Processing of interferometric fringe scanner data for theAXAF/TMA x-ray telescope
Paul Glenn and Andrea Sarnik
An interferometric fringe scanner was used for the Technology Mirror Assembly (TMA) x-ray telescope, builtby Perkin-Elmer as a technology demonstration for the Advanced X-Ray Astrophysics Facility (AXAF). Wediscuss advanced data processing features, implemented during a follow-on project to improve the mirror'smidfrequency errors still further. Data processing techniques include interleaving of multiple scans, andoptimal smoothing and interpolation of the interleaved data onto a uniform grid. We discuss the underlyingmathematics behind the processes, the motivation in terms of recovering the highest possible spatialfrequencies from the data, and some sample results.
1. Introduction
The technique of fringe scanning has existed in prin-ciple since the beginning of interferometric opticalmetrology. When a metrologist draws a set of straightlines across an interferogram, marks off fringe centers,and measures their separations, he is performingfringe scanning. This process has been automatedand made much more accurate using such techniquesas microdensitometer scanning of the interferogramunder computer control.1 In the original Advanced X-Ray Astrophysics Facility (AXAF) Technology MirrorAssembly (TMA) project, a fringe scanning instru-ment was developed which scanned live fringes.2 Thisinstrument used standard test plate techniques, utiliz-ing a toroidal test plate to serve as a reference against ameriodional slice of the near cylindrical mirror. Thefringe scanner gave very good accuracy and spatialfrequency response.2 3 However, with the initiation ofthe recent repolishing effort of the TMA, even tightergoals for accuracy and frequency response were set.This led to improvements in both the hardware andthe data analysis. In this paper, we describe somespecific aspects of the improved data analysis. Wediscuss the motivation for the improvements in terms
Paul Glenn is with Bauer Associates, Inc., 21 Thomas Road,Wellesley, Massachusetts 02181, and A. Sarnik is with Perkin-ElmerCorporation, 100 Wooster Heights Road, Danbury, Connecticut06810.
of accuracy and frequency response and show somesample results.
II. Summary of Basic Fringe Scanning Data Analysis
Figure 1 is a conceptual view of the TMA fringescanner, showing the near cylindrical test piece and thetoroidal test plate. A fringe pattern is generated be-tween the two surfaces and is photometrically scannedas shown. The resulting set of brightness data is pro-cessed by a computer to generate a series of fringecenter locations. (In this paper, we do not elaborateon the process of accurately determining the locationof each fringe center.) By the nature of the interfero-metric test, each of these fringes corresponds to asuccessive optical path difference (OPD) of one wave-length, or a surface height change of one-half wave-length. Figure 2 pictures this process and shows theresult of removing piston and tilt from the raw data.In this final result, the data points are not equallyspaced in either the OPD or in the scan coordinate.
Our improvements in the basic process outlinedabove were motivated by a need for increased accuracyand for increased sensitivity to higher spatial frequen-cies. Both of these needs could be met by collectingmultiple scans of data. However, to take full advan-tage of the multiple scans, we needed to improve theinterleaving of the scans into a single data set, and toimprove the interpolation of the single data set onto ausable, uniform grid. We discuss each of these pro-cesses in turn in the following sections.
Ill. Improvements in the Interleaving of Scans
Previously, we combined multiple data sets by firstinterpolating each individual data set onto a 1-mmgrid and then averaging the grids. This approach does
Fig. 1. Conceptual view of the TMA fringe scanner that measuresaxial mirror profiles.
6x
t;D 5, I_: 4, -
' 3x
2x - - - - -2 -k-
S1OTH SURFACE
PISTON AND TILT REMOVEDFROM THE SCAN
SCANDIRECTION
Fig. 2. Basic interpretation of fringe scanner data. Data points areequally spaced in the optical path difference (ODP) but are unequal-
ly spaced along the scan.
not preserve all the higher spatial frequency informa-tion available from data sets which have differingamounts of piston and tilt, i.e., interleaved fringes.This was demonstrated, for example, in a computersimulation by Van Speybroeck of the SmithsonianAstrophysical Observatory, showing that the calculat-ed correlation length could be up to -50% larger thanthe real correlation length, for the limiting case of thereal correlation length being the same order as thefringe spacing. (If the real correlation length is muchlonger than the fringe spacing, the previous algorithmworks very well, as expected.)
In the improved interleaving algorithm, we combinethe scans before interpolating the results onto a uni-form grid. This preserves the higher spatial frequencyinformation to the fullest extent possible. Figure 3shows the problem of interleaving conceptually. It isintuitively clear how to combine the noiseless scansillustrated. However, when there is noise present, theproblem becomes more difficult. Intuitively, onewould want to minimize the jaggedness of the finalinterleaved scan. Below we describe the mathematicsbehind this intuitive approach.
The improved algorithm is a least-squares approachwhereby the added pistons and tilts of all scans (otherthan the first scan) are adjusted to minimize the mean-square calculated OPD between closely neighboringpoints from different scans. The term closely neigh-boring means that the requirement that two neighbor-ing points have the same OPD is strongest when thepoints lie on top of each other, and weakest when thepoints are far apart. Analytically, this can be writtenas follows:
Fig. 3. Intuitive approach to interleaving. Two hypothetical scansare shown, where the sampled points of one scan lie midway betweenthe sampled points of the other. First the scans are shown inter-leaved using obviously incorrect relative piston and tilt. Then, the
intuitively correct interleaving is shown.
NSCN-1 NSCN NFRN
n=I m=(n+l) i=1
X (yn,i - Ymj) 2Wni;mj, (1)
where S = the (least-squares) loss function to be mini-mized,
NSCN = number of scans to be combined,NFRN = number of fringe centers [this number will
actually vary from scan to scan, although itis shown as a constant in Eq. (1)],
Yni = calculated OPD value (including added pis-ton and tilt) from the nth scan at the ithfringe center, and
Wn,i;mj = a weighting function described above, whichis maximum when the two fringe centers(scan n, fringe i, and scan m, fringe j) lie ontop of each other, and minimum when theyare far apart.
In short, Eq. (1) states mathematically the loss func-tion which we would like to minimize as described inthe previous paragraph. To do this, we need to definethe weighting function W, and the parameters whichwe can vary in the minimization process.
We define W to be the correlation function, as fol-lows:
Wni;mj = corr(xn~i - Xmi), (2)
where corr = the correlation function of the OPD data,as a function of shear distance (the calcu-lation of corr is covered below), and
Xn = x (axial position) value from the nth scanat the ith fringe center.
The calculation of the correlation function, -corr, ismost easily handled after interpolating the data to auniform grid. This is done iteratively, as follows.First, the data are combined in the previous method(i.e., first linearly interpolating and then averaging),and the correlation function is calculated. This corre-lation function is then used as input to the fittingroutine which minimizes the loss function in Eq. (1).Then, with the pistons and tilts optimized, we recom-bine the data sets. This will result in a single conglom-
erate data set on a uniform grid, which can then beused to recalculate corr. The process can now berepeated, using the new estimation of corr, and recal-culating the optimum pistons and tilts. The process isstopped when the pistons and tilts have converged totheir final values. The result is an optimally inter-leaved set of scans.
To finish the statement of the problem, we need todefine the parameters which will be varied. This isdone by expressing the calculated values of y,,i asfollows:
y.,i = yO.,i + An + Bnx.,i, (3)
where yi = measured OPD value (nth scan, ithfringe center),
An = the piston to be added to the nth scan,and
Bn = the tilt to be added to the nth scan.We arbitrarily set Al = B1 = 0, which means that we arenailing down the first scan and moving all the otherswith respect to it. (We could just as easily have pickedany other scan to nail down.)
Combining Eqs. (1), (2), and (3), we have
S = (2) E [(Y0n + An + BnXnbi)nmii
- (Yomj + Am + Bmxmj)12
X corr(Xn,i - Xmj). (4)
Eq. (4), then, is an expression for the scalar quantity Sin terms of the known values Y00,j, i, and corr, andthe free parameters An and Bn. The equations whichexpress the minimization of S with respect to An andBn are
dS/dA = dS/dBn = 0 for all n > 1. (5)
Equations (5), then, are a set of linear equations in therelative pistons and tilts of the various scans. Theirsolution determines the interleaving which minimizesthe jaggedness of the final scan, as we sought intuitive-ly. The actual algebraic solution of Eqs. (5) isstraightforward but tedious. It was implemented insoftware routines which were added to the originalfringe scanner processing program. In Sec. V, wepresent some qualitative results of using the improvedinterleaving, as well as the improved filtering/interpo-lation discussed below.
IV. Improvements in the Interpolation to a Uniform Grid
Previously, we interpolated a scan of raw data onto auniform grid by using the nearest neighboring datapoint on either side of each grid point. This approachdoes not fully exploit the noise reduction advantages oftaking multiple scans of data. In particular, for exam-ple, if there are several measurement points in betweentwo neighboring grid points, it would clearly be advan-tageous to use some sort of weighted average of allthese points in the interpolation process. Thus, we seethat the interpolation process is closely associated with
the filtering process which reduces measurementnoise.
The filtering and interpolation process which wearrived at is based on standard probability theory andoptimal estimation theory.4 The process involves thefollowing three main steps:
(1) Apply an optimal filter to the data in the for-ward direction along the scan (i.e., along increasing xvalues). Applying an otpimal filter means that thebest estimate for each data point is a combination ofthe knowledge gained from all the previous points, andthe knowledge gained from the new measurement atthe point. (This filtering process can give results notonly at the points themselves, but also at any points inbetween, including the uniform grid which we willinterpolate onto.)
(2) Apply the same type of optimal filter to the rawdata, but this time proceeding in the backward direc-tion (i.e., along decreasing x values).
(3) At each point (either a data point or the uniformgrid which we are interpolating onto), optimally com-bine the forward and backward filter results to give thebest possible (i.e., optimal) estimate.
The filter implied in steps 1 and 2 and the optimalcombining implied in step 3 are discussed further be-low.
A. Kalman Filtering of the Raw Data
The principle behind the Kalman filter can be sum-marized in three steps:
(1) At any given point, predict the OPD value thatyou should measure, based solely on the values youhave measured at the previous points.
(2) Recall the value actually measured by the fringescanner at this point.
(3) Form the optimal linear combination of thesetwo OPD values, to give the true best estimate of OPD.(The fact that we are forming a linear combinationmeans that the Kalman filter is a linear filter. Howev-er, it can be shown that, if the surface height errors andthe measurement noise errors are both Gaussian ran-dom processes, there is no nonlinear filter which couldimprove on this filter.)
Step 1 is the heart of the filter. It can be picturedconceptually by thinking of how a ship might estimateits position if the sky became cloudy-it would recallthe previous position estimates gathered when the skywas clear, plot a course from them, and extrapolate itsnew position assuming the course was unchanged inthe meantime. The mathematical definition of step 1,naturally, is much more formidable. It is complicatedand depends on how many previously measured pointsare utilized. We refer the interested reader to Ref. 4for more detailed discussions. For the TMA project,we utilized two previous points in defining the expect-ed OPD. This required that a third-order matrix beinverted at each point. Calculation time for the caseof using three previous points would have been prohib-itive and the advantage small.
Step 2, in terms of our ship picture, corresponds tothe sky clearing and a new measurement becoming
available. (Such information would always be wel-come. However, even without it, we can still estimateour position or OPD. This is what we need to do at theuniform grid coordinates which do not happen to coin-cide with a measured fringe center location.)
Step 3, again in terms of our ship picture, corre-sponds to deciding how much credence to give to theexpected position, and how much to the measuredposition. Clearly, each of the two values has somevalidity, and should therefore be considered. Optimalestimation theory tells us4 that, if we have two esti-mates of a parameter, Yi and Y2, with associated errorvariances V1 and V2, the optimal estimate is given by
(6)Yoptimal = (Y1V 2 + Y2 V1)/(V1 + V2),
with associated error variance
Voptimai = 11(1/Vl + 1/V2)-
The Kalman filtering analysis alluded to above tells usthe error variance of the predicted OPD, while aknowledge of our measurement accuracy tells us theerror variance of the measured OPD. Equation (6),then, defines the optimal combination alluded to instep 3, while Eq. (7) gives us the resulting error vari-ance. [Note from Eq. (7) that Voptimal is always smallerthan either of the component variances.]
B. Optimal Combining of the Two Filtered Data Sets
Equation (6), which we used to combine predictedand measured OPD values, can be used in exactly thesame way to combine the forward filtered OPD valuewith the backward filtered value. This is because wealready have the error variances calculated for each ofthe two values. Then, Eq. (7) can be used to give us afinal error variance in our optimal estimate. This,then, completes the filtering and interpolation opera-tion. In the following section, we briefly discuss thekind of improvement we achieved by using the im-proved interleaving and interpolation algorithms.
V. Results
We ran comparative test cases to characterize theeffectiveness of the interleaving and the filtering andinterpolation software. In the case of the interleaving,we concentrated on running analytical test cases,where we input artificial scans with artificial (butknown) pistons and tilts added. We found that theaccuracy of the software in recovering the rigid bodymotions of the scans was of the order of 0.001 ,im,which easily meets our goals.
In the case of the filtering and interpolation soft-ware, we ran artificial test cases, but we also analyzedthe results on real data and compared the results withthose of the previous two-point (linear) interpolation.We wanted to be sure that the software improved therepeatability of the interpolation, without sacrificingany of the higher frequency surface data.
As a test of the algorithm performance, the followingtest was run. Ten scans were taken over the full aper-ture of the TMA parabola (400 mm), with a nominalfringe spacing of -1 mm. For each scan, the fringe
z
LiiJ
* ju Log. 20.00 30. 00 .o So.o0
oL / SCANcy 0 / COORDCL1 / ( M x 1 0 )
,I
Fig. 4. Average surface profile using the previous linear interpola-tion method.
(nz00'
-Jt_4
LiL01
CL
Li
0.00SCAN
COORD( M x 1 0 )
Fig. 5. Average surface profile using the new Kalman filteringmethod.
minima and maxima were found and then interpolatedonto a grid evenly spaced at 1-mm intervals. In case 1,the scans were interpolated using the simple linearinterpolation algorithm which we had used in the pre-vious TMA program. In case 2, the scans were inter-polated using the Kalman filtering technique de-scribed in this paper. The ten interpolated scans werethen averaged. At each point, the average and the rmsdeviation of the ten scans were computed. Figures 4and 5 show the average surface profile for the linearand Kalman interpolation, respectively. Figures 6and 7 show the rms deviation at each point. As shownin Figs. 6 and 7, the Kalman interpolation algorithmproduced an improvement of approximately a factor of3 in the average rms deviation.
To ascertain that real high frequency figure errors(in particular, the predicted 4-mm upper midfre-
o e10.00 20.00 30.00. 40.00SCAN COORDINATE (NMlx10)
Fig. 6. Root mean square (rms) scan-to-scan variation using theprevious linear interpolation method (average variation =
0.0034 jlm).
00 Ib.00 20.00 30.00. 40.00SCAN COORDINATE (x10)
50. 00
Fig. 7. Root mean square (rms) scan-to-scan variation using thenew Kalman filtering method (average variation = 0.0011 um).
1 e Ad PER IC:I
1 r1 PER7. n
I
LINERR :N'ERP:LR7
':N
I I
I lEI I A MR \: T R II I
-3 -2 - ILOG (REQUENCY ( I /11 ) '
0
Fig. 8. Integrated power spectral density (PSD) function usingfirst the previous linear interpolation method and then the new
Kalman filtering method.
quency errors) are not being filtered out, the powerspectral density (PSD) of the average files was com-puted. Figure 8 shows plots for the two cases on a log-log scale of the integrated PSD. The integrated PSD,F(f), is given as
F(f) = | PSD(f')df', (8)
wheref = the spatial frequency argument of the inte-grated PSD, and
fmax = the maximum spatial frequency consideredin the analysis.
On both plots, there is an abrupt change at a fre-quency of [1/(4 mm)], and then another smaller changeat a frequency of [1/(8 mm)]. The occurrence in bothplots indicates that this feature is not being filteredout. Nonetheless, the plot of the Kalman filtered datahas a lower total integrated PSD, which indicates thatthe Kalman filtering is doing a better job of filteringout the undesired high frequency noise.
VI. Summary, Conclusions
We have discussed the nature of fringe scanning asan interferometric metrology technique and have out-lined some inherent problems in terms of samplingdensity, and filtering and interpolating to a usable,uniform grid. In the TMA repolishing program, wehave developed and tested the algorithms and softwareto solve these problems, by interleaving multiple scansbefore interpolation, and by using an optimum linearKalman filter for the filtering and interpolation.Tests show very good results, with interleaving accura-cies of the order of 0.001 gm, and repeatability im-provements of the order of a factor of 3, with faithfulcharacterization of the higher spatial frequency sur-face errors. The software is currently being used onthe TMA repolishing effort, during which the mirrorsurfaces should be driven to even higher qualities thanbefore.
This material was presented as paper 830-42 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.
References
1. A. Slomba and L. Montagnino, "Subaperture Testing for Mid-Frequency Figure Control on Large Aspheric Mirrors," Proc. Soc.Photo-Opt. Instrum. Eng. 429,114 (1983).
2. A. Slomba, R. Babish, and P. Glenn, "Mirror Surface Metrologyand Polishing for AXAF TMA," Proc. Soc. Photo-Opt. Instrum.Eng. 597, 40 (1985).
3. P. Glenn and A.. Slomba, "Derivation of Requirements for Sur-face Quality and Metrology Instrumentation for AXAF TMA,"Proc. Soc. Photo-Opt. Instrum. Eng. 597, 55 (1985).
