Interferometric measurement of near-cylindricalsurfaces with high amplitude resolution

Alfred Gatzweiler and Holger Glatzel

Subaperture interferometric measurements of highly curved surfaces with a shape close to a cylinderhave been performed in a normal-incident setup that consists of a Fizeau interferometer in combinationwith a plano–concave cylindrical lens. Since the field of view in the circumferential direction is limitedby spherical aberration, the optical components were designed to minimize spherical aberration. Forreference measurements a second plano–convex cylindrical lens was used. The subaperture setup leadsto three-dimensional surface maps of the objects under test. To eliminate the influence of residualgeometric aberrations, rectangular polynomials have been fitted and subtracted from the raw data. Fordeformations with spatial wavelengths below 30 mm, a rms amplitude resolution of 1 nm and a rmsamplitude accuracy of 3 nm were achieved. Measurements on Wolter-type-I mirror shells are discussedin detail.Key words: Interferometry, metrology, x-ray optics, cylindrical surfaces. r 1995 Optical Society of

America

1. Introduction

The x-ray telescope XMM 1x-ray multimirror2 of theEuropean Space Agency has a Wolter-type-I grazing-incidence optical design and is composed of threemodules. Each module consists of 58 nested mirrorshells.1,2 The shells are 600 mm long and diametersrange from 300 to 700 mm. To meet the weightrequirement of 1 kg@mirror, carriers made of carbonfibers embedded in an epoxy resin matrix 1Carbonfiber reinforced plastic23 were epoxy replicated froman Al@Ni master piece.4–7 The walls of the mirrorsare only between 0.6 and 1.2 mm thick. Since theepoxy resin and the carbon fibers expand differentlyunder changes of moisture or temperature, a print-through of the carbon fiber structure on the mirror’ssurface was expected.4,8 To monitor this effect, ameasuring technique, which fulfills the followingcriteria, had to be developed:

1a2 amplitude resolution and accuracy of a few

When this research was done both authors were with Carl ZeissCompany, D-73446 Oberkochen, Germany. A. Gatzweiler is nowwith the RRT Company, Aachener Strasse 1349, D-50859 Koln,Germany.Received 23 November 1994; revised manuscript received 22

June 1995.0003-6935@95@317207-06$06.00@0.

r 1995 Optical Society of America.

nanometers for spatial wavelengths ranging from afew millimeters to several tens of millimeters,

1b2 determination of the three-dimensional topol-ogy to gain insight of the printthrough phenomena,

1c2 contactless because of the soft and flimsyshells, and

1d2 short measuring time.

Since the flimsy shells pick up small amounts ofacoustic noise, the influence of vibrations has to bereduced by averaging over a large number of measure-ments.

2. Optical Test Setup

To achieve the high amplitude resolution in spite oftest object vibrations, the chosen instrument was astandard Carl Zeiss interferometer, DIRECT100,9which uses light with a wavelength of 632.8 nm. Itsstroboscopelike data sampling makes the instrumentnearly insensitive to vibrations and air turbulences.The chosen subaperture normal-incidence setup,which was first applied by Geary,10–14 is displayed inFig. 1. To access the inner surface of the tube-shaped optic, a flat folding mirror is necessary.Since the subaperture shape of the mirrors is close toa cylinder,2 themeasuringwave front could be adaptedto the shape of the test surface by a cylindrical lens,which is positioned with its focal line in the rotationalaxis of the cylindrical test surface 1see Fig. 22.14With two cylindrical lenses with focal lengths of 2100

1 November 1995 @ Vol. 34, No. 31 @ APPLIED OPTICS 7207

and 2150 mm, all the XMM mirror shells with radiiranging from 150 to 350 mm could be measured.The field of view is limited by the aperture of theprecision cylindrical lenses. Their geometric dimen-sions of 60 mm 3 26 mm lead to subaperture widthsof from approximately 60 mm 3 30 mm to 60 mm 390 mm, depending on the radius of the surface undertest and the focal length of the cylindrical lens. Theoptics of the interferometer could be adjusted to afield of view of 62 mm.Because of the afocal arrangement of the lens and

test object 1F8 5 M2, the wave front reflects on itselfwhen it is used to pick up the surface information ofthe test object as an optical path difference. After itpasses through the cylindrical lens a second time, thewave front is nearly plane again and the interferencewith the plane reference wave front can be evaluated.In addition to the deformations of the test surface, theinterference pattern picks up the surface deforma-tions of the optical components, principally those of

Fig. 1. Principal test configuration with a Fizeau interferometerand x-ray optics.

Fig. 2. Two orthogonal principal intersections of the optical testsetup.

7208 APPLIED OPTICS @ Vol. 34, No. 31 @ 1 November 1995

the cylindrical lens, and the imaging aberrations ofthe optical system.In the following sections we discuss the measure-

ments undertaken to minimize these errors and theachieved resolutions and accuracies.

3. Imaging Aberrations

The imaging aberrations of the optical system, consist-ing of the interferometer’s optics, the cylindrical lens,and the test surface, cause the reflected wave front tobe curved. Three aberration types dominate:

1a2 a tilt of the focal line of the cylindrical lens andthe rotational axis of the cylindrical test surface,

1b2 a decenter of the focal line of the cylindricallens and the rotational axis of the test object 1defocus2,and

1c2 spherical aberration of the optical system con-sisting of the cylindrical lens and the test object.

The contribution of the interferometer’s optics can beneglected. Since we are dealing with cylindricalinstead of spherical symmetry, the term cylindricalaberration would be more suitable.The aberration tilt and defocus 31a2 and 1b24 can be

reduced by a careful alignment of the components.Since the spherical aberration of the cylindrical lensis high 1see Fig. 42, its optical parameters had to becarefully adapted to the test object to avoid a reduc-tion of the measured field on the test surface byspherical aberration. If the wave-front slope causesa fringe pattern that cannot be resolved by theinterferometer’s detector, which is a charge-coupleddevice 1CCD2, a measurement at this location is nolonger possible.15 The parameter that limits thefield of view is the distance between fringes in theplane of the detector, which must not be lower thantwice the pixel distance on the CCD detector.The radii of the mirror surfaces range from 150 to

350 mm. With two commercially available plano–concave lenses it was possible to measure all thesurfaces with tolerable imaging aberrations. Thefull apertures of the cylindrical lenses could be used.To optimize spherical aberration, the flat surface ofthe cylindrical lens has to face the surface under test1see Fig. 22.16 The measurable fields on the testsurfaces were 60 mm in the axial direction andbetween 26 and 90 mm in the circumferential direc-tion.Even though we tried to minimize the imaging

aberrations by a careful alignment and a suitableoptical design, they still dominate the interferogram.Figure 3 displays a typical three-dimensional profilederived from the interferogram. A circumferentialcut shows the typical shape caused by the influence ofdefocus and spherical aberration 1Fig. 42. To gain abetter view of the deformations in the millimeterrange, these aberrations were removed from the data.Zernike polynomials do not represent an appropriateset of functions for a fitting procedure, since thegeometry of the discussed configuration has no rota-

tional symmetries. It would be necessary to increasethe degrees of freedom and more information wouldbe lost through the fitting routine. Furthermore,since Zernike polynomials are defined only on el-lipses, the data at the edges of the rectangular objectfield would be lost completely. For this reason thefollowing nonorthogonal set of polynomials was fittedto the data and subtracted. The Cartesian coordi-nates on the surface of the test object are representedby x and y:

Pi, j1x, y2 5 xiyj with i 1 j # n, 112

where n is the order of the polynomial. To be able tocompensate spherical aberrations, polynomials of theorder of n 5 4 had to be applied.Figure 5 shows the function fitted to the raw data of

a typical surface 1Fig. 32. The data after subtractionof the fitted function are displayed in Fig. 6. Anundesired side effect of any fitting technique is a lossof information on the global shape of the test surface.Aone-dimensional Fourier expansion showed that thefirst two Fourier components with wavelengths equalto the profile length and half of the profile length arestrongly reduced by a fit of polynomials to the fourthorder. Since imaging aberrations are not present inthe axial direction, untreated axial profiles with fullinformation over the entire spatial wavelength spacecan be used for optical performance prediction.Fortunately, for Wolter optics with a grazing angle ofincidence, deformations in the circumferential direc-tion can be neglected.

4. Instrument Calibration

Beside the deformations of the test surface t1x, y2, theinterferometric measurementmt1x, y2 picks up the

Fig. 4. Profile in the direction of mean intersection plane IIshowing a combination of defocus and spherical aberration.

Fig. 3. Raw data of a measurement on a Wolter-type-I mirrorshell. The reference data have already been subtracted. In thispseudo-three-dimensional plot, the cylindrical axis is directedhorizontally.

deviations of the various optical components, princi-pally those of the cylindrical lens, from their idealgeometry o1x, y2:

mt1x, y2 5 o1x, y2 1 t1x, y2. 122

To determine o1x, y2 an object r1x, y2 with deformationssmaller than the desired amplitude accuracy has toreplace the test object:

mr1x, y2 5 o1x, y2 1 r1x, y2. 132

To ensure that o1x, y2 is constant in both measure-ments, the influence of the test object and the refer-ence object on the global shape of the reflected wavefront has to be the same. Eliminating the unknowno1x, y2 in Eqs. 122 and 132 leads to

t1x, y2 5 mt1x, y2 2 mr1x, y2 1 r1x, y2. 142

The accuracy with which t1x, y2 can be determined islimited by the quality of the reference object. Theaccuracy can be increased by randomly moving thereference object 3r1x, y2 = ri1x, y24 between several refer-ence measurements mi1x, y2. Again the global shapeof the reflected wave front has to remain unaltered forall the reference measurements:

m11x, y2 5 o1x, y2 1 r11x, y2,

···mi1x, y2 5 o1x, y2 1 ri1x, y2, i 5 1, 2, . . . ,

···mn1x, y2 5 o1x, y2 1 rn1x, y2. 152

For cylindrical optics the degrees of freedom are

Fig. 5. Plot showing the set of polynomials that has been fitted tothe data of Fig. 3.

Fig. 6. After subtraction of the fitted polynomials, midwave-length deformations, in this case the printthrough of the fiberstructure, remain.

1 November 1995 @ Vol. 34, No. 31 @ APPLIED OPTICS 7209

limited to translations in the direction of the cylindri-cal axis 1intersection I in Fig. 22 and 180-deg rotationsaround the optical axis.An averaging of n measurements reduces the sys-

tematic error of the reference measurement:

t1x, y2 5 mt1x, y2 21

n oi51

n

mi1x, y2 11

n oi51

n

ri1x, y2. 162

To quantify the improvement, we define s as the rmsvalue after subtracting the polynomial fit fp describedin Section 3:

s1 f 22 [ 7 f 2 fp@f 2 fp8 172

with 7 @ 8 representing the scalar product:

7 f@g8 51

LxLye0

Lx e0

Ly

f1x, y2g1x, y2dxdy, 182

with f and g representing the arbitrary functions andLx and Ly as the dimensions of the field of view.The rms s value of the remaining systematic error

of the measurement

1

n oi51

n

ri1x, y2

can be estimated by

s31n oi51

n

ri1x, y24 >1

Œns1r12. 192

There are two prerequisites for this estimate. First,the functions ri have to be nearly orthogonal:

7ri@rj8 9 7rj@ri8 for all i fi j. 1102

Second, the rms values of all the contributions of thereference object s1ri2 have to be approximately thesame:

s1r12 > s1r22 . . . > s1ri2 . . . > s1rn2. 1112

These prerequisites can be checked by forming thedifferences ri 2 ri11, ri11 2 ri12, ri11 2 ri13 in Eq. 142,forming six scalar products of these differences andsolving the obtained system of equations for 7ri@rj8.It is our experience that reference measurementstaken from a surface with a statistically randomtopology fulfill these prerequisites. However, thewell-known message of Eq. 192 is that, by moving thereference object between measurements and averag-ing over nmeasurements, the accuracy of the calibra-tion procedure can be improved by a factor of Œn. If asingle reference surface is used to create the calibra-tion file, it should be moved by distances greater thanthe correlation length to earn the full Œn averagingeffect. To avoid the procurement and adjustment ofa reference object, the real test surface itself may beused as a reference surface. A large number n of

7210 APPLIED OPTICS @ Vol. 34, No. 31 @ 1 November 1995

reference measurements is then necessary to reducethe systematic error of the measurement sufficiently.For calibrating our instrument, a commercially

available plano–convex cylindrical lens with its flatsurface coated by a reflecting layer was used 1see Fig.72. The combination of a convex cylindrical surfaceand reflecting flat surface in a single componentsimplifies the alignment. To estimate the quality ofthe reference object, two measurements were per-formed. The reference object was rotated by 180°around the optical axis between measurements:

m11x, y2 5 o1x, y2 1 r11x, y2,

m21x, y2 5 o1x, y2 1 r21x, y2. 1122

The subtraction of the two measurements and theprerequisites of Eq. 192 lead to

s1r12 > s1r22 >1

Œ2s1m1 2 m22. 1132

We found the rms value of the reference object to be4.2 nm after we subtracted a fourth-order polynomialfit. Therefore, according to Eq. 192, the averaging oftwo reference measurements was sufficient to achievean absolute accuracy of 3 nm rms.

5. Amplitude and Lateral Resolution

The amplitude resolution of the measuring techniquewas limited by three effects:

1a2 Air turbulences and thermal convection cur-rents; a long path of rays enhances this effect.

1b2 Mechanical vibrations of the flimsy shell, intro-duced by ground vibrations and acoustic noise. Todampen ground vibrations the entire setup wasmounted on an air-bearing isolated table.

1c2 Speckle interferences caused by local inhomoge-neities of the optical components.

We define the amplitude resolution Ares as the rmss value 3for the definition see Eq. 1924 of the differenceof two subsequently performed measurements ma

Fig. 7. Course of rays in the reference test setup. The cylindri-cal lens has to be tested. The plano–convex reference lens with itsreflective back surface serves as a reference object.

andmb divided by Œ2:

Ares [1

Œ2s1ma 2 mb2. 1142

Since air turbulences andmechanical vibrations showa random time dependence, the contribution of theseeffects could be reduced by averaging over a largenumber of subsequent measurements. It takes only1 min for the Carl Zeiss interferometer, DIRECT100,to acquire and process 1000 measurements. Since aportion of the speckle interferences varied with timein a nonrandom way, they could not be eliminated byaveraging or by calibration measurements. There-fore, the interferometer’s optics offer a scan mode, inwhich the internal paths of rays are altered during anaveraging measurement without limiting the lateralresolution. A Fourier expansion of the difference oftwo subsequently measured axial profiles 1ma 2 mb2shows the spatial frequency dependence of the repeat-ability.Figure 8 displays the power spectral density 1PSD2

in the axial direction of a repeatability profile 1ma 2 mb2as a function of spatial frequency f. The PSD isdefined such that

PSD1 f 2 [ 1⁄2A21 f 2L, 1152

where A1 f 2 symbolizes the Fourier amplitude of spa-tial frequency f andL is the length of the axial profile.6Then the s2 variance of the Fourier expanded profileis the integral over the PSD:

s2 5 ef0

`

PSD1 f 2df, 1162

where f0 5 1@L.By averaging 2000 measurements and operating in

the scan mode, an amplitude resolution of 1 nm rmswas achieved. Since fitted polynomials were sub-tracted, this resolution takes into account only thosespatial wavelengths that are smaller than 30 mm.In the spatial wavelength range between 20 and 60mm, strong features, which are caused by thermalconvection currents, can be observed 1compare Fig.

Fig. 8. Power spectral density in the axial direction of a repeatabil-ity profile.

82. Without subtracting the fourth-order fit, the am-plitude resolution measured between 2 and 6 nm rms.The sharp peak in the PSD of Fig. 8 is caused by aslight printthrough of the interference pattern. Atilt of the reference plane with respect to the mea-sured object creates a high number of fringes, which isnecessary for the measurement technique applied bythe DIRECT100 interferometer.9 The lateral resolu-tion is limited by the optical resolution of the instru-ment. The cutoff of the modulation transfer functioncan be seen in Fig. 8 and is located at a spatialwavelength of 0.7 nm.

6. Summary

A technique for fast, noncontacting, three-dimen-sional measurements of near-cylindrical optics with ahigh amplitude resolution and accuracy has beendeveloped to observe the printthrough of the carbonfiber structure of lightweight grazing-incidence optics.The European Space Agency has selected this tech-nique for the production phase of the x-ray spectros-copy mission XMM. The main components of thenormal-incidence interferometric setup were a CarlZeiss interferometer 1DIRECT1002 and a cylindricallens for illumination. The influence on the ampli-tude resolution by random vibrations and air turbu-lences could be reduced by averaging over a largenumber 120002 of measurements. This was possiblebecause of the high data sampling rate of theDIRECT100 interferometer. The contribution ofspeckle interferences was suppressed by altering theinternal paths of rays during an averaging cycle.A reproducibility of 1 nm rms for spatial wavelengthssmaller than 30 mm was achieved on flimsy mirrorshells with a wall thickness of 1 mm. The lateralresolution was limited to 0.7 mm by the opticalresolution of the instrument. To improve the ampli-tude accuracy, the instrumental errors were deter-mined and subtracted from the data. By using ahigh-quality reference object and averaging over tworeference measurements, an absolute accuracy of 3nm rms was achieved for spatial wavelengths smallerthan 30 mm.

This research was performed under the EuropeanSpace Agency contract 8767@90@NL@CC. We thankseveral colleagues for their support and especially R.Debitsch, B. Dorband, and A. Matthes at Carl ZeissCompany for their contributions.

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