Direct comparison of mechanical and optical measurements of the ...
Direct comparisonof mechanicaland optical measurements of the finish of precisionmachinedopticalsurfaces E. L. Church U.S.Army Armament Research and DevelopmentCenter Dover, New Jersey 07801 -5001 T. V. VoÉurger NationalBureauof Standards Gaithersburg,Maryland20899 J. C. Wyant OpticalSciencesCenter University of Arizona Tucson,Arizona85721
Abstract. This paper comparestwo methodsof measuringthe fin¡sh of precisionmachinedopticalsurfaces:the older,well-establishedmechanicalstylus gauge and a recently developedoptical gauge using interference microscopy. Resultsare found to be in good quantitativeagreementfor both randomand periodicsurfacefeatures, providedthat appropriatefiltering proceduresare included¡n the data analysisto accountfor the differingtransferf unctionsand bandwidthsof the two measurementtechniques.Theseresultsaffirm the use of these techniquesfor the quant¡tativemeasurementand specificationof machinedopticalsurfaces. Subject terms:surfacemetrology;precisíonmachining;diamondturning; single-point machining;surfacefinish;profilometry;machinedoptics. OpticalEngineeríng24(3).388-395(May/June | 985).
CONTENTS I. Introduction 2. Linear measurements 3. Extremeand effectivebandwidthlimits 4. Transferfunctions 5. Data processing 6. Resultsfor the germaniumsurfaceó 7. Resultsfor the siliconsurface 8. Quantitativefinish parameters 9. Discussion 10. Summaryand conclusions ll. Acknowledgments 12. Appendix A: effectsof surfacælayers 13. Appendix B: samplerotation 14. References I. INÎRODUCTION This paper discussesthe analysisof surfaceprofile measurements and comparesmechanicaland optical stylus measurements of two typesof test surfaces.It confirmsand extendsthe resultsof a paper on the samesubjectpublishedpreviously.t The mechanicalstylus measurements were made using the digitized Talystep* at the National Bureau of Stand¿rdsin ôaitheriburg, Maryland.z'¡This instrumentmeasuresthe surfaceprofile by drawinga fine diamond-tippedstylusacrossthe surfaceand converting its verticalmotion into an electricalsignalby an electromechanical transducer.This is the sameinstrumentusedearlier,rexceptfor cleaningof the lead screwand replacementof the stylustip. Gñ-in ircr. of commercialequipmentarc identifiedin this paperto specifyexperimental?rocedurcs. In nocasedoessuchidentificationimply rccógliitionofendórseårent by the Departmentof Defenseor the Departmentof Commerce. 'vited PaperME-103receivedNov. 15,t984; revisedmanuscriptreceivedFeb. I 3. l9g5: :eptedfor publicarionFeb.I 5, 1985;receivedby Managingllditor Feb.2g, | 9S5.Thi; ¿p9r rs a revisionof Paper508-13which was presentedat the SPIE conferenceon ProductionAspectsofSinglePoint MachinedOptics,Aug.23-24,1984,SanDiego,CA. t h€_papjrpresen-ted rhereappeårs(unrefereed)in SPIE ProceedingsVol. 509.e 1985Sæiety of Photo-OpìicalInstrumenrarionEngineers.
388
/ OFflCAL ENGINEERING/ May/June 1 985 ,/ Vot. 24 No. 3
The optical stylusmcasurements weremadeusinga trVykomodel l(XX)noncontactingprofiling microscope.,rThis instrument measuresthe surfaceprofile by determiningthe phasevariationsof the light reflectednormally from the surfaceand convertingthem into heightvariationsusing
z(x)=f ø*l ,
(t)
whereÀ = 0.6328¡rm is the meanmeasuringwavelength.As previously,l measurements weremadeat Wyko Optical, Inc. (Tucson, Arizona) usinga stock instrumentwith a 20X objective. Raw profile data from each measurementwere recordedand analyzedat BrookhavenNational laboratory (Upton, New york) using routinesdevelopedunder the aegisofthe National Synchrotron Light $6r¡¡çe.1.5.0 Resultsare discussedherein termsof two profile statistics:the power spectraldensityand bandwidth-limited valuesof the root-mean-square (rms) roughness.Thesewerechosen becauseof their direct relationshipwith functional performanceof the surfaceswhenusedin an optical system.The first is a one-to-one mapping of the scattered-lightintensity,and the secondis directly relatedto the total integratedscatter(TIS) or Strehl factor.s.r The test surfaces,germanium and silicon disks supplied by Pneumo Precision,Inc. (Keene,New Hampshire),were nominally flat, finished by single-point fly cutting. Machined surfaceswerè chosenbecause.they exhibit both randomand periodicfeaturesand becauseprofile data taken acrossthe lay of suchsurfacesare more directly interpretedthan thoseof polishedsurfaces.5-s The surfacesusedhad finishesõf the order of 50to | 00 Å , chosen to fall comfortably within the measurementrangesof the instrumentsratherthan to representthe stateof the art of finishing.A discussionofstate-of-the-artsurfacesis givenin anotherpaperin this collection.cAlthough the two setsof test surfaceswêremachined under similar conditions,their surfacetextureswerevery different: thefinish of thegermaniumwasmainly in theform of low-frequency randomroughness whilethe siliconshowedhigh-frequency periodic
MACHINEDOPTICALSURFACES OF THE FINISHOF PRECISION AND OPTICALMEASUREMENTS OF MECHANICAL DIRECTCOMPARISON
-1
îl
.1
- ,l l - l _ ,l ^ _ ,
I
Fig.3. Trander function¡ for l¡noarmoatur€m€ntr.
Z*."rur"¿(x) = M(x)*Zo6¡.",(x) ,
(3)
or, equivalently,a simplemultiplication in frequencyspace: Z,,,"".ur"¿(Ð :
Fig. l. Nomarskim¡crophotographof th€ gefman¡umtost surfaco.Th€ long dimensionin the photographi¡ 240 pm.
t*¡(0zoujot(Ð ,
wherethe overbardenotesthe Fourier transform. M(x) is then the measurementimpulse response,and M(f) is its transfer function. Becauseof the simplemultiplicativeform of Eq. (4), it is convenient systemsin the frequencydomain. to comparemeasurement We take the mechanicalgaugeto behaveas a linear measuring instrumentover its operatingrangeas definedin the following section. The optical gaugemay be shownto behaveas a linear instrument in the smooth-surfacelimit
ffi..,,
Fig.2. Nomarckimicrophotographof the s¡licontest ¡urface.The bands conerpond to the 3.4 pm tool feed.
roughness. As discussed below, this complementarity provides a sensitivetest of our understanding of the measurementprocess. Figures I and2 are Nomarski microphotographs of the germanium and silicon surfaces,respectively.The tool marks on the germanium surface are fairly random, while the striking bands in the photograph of the silicon surface clearly show the 3.4 pm tool-feed spacing.Thesedifferencesin texture are also very conspicuousto the eye:the germanium appearssmooth, while the silicon appearsrough and displays diffraction colors.
2. LINEAR MEASUREMENTS The measurement process may be viewed as a mathematical operation: Zmeasured:MZobject'
(5)
roughness.Detailswill be pubwhereZrn . is the root-mean-square lishedelsewhere. Figure 3 is a sketchof canonicalforms of measurementtransfer functions.The perfectcurve,A, is unity forall frequencies-a practical impossibility.Curve B is the ideal curve-unity over as wide a rangeas possibleand zerooutside.Curve C is a realisticcurve-the measurementis sensitiveto a more-or-lesswelldefined range of frequencieswith a frequency{ependenttransferfunction. CurveC can be transformedinto a practical approximation of curve B by careful hardwaredesignor, as discussedbelow, by processingthe measurement data in subsequentsoftware. situationwould occurwhenthefrequency The ide¿lmeasurement content of the object being measuredfalls completely within the bandpassof the measurementapparatus,in which casethe measurementresult would be transparentto the techniqueused.ln the caseof surfaceroughness,the oppositesituation holds: the roughness is spread over a wide range of surfacewavelenghs-from atomic diametersto the diameterof the workpiece-wider than the bandpassof any singlemeasurementtechnique.In that case,the comparisonand interpretation of measurementsmust necessarily take into accounttheir differing transferfunctions. the bandwidthlimits of the two The followingsectiondiscusses instrumentsusedhere,and following that, theforms of their transfer functions. 3. EXTREME AND EFFECTIVE BANDTVIDTH LIMITS aredeterminedby the The bandwidthlimits of profile measurements total tracelengthand the samplinginterval. If the trace lengthL is sampledat N equallyspacedpoints,the extremesurfacefrequencies are includedin the measurement
Q) : ¡(e¡treme) 'mln
where M is a measurementoperatorandZ,in our case,is the surface profile. A good measurement is one for which M is in the unit operationover as wide an operatingrangeas possible. The most desirablemeasurementsare linear; that is, sine waves are measuredwith no harmonicdistortion. In that case,Eq. (2) takes the form of a convolution in real space:
(4)
|
,
L
f$¡lreme¡ : fNyquisr=
(ó) N
I
The effectivevalueslie within theselimits byamounts that dependon / MaY/June 1985,2Vol. 24 No. 3,2 389 OPTICALENGINEERING
\MYANT CHURCH,VORBURGER,
TABTE l. Measurement Paramstsror
Numberof SampledPoints N ¿IOOO , 1024
lnstrumont Mechanical Optical
Totaltrace length L
Samplinginterval L,/N
1500 665
0.375 0.649
Effective1/f.", 4L/N 1.5 2.6
Effeaive 1,/f'''¡n L/2
750 333
rAll d¡stanc€s¡n mícrometers
detailsof the measurementprocess. The effectivelow-frequencycutoff is determinedby the fact that the raw profile data must be detrendedto removespuriouspiston (constant),tilt (linear),and possiblypower(quadraticlcontributions process.This is usuallydoneby introducedduring the measurement polynomialfitted to eachdata set.It can subtractinga least-squares be shownt that this operation is equivalentto a high-passfiltering operation that eliminatesthe frequenc] f : 0 and attenuatesthe frequencyf -- | lL, but passeshigher frequenciesessentiallyunaffected.For this reason,wechoosetheeffectivelow-frequencylimit to be twice the extremevaluegiven in Eq. (6). The effectivehigh-frequencylimit is determinedby the fact that any signallying abovethe Nyquist frequency[Eq.(ó)] is not eliminated,but appearsas an alias within the measurementrangeat the frequency falias:2f¡r'u¡.1 -f
.
(7)
Welldesignedmeasurement systemsincludean antialiasingfilter to attenuate or eliminate frequenciesabove the Nyquist frequency beforethe samplingoperationin order to eliminatesucheffects.It is the propertiesof theseantialiasingmechanismsthat determinethe effectiveupper frequencylimit. In the case of the mechanicalgauge, antialiasing filtering is accomplishedby the electrical and mechanicalproperties of the stylusinstrument.Its transferfunction hasbeenmeasuredsand was found to follow closelythat of a simplelow-passfilrer:
M(Ð :
t l +(fdJ
(8)
The value of do, the spatialwavelenglhfor 50Voamplitude attenuation, is O.862pm.The finite sizeof the stylustip also attenuatesthe measurementof short surfacewavelenghs but does not place a significantlimitation on the presentstudysinceits measuredwidth is only of the order of 0.5 pm. In particular,the estimatedtip curvature is significantly greater than the rms curvature of the test surfaces obtained by evaluatingthe fourth momentsof their profile power spectra.The procedurefor evaluatingthe latter is discussedin Ref. l. In the caseof the optical gauge,antialiasingis accomplishedby thefinite resolvingpowerof its optical microscopeand the sampling apertureofthe photodiodearray in its focal plane,asdiscussedin the following section. To stay within the bandwidth limits imposed by the various filtering operations,wetaketheeffectiverangeof surfacefrequencies to lie a factor of two within the extremevaluesgivenin Eq. (6); that is ¡{effect) -min -
L , L I
(e) N
. :_. f' ( e l t e c t ) : : f - ^Nyqulst . max 2 4L Numerical values of these frequencies and other parameters of the measurementsdiscussedbeloware given in Table I. Theseshow that although the bandwidth limits of the two measurementtechniques are different, so that raw measurementsare not directly comparable, there is a considerable region of overlap over which quantitative comparisons can be made. 3W
/ OPflCAL ENGINEERING / May/June 1985 ,zVol. 24 No. 3
4. TRANSFERFUNCTIONS The behavior of the transfer function of the mechanicalmeasurement at its low- and high-frequencylimits hasbeendiscussed above and in Ref. 5. We take its value betweenthe effectivelimits to be unity. A more detailedanalysiswould include effectsof a variable transferfunction in a mannersimilar to that discussedbelowfor the optical gauge. The shapeof the transer function of the optical gaugeis determinedby threeprincipal factors:thef¡nitetemporalcoherence of the light used,thepropertiesof its opticalsystem,and thefinite pixel size ofthe photodiodearray in its focalplane.The total transferfunction of the measurement is then the product of threefactors: t"t(Ð :
ú"oh"r"n""úopticsúarray
(t0)
We now examinethesefactorsindividually. The instrumentusesfìnite bandwidthlight to eliminateinterferenceeffects:À : 0.6328t 0.0200¡rm. The parametèrdescribing thecontrastattenuationdueto a wavelengthspreadof AÀabout À is
(l t)
*."*
However,the surfacesconsideredheresatisfythe Rayleighsmoothnesscondition, Eq.(5), in whichcasecoherence effectsarenegligible. That is, for our measuremènts M"oh"r"no(Ð :
I
(t2)
The phase-to-intensitytransfer function of the optical system dependson opticaland structuraldetailsthat arenot readilyaccessible. For the presentpurposes,we model this as a simple triangle function: úopti"r(Ð = l -fdr".o¡,
(13)
where 4 À dresot:;2NA
(14)
and NA is the numerical aperture of the objective.For the 20X objectiveusedin the presentmeasurements, NA : 0.4,from which it followsthat dr.ro¡ : I ,¡m. Equation(13)is simplythe textbook form of the intens'iiy+o-intensitymodulationtransferfunction of a simplelenslinearizedby extrapolatingits tangentat zerofrequency, which accountsfor the factor of 4l n An obviousrefinementwould be to usethe full algebraicform appropriatefor the annularaperturer0 resulting from the presenceof the referencemirror in the microscopeobjective. Eachpixelofthe photodiodearrayin theopticalgaugeintegrates the intensityof the imageoveran essentialty squareaperturewith a sidelengthof approximatelyL/N : 0.65¡rmwhenprojectedonto the surfacebeingmeasured. In the instrumentusedhereeachmeasurementpoint corresponds to the averageof two adjacentpixels' The resultingarray transferfunctionis then
-----' MACHINEDOPflCALSURFACES OF PRECISION OF THE FTNISH OF MECHANICALAND OMCAL MEASUREMENTS DIRECTCOMPARISON
ú"..r{Ð =
sin(2zrfLlN) 2¡rfllN
(t5)
This function is unity at low frequenciesand goesthrough its first zÊroat the Nyquist frequencyof the array.* limit (TableI), thethreeamplitudeAt theeffeótivehigh-irequency faõtorsin pq. (lO) havethe values1,0'615,and transfer-function givinga total attenuationof0.392'At the surface 0.637,respectivety, wau"tengittof 3.4 ¡rm-the period of the tool marks in Fig' 2-the cor..rpoiding factorsare l, 0.706,and 0-777' gvrnga total attenuation oi O.S¿g.lttenuationsof this magnituderequirecorrection" To compensatefor the nonconstanttransferfunction of the optical gauge,ïe include a restorationfilter in the data analysisthat conJistiof the frequencylomain inversefilter
ú¡n""rs.(Ð: tvto],,.r{Ðv-Lr(Ð
(tó) germanium rurface.
@e
In principle,this filter can be usedat freguenciesabovethe effective maximum given in Tabte I. However, it divergesat the extreme frequencytEq. (ó)1,which correspondsto a surf¿cewavelengthof 1.3 pm, änì ås a practical matter, we cut it off at 2 r¿mto avoid tpuiiour noiseeffeìts. At that point the total amplitude restoration to a powerrestorationfactorof20'93' fäctoris4.575,corresponding 5. DATA PROCESSING Raw data from eachinstrumentare first detrendedby subtractinga least-squaresquadratic fitted to the individual data sets' The detrendedprofiles(residuals)are then plotted to givethe unrestored profiìe.Theseprofilesare then passedthroughthe restoration finish -which involves Fourier transformation using the FFT algofilter, rithm, multiplication by the restorationfilter, and inversetransformation. Thé resultingrestoredprofilesare also plotted' In the mechanicalcase,the restoration filter involves simple multiplicationsby unity, and the restoredand unrestoredprofilesare identícal.In the óptical case,restorationinvolvesmultiplication by the inversefrlter in Eqs. (13) through (ló) for frequenciesup to . 0.5 pm-l and by zerofor higherfrequencies. 'ihe restoredresidualsare then multiplied by a Hamming data windowand the periodogramspectralestimateisformedby computing the squaremagnitudeofthe Fourier transform ofthe product' Tñi. rp"ót.u- is ihen plotted over the extreme range of surface frequenciesspannedby the measurementtechniqueinvolved' ilandwidt'h-timitedistimatesof the rms surfaceroughness,slope, and curvatureare then computedby evaluatingweightedintegrals (sums)ofthe spectrumoversèlectedrangesofsurfacefrequency'It is ihe comparisoì of thesequantities,measuredover the sameranges for eachìnstrument,that offersthe most sensitivetestof the equivaof the different measurementtechniques. lence ' for the operationsdescribedabove The mathematicalexpressions are givenin Ref. l. 6. RESULTS FOR THE GERMANIUM SURFACES Figures4 and 5 show the raw and restoredoptical profiles^ofthe gelnanium test surface shown in Fig. l. Only the first 500 ¡rm of iO.Smm) section'ofthe profile is shown,although the full range ùOS,rtn was usedin theãnalysis.The vertical scalecorrespondsto t60b Å. The two profilesare essentiallyidentical,althoughclose examination showi that the restoredprofile doeshavegreaterfine structure. Figure 6 is a sectionof a mechanicalprofilg of the samesurface plottedon the samescaleas in Figs.4and 5. Althougha one-to-one òomparisonwith the precedingprofilesis impossiblesincethey were takenat differentpointson the surface,it is clearthat thereis a strong "statistical"simiúrity betweenthe mechanicaland opticaldata' In profileappearsto showevengreater detail,however,themechanical * Eqution ( I 5) differs from Eq. ( | 4) in Ref. I by the presenceoftheaspcct ratio of 2 in the arsument oi the sinc function to account for the two-pixel averaglng menttoned aDove'
Fiõ3:Tottõred
ven¡on of tho opt¡cal profile in Fig. 4.
I
a a N
Fig. 6. Mechanicalprofile of tfre germanium ¡urfaco.
fine structurethan doesthe restorçdoptical profile. The proper way to appreciatethesedifferencesis to seehow the ipowei" is the distributed among different surfacefreroughnéss queicies, thãt is, by looking at the spectraldistributions of the profiles. ' ofthe Figures7, 8, and 9 are logJogplots of the periodograms three-profilesshown in Figs.4, 5, and !, respectively.The vertical and the horiscaleiuns betweenl0-7 and l0-l ¡¡m+l (ó decades), scalerunsfrom l0-3 to 2 pm-l (3*decades)-These zontalfrequency / Mav/Junø1985,/ Vol.24 No'3 / 391 ENGINEERING OPTICAL
\MYANT CHURCH,VORBURGER,
The spectrallinesat a wavelengthof about 3 ¡.rmthat appearin each spectrumare the fundamentálof the tool marks'periodicity, which is hard to discernin the Nomarski microphotographand is unrecognizablein the surfaceprofiles.Thesespectrallinesareof the order of two or threefrequencybins wide,the natural linewidth for the Hamming data window. The additional lines at 1.9 pm in the optical spectrumand at 0.9 ¡rmin the mechanicalspectrumaretaken to be spurioussincethey do not appearin other profilesof the same surface.
7. Periodogramof the raw opticrl profile in Fig, 4.
8. Periodogramof the rortorod opt¡cal profile ¡n
n
H o
2
è
profilein Fig.6. Fig.9, Periodogram of the mechanical wide rangesare requiredto displaythe full characterofthe structure found in thesesurfaces. The three spectraare very similar except at high frequencies, which, in the germaniumsample,representonly a small fraction of Ìstotal roughness. Figures7 and8 showtheimportantamplification and its ultimate ;ffectsof the restorationfilter at shortwavelengths cutoff at 2 ¡rm.The mechanicaldata,whichextendto a wavelengthof with the restoredthan with 0.75 ¡tm,areclearlyin betteragreement opticaldataovercommonfrequencyranges. the unrestored ENGINEERING / May/June1985/ Vol.24 No.3 392 / OPTICAL
7. RESULTS FOR THE SILICON SURFACE Figuresl0 and I I show the raw and restoredoptical profilesof the silicon surfaceshown in Fig. 2. In this case,only the firsl lfi) ¡rm sectionofthe profile is shown,althoughagainthefull óó5¡rmofdata wereusedin the analysis.The verticalscalesare again tó00 Ã. The conspicuousperiodicitiesin the profiles correspondto the approximately 3 ¡¿mtool feed. However, there are two obvious differencesin the profiles: the restoredversion has a larger amplitude,and the shapeof the oscillationsis smoother.ln particular,the ofthe peaksin the notch that frequentlyappearson theleadingedges unrestoredprofile is missingin the restoredversion. Figure 12is a mechanicalprofile of the samesurfaceplotted on the samescales.Superficially,it is very different from either of the optical profiles; it has a much larger peak-to-valleydistance,the peaks and valleys are sharper, and a distinct intermediatepeak appearsbetweenthe major ones.To appreciatethe sourceof these differences,we examinethe spectraof theseprofiles. FiguresI 3, I 4, and I 5 arethe periodogramsof the profilesshown in Figs. 10, I l , and 12, respectively,on the samescalesas the earlier spectra. FigureI 3 showsa pair ofvery sharpand intenselinesthat arethe fundamentaland fïrstharmonicof the3 ¡rmtool feed.In Fig. 14,the first harmonicis cut off by the 2 pm cutoff of the restorationfilter, leaving only the fundamental.This accountsfor the simlutaneous amplification and smoothingof the shapeof the oscillationsin Fig. the I l. It doesnot appearas a smoothsinusoidin Fig. I I because plotting routine usesa linear rather than a Whittaker interpolation schemefor connectingthe discretedata points. The mechanicaldata, Fig. 15,showthe intensefundamentaland the first harmonic of the tool feed,as well as the secondand third harmonics. It is the presenceof these higher harmonics in the that lead to the greateramplitude and mechanicalmeasurements detail of the profile in Fig. 12. The peaksof the mechanicallinesare of the order of two to four frequencybins wide, which is again comparablewith the natural Hamming width. However,they have considerablybroader bases than do the optical lines,due, presumably,to the fact that the involve a dynamicmeasurementsystem, mechanicalmeasurements which is subjectto vibration and drive-ratefluctuations. Comparisonof the threespectraovercommonfrequencyranges showsclearly that the restoredoptical spectrumis in much better agreementwith the mechanicaldata than is its unrestoredversion. t. QUANTITATIVE FINISH PARAMETERS Surfaceprofilesand spectraprovidevaluablecomplementaryviews of surfacetopography.Profilescan revealextraneousspikesor pits in the measurementthat would be difficult to recognizein the frequencydomain.On theotherhand,spectracanrevealsmallperiodicities that are invisiblein the profile tracings.Theserepresentations, however,areprimarilyvisualand quantitative.A quantitativecomparisoncanbe madeby examiningfinish parametersevaluatedfrom the profile spectraover the samerangesof surfacefrequencies. Table II lists a number of such parametersderivedfrom the presentseriesof measurements. Threetypesof measurements of three surfacesare compared(top to bottom): unrestoredoptic¿!, of two germanium restoredoptical,and mechanical measurements surfacesand one silicon surface. The parameters compared(left to right)arethe periodof thetool
DIRECTCOMPARISON OF MECHANICAL AND OPTICALMEASUREMENTS OF THE FINISHOF PRECISION MACHINEDOPT|CALSURFACES
E d
t
o a
a E
z N
z G
Fig. 1O. Raw optical profile of the rilicon ¡urface.
Fig. I 3. Periodogramof the raw optical file in Fig. 1O.
Þ Ë
E
T
E
N
, À
=
Fig. 11. Re¡tored ver¡ion of tùe optical profile in Fig. lô.
t o
È
E
C
Fig. 14, Periodogramof tho ro.torod opt¡crl profilo in Fig. l l.
É ¿ F
a G a N
o c o
C z c
Fig. 12. Mechanical prof¡le of the rilicon ¡urfaco.
Fig. 15. Periodogramof the mochanicalprofile in Fig. 'l 2.
marks, the amplitude of their fundamentals,and the rms valuesof the surfaceroughnessobtainedby integratingthe spectralestimates betweenthe indicated rangesof surfacewavelengihs:2 to lZ pm, 12to 333pm, and 2 to 333 pm. The 2 ¡rm limit iJ the cutoff of the opticalrestoration filter, l2 pm is thenominalupperlimit of conventionalTIS measurements madewith normal-incidence HeNelight,and 133 ¡m is the effectiveupper wavetengthlimit imposedby the quadratic detrendingof the optical data. The last column of Table II givesthe wide-openrms roughness values,that is, the nonequivalent valuesobtainedby analysisof the
unwindowedprofile data overthe extremerangeof surfacefrequencies determined by the measurementproceduresthemselvei(cf. Table I). The numbersin parentheses in the mode column are ih" numbersof independentmeasurements usedto generatethe average valuesand standarddeviationsgivenin the table. 9. DISCUSSION The optical and mecha'nical measurements of the 3 pm tool feedare in excellentagreementfor eachsample,althoughtherearesignificant OPTICALENGTNEERTNG / Mav/June 19Bb ,/ Vot. 24 No. 3 ,z 393
WYANT CHURCH.VORBURGER.
TABLE ll. Finish Parametert Line position Mode ,¿m
Amplitude fundamental
#1 GERMANIUMSURFACE
10.3+ O.8 12.8+ 0.8 13.7t O.3
Raw optical (61 Reslor€d opt¡csl (6) Mechanical(2)
2.759r0.008 2.759+0.008 2.703tO.OO2
1 . 2 +O . 1 2.7t O.2 2.61 0.1
Raw optical (61 Restored opt¡cal(6) Mechanical(11
3.03510.021 3.035f0.021 2.977
GERMAN¡UMSURFACE#2 8.4+ 0.9 1.91 O.1 11.Ot 1.O 3.8r O.2 10.6 4.1
Raw opt¡cal (2) Restored opt¡cal(2) Mechanical(3)
3.4o,2tO.O21 3.¿to2fo.o21 3.345+0.035
37.4X. 4.4 68.1t 7.6 7 5 . 1+ 1 1 . 0
SILICONSURFACE 39.2+ 4.0 70.at 7.o 8 4 . 1t 1 1 . 6
variations.The 3 Â amplitudesof the fundamensurface-to-surface tal on the germaniumsurfaceare also in very good agrgementafter restoratio;, and are comparablewith thi value of 2 ,{ computed from the tool radius. It is noteworthy that this periodic surface feature with atomic dimensionsis observedwith a signal-to-noise ratio ofroughly 50 to l. The varióus bandwidth-limitedroughnessvaluesof the germanium surfaces,includingthe wide-openvaluesin the lastcolumn,are in good agreementwith or without restoration sincethe finish of theie surfacesis dominatedby long surfacewavelengths.Resultsfor the silicon surface,however,are quite differenurestorationis essential to getagreementfor the high-frequency(zto 12pm) and broadband (2 to 333 ¡rm) values since these are dominated by the tool marks. high-fràquency -The importanceof comparingresultsoverequalfrequencyranges . showndiamaticallyin the lastcolumn of thesilicon data. Restoration is important, bui evenso,the mechanicalroughnessis still twice the optical value.The sourceof this differencelies not in the meaof surementtechniques,but in the"apples-and-oranges'comparison data involvingdifferent bandwidths. 10. SUMMARY AND CONCLUSIONS This paper discussessurface profile measurementsin general, in particular, and compares mechánicaland optical measurements resultsobtained with two typesof test surfaces:one dominated by low-frequencyrandom roughnessand the other by high-frequency 'periodicroughness. The comþarison is made over that overlapping region of frequency-"mpiitudespacein which both measurementtechniquesare únearând havea commonbandpass.The mechanicalmeasurement is taken to havea constanttransferfunction over its effectivebandnonconstantresponse width, whiletheopticalmeasurementinvolvesa that attenuateshigh surfacefrequencies.To compensatefor this,a first-order modelõf the transferfunction of the optical instrumentis usedto designa simplerestorationfilter that hasbeenimplemented in softwarecompatiblewith the commercialinstrument. Qualitativeandqr¡antitativecomparisonsshowtheimportance.of of surfacescontainrestòrationeffectsfor broadbandméasurements for the studyoftool marks, roughness, ing significanthigh-frequency an¿ fór ttreextractionoffinish information correspondingto largeangle -Th.surfacescattering. betweenthe results of mechanicaland optical "g.".ment sensitiveto mechanicalpropertiesat immense measurem€nts-one pressure and the other to electromagneticpropertiesat optical fre' rencies-is extremelygratifying,and affirms the useof thesetech,. ren€s for the quaniitative measurementand specificationof machinedoptical surfaces. the effectsof surfacelayersin the optical Appendix A discusses -."tut -.nts, and Appendix B addressesthe effects of surface gg4 / OPTICAL / Mav/Junel985'/Vol' 24 No'3 ENGINEERING
rms
fms wide open
A
Ä
{ri =rrlåg3 ¡.m) (f-1=2-333 pml
(f-1=2-12 rm)
32.1X3.0 3 3 . 1 +3.0 34.0r 3.4
33.7+ 2.t 35.5t 2.6 36.7+ 3.3
35.4t 2.8 36.4X 2.8 37.4t 3.O
36.3+ 3.7 373t 3.8 34.7
37.3f 3.6 38.9i 3.7 36.3
38.1+ 2.6 39.3+ 3.6 37.8
zò.t+3.5
44.6+ 1.9 74.1+ 5.8 88.O+11.O
2 1 . 3 i 3.4 25.8+. 0.5
51.1+ O.7 74.2t 5.7 145.7*.14.O
rotation on linear profile measurements. 11. ACKNOWLEDGMENTS The test surfaceswereprovided by R. Clark at Pneumo Precision, Inc. C. Giauque assisiedwith the Talystep measurementsat the National Buréauof Standards,and C. Koliopoulos and S' Iange at Wyko, Inc- At Brookhaven made the optical me.asurements National laboratory,'P. Takacsprovided valuableassistanceand discussions,H. Berry carried through the data analyses,and W' Marin, Jr. preparedthe Nomarskimicrophotographs.D. Aspnesat Bell CommunicationsResearch,Inc. provided information on surfacelayers.This work wassuppportedin part by the Departmentof Energy(E.L.C.) and the Departmentof Commercc(T.V'V')' 12, APPENDIX A: EFFECTS OF SURFACE LAYERS The optical gaugemeasuresthe phaseof the light reflectedfrom the surfaè bein! eiamined. To convert phaseinto height we requirea model. Pq. (t) is basedon the model that the surfaceis a simple interface:air aboveand materialbelow. Realsurfacesinvariablyinvolvesomesort of chemicalorphysical layersdepositedduring or after the machiningprocess.In interpretin termsof Eq. (l) we havemadethe ing the piesentmeasuiements of theselayersare negligible'To effects that the irñplicit assumption complicatedphysicalmodet a more slightly a examine we ttris ¡usiify structure. simpleone-layer ihe additiãn of a layer of thicknesst to a baresubstrateshiftsits apparentheight,as determinedby Eq- (l)' accordingto
ztay.*Å : zbar.+
À *
ú(t) '
(17)
wherery'(t)is the phaseof
a
16¡ * r¡2exp(-iqt)
exP(*iq")
I * rorrt2exp(-iq¡)
r02
Here 4zr 9a A;N"t
'
(le)
and
Nb-Nu
rab: lfr\
(20)
a to b is the normal-incidence amplitude reflection coefficient of the
MACHINEDOPTICALSURFACES AND OFTICALMEASUREMENTS OF THE FINISHOF PRECISION OIRECT OF MECHANICAL COMPARISON
interface,wherethesubscriptsare Q: air, I : layer,and2: substrate. In the thin-film limit, q ( I, t t -N,2
(2t)
a:l*i+zr;¡$+....
.
. \'
d(d,R) : ,
(22)
rtrhere,for simplicity, we havetaken the index of air to be unity*Surfacelayerson germaniumand siliconconsistofabout 25 Ã of oxide plus alayer of organicmaterialwith N- 1.5whosethickness zeroafter cleaningto somethingof the order variesfrom ess.¡:ntially ofanother 25 Ä after storageand handling.Substitutingreasonable Valuesfor the various indicesinto Eq. (21), we find that the coefficient multiplyingt(x) lies between0.6 and 0.8. It follows that the magnitudeof the layer coqtribution to the apparentoptical height could amount to l5 to 40 ,{, which is a signifircantamount. The analogousexpressionfor mechanicalmeasurements obviouslydependson the mechanicalrather than on the optical propertiesof the layer. If we write Z6r"n¿(x):Z6ur"(x)*kt(x)+...,
d(0,R) :
+ [ ( R * a ¡ z - ( R s i n e ¡ 2 1 %- R c o s t i ,
(24)
whereR is the machiningradius.In the limit of large R/d,
If the indicesof refractionN are real, I -N;z Z 1 ^ r " n 6 ( x ) : Z 6 ^ n ( x )+ T : ¡ 7 t ( x ) * . . .
direction, a given surface wavelength d appears at the longer wavelength:
(23)
thereare two obviousextemes:k : I for a "hard" layerand k : 0 for a completely.*soft"one. The critical quantity is not the total magnitude of the layer correctionsin either Eqs. (22) or (23) but the magnitudeof their fluctuationsalong the proftle and their correlation with the height fluctuationsof the substrate.A constantlayer thickness,for example, would simply shift the apparentsubstrateprofile by a constant amount, which would be unobservableeither optically or mechanically. The good agreementbetweenthe optical and mechanicalmeasurementsobservedin the presentexperimentsis attributed to the fact that over the surface-wavelength region explored, the layer tracks the substrate,and any differential effect is small and submergedin the experimentalerror. However,asmeasurement accuracyimprovesand finer and finer interpretationsof measurementdata are attempted, one must be alert to the fact that mostsurfacesare not simplestructuresand that surfacelayerscould lead to significanteffects. 13. APPENDIX B: SAMPLE ROTATION The text considersthe casein which the direction of the surfacelay (tool marks)is perpendicularto thedirection of the profile measurement. If the sampleis rotated through an angle 0 away from that
d
'
(2s)
"or0 which is valid for all rotation anglesexceptthoseveri near90". In this way the apparentwavelengthcan easilybe stretchedby a factor of l0 (rotationof 84.3'). The transferfunction of the optical gaugeis (presumably)circularly symmetricand thereforeindependentof the samplerotation. That is, the frequencyappearingin Eq. (13) is the true surface frequencyf = l / d ratherthan theapparentvaluegivenby Eqs.(24) or (25). However,sincethat transfer function vanishesfor wavelengthslessthan I ¡rmfor the20X instrument,the presentstretching techniquecannotgiveusefulinformation about surfacewavelengths shorterthan that value. The transfer function of the array doesdependon the angleof rotation sinceits pixelsare not circularly symmetric.As discusscdin the text, the pixelsareessentiallysquaresthat are summedpairwise to give a rectangularform with an aspectratio of A = 2. That quantity appearsasa factor in thecommonargumentin the numerator and denominatorof Eq.( l5). In thecaseof a rotatedsample,that factor A mustbe replacedby A[cos0 * Asind]-r. Samplerotation is usefulfor zoomingin on high-frequencypro{ile features(suchasthefine structureof the3.4¡rmlinesin thesilicon for examiningwavelengthcontributionslying betweenthe surfacæ), I ¡rmmicroscopecutoff and the2 ¡rmfilter cutoff, and for identifying aliasedlines in the spectrumsincethey will move to higher rather than to lower frequencies as the sampleis rotatedfrom d = 0.
14. REFERENCES l. E. L. Church, i¡ hecision SurfaceMetrology, JamesC. Wyant, cd., Proc. SPIE 429, 105(1983). 2. E. C. Teaguc,Nat. Bur. Stand.(U.S.)TechnicalNote 902 (197Q. 3. T. V. Vorburger,E. C. Tcaguc,and F. E. Scire,Dimensions/NBs62(l l), lE (r97E). 4. J. C. Wyant, C. L. Koliopolis, B. Bhushan,and O. E. George,Am. Soc. Lub. Engrs.Trans.27,l0l (1984). 5. E. L. Church, M. R. Howells,and T. V. Vorburger,in ReflectingOptics for SynchrotronRadiation,Malcom R. Howclls,cd., Proc. SPIE 315, 202 (t982). 6. E. L. Church and H. C. Berry, Wear 83, 189(1982). 7. E.L. Church, i¡ PrecisionSurfaceMetrology, JamesC. Wyant, ed., Proc.SPIE 429,86 (19E3). E. E. L. Church, H. A. Jenkinson,and J. M. Tavada,Opt. Eng. 16(4),360 (1977). 9. E. L. ChurchandP.Z. Takacs,Opt. Eng.24{3),(1985). t0. E. L. O'Neill, J. Opt. Soc.Am. 46, 285(1950.
OPTICALENGINEERING / May/June 1985 / Vol. 24 No. 3 ,z 395
Abstract. This paper comparestwo methodsof measuringthe fin¡sh of precisionmachinedopticalsurfaces:the older,well-establishedmechanicalstylus gauge and a recently developedoptical gauge using interference microscopy. Resultsare found to be in good quantitativeagreementfor both randomand periodicsurfacefeatures, providedthat appropriatefiltering proceduresare included¡n the data analysisto accountfor the differingtransferf unctionsand bandwidthsof the two measurementtechniques.Theseresultsaffirm the use of these techniquesfor the quant¡tativemeasurementand specificationof machinedopticalsurfaces. Subject terms:surfacemetrology;precisíonmachining;diamondturning; single-point machining;surfacefinish;profilometry;machinedoptics. OpticalEngineeríng24(3).388-395(May/June | 985).
CONTENTS I. Introduction 2. Linear measurements 3. Extremeand effectivebandwidthlimits 4. Transferfunctions 5. Data processing 6. Resultsfor the germaniumsurfaceó 7. Resultsfor the siliconsurface 8. Quantitativefinish parameters 9. Discussion 10. Summaryand conclusions ll. Acknowledgments 12. Appendix A: effectsof surfacælayers 13. Appendix B: samplerotation 14. References I. INÎRODUCTION This paper discussesthe analysisof surfaceprofile measurements and comparesmechanicaland optical stylus measurements of two typesof test surfaces.It confirmsand extendsthe resultsof a paper on the samesubjectpublishedpreviously.t The mechanicalstylus measurements were made using the digitized Talystep* at the National Bureau of Stand¿rdsin ôaitheriburg, Maryland.z'¡This instrumentmeasuresthe surfaceprofile by drawinga fine diamond-tippedstylusacrossthe surfaceand converting its verticalmotion into an electricalsignalby an electromechanical transducer.This is the sameinstrumentusedearlier,rexceptfor cleaningof the lead screwand replacementof the stylustip. Gñ-in ircr. of commercialequipmentarc identifiedin this paperto specifyexperimental?rocedurcs. In nocasedoessuchidentificationimply rccógliitionofendórseårent by the Departmentof Defenseor the Departmentof Commerce. 'vited PaperME-103receivedNov. 15,t984; revisedmanuscriptreceivedFeb. I 3. l9g5: :eptedfor publicarionFeb.I 5, 1985;receivedby Managingllditor Feb.2g, | 9S5.Thi; ¿p9r rs a revisionof Paper508-13which was presentedat the SPIE conferenceon ProductionAspectsofSinglePoint MachinedOptics,Aug.23-24,1984,SanDiego,CA. t h€_papjrpresen-ted rhereappeårs(unrefereed)in SPIE ProceedingsVol. 509.e 1985Sæiety of Photo-OpìicalInstrumenrarionEngineers.
388
/ OFflCAL ENGINEERING/ May/June 1 985 ,/ Vot. 24 No. 3
The optical stylusmcasurements weremadeusinga trVykomodel l(XX)noncontactingprofiling microscope.,rThis instrument measuresthe surfaceprofile by determiningthe phasevariationsof the light reflectednormally from the surfaceand convertingthem into heightvariationsusing
z(x)=f ø*l ,
(t)
whereÀ = 0.6328¡rm is the meanmeasuringwavelength.As previously,l measurements weremadeat Wyko Optical, Inc. (Tucson, Arizona) usinga stock instrumentwith a 20X objective. Raw profile data from each measurementwere recordedand analyzedat BrookhavenNational laboratory (Upton, New york) using routinesdevelopedunder the aegisofthe National Synchrotron Light $6r¡¡çe.1.5.0 Resultsare discussedherein termsof two profile statistics:the power spectraldensityand bandwidth-limited valuesof the root-mean-square (rms) roughness.Thesewerechosen becauseof their direct relationshipwith functional performanceof the surfaceswhenusedin an optical system.The first is a one-to-one mapping of the scattered-lightintensity,and the secondis directly relatedto the total integratedscatter(TIS) or Strehl factor.s.r The test surfaces,germanium and silicon disks supplied by Pneumo Precision,Inc. (Keene,New Hampshire),were nominally flat, finished by single-point fly cutting. Machined surfaceswerè chosenbecause.they exhibit both randomand periodicfeaturesand becauseprofile data taken acrossthe lay of suchsurfacesare more directly interpretedthan thoseof polishedsurfaces.5-s The surfacesusedhad finishesõf the order of 50to | 00 Å , chosen to fall comfortably within the measurementrangesof the instrumentsratherthan to representthe stateof the art of finishing.A discussionofstate-of-the-artsurfacesis givenin anotherpaperin this collection.cAlthough the two setsof test surfaceswêremachined under similar conditions,their surfacetextureswerevery different: thefinish of thegermaniumwasmainly in theform of low-frequency randomroughness whilethe siliconshowedhigh-frequency periodic
MACHINEDOPTICALSURFACES OF THE FINISHOF PRECISION AND OPTICALMEASUREMENTS OF MECHANICAL DIRECTCOMPARISON
-1
îl
.1
- ,l l - l _ ,l ^ _ ,
I
Fig.3. Trander function¡ for l¡noarmoatur€m€ntr.
Z*."rur"¿(x) = M(x)*Zo6¡.",(x) ,
(3)
or, equivalently,a simplemultiplication in frequencyspace: Z,,,"".ur"¿(Ð :
Fig. l. Nomarskim¡crophotographof th€ gefman¡umtost surfaco.Th€ long dimensionin the photographi¡ 240 pm.
t*¡(0zoujot(Ð ,
wherethe overbardenotesthe Fourier transform. M(x) is then the measurementimpulse response,and M(f) is its transfer function. Becauseof the simplemultiplicativeform of Eq. (4), it is convenient systemsin the frequencydomain. to comparemeasurement We take the mechanicalgaugeto behaveas a linear measuring instrumentover its operatingrangeas definedin the following section. The optical gaugemay be shownto behaveas a linear instrument in the smooth-surfacelimit
ffi..,,
Fig.2. Nomarckimicrophotographof the s¡licontest ¡urface.The bands conerpond to the 3.4 pm tool feed.
roughness. As discussed below, this complementarity provides a sensitivetest of our understanding of the measurementprocess. Figures I and2 are Nomarski microphotographs of the germanium and silicon surfaces,respectively.The tool marks on the germanium surface are fairly random, while the striking bands in the photograph of the silicon surface clearly show the 3.4 pm tool-feed spacing.Thesedifferencesin texture are also very conspicuousto the eye:the germanium appearssmooth, while the silicon appearsrough and displays diffraction colors.
2. LINEAR MEASUREMENTS The measurement process may be viewed as a mathematical operation: Zmeasured:MZobject'
(5)
roughness.Detailswill be pubwhereZrn . is the root-mean-square lishedelsewhere. Figure 3 is a sketchof canonicalforms of measurementtransfer functions.The perfectcurve,A, is unity forall frequencies-a practical impossibility.Curve B is the ideal curve-unity over as wide a rangeas possibleand zerooutside.Curve C is a realisticcurve-the measurementis sensitiveto a more-or-lesswelldefined range of frequencieswith a frequency{ependenttransferfunction. CurveC can be transformedinto a practical approximation of curve B by careful hardwaredesignor, as discussedbelow, by processingthe measurement data in subsequentsoftware. situationwould occurwhenthefrequency The ide¿lmeasurement content of the object being measuredfalls completely within the bandpassof the measurementapparatus,in which casethe measurementresult would be transparentto the techniqueused.ln the caseof surfaceroughness,the oppositesituation holds: the roughness is spread over a wide range of surfacewavelenghs-from atomic diametersto the diameterof the workpiece-wider than the bandpassof any singlemeasurementtechnique.In that case,the comparisonand interpretation of measurementsmust necessarily take into accounttheir differing transferfunctions. the bandwidthlimits of the two The followingsectiondiscusses instrumentsusedhere,and following that, theforms of their transfer functions. 3. EXTREME AND EFFECTIVE BANDTVIDTH LIMITS aredeterminedby the The bandwidthlimits of profile measurements total tracelengthand the samplinginterval. If the trace lengthL is sampledat N equallyspacedpoints,the extremesurfacefrequencies are includedin the measurement
Q) : ¡(e¡treme) 'mln
where M is a measurementoperatorandZ,in our case,is the surface profile. A good measurement is one for which M is in the unit operationover as wide an operatingrangeas possible. The most desirablemeasurementsare linear; that is, sine waves are measuredwith no harmonicdistortion. In that case,Eq. (2) takes the form of a convolution in real space:
(4)
|
,
L
f$¡lreme¡ : fNyquisr=
(ó) N
I
The effectivevalueslie within theselimits byamounts that dependon / MaY/June 1985,2Vol. 24 No. 3,2 389 OPTICALENGINEERING
\MYANT CHURCH,VORBURGER,
TABTE l. Measurement Paramstsror
Numberof SampledPoints N ¿IOOO , 1024
lnstrumont Mechanical Optical
Totaltrace length L
Samplinginterval L,/N
1500 665
0.375 0.649
Effective1/f.", 4L/N 1.5 2.6
Effeaive 1,/f'''¡n L/2
750 333
rAll d¡stanc€s¡n mícrometers
detailsof the measurementprocess. The effectivelow-frequencycutoff is determinedby the fact that the raw profile data must be detrendedto removespuriouspiston (constant),tilt (linear),and possiblypower(quadraticlcontributions process.This is usuallydoneby introducedduring the measurement polynomialfitted to eachdata set.It can subtractinga least-squares be shownt that this operation is equivalentto a high-passfiltering operation that eliminatesthe frequenc] f : 0 and attenuatesthe frequencyf -- | lL, but passeshigher frequenciesessentiallyunaffected.For this reason,wechoosetheeffectivelow-frequencylimit to be twice the extremevaluegiven in Eq. (6). The effectivehigh-frequencylimit is determinedby the fact that any signallying abovethe Nyquist frequency[Eq.(ó)] is not eliminated,but appearsas an alias within the measurementrangeat the frequency falias:2f¡r'u¡.1 -f
.
(7)
Welldesignedmeasurement systemsincludean antialiasingfilter to attenuate or eliminate frequenciesabove the Nyquist frequency beforethe samplingoperationin order to eliminatesucheffects.It is the propertiesof theseantialiasingmechanismsthat determinethe effectiveupper frequencylimit. In the case of the mechanicalgauge, antialiasing filtering is accomplishedby the electrical and mechanicalproperties of the stylusinstrument.Its transferfunction hasbeenmeasuredsand was found to follow closelythat of a simplelow-passfilrer:
M(Ð :
t l +(fdJ
(8)
The value of do, the spatialwavelenglhfor 50Voamplitude attenuation, is O.862pm.The finite sizeof the stylustip also attenuatesthe measurementof short surfacewavelenghs but does not place a significantlimitation on the presentstudysinceits measuredwidth is only of the order of 0.5 pm. In particular,the estimatedtip curvature is significantly greater than the rms curvature of the test surfaces obtained by evaluatingthe fourth momentsof their profile power spectra.The procedurefor evaluatingthe latter is discussedin Ref. l. In the caseof the optical gauge,antialiasingis accomplishedby thefinite resolvingpowerof its optical microscopeand the sampling apertureofthe photodiodearray in its focal plane,asdiscussedin the following section. To stay within the bandwidth limits imposed by the various filtering operations,wetaketheeffectiverangeof surfacefrequencies to lie a factor of two within the extremevaluesgivenin Eq. (6); that is ¡{effect) -min -
L , L I
(e) N
. :_. f' ( e l t e c t ) : : f - ^Nyqulst . max 2 4L Numerical values of these frequencies and other parameters of the measurementsdiscussedbeloware given in Table I. Theseshow that although the bandwidth limits of the two measurementtechniques are different, so that raw measurementsare not directly comparable, there is a considerable region of overlap over which quantitative comparisons can be made. 3W
/ OPflCAL ENGINEERING / May/June 1985 ,zVol. 24 No. 3
4. TRANSFERFUNCTIONS The behavior of the transfer function of the mechanicalmeasurement at its low- and high-frequencylimits hasbeendiscussed above and in Ref. 5. We take its value betweenthe effectivelimits to be unity. A more detailedanalysiswould include effectsof a variable transferfunction in a mannersimilar to that discussedbelowfor the optical gauge. The shapeof the transer function of the optical gaugeis determinedby threeprincipal factors:thef¡nitetemporalcoherence of the light used,thepropertiesof its opticalsystem,and thefinite pixel size ofthe photodiodearray in its focalplane.The total transferfunction of the measurement is then the product of threefactors: t"t(Ð :
ú"oh"r"n""úopticsúarray
(t0)
We now examinethesefactorsindividually. The instrumentusesfìnite bandwidthlight to eliminateinterferenceeffects:À : 0.6328t 0.0200¡rm. The parametèrdescribing thecontrastattenuationdueto a wavelengthspreadof AÀabout À is
(l t)
*."*
However,the surfacesconsideredheresatisfythe Rayleighsmoothnesscondition, Eq.(5), in whichcasecoherence effectsarenegligible. That is, for our measuremènts M"oh"r"no(Ð :
I
(t2)
The phase-to-intensitytransfer function of the optical system dependson opticaland structuraldetailsthat arenot readilyaccessible. For the presentpurposes,we model this as a simple triangle function: úopti"r(Ð = l -fdr".o¡,
(13)
where 4 À dresot:;2NA
(14)
and NA is the numerical aperture of the objective.For the 20X objectiveusedin the presentmeasurements, NA : 0.4,from which it followsthat dr.ro¡ : I ,¡m. Equation(13)is simplythe textbook form of the intens'iiy+o-intensitymodulationtransferfunction of a simplelenslinearizedby extrapolatingits tangentat zerofrequency, which accountsfor the factor of 4l n An obviousrefinementwould be to usethe full algebraicform appropriatefor the annularaperturer0 resulting from the presenceof the referencemirror in the microscopeobjective. Eachpixelofthe photodiodearrayin theopticalgaugeintegrates the intensityof the imageoveran essentialty squareaperturewith a sidelengthof approximatelyL/N : 0.65¡rmwhenprojectedonto the surfacebeingmeasured. In the instrumentusedhereeachmeasurementpoint corresponds to the averageof two adjacentpixels' The resultingarray transferfunctionis then
-----' MACHINEDOPflCALSURFACES OF PRECISION OF THE FTNISH OF MECHANICALAND OMCAL MEASUREMENTS DIRECTCOMPARISON
ú"..r{Ð =
sin(2zrfLlN) 2¡rfllN
(t5)
This function is unity at low frequenciesand goesthrough its first zÊroat the Nyquist frequencyof the array.* limit (TableI), thethreeamplitudeAt theeffeótivehigh-irequency faõtorsin pq. (lO) havethe values1,0'615,and transfer-function givinga total attenuationof0.392'At the surface 0.637,respectivety, wau"tengittof 3.4 ¡rm-the period of the tool marks in Fig' 2-the cor..rpoiding factorsare l, 0.706,and 0-777' gvrnga total attenuation oi O.S¿g.lttenuationsof this magnituderequirecorrection" To compensatefor the nonconstanttransferfunction of the optical gauge,ïe include a restorationfilter in the data analysisthat conJistiof the frequencylomain inversefilter
ú¡n""rs.(Ð: tvto],,.r{Ðv-Lr(Ð
(tó) germanium rurface.
@e
In principle,this filter can be usedat freguenciesabovethe effective maximum given in Tabte I. However, it divergesat the extreme frequencytEq. (ó)1,which correspondsto a surf¿cewavelengthof 1.3 pm, änì ås a practical matter, we cut it off at 2 r¿mto avoid tpuiiour noiseeffeìts. At that point the total amplitude restoration to a powerrestorationfactorof20'93' fäctoris4.575,corresponding 5. DATA PROCESSING Raw data from eachinstrumentare first detrendedby subtractinga least-squaresquadratic fitted to the individual data sets' The detrendedprofiles(residuals)are then plotted to givethe unrestored profiìe.Theseprofilesare then passedthroughthe restoration finish -which involves Fourier transformation using the FFT algofilter, rithm, multiplication by the restorationfilter, and inversetransformation. Thé resultingrestoredprofilesare also plotted' In the mechanicalcase,the restoration filter involves simple multiplicationsby unity, and the restoredand unrestoredprofilesare identícal.In the óptical case,restorationinvolvesmultiplication by the inversefrlter in Eqs. (13) through (ló) for frequenciesup to . 0.5 pm-l and by zerofor higherfrequencies. 'ihe restoredresidualsare then multiplied by a Hamming data windowand the periodogramspectralestimateisformedby computing the squaremagnitudeofthe Fourier transform ofthe product' Tñi. rp"ót.u- is ihen plotted over the extreme range of surface frequenciesspannedby the measurementtechniqueinvolved' ilandwidt'h-timitedistimatesof the rms surfaceroughness,slope, and curvatureare then computedby evaluatingweightedintegrals (sums)ofthe spectrumoversèlectedrangesofsurfacefrequency'It is ihe comparisoì of thesequantities,measuredover the sameranges for eachìnstrument,that offersthe most sensitivetestof the equivaof the different measurementtechniques. lence ' for the operationsdescribedabove The mathematicalexpressions are givenin Ref. l. 6. RESULTS FOR THE GERMANIUM SURFACES Figures4 and 5 show the raw and restoredoptical profiles^ofthe gelnanium test surface shown in Fig. l. Only the first 500 ¡rm of iO.Smm) section'ofthe profile is shown,although the full range ùOS,rtn was usedin theãnalysis.The vertical scalecorrespondsto t60b Å. The two profilesare essentiallyidentical,althoughclose examination showi that the restoredprofile doeshavegreaterfine structure. Figure 6 is a sectionof a mechanicalprofilg of the samesurface plottedon the samescaleas in Figs.4and 5. Althougha one-to-one òomparisonwith the precedingprofilesis impossiblesincethey were takenat differentpointson the surface,it is clearthat thereis a strong "statistical"simiúrity betweenthe mechanicaland opticaldata' In profileappearsto showevengreater detail,however,themechanical * Eqution ( I 5) differs from Eq. ( | 4) in Ref. I by the presenceoftheaspcct ratio of 2 in the arsument oi the sinc function to account for the two-pixel averaglng menttoned aDove'
Fiõ3:Tottõred
ven¡on of tho opt¡cal profile in Fig. 4.
I
a a N
Fig. 6. Mechanicalprofile of tfre germanium ¡urfaco.
fine structurethan doesthe restorçdoptical profile. The proper way to appreciatethesedifferencesis to seehow the ipowei" is the distributed among different surfacefreroughnéss queicies, thãt is, by looking at the spectraldistributions of the profiles. ' ofthe Figures7, 8, and 9 are logJogplots of the periodograms three-profilesshown in Figs.4, 5, and !, respectively.The vertical and the horiscaleiuns betweenl0-7 and l0-l ¡¡m+l (ó decades), scalerunsfrom l0-3 to 2 pm-l (3*decades)-These zontalfrequency / Mav/Junø1985,/ Vol.24 No'3 / 391 ENGINEERING OPTICAL
\MYANT CHURCH,VORBURGER,
The spectrallinesat a wavelengthof about 3 ¡.rmthat appearin each spectrumare the fundamentálof the tool marks'periodicity, which is hard to discernin the Nomarski microphotographand is unrecognizablein the surfaceprofiles.Thesespectrallinesareof the order of two or threefrequencybins wide,the natural linewidth for the Hamming data window. The additional lines at 1.9 pm in the optical spectrumand at 0.9 ¡rmin the mechanicalspectrumaretaken to be spurioussincethey do not appearin other profilesof the same surface.
7. Periodogramof the raw opticrl profile in Fig, 4.
8. Periodogramof the rortorod opt¡cal profile ¡n
n
H o
2
è
profilein Fig.6. Fig.9, Periodogram of the mechanical wide rangesare requiredto displaythe full characterofthe structure found in thesesurfaces. The three spectraare very similar except at high frequencies, which, in the germaniumsample,representonly a small fraction of Ìstotal roughness. Figures7 and8 showtheimportantamplification and its ultimate ;ffectsof the restorationfilter at shortwavelengths cutoff at 2 ¡rm.The mechanicaldata,whichextendto a wavelengthof with the restoredthan with 0.75 ¡tm,areclearlyin betteragreement opticaldataovercommonfrequencyranges. the unrestored ENGINEERING / May/June1985/ Vol.24 No.3 392 / OPTICAL
7. RESULTS FOR THE SILICON SURFACE Figuresl0 and I I show the raw and restoredoptical profilesof the silicon surfaceshown in Fig. 2. In this case,only the firsl lfi) ¡rm sectionofthe profile is shown,althoughagainthefull óó5¡rmofdata wereusedin the analysis.The verticalscalesare again tó00 Ã. The conspicuousperiodicitiesin the profiles correspondto the approximately 3 ¡¿mtool feed. However, there are two obvious differencesin the profiles: the restoredversion has a larger amplitude,and the shapeof the oscillationsis smoother.ln particular,the ofthe peaksin the notch that frequentlyappearson theleadingedges unrestoredprofile is missingin the restoredversion. Figure 12is a mechanicalprofile of the samesurfaceplotted on the samescales.Superficially,it is very different from either of the optical profiles; it has a much larger peak-to-valleydistance,the peaks and valleys are sharper, and a distinct intermediatepeak appearsbetweenthe major ones.To appreciatethe sourceof these differences,we examinethe spectraof theseprofiles. FiguresI 3, I 4, and I 5 arethe periodogramsof the profilesshown in Figs. 10, I l , and 12, respectively,on the samescalesas the earlier spectra. FigureI 3 showsa pair ofvery sharpand intenselinesthat arethe fundamentaland fïrstharmonicof the3 ¡rmtool feed.In Fig. 14,the first harmonicis cut off by the 2 pm cutoff of the restorationfilter, leaving only the fundamental.This accountsfor the simlutaneous amplification and smoothingof the shapeof the oscillationsin Fig. the I l. It doesnot appearas a smoothsinusoidin Fig. I I because plotting routine usesa linear rather than a Whittaker interpolation schemefor connectingthe discretedata points. The mechanicaldata, Fig. 15,showthe intensefundamentaland the first harmonic of the tool feed,as well as the secondand third harmonics. It is the presenceof these higher harmonics in the that lead to the greateramplitude and mechanicalmeasurements detail of the profile in Fig. 12. The peaksof the mechanicallinesare of the order of two to four frequencybins wide, which is again comparablewith the natural Hamming width. However,they have considerablybroader bases than do the optical lines,due, presumably,to the fact that the involve a dynamicmeasurementsystem, mechanicalmeasurements which is subjectto vibration and drive-ratefluctuations. Comparisonof the threespectraovercommonfrequencyranges showsclearly that the restoredoptical spectrumis in much better agreementwith the mechanicaldata than is its unrestoredversion. t. QUANTITATIVE FINISH PARAMETERS Surfaceprofilesand spectraprovidevaluablecomplementaryviews of surfacetopography.Profilescan revealextraneousspikesor pits in the measurementthat would be difficult to recognizein the frequencydomain.On theotherhand,spectracanrevealsmallperiodicities that are invisiblein the profile tracings.Theserepresentations, however,areprimarilyvisualand quantitative.A quantitativecomparisoncanbe madeby examiningfinish parametersevaluatedfrom the profile spectraover the samerangesof surfacefrequencies. Table II lists a number of such parametersderivedfrom the presentseriesof measurements. Threetypesof measurements of three surfacesare compared(top to bottom): unrestoredoptic¿!, of two germanium restoredoptical,and mechanical measurements surfacesand one silicon surface. The parameters compared(left to right)arethe periodof thetool
DIRECTCOMPARISON OF MECHANICAL AND OPTICALMEASUREMENTS OF THE FINISHOF PRECISION MACHINEDOPT|CALSURFACES
E d
t
o a
a E
z N
z G
Fig. 1O. Raw optical profile of the rilicon ¡urface.
Fig. I 3. Periodogramof the raw optical file in Fig. 1O.
Þ Ë
E
T
E
N
, À
=
Fig. 11. Re¡tored ver¡ion of tùe optical profile in Fig. lô.
t o
È
E
C
Fig. 14, Periodogramof tho ro.torod opt¡crl profilo in Fig. l l.
É ¿ F
a G a N
o c o
C z c
Fig. 12. Mechanical prof¡le of the rilicon ¡urfaco.
Fig. 15. Periodogramof the mochanicalprofile in Fig. 'l 2.
marks, the amplitude of their fundamentals,and the rms valuesof the surfaceroughnessobtainedby integratingthe spectralestimates betweenthe indicated rangesof surfacewavelengihs:2 to lZ pm, 12to 333pm, and 2 to 333 pm. The 2 ¡rm limit iJ the cutoff of the opticalrestoration filter, l2 pm is thenominalupperlimit of conventionalTIS measurements madewith normal-incidence HeNelight,and 133 ¡m is the effectiveupper wavetengthlimit imposedby the quadratic detrendingof the optical data. The last column of Table II givesthe wide-openrms roughness values,that is, the nonequivalent valuesobtainedby analysisof the
unwindowedprofile data overthe extremerangeof surfacefrequencies determined by the measurementproceduresthemselvei(cf. Table I). The numbersin parentheses in the mode column are ih" numbersof independentmeasurements usedto generatethe average valuesand standarddeviationsgivenin the table. 9. DISCUSSION The optical and mecha'nical measurements of the 3 pm tool feedare in excellentagreementfor eachsample,althoughtherearesignificant OPTICALENGTNEERTNG / Mav/June 19Bb ,/ Vot. 24 No. 3 ,z 393
WYANT CHURCH.VORBURGER.
TABLE ll. Finish Parametert Line position Mode ,¿m
Amplitude fundamental
#1 GERMANIUMSURFACE
10.3+ O.8 12.8+ 0.8 13.7t O.3
Raw optical (61 Reslor€d opt¡csl (6) Mechanical(2)
2.759r0.008 2.759+0.008 2.703tO.OO2
1 . 2 +O . 1 2.7t O.2 2.61 0.1
Raw optical (61 Restored opt¡cal(6) Mechanical(11
3.03510.021 3.035f0.021 2.977
GERMAN¡UMSURFACE#2 8.4+ 0.9 1.91 O.1 11.Ot 1.O 3.8r O.2 10.6 4.1
Raw opt¡cal (2) Restored opt¡cal(2) Mechanical(3)
3.4o,2tO.O21 3.¿to2fo.o21 3.345+0.035
37.4X. 4.4 68.1t 7.6 7 5 . 1+ 1 1 . 0
SILICONSURFACE 39.2+ 4.0 70.at 7.o 8 4 . 1t 1 1 . 6
variations.The 3 Â amplitudesof the fundamensurface-to-surface tal on the germaniumsurfaceare also in very good agrgementafter restoratio;, and are comparablewith thi value of 2 ,{ computed from the tool radius. It is noteworthy that this periodic surface feature with atomic dimensionsis observedwith a signal-to-noise ratio ofroughly 50 to l. The varióus bandwidth-limitedroughnessvaluesof the germanium surfaces,includingthe wide-openvaluesin the lastcolumn,are in good agreementwith or without restoration sincethe finish of theie surfacesis dominatedby long surfacewavelengths.Resultsfor the silicon surface,however,are quite differenurestorationis essential to getagreementfor the high-frequency(zto 12pm) and broadband (2 to 333 ¡rm) values since these are dominated by the tool marks. high-fràquency -The importanceof comparingresultsoverequalfrequencyranges . showndiamaticallyin the lastcolumn of thesilicon data. Restoration is important, bui evenso,the mechanicalroughnessis still twice the optical value.The sourceof this differencelies not in the meaof surementtechniques,but in the"apples-and-oranges'comparison data involvingdifferent bandwidths. 10. SUMMARY AND CONCLUSIONS This paper discussessurface profile measurementsin general, in particular, and compares mechánicaland optical measurements resultsobtained with two typesof test surfaces:one dominated by low-frequencyrandom roughnessand the other by high-frequency 'periodicroughness. The comþarison is made over that overlapping region of frequency-"mpiitudespacein which both measurementtechniquesare únearând havea commonbandpass.The mechanicalmeasurement is taken to havea constanttransferfunction over its effectivebandnonconstantresponse width, whiletheopticalmeasurementinvolvesa that attenuateshigh surfacefrequencies.To compensatefor this,a first-order modelõf the transferfunction of the optical instrumentis usedto designa simplerestorationfilter that hasbeenimplemented in softwarecompatiblewith the commercialinstrument. Qualitativeandqr¡antitativecomparisonsshowtheimportance.of of surfacescontainrestòrationeffectsfor broadbandméasurements for the studyoftool marks, roughness, ing significanthigh-frequency an¿ fór ttreextractionoffinish information correspondingto largeangle -Th.surfacescattering. betweenthe results of mechanicaland optical "g.".ment sensitiveto mechanicalpropertiesat immense measurem€nts-one pressure and the other to electromagneticpropertiesat optical fre' rencies-is extremelygratifying,and affirms the useof thesetech,. ren€s for the quaniitative measurementand specificationof machinedoptical surfaces. the effectsof surfacelayersin the optical Appendix A discusses -."tut -.nts, and Appendix B addressesthe effects of surface gg4 / OPTICAL / Mav/Junel985'/Vol' 24 No'3 ENGINEERING
rms
fms wide open
A
Ä
{ri =rrlåg3 ¡.m) (f-1=2-333 pml
(f-1=2-12 rm)
32.1X3.0 3 3 . 1 +3.0 34.0r 3.4
33.7+ 2.t 35.5t 2.6 36.7+ 3.3
35.4t 2.8 36.4X 2.8 37.4t 3.O
36.3+ 3.7 373t 3.8 34.7
37.3f 3.6 38.9i 3.7 36.3
38.1+ 2.6 39.3+ 3.6 37.8
zò.t+3.5
44.6+ 1.9 74.1+ 5.8 88.O+11.O
2 1 . 3 i 3.4 25.8+. 0.5
51.1+ O.7 74.2t 5.7 145.7*.14.O
rotation on linear profile measurements. 11. ACKNOWLEDGMENTS The test surfaceswereprovided by R. Clark at Pneumo Precision, Inc. C. Giauque assisiedwith the Talystep measurementsat the National Buréauof Standards,and C. Koliopoulos and S' Iange at Wyko, Inc- At Brookhaven made the optical me.asurements National laboratory,'P. Takacsprovided valuableassistanceand discussions,H. Berry carried through the data analyses,and W' Marin, Jr. preparedthe Nomarskimicrophotographs.D. Aspnesat Bell CommunicationsResearch,Inc. provided information on surfacelayers.This work wassuppportedin part by the Departmentof Energy(E.L.C.) and the Departmentof Commercc(T.V'V')' 12, APPENDIX A: EFFECTS OF SURFACE LAYERS The optical gaugemeasuresthe phaseof the light reflectedfrom the surfaè bein! eiamined. To convert phaseinto height we requirea model. Pq. (t) is basedon the model that the surfaceis a simple interface:air aboveand materialbelow. Realsurfacesinvariablyinvolvesomesort of chemicalorphysical layersdepositedduring or after the machiningprocess.In interpretin termsof Eq. (l) we havemadethe ing the piesentmeasuiements of theselayersare negligible'To effects that the irñplicit assumption complicatedphysicalmodet a more slightly a examine we ttris ¡usiify structure. simpleone-layer ihe additiãn of a layer of thicknesst to a baresubstrateshiftsits apparentheight,as determinedby Eq- (l)' accordingto
ztay.*Å : zbar.+
À *
ú(t) '
(17)
wherery'(t)is the phaseof
a
16¡ * r¡2exp(-iqt)
exP(*iq")
I * rorrt2exp(-iq¡)
r02
Here 4zr 9a A;N"t
'
(le)
and
Nb-Nu
rab: lfr\
(20)
a to b is the normal-incidence amplitude reflection coefficient of the
MACHINEDOPTICALSURFACES AND OFTICALMEASUREMENTS OF THE FINISHOF PRECISION OIRECT OF MECHANICAL COMPARISON
interface,wherethesubscriptsare Q: air, I : layer,and2: substrate. In the thin-film limit, q ( I, t t -N,2
(2t)
a:l*i+zr;¡$+....
.
. \'
d(d,R) : ,
(22)
rtrhere,for simplicity, we havetaken the index of air to be unity*Surfacelayerson germaniumand siliconconsistofabout 25 Ã of oxide plus alayer of organicmaterialwith N- 1.5whosethickness zeroafter cleaningto somethingof the order variesfrom ess.¡:ntially ofanother 25 Ä after storageand handling.Substitutingreasonable Valuesfor the various indicesinto Eq. (21), we find that the coefficient multiplyingt(x) lies between0.6 and 0.8. It follows that the magnitudeof the layer coqtribution to the apparentoptical height could amount to l5 to 40 ,{, which is a signifircantamount. The analogousexpressionfor mechanicalmeasurements obviouslydependson the mechanicalrather than on the optical propertiesof the layer. If we write Z6r"n¿(x):Z6ur"(x)*kt(x)+...,
d(0,R) :
+ [ ( R * a ¡ z - ( R s i n e ¡ 2 1 %- R c o s t i ,
(24)
whereR is the machiningradius.In the limit of large R/d,
If the indicesof refractionN are real, I -N;z Z 1 ^ r " n 6 ( x ) : Z 6 ^ n ( x )+ T : ¡ 7 t ( x ) * . . .
direction, a given surface wavelength d appears at the longer wavelength:
(23)
thereare two obviousextemes:k : I for a "hard" layerand k : 0 for a completely.*soft"one. The critical quantity is not the total magnitude of the layer correctionsin either Eqs. (22) or (23) but the magnitudeof their fluctuationsalong the proftle and their correlation with the height fluctuationsof the substrate.A constantlayer thickness,for example, would simply shift the apparentsubstrateprofile by a constant amount, which would be unobservableeither optically or mechanically. The good agreementbetweenthe optical and mechanicalmeasurementsobservedin the presentexperimentsis attributed to the fact that over the surface-wavelength region explored, the layer tracks the substrate,and any differential effect is small and submergedin the experimentalerror. However,asmeasurement accuracyimprovesand finer and finer interpretationsof measurementdata are attempted, one must be alert to the fact that mostsurfacesare not simplestructuresand that surfacelayerscould lead to significanteffects. 13. APPENDIX B: SAMPLE ROTATION The text considersthe casein which the direction of the surfacelay (tool marks)is perpendicularto thedirection of the profile measurement. If the sampleis rotated through an angle 0 away from that
d
'
(2s)
"or0 which is valid for all rotation anglesexceptthoseveri near90". In this way the apparentwavelengthcan easilybe stretchedby a factor of l0 (rotationof 84.3'). The transferfunction of the optical gaugeis (presumably)circularly symmetricand thereforeindependentof the samplerotation. That is, the frequencyappearingin Eq. (13) is the true surface frequencyf = l / d ratherthan theapparentvaluegivenby Eqs.(24) or (25). However,sincethat transfer function vanishesfor wavelengthslessthan I ¡rmfor the20X instrument,the presentstretching techniquecannotgiveusefulinformation about surfacewavelengths shorterthan that value. The transfer function of the array doesdependon the angleof rotation sinceits pixelsare not circularly symmetric.As discusscdin the text, the pixelsareessentiallysquaresthat are summedpairwise to give a rectangularform with an aspectratio of A = 2. That quantity appearsasa factor in thecommonargumentin the numerator and denominatorof Eq.( l5). In thecaseof a rotatedsample,that factor A mustbe replacedby A[cos0 * Asind]-r. Samplerotation is usefulfor zoomingin on high-frequencypro{ile features(suchasthefine structureof the3.4¡rmlinesin thesilicon for examiningwavelengthcontributionslying betweenthe surfacæ), I ¡rmmicroscopecutoff and the2 ¡rmfilter cutoff, and for identifying aliasedlines in the spectrumsincethey will move to higher rather than to lower frequencies as the sampleis rotatedfrom d = 0.
14. REFERENCES l. E. L. Church, i¡ hecision SurfaceMetrology, JamesC. Wyant, cd., Proc. SPIE 429, 105(1983). 2. E. C. Teaguc,Nat. Bur. Stand.(U.S.)TechnicalNote 902 (197Q. 3. T. V. Vorburger,E. C. Tcaguc,and F. E. Scire,Dimensions/NBs62(l l), lE (r97E). 4. J. C. Wyant, C. L. Koliopolis, B. Bhushan,and O. E. George,Am. Soc. Lub. Engrs.Trans.27,l0l (1984). 5. E. L. Church, M. R. Howells,and T. V. Vorburger,in ReflectingOptics for SynchrotronRadiation,Malcom R. Howclls,cd., Proc. SPIE 315, 202 (t982). 6. E. L. Church and H. C. Berry, Wear 83, 189(1982). 7. E.L. Church, i¡ PrecisionSurfaceMetrology, JamesC. Wyant, ed., Proc.SPIE 429,86 (19E3). E. E. L. Church, H. A. Jenkinson,and J. M. Tavada,Opt. Eng. 16(4),360 (1977). 9. E. L. ChurchandP.Z. Takacs,Opt. Eng.24{3),(1985). t0. E. L. O'Neill, J. Opt. Soc.Am. 46, 285(1950.
OPTICALENGINEERING / May/June 1985 / Vol. 24 No. 3 ,z 395
