Performance prediction of the AXAF Technology MirrorAssembly using measured mirror surface errors

Paul Glenn, Paul Reid, Albert Slomba, and Leon P. Van Speybroeck

We have developed a math model relating the measured parameters of the Technology Mirror Assembly

(TMA) to its final performance. This scalar scattering model is valid for large and small amplitude features.

It allows the user to specify power spectral densities and/or autocovariance functions within any spatialbandwidth, including microroughness. We present new TMA data in the bandwidth of -0.1-1000 mm ,

predicting performance and comparing them with x-ray test data. We also account for assembly, alignment,

and particulate contamination. Finally, we comment on improved performance expected after repolishing.

1. Introduction

The Technology Mirror Assembly (TMA) forNASA's Advanced X-Ray Astrophysical Observatory(AXAF) was successfully fabricated and tested. Pre-dictions of performance have been made based onmeasured surface quality,12 and the x-ray perfor-mance has been measured and compared with the pre-dictions. 2 3 Although the TMA surfaces largely mettheir quality goals, differences between the predictedand the measured performance led to the postulationof surface errors that were incompletely sampled bythe spatial frequency bandpasses of the various me-trology instruments.2 This in turn led to the initiationof a repolishing program to improve the surface qualitystill further.

As part of the repolishing program, Perkin-Elmerdeveloped an improved math model and computersoftware to predict performance for a wider variety ofsurface errors. In this paper, we summarize the mathmodel and give the performance predictions for thenewest surface measurements using metrology instru-mentation with increased bandpass capabilities. Wealso cover the integration of the effects of dust scatter.Finally, we comment on our preliminary observationson the efficacy of the repolishing using the improvedmetrology and polishing instrumentation.

Paul Glenn is with Bauer Associates, Inc., 21 Thomas Road,Wellesley, Massachusetts 02181; L. P. Van Speybroeck is withSmithsonian Astrophysical Observatory, 60 Garden Street, Cam-bridge, Massachusetts 02138; P. Reid is with Perkin-Elmer Corpora-tion, 100 Wooster Heights Road, Danbury, Connecticut 06810; andA. Slomba is with United Technologies Optical Systems, P.O. Box109660, West Palm Beach, Florida 33410.

Received 9 October 1987.0003-6935/88/081539-05$02.00/0.© 1988 Optical Society of America.

II. Improved Math Model

Both the improved and the previous math modelshave the same starting point, which is the scalar scat-tering theory of Beckmann and Spizzichino.4 [Thistheory has been successfully used in many other ana-lyses and computer codes, such as NASA's OpticalSurface Analysis Code (OSAC).5 ,6] The theory itself isvalid for both small and large amplitude surface errors.It is good for both polarizations, and in fact matchesthe vector theory results for small scattering angles.The theory can be summarized as follows7 :

dP/dQtotai = (Strehl dP/dQspec) + dP/dQpe, 0 dP/dQscat; (1)

Strehl is the conventional Strehl ratio given by

Strehl = exp[-2(ko sina)]2 ; (2)

dP/dQspec is the intensity pattern of the system if therewere no scattering (e.g., an Airy pattern, or a Gaussianprofile caused by large scale geometric errors such asalignment and full aperture axial slope); and dP/dQscatis the scattered intensity pattern to be defined, whichwould arise from a system with dP/dQspec being aninfinitely narrow delta function. The 0 denotes aconvolution operation. Finally, k = 27r/wavelength isthe wavenumber, a is the grazing angle, and is therms equivalent surface error. (In this context, rmsequivalent surface error means the rms for a singlesurface representing the entire system, which in thecase of uncorrelated errors on two mirrors, would be21/2 times the rms on a single mirror.)

Detailed development of Eq. (1) in terms of a full2-D surface model was summarized previously.67

Such a development was used previously in TMA butwas extremely tedious and difficult to run in software.The advance in the improved math model relates to aparticular approximation made in the full 2-D theory,as summarized below.

In the current implementation of the math model,no approximations are made concerning the magni-tude of the surface errors. However, one importantapproximation is made, which makes the analysis easy

15 April 1988 / Vol. 27, No. 8 / APPLIED OPTICS 1539

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Fig. 1. Typical grazing incidence scatter profile from a single longi-tudinal mirror strip, showing the severe elongation in the tangentialdirection. Because of the elongation, the integral over a circularregion can be approximated by an integral over an infinite stripwhich is inscribed by the circle. (This approximation holds as longas the circle in question is large enough that, if its size were multi-plied by the reciprocal of the grazing angle, it would encircle a largeportion of the energy. The excellent accuracy of this approximation

for TMA is discussed in the text.)

and efficient, but which restricts it to grazing incidencesystems (practically speaking, systems with grazingangles smaller than several degrees). As we discussbelow, the approximation comes in integrating theintensity over a cone of scattered angles to arrive at theencircled energy (EE).

It is convenient to think of the x-ray system as asingle mirror (with the composite errors of both realmirrors) which is composed of a series of barrel stavesor thin longitudinal strips. If each stave has the samesurface characteristics, the EE of the system is natural-ly the same as the EE one would get from consideringonly a single stave. To examine the EE of a singlestave, we consider the scatter in the sagittal and tan-gential directions. If we define the (small) scatteringangles in the two directions as s and t, respectively, theEE for an angular radius r is given by

EE(r) = o2 +t')l'2j r dsdt dP/dQtota1(s,t). (3)

This expression involves no approximations, otherthan that the scatter angles must be much smaller thana radian. It is the basis of the math model used on theoriginal TMA program. Unfortunately, it involvesvery time-consuming and difficult 2-D computer cal-culations and is difficult to implement for arbitrarydescriptions of the surface errors. However, a veryuseful approximation comes in noting that the scatterpattern from a single barrel stave in a grazing incidencesystem is very much lengthened in the tangential di-rection by a factor of (1/a). In other words, the lines ofconstant scattered intensity are extremely eccentricellipses, rather than circles. The major axes of theseellipses point in the tangential scatter direction.Thus, it is a reasonable approximation to change thecircular limits of integration in Eq. (3) into an infinite

strip of integration. The two regions of integration areshown in Fig. 1, which also shows some lines of con-stant scattered intensity. The approximate EE in thisnew region of integration is given by

EE(r) = J dt J ds dP/dQtota1(st) (4)

This result can be expected to be accurate down toangles that are a factor of (a) smaller than angles whichcontain a large portion of the EE. In other words, forexample, if for a certain x-ray energy a large portion ofthe EE is contained in a circle with a diameter of 115sec of arc, Eq. (4) can be expected to give accurateresults all the way down to 1 sec of arc (since a = 1/115for TMA). If a large portion of the EE is contained in acircle with a diameter of 1 sec of arc, Eq. (4) can beexpected to give accurate results all the way down to(1/115) sec of arc.

Equation (4) is much easier to evaluate than Eq. (3),since it is 1-D (assuming the infinite integral over s canbe found in closed form, which it can). Without goingthrough all the algebra, the final reduction of Eq. (4) toa simple 1-D expression is

EE(r) = Strehl dt [dP/dtpecr

+ FT(FT(dP/dt.peC)exp[4kg(XX/a)] - 1)], (5)

where FT denotes a Fourier transform, dP/dtspec is the1-D specular image (i.e., with no scattering) from asingle stave, g(x) is the 1-D autocovariance (ACV) (inthe longitudinal direction) of the equivalent singlesurface, and X is the dimensionless variable which isthe Fourier transform conjugate of the tangential scat-ter angle t.

Eq. (5), then, summarizes the improved math model.It conveniently expresses the EE in terms of 1-D sur-face characteristics which are directly measurable bythe TMA metrology instrumentation. The theory hasthe advantage that the EE rigorously approaches unityas the angle grows larger, which means that inaccuracyin one angular range must be compensated for by theopposite inaccuracy in a different range.

Ill. Software Implementation

We have implemented the improved math model ina software package called EEGRAZ (encircled energyfor grazing incidence systems). We have run manytest cases and have compared the results with thosefrom other packages, including OSAC, the previousTMA prediction package, and a package in use by VanSpeybroeck at the Smithsonian Astrophysical Obser-vatory. Agreement was uniformly excellent, usuallyof the order of one part in 104.

The advantages of using EEGRAZ in comparison withany of the other packages are its ability to utilizedirectly the measured 1-D surface characterizations,the wide variety of surface characterizations it canutilize, its user friendliness, and its short calculationtime. The following list summarizes the system pa-rameters which the user specifies to EEGRAZ:

1540 APPLIED OPTICS / Vol. 27, No. 8 / 15 April 1988

(1) Specular beam characteristics (a 1-D Gaussianwith arbitrary width, including zero).

(2) Scattering halo characteristics: these are im-plicit in the user's definition of the surface errors,which can be any combination of the following:

(a) Gaussian autocovariance (ACV) functions.(b) Exponential ACVs.(c) Gaussian power spectral density (PSD) func-

tions, with arbitrary center spatial frequency and la-spatial frequency radius.

(d) Piecewise defined log-log PSD.(e) A high frequency surface rms value, with fre-

quencies which are higher than any that would scatterinto the angular scattering range of interest.

(3) X-ray energy.(4) Grazing angle.(5) Angular scattering range of interest.In the following section, we briefly summarize some

of the new TMA surface data collected, and then givethe performance prediction results obtained from run-ning EEGRAZ.

IV. New TMA Surface Data

Table I summarizes the applicable surface measure-ment parameters. There we show the values originallyreported, as well as the new values measured with theimproved instrumentation. (All the values corre-spond to the TMA before the repolishing was begun.)Although the originally stated intent of the repolishingeffort was to reduce the newly measured upper midfre-quency errors, we see from Table I that there werenewly measured errors in both the low and the high

Table 1. Tabulation of Old and New TMA Surface Measurementsa

Parameter Old value New value

Microroughness 9-A rms

Upper midfrequency N/A

Midfrequency

Low frequency

Parab: 38-A rms,5-mm corrlength

Hyperb: 28-A rms,9-mm corrlength

>0.6-sec of arc core,from low freq, assyand alignment

13.5-A rms (up to 16A possible, incl inmodel)

23-A rms, Gaussian,center freq = 0.25mm', 1-a radius =0.015 mm-1

Same as old data

3-sec of arc core

Notes:(1) Microroughness increased due to bandlimit considerations (im-

portant for high energy)(2) Upper midfrequency added due to improved instrumentation(3) Low frequency increase includes assembly and alignment defor-

mations, as well as correction of a metrology calibration incon-sistency in the original data (important for low energy)

a Old data were measured before x-ray testing and were previouslyreported. New data were measured with improved instrumentationbefore the start of the repolishing effort.

frequency regimes as well. As we discuss in Sec. VI,these errors are at least as important as the uppermidfrequency errors in accounting for the measuredx-ray performance. The newly measured low frequen-cy error is accounted for primarily by the finding of ametrology calibration inconsistency found in the origi-nal data, which has since been corrected. The newlymeasured high frequency error is accounted for bybandpass and filtering considerations in the old andnew instrumentation.8

Before discussing in Sec. VI the performance predic-tions using the data in Table I, we first briefly summa-rize our efforts aimed at approximately integrating theresults of a detailed dust scatter analysis.

V. Integration of Dust Scatter Analysis

A significant amount of particulate contaminationwas found on the TMA mirror surfaces. Van Spey-broeck of SAO performed a very detailed analysis, 9

which characterized both the contaminants and theireffect on performance. Since we found that the effecton performance was not overwhelming, we do not gointo the characterization in detail. Figures 2 and 3,however, show the intensity distribution and encircledenergy for the lowest and highest x-ray test energies.The effects are comparable and non-negligible, scat-tering approximately one-third of the energy into aregion of a couple of seconds of arc. The integration ofthis data into the EEGRAZ performance predictions isdiscussed below.

The fact that the dust scattering has a prominentcentral core containing not much of the total energywould make a rigorous analysis difficult. This is be-cause the required convolution would have to extend agreat distance to encompass all the energy. However,at these distances, the scattered intensity is many or-ders of magnitude smaller than at the core, whichwould lead to numerical difficulties in the convolutionoperation.

The approach, therefore, was to characterize thedust scatter as a sum of a narrow Gaussian, and what-ever was leftover. The narrow Gaussian portion wasconvolved with the rest of the TMA image core, withthe result that no noticeable change occurred. In fact,to this level of approximation, we simply ignored anychange in the overall TMA core size due to the smallcore of the dust scatter.

The convolution of the leftover part of the dustscatter was the real issue. To accomplish this withreasonable accuracy, without having to use 2-D doubleprecision fast Fourier transform (FFT) or other tech-niques, we used the following approach.

We assumed that the distribution of energy far out-side the 20-sec of arc level was irrelevant-only theamount of scattered energy mattered. In other words,the shape of the distribution was relevant only in theapproximate neighborhood of the image size in ques-tion. Over this region, it appeared that the encircledenergy curves looked much like those from a (2-D)Gaussian image. However, rather than try to fit thecurves to their Gaussian counterparts (accepting the

15 April 1988 / Vol. 27, No. 8 / APPLIED OPTICS 1541

TMA DUST - NORMALIZED EE AND INTENSITYE = 1.4 9 kV

0.9

0.9

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

RADIUS (ARC SECONDS)13 ENCIRCLED ENERGY + INTENSITY

Fig. 2. Sample dust scatter result (x-ray energy =scatter fraction = 0.289).

1.49 keV, total

volved encircled energy, without having to fit eithercurve to a Gaussian, and without having to perform adetailed convolution.

The slight inaccuracies in this approach seem justifi-able, given the real problems associated with a morerigorous solution, and given the uncertainties in thecontamination measurements. (Van Speybroeck ofSAO has stated that the dust scatter results he pub-lished are the most extreme guess, and could be toohigh by as much as a factor of 2.9)

In summary, we broke the dust scatter into a narrowGaussian and whatever was leftover. The narrowGaussian caused no noticeable change in the TMAprofile when convolved with it. The part that wasleftover, though, was wide angle enough to cause achange. The necessary convolution was approximat-ed using the method described above and Eq. (6). Theresults of these two convolution operations were ap-propriately weighted and added, giving the results dis-cussed below.

0.9

0.0

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04

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0 0.2 0.4 0.6 0.6 1 1.2 1 4 1.6 1.8 2

RADIUS (ARC SECONDS)1 ENCIRCLED ENERGY Y INTENSITY

Fig. 3. Sample dust scatter result (x-ray energy = 6.40 keV, totalscatter fraction = 0.338).

necessary inaccuracies and difficulties), we used theencircled energy values directly, as follows:

When two Gaussians are convolved, the encircledenergies of the component Gaussians and the resultingGaussian are well known. It is simply a matter ofalgebra, then, to express the encircled energy of theconvolved result at any angle, in terms of the encircledenergies of the component Gaussians at the samepoint. This gives the result

EE,.,v = 1 - expiln( - EEI)

* ln( - EE2)/ln[(l - EE1 )(1 - EE2 )]I, (6)

where EEconv is the encircled energy of the convolvedresult, and EE1 and EE2 are the encircled energies ofthe two component curves at the same point.

Equation (6), then, was the foundation of the meth-od. By assuming that the dust scatter curve and theTMA intensity curve without dust are both reasonablyGaussianlike in the neighborhood of the image in ques-tion, Eq. (6) gives a method for calculating the con-

VI. Performance Prediction Results

Figures 4, 5, and 6 graphically illustrate the perfor-mance prediction results from running EEGRAZ andintegrating the dust scatter. Several conclusions canbe drawn, as discussed below.

First, it is apparent that the increased low frequencyerror is crucial in explaining the low energy small-angleperformance. And second, the increased high fre-quency error is crucial in explaining the high energylarge-angle performance. (In fact, as shown, an evenhigher value than measured reproduces the test data'noticeably better still. Although not shown, the use ofthe old value of 9 A gives correlations which are muchworse.) Although we do not show the results of delet-ing the newly measured upper midfrequency errorsfrom the prediction, the effect was generally smallerthan either the low frequency or the high frequencyeffects. Finally, we see that integrating the dust scat-ter data generally gives better correlation than notintegrating it. (As previously alluded to, Van Spey-broeck felt that the dust analysis was probably toopessimistic, by perhaps a factor of 2. The integrationshown here indeed used only half of the dust scattercalculated and gave the best correlation with the testdata.)

All in all, the performance predictions using thenewly measured surface parameters and the integra-tion of the dust analysis show excellent agreement withthe measured x-ray performance data.

VII. Summary, Comments on the Repolishing

We have described an improved math model beingused to predict TMA encircled energy performance.The model is valid for high and low amplitude surfaceerrors of essentially any spatial frequency. It has beenimplemented in EEGRAZ, a user-friendly softwarepackage which is extremely flexible in the ways itallows the user to specify the surface errors. We usedEEGRAZ to predict performance based on expandedbandwidth measurements of the TMA mirrors. Based

1542 APPLIED OPTICS / Vol. 27, No. 8 / 15 April 1988

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diam image circle. Order of hatched bars: (1) x-ray data; (2) 0.6-secof arc core, 13.5-A roughness; (3) 3.0-sec of arc core, 13.5-A rough-ness; (4) 3.0-sec of arc core, 16.1-A roughness; (5) 3.0-sec of arc core,

16.1-A roughness, plus dust.

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0.7

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0.4

0.3

0.2

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1.49 2.98 4.95 6.4

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Fig. 5. Comparison of predicted encircled energies in a 8-sec of arc

diam image circle. Order of hatched bars: (1) x-ray data; (2) 0.6-secof arc core, 13.5-A roughness; (3) 3.0-sec of arc core, 13.5-A rough-ness; (4) 3.0-sec of arc core, 16.1-A roughness; (5) 3.0-sec of arc core,

16.1-A roughness, plus dust.

on the new measurements, as well as the finding andcorrecting of a metrology calibration inconsistency inthe old data, we made performance predictions thatagree very closely with the measured x-ray data. Inte-gration of the results of an analysis of dust scatterimproved the agreement still further.

The performance prediction success gives confi-dence in the use of the repolishing effort. That effortis currently under way, although considerable workremains. Preliminary data are promising in severalrespects. First, the correction of the effects of themetrology calibration inconsistency is proceeding rap-idly. Second, microroughness values of 10-A rms andbelow are being routinely achieved with modifiedsmoothing action. And finally, the upper midfre-quency errors (for which the repolishing effort wasoriginally targeted) appear to be reduced to negligiblelevels by a reasonable amount of smoothing. The

0.6 __

1.49 2.98 4.95 6.4

X-RAY ENERGY (kV)

Fig. 6. Comparison of predicted encircled energies in a 20-sec of arcdiam image circle. Order of hatched bars: (1) x-ray data; (2) 0.6-secof arc core, 13.5-A roughness; (3) 3.0-sec of arc core, 13.5-A rough-ness; (4) 3.0-sec of arc core, 16.1-A roughness; (5) 3.0-sec of arc core,

16.1-A roughness, plus dust.

simultaneous convergence in all these spatial frequen-cy regimes will again be a challenge, but preliminaryindications are promising.

This material was presented as paper 830-43 at theConference on Grazing Incidence Optics for Astro-nomical and Laboratory Applications, sponsored bySPIE, the International Society for Optical Engineer-ing, 17-19 Aug. 1987, San Diego, CA.

References1. P. Glenn, A. Slomba, and R. Babish, "TMA Mirror Quality Re-

quirements and Achievements," Proc. Soc. Photo-Opt. Instrum.Eng. 640, 45 (1986).

2. L. Van Speybroeck et al., "Correspondence Between AXAF TMAX-Ray Performance and Models Based Upon Mechanical andVisible Light Measurements," Proc. Soc. Photo-Opt. Instrum.Eng. 597, 20 (1985).

3. D. A. Schwartz et al., "X-Ray Testing of the AXAF TechnologyMirror Assembly (TMA) Mirror," Proc. Soc. Photo-Opt. In-strum. Eng. 597, 10 (1985).

4. P. Beckmann and A. Spizzichino, The Scattering of Electromag-netic Waves from Rough Surfaces (Pergamon, New York, 1963).

5. R. Noll, P. Glenn, and J. Osantowski, "An Optical Surface Analy-sis Code (OSAC)," Proc. Soc. Photo-Opt. Instrum. Eng. 362, 78(1982).

6. P. Glenn, "Space Telescope Performance Prediction Using theOSAC Code," Opt. Eng. 25, 1026 1986); Proc. Soc. Photo-Opt.Instrum. Eng. 571, 164 (1985).

7. R. Noll and P. Glenn, "Mirror Surface Autocovariance Functionsand Their Associated Visible Scattering," Appl. Opt. 21, 1824(1982).

8. P. Reid and P. Glenn, "Measurement of Micro Roughness andEffects of Detector Bandwidth and Finite Width," Proc. Soc.Photo-Opt. Instrum. Eng. 830, (1987), in press.

9. L. Van Speybroeck, "Dust," Smithsonian Astrophysical Observa-tory Internal Memorandum (27 Mar. 1987).

15 April 1988 / Vol. 27, No. 8 / APPLIED OPTICS 1543

IMAGE DIAMETER = 1 ARC SECOND