptics is therefore used to analyze wave-front errorsof a size roughly equivalent to or smaller than thex-ray wavelength. Wave-front errors are twice thesize of the projected surface errors, which are smallerthan the actual surface errors by a factor of the sineof the grazing angle. Thus, to effectively analyze allfabrication errors, we must employ two different ap-proaches to account for the full range of error spatialfrequency.

We also have attempted to fabricate a high-resolution Wolter mirror by vacuum replication.7,15

In our method a master mandrel was accuratelyshaped into a reverse form by grinding and polishing.A Wolter mirror made of glass was then replicatedfrom the master mandrel, and its inner surface waspolished. In this case figure errors were the result ofthe mandrel machining, and figure error deviationwas approximately 200 nm peak to peak, while sur-face roughness was determined by the final polishing,and rms value of surface roughness was approxi-mately 0.3 nm.

In this paper we present analyses of the influenceof fabrication errors on this Wolter mirror. We di-vide fabrication errors into two categories, that is,figure errors and surface roughness, according totheir spatial frequency. We study the influences ofthe fabrication errors on the mirror’s imaging perfor-mance by using ray-tracing calculations, the firstbased on geometrical optics, and the second takingdiffraction scattering into account. These calculatedresults are compared with experimental data to con-firm their reliability. We then derive a tolerancecriterion for fabrication errors from these analyses.Finally, we discuss problems in current fabricationtechniques and changes that might be made in boththe design and the fabrication process to improveWolter mirror imaging performance.

2. Basic Theory and Theoretical Estimations

We analyzed a Wolter mirror designed for soft-x-raymicroscope application ~Fig. 1!.

An error spectrum can be obtained by conversionfrom the fabrication error surface profile ε~z! througha Fourier transform, as follows:

ε~z! 5 (n51

` Fan sinSnpzL D 1 bn cosSnpz

L D]5 (

n51

`

a~n!sin~2pvz 1 f!, (1)

058 APPLIED OPTICS y Vol. 37, No. 34 y 1 December 1998

where a~n! is the amplitude of the fabrication errorspectrum, L is the mirror length, n is an integer, andn is the spatial frequency of the fabrication error.The spatial frequency n is related to the spatial wave-length l of the fabrication error according to the fol-lowing equation:

n 51l

5n

2L. (2)

As mentioned above, fabrication errors a~n! may bedivided into figure errors and surface roughness ac-cording to their spatial frequency n.

A. Figure Error Analysis by Geometrical-Optics Theory

In general, the effect on imaging performance of anaxial error, that is, an error along the optical axis, ismuch more serious than that of a lateral error, thatis, an error occurring perpendicular to the axis.10,16

Therefore we discuss only the influence of an axialerror.

Rays incident to the mirror are deflected by a figureerror. The reflected rays arrive at displaced posi-tions on the image plane. The displacement of thedeflected ray on the image plane is given by 2Duxf,where Dux is the slope of the figure error at the inci-dent position and f is the distance between the mirrorand the image plane ~Fig. 2!. The displacement Dxof the deflected ray on the objective plane, which isconverted from that on the image plane, is expressedas

Dx 52Duxf

M, (3)

here M is the magnification of the Wolter mirror.hen we place a sine-wave figure error, ε~z! 5

Fig. 1. Schematic diagram of our Wolter mirror. The magnifi-cation and grazing-incidence angles are M 5 32 and u 5 40 mrad,espectively. This mirror was designed for 4.5-nm x rays.

Isa

fme

rwa

M

Ttrmne

wgrt

ql

wl

S

a~n!sin~2pnz!, on our Wolter mirror, the slope Dux isgiven by

Dux 5dεdz

5 2pna~n!cos~2pnz!. (4)

The image degradation caused by this figure error is

Dx 5 2pn 3 a~n!f

Mcos~2pnz!. (5)

Therefore the degradation is proportional to the prod-uct of the spatial frequency n and the amplitude a~n!.n other words, the degradation is determined by thelope, which is determined by the frequency and themplitude of the fabrication error.To estimate the magnitude of the influence on per-

ormance of a typical figure error in the case of ourirror ~shown in Fig. 1!, we assumed a sine-wave

rror of n 5 1 mm21 and a~n! 5 10 nm. Sinceucos~2pnz!u # 1, Du is

262.8 mrad # Du # 62.8 mrad.

Thus image degradation in this case, with the doublereflection of the Wolter mirror being taken into ac-count, is given by

uDxu # 2 3 2Duf

M> 11 mm. (6)

Therefore the rays deflected by this figure error focusinto a spot with an 11-mm radius on the object planeand produce a bell-shaped point-spread function.

B. Surface Roughness Analysis by Diffraction Theory

Surface roughness causes scattering, which degradesthe imaging performance of the Wolter mirror.When a wave-front aberration due to surface rough-ness is roughly equivalent to or smaller than thex-ray wavelength applied, the resulting scatteringmust be analyzed by diffraction theory.

In the case of the Wolter mirror, wave-front aber-ation due to surface roughness is given by 2s sin u,here s is the rms height of the surface roughnessnd u is the grazing-incidence angle. According to

Fig. 2. Configuration and symbol expressions for figure erroranalysis.

areshal’s criterion16, we must reduce the wave-front aberration to less than ly14 to obtain adequateimaging performance. Therefore we must reducesurface roughness according to the following relation:

s ,l

141

2 sin u

1

Î25

l

28Î2 sin u. (7)

his relation must be divided by the square root of 2o account for the double reflections of a Wolter mir-or. When this criterion is applied to our Wolterirror, we obtain s , 1.3 nm. If the surface rough-ess is less than this value, we can ignore the influ-nce of scattering that is due to surface roughness.If the surface roughness is larger than this value, itill cause significant scattering. The scattering an-le and intensity are determined by the surfaceoughness spatial frequency and amplitude, respec-ively.

The scattering angle is related to the spatial fre-uency of the surface roughness according to the fol-owing grating equation:

nl 5 d~cos us 2 cos ui!, (8)

here n is the diffraction order; l is the x-ray wave-ength; d is the grating pitch; and ui and us are the

grazing angles of incidence and diffraction, respec-tively ~Fig. 3!.

To estimate the displacement Dx, due to scattering,from the ideal focusing position on the object plane,we define the scattering angle Dus as follows:

us 5 ui 1 Du. (9)

ubstituting this equation into Eq. ~8!, we obtain

nl 5cos~ui 1 Dus! 2 cos ui

n. (10)

Using the approximations of cos Dus > 1 and sin Dus> Dus and limiting to n 5 1, we obtain

Dus 5 2ln

sin ui. (11)

Fig. 3. Configuration and symbol expressions for the surfaceroughness analysis.


fwf

c

8

When Eq. ~11! is applied to our Wolter mirror, thedisplacement Dx is converted from the image plane tobe

Dx 5Dus fM

. (12)

To estimate the magnitude of the influence of sur-ace roughness on our Wolter mirror’s performance,e assumed surface roughness of a typical spatial

requency of 10 mm21. We obtained Dus 5 0.75mrad and Dx 5 32 mm at the x-ray wavelength l 5 3nm. Thus scattered rays that were due to this high-frequency surface roughness spread out beyond a 32-mm-radius circle and overlay the resulting image asbackground noise.

The scattering intensity distribution is propor-tional to the power spectral density ~PSD!, W~p!, ofthe surface roughness, which is the square of thespectrum amplitude, as follows13,17:

dIdus

52k3

pIR sin ui sin2 us W~p!

516p2

l3 IR sin ui sin2~ui 1 Dus!WS22p sin u

lDusD ,

~k 5 2pyl!. (13)

By use of approximations of cos Dus > 1 and sin Dus >Dus, the scattering intensity is given by

dIdus

>16p2

l3 IR~sin3 ui 1 2 sin2 ui cos uiDus!W~p!. (14)

This relation indicates that the scattering intensity isnegligibly small when W~p! is much smaller than thecube of the wavelength l. Note that W~p! in thispaper is expressed as a one-dimensional PSD. Ad-ditionally, the PSD of the surface roughness can bederived from the scattering intensity profile by use ofrelation ~14!.

3. Ray-Tracing Analyses and Experiments

A. Figure Errors

Using ray-tracing calculations, we analyzed the in-fluences of two types of figure error on the imagingperformance of our Wolter mirror. The first was asine-wave type of figure error, and the second was theresidual figure error as measured in our fabricatedmirror. The sine-wave error that we selected corre-sponded to a peak in the error spectrum that in turncorresponded to the periodic structure produced byour mirror machining procedure. The residual fig-ure error of our mirror was analyzed for comparisonwith experimental results.

Based on these ray-tracing calculations, resolutionwas measured and images were simulated to evalu-ate the corresponding image degradation resultingfrom the figure errors. We experimentally evalu-ated the actual influence of figure error on the imag-


ing performance of our fabricated Wolter mirror bymeasuring resolution and by assessing image quality.To compare our calculated results with the experi-mental results, the knife-edge resolution criterionwas adopted as our standard for its ease of applica-tion. By this criterion, resolution is defined as thewidth between the positions of 25% and 75% of themaximum intensity in the intensity profile of theknife-edge image.18,19 The actual images experi-mentally obtained were directly compared with oursimulated images.

1. Ray-Tracing CalculationsResolution evaluation by ray-tracing calculation wasperformed according to the following procedure.First, a spot diagram was obtained based on ray-tracing calculations for our Wolter mirror with eachtype of figure error. Next, the line-spread function~LSF! was derived from this spot diagram. By inte-gration of the LSF the intensity profile of the theo-retically projected knife-edge image was obtainedand its resolution calculated.

Imaging simulation was performed by numerousray-tracing calculations for the rays that would beemitted through the open areas of a patterned tem-plate. We derived the resulting pattern image bymapping the ray distribution on the image plane.

All ray-tracing calculations were performed withcustom software that we have written for analyzingthe influence of the fabrication error on the Woltermirror.

2. Experimental MethodsWe evaluated the performance of our fabricatedWolter mirror by knife-edge resolution measure-ments and by actual imaging. The evaluation sys-tem employed is shown in Fig. 4. The x rays emittedfrom a plasma produced by a Nd:YAG laser beampassed through an x-ray filter ~Ti: 1.0 mm! and wereollected by a 1y4 Wolter-type condenser mirror.

The test sample was illuminated by this condensermirror. A Si knife-edge and a Au line and spacepattern on SiN film were used as test samples. Thex rays passing through the sample were enlarged byour Wolter mirror under evaluation, and images were

Fig. 4. Schematic diagram of the evaluation system for x-rayimaging performance.

pt

oweod

3FaslaTttttert

w

wficrt

7mttw

ptiirwc

rspdoiepmp

etff0

obtained by a zooming tube x-ray detector.20 Therojected x-ray images of the test samples were usedo evaluate our Wolter mirror.

An anisotropically etched Si edge was used as anptical knife edge for our test sample. It was coatedith Au to prevent x rays from penetrating it. Knife-

dge resolution was evaluated from an intensity profilef the knife-edge image. The LSF was also derived byifferentiation of this intensity profile.

. Resultsigure 5 shows the calculated resolution contour maps a function of the amplitude and frequency of theine-wave figure errors. As mentioned above, reso-ution is proportional to the product of the amplitudend the spatial frequency of a sine-wave figure error.he contour lines shown in Fig. 5 also represent theolerance criterion for sine-wave figure errors. Aarget resolution can be achieved if the error spec-rum of wavelike figure errors is kept below the con-our line corresponding to that resolution. Forxample, to achieve 0.1-mm resolution, we need toeduce the error spectrum to beneath the 0.1-mm con-our line in Fig. 5.

From Eq. ~5!, the resolution determined by a sine-ave error is represented as

d 5 kna~n! f

M, (15)

here d is the knife-edge resolution and k is a coef-cient. The coefficient k 5 9.3 was derived from thealculated resolution contour map for our Wolter mir-or. Thus the tolerance criterion for achieving a cer-ain resolution d is represented as

a~n! ,dM

9.3nf. (16)

Figure 6 shows the figure error profile for our fab-ricated Wolter mirror experimentally measured witha custom-built ultraprecision instrument.21 These

Fig. 5. Resolution contour map for various amplitudes and fre-quencies of sine-wave errors obtained by ray-tracing analyses.

error measurement data were sampled at 0.1-mmpitch intervals and were applied to ray-tracing cal-culations.

Ray tracing was used to calculate the LSF and theknife-edge profile @Figs. 7~a! and 7~b!, respectively#.We obtained, on the basis of this knife-edge profile, aresolution of 1.29 mm through the previously ex-plained calculations. The experimentally obtainedLSF and knife-edge profile are shown in Figs. 7~c! and~d!, respectively. The knife-edge resolution of 1.18m was derived from Fig. 7~d!. Both the experimen-

ally obtained resolution and the experimentally ob-ained shape of the LSF quantitatively agree wellith those acquired by ray-tracing calculations.Figures 8~a! and 8~b! are images of a line and space

attern obtained by ray-tracing calculations based onhe figure error profile shown in Fig. 6 and by exper-ment, respectively. The experimental image is sim-lar to the calculated image. Since our theoreticalesults agreed with those obtained experimentally,e believe that our ray-tracing calculations can be

onsidered reliable.

B. Surface Roughness

1. Analysis MethodsIn our modified ray-tracing analysis of surface rough-ness, we incorporated the effect of scattering into theray-tracing calculation by both specularly and dif-fusely reflecting rays from a Wolter mirror accordingto a scattering angular distribution.22 In this calcu-lation some of the rays incident to the Wolter mirror’sfirst mirror ~the hyperboloid mirror! are specularlyeflected while others are diffusely scattered. Thecattering’s angular distribution determines therobability of the rays’ being reflected specularly oriffusely. All the rays are again reflected specularlyr diffusely by the second mirror ~the ellipsoid mirror!n the same manner. In our calculations we consid-red only rays diffusely reflected in the incidentlane, since the image degradation that they cause isuch worse than that caused by rays scattered per-

endicular to the incident plane.In our modified ray-tracing calculations, we used

xperimentally obtained scattering angular distribu-ions measured with a glass surface. The glass sur-ace had been polished in the same manner as ourabricated Wolter mirror and had approximately.3-nm rms surface roughness as measured with a

Fig. 6. Figure error profile measured for our Wolter mirror.


t

8

WYKO Corp. TOPO-3D microsurface measurementsystem.

The scattering measurements were performed withan x-ray reflectometer ~Fig. 9!. The apparatus con-sisted of an x-ray generator, a paraboloid x-ray colli-mator mirror, two aperture slits, a sample stage, u and2u stages, and a gas flow proportional counter as adetector—all mounted inside a vacuum chamber.The collimator mirror reflected the x-ray beam fromthe generator, which then passed through the two ap-erture slits to arrive incident to the sample surface.Scattered x rays were measured by the gas flow pro-portional counter mounted on the 2u stage. All thescattering measurements were performed with C ka xrays ~l 5 4.4 nm!.

The angular distribution of the scattering was ob-tained by subtraction of the normalized incident-

Fig. 7. Comparison of calculated and experimental results: ~atracing, ~c! LSF obtained experimentally, and ~d! knife-edge profil

Fig. 8. Images of a line and space pattern obtained ~a! by ray-racing calculation, and ~b! by experimental x-ray imaging. Both

the line and the space widths equal 0.45 mm.


beam angular distribution from the reflected beamangular distribution. The angular region in whichscattering can be measured is limited on both theinside and the outside by the incident beam’s angularwidth and by detector noise, respectively.17 Thusthe measurable spatial frequency of the surfaceroughness corresponding to the scattering angle islimited to a certain region. We therefore measuredscattering angular distributions in three differentspatial frequency regions by changing the incidenceangles according to Eq. ~11!. The scattering angulardistributions obtained were converted to the scatter-ing angular distributions that would result underactual Wolter mirror scattering conditions and werethen used in the modified ray-tracing calculations.

2. ResultsWe obtained scattering angular distributions corre-sponding to three spatial frequency regions: ~I! The

obtained by ray tracing, ~b! knife-edge profile obtained by rayained experimentally.

Fig. 9. Schematic diagram of the reflectometer used in the scat-tering measurements.

! LSFe obt

4

mFkrufsxssib

ibt

twIsisi

i

region from 1.75 to 10.9 mm21, which was measuredat the grazing incidence of 0.4°; ~II! the region from.17 to 22.7 mm21, which was measured at 1.0°; and

~III! the region from 19.6 to 52.6 mm21, which wasmeasured at 2.5°. Figure 10 shows scattering pro-files obtained for region II. PSD’s for the three dif-ferent spatial frequency regions of surface roughnesswere derived from their scattering profiles ~Fig. 11!.

The scattering angular distributions used in theodified ray tracing were derived from these PSD’s.igures 12~a! and 12~b! are a calculated LSF and anife-edge profile, respectively, derived by modifieday-tracing calculations. These were obtained byse of the scattering angular distribution derivedrom the scattering profile for region II. Since thecattering has caused no change in the profile of the-ray spot, the resolution will also not be affected bycattering. Although the background noise is toomall to appear in Fig. 12, the scattering will slightlyncrease this noise, which will affect image contrastut not resolution.Figure 13 shows simulated 1000-lineymm grating

mages that were obtained by modified ray tracingoth with and without scattering. Compared withhe perfect, unscattered image shown in Fig. 13~a!,

Fig. 10. Example of scattered and incident beam profiles exper-imentally obtained. ~a! shows the complete profiles. ~b! shows amagnification of the high angle region in which scattering mea-surements reached the detector noise level. These profiles weremeasured with a polished Pyrex glass sample at a grazing-incidence angle of 1.0°.

he image contrasts in Figs. 13~b!, 13~c!, and 13~d!,hich are images affected by scattering from regions

, II, and III, respectively, are degraded because of thecattering. However, the fine structure of the grat-ng can be recognized. The contrast degradationeen in Figs. 13~b! and 13~c! is not so serious, but thatn Fig. 13~d! is significant.

The influence of surface roughness that can be seenn our scattering analysis, especially in the case of

Fig. 11. PSD functions derived from the scattering profiles. Thecurves correspond to the grazing-incidence angles of ~I! 0.4°, ~II!1.0°, and ~III! 2.4°, respectively. The rms values of the surfaceroughness calculated from the PSD function are also indicated.

Fig. 12. Results obtained by modified ray tracing: ~a! LSF, and~b! knife-edge profile.


wosn

sttLatrt1se

wtc0attitfe

li

8

Fig. 13~d!, seems to be greater than that actuallyshown in Fig. 8~b!. Although the results obtained

ith our scattering analysis do not agree well withur experimental observations, we can conclude thatcattering may cause an increase in backgroundoise, which reduces image contrast.Table 1 summarizes the results obtained by our

cattering analyses. The modulation transfer func-ion values are modulations at a 1000-lineymm spa-ial frequency and were derived from the calculatedSF. Scattering due to surface roughness regions IInd III did not degrade resolution but reduced con-rast slightly. Scattering due to surface roughnessegion I greatly degraded both resolution and con-rast. The surface roughness values given in Tablewere derived from PSD’s that were derived from our

cattering measurements. Surface roughness great-r than ly14 ~i.e., 1.3 nm!, as seen in region I, seriously

degraded imaging performance. Although the rmsroughness obtained from region I is close to ly14, thePSD components beyond ly14 strongly degrade itsimaging performance.

Fig. 13. Influence of scattering as seen in simulated images basedon modified ray tracing: ~a! perfect image uninfluenced by scat-tering, ~b! image influenced by region III scattering, ~c! imageinfluenced by region II scattering, and ~d! image influenced byregion I scattering.

Table 1. Summary of Our Theoretical and Experimental Analyses ofthe Effect of Scattering

Region

GrazingAngle~deg!

SpatialFrequency

~mm21!

rmsRoughness

~nm! MTFa

Knife-EdgeResolution

~mm!

I 0.4 1.75–10.9 1.5 0.24 9.08II 1.0 4.17–22.7 0.5 0.84 0.0III 2.5 19.6–52.6 0.08 0.99 0.0

aModulation transfer function.


4. Discussion

Figure 14 illustrates a summary of our analyses. Itis overlaid with the various residual error spectraobtained with our fabricated Wolter mirror, Mare-shal’s criterion16, CD, and the tolerance criterion, AB,

hich we have derived. The intersection point ofhe lines AB and CD is defined as E. The toleranceriterion was determined by the contour line for.1-mm resolution derived by geometrical ray-tracingnalyses. As such, this figure represents the rela-ionship between the fabrication error spectrum andhe tolerance criterion for fabrication error for achiev-ng 0.1-mm resolution. The complete tolerance cri-erion covering the full range of error spatialrequency consists of the two lines, AE and ED, in therror spectrum.According to ray-tracing analysis, which is valid for

ow-frequency error, the rays scattered by such errorsn the A–E–C area of Fig. 14 focus onto an acceptable

region on the image plane corresponding to the targetresolution. Therefore fabrication errors in the areaof A–E–C are permitted even though their spectraexceed Mareshal’s criterion.

In contrast, the other line, CD, determined byMareshal’s criterion has a constant value for all highspatial frequencies. The error spectrum compo-nents in the area of B–E–D are also acceptable eventhough they exceed the criterion defined by ray-tracing analysis. According to Mareshal’s criterion,wave-front aberration due to high-spatial-frequencyerrors in this B–E–D area remains in the acceptablerange, and the x-ray wave front is able to form anideal focusing spot, referred to as an Airy pattern.

In our figure error analyses, we have treated figureerror as consisting of a single sine-wave error. Al-though an actual figure error spectrum consists oftwo or more peaks, that is, it is a composite of severalsine-wave figure errors, our tolerance criterion is still

Fig. 14. Relation between results derived from theoretical anal-yses ~lines AB and CD! and the measured spectra of residualmanufacturing errors.

useful. In the case of an error spectrum consisting ofseveral peaks, imaging performance is primarily de-termined by the size of the largest peak relative to thetolerance criterion. In Fig. 14 the actual figure errorspectrum of our fabricated Wolter mirror consists ofmany peaks. Since the largest peak in this errorspectrum exceeds the resolution contour line thatwould correspond to 1 mm ~indicated as a dashed linein Fig. 14!, we believe that this peak limits resolutionby approximately 1 mm. Therefore the imaging per-formance of our Wolter mirror was determinedmainly by figure error.

In comparing the fabrication accuracy of ourWolter mirror with the tolerance criterion for 0.1-mmresolution, we find that fabrication errors at approx-imately the 0.3-mm21 spatial frequency, near pointE, show the greatest difference from the criterion.We therefore believe that the error spectrum regionnear this intersection point E is the most responsiblefor serious degradation of our Wolter mirror’s imag-ing performance. This region has been referred toby other researchers as the midspatial frequency re-gion.23,24

There are two methods clearly available by whichwe may improve the imaging performance of ourWolter mirror. First, we must bring residual fabri-cation errors to within the tolerance criterion by re-ducing their amplitude, and, second, we must relaxthe fabrication tolerance criterion itself. Since thereare many peaks in the error spectrum near the in-tersection point of the two tolerance lines, we mustdevelop fabrication techniques to reduce these par-ticular peaks. By contrast, to enlarge the tolerancecriterion, we can select other design parameters ac-cording to relation ~7! and ~16!. By selecting ashorter focal length andyor a larger magnification wecan move the oblique line AB in Fig. 14, determinedby geometrical optics, toward the upper right. Byselecting a smaller grazing angle we can move thehorizontal line CD in Fig. 14, determined by diffrac-tion theory, upward. Both approaches will effec-tively relax the tolerance criterion. However, byselection of such design parameters the inner diam-eter of the mirror becomes smaller, and, accordingly,its fabrication becomes more difficult. Furthermore,the effective collecting area becomes smaller, that is,less efficient, and the focal plane plate scale becomeslarger. Equipment employing such optics wouldhave to be larger to accommodate this situation.Therefore the ideal tolerance criterion must be bal-anced with the ease of practical fabrication.

The analytical methods that we have discussedhere, including the determination of a fabrication tol-erance criterion as well as design optimization, can beapplied not only to the Wolter mirror but also to otherx-ray optical systems.

5. Conclusions

Our analyses of figure error determined that resolu-tion was proportional to the product of the amplitudeand the spatial frequency of the error. The resultsthat we obtained by geometrical ray-tracing analyses

were confirmed by comparison with experimental re-sults. Scattering due to surface roughness, charac-terized by high spatial frequency, was found to causeimage contrast degradation but did not significantlyaffect the resolution of the mirror.

A tolerance criterion for Wolter mirror fabricationwas derived from these results, and the residual fab-rication error of our Wolter mirror was comparedwith it. The error spectrum near the intersectionpoint of the tolerance criterion lines was found to bethe most serious cause of image degradation.

On the basis of our results, we have determinedthat Wolter mirror imaging performance may be ef-fectively improved by development of improved fab-rication techniques and by careful selection ofoptimum design parameters.

The analysis methods presented here are applica-ble not only to the Wolter mirror but also to otherx-ray optical systems.

References1. H. Wolter, “Spiegelsysteme streifenden Einfalls als abbildende

Optiken fur Rontgenstrahlen,” Ann. Phys. 10, 94–114 ~1952!.2. H. Wolter, “Verallgemeinerte Schwarzschildsche Spiegel-

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influence of fabrication errors on wölter mirror imaging performance

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