AO_41_19_3853

Optical interferometry for measurement of thegeometric dimensions of industrial parts

Peter de Groot, Jim Biegen, Jack Clark, Xavier Colonna de Lega, and David Grigg

We describe an instrument for the measurement of surface flatness, parallelism, and size �thickness� ofplane-parallel parts in a single measurement to 1� gauge capability of 0.02, 0.03, and 0.06 �m, respec-tively. A low-coherence IR profiler viewing both sides of the part simultaneously, believed to be novel,accommodates a wide variety of industrial surface finishes, including machined, ground, or lapped parts,with a 75-mm field of view and 15,000 pixels per side. A heterodyne laser displacement gauge togetherwith an integrated zeroing system allows for a range of part sizes from 0 to 100 mm. © 2002 OpticalSociety of America

OCIS codes: 120.3180, 120.3940, 150.3040, 120.6650, 150.6910.

1. Introduction

As in other areas of technology development, efficientand precise mechanical systems require ever-improving metrology. Valves, injectors, and pumpsincorporate assemblies of plane-parallel disks tightlycontrolled for flatness, thickness �also called partsize�, and parallelism. It is not unusual to cite tol-erances of the order of 1 �m in the fuel injector in-dustry, a level of precision that is fast approachingthat of optical components.

Optical interferometry is a logical choice for im-proving metrology on precision-engineered parts,given the long history of optical surface form testingto the nanometer uncertainty level. However, acomplete optical validation of geometric dimensionsrequires several innovations. First we require anoptical profiler that is at the same time robust, fast,high precision, free of interference fringe-order am-biguity, adapted to ground and lapped metal sur-faces, and able to view complex forms withoutshadowing. Next we need to examine the test partsimultaneously from different viewing angles usingtwo or more such profilers. Each profiler measuresthe absolute positions of surface points in three di-mensions with respect to a coordinate system local to

The authors are with the Zygo Corporation, 21 Laurel BrookRoad, Middlefield, Connecticut 06455. P. de Groot’s e-mail ad-dress is [email protected].

Received 10 October 2001; revised manuscript received 4 Janu-ary 2002.

0003-6935�02�193853-08$15.00�0© 2002 Optical Society of America

each profiler. Finally, we need to locate the variousprofilers with respect to each other in a common co-ordinate system so as to determine geometric rela-tionships.

Here we describe an optical metrology systemdeveloped to the level of a commercial instrument,presently in use for production verification of di-mensions and tolerances on plane-parallel,precision-engineered engine components up to 75mm in diameter and 100 mm in length.1 The sys-tem simultaneously measures part size, surfaceparallelism, and surface form with a 1� gauge re-peatability and reproducibility of 0.06, 0.03, and0.02 �m, respectively. The dimensional calcula-tions involve some 30,000 data points acquired andprocessed within 10 s to create three-dimensionalimages such as are shown in Fig. 1. We describethe general principles of the instrument, the en-abling technologies, and the results of performancetesting.

2. Relational Measurements by Use of Dual OpticalProfilers

The fabrication of precision-engineered industrialcomponents is governed by standard practice in geo-metric dimensioning and tolerancing �GDT�, whichdescribes allowable deviations in surface form as wellas the relationships between part surfaces such asangle and separation.2 A significant number ofparts having challenging GDT requirements arenominally plane parallel. Figure 2 clarifies typicaldimensions for parts of this type. The flatness tol-erance specifies a zone defined by two parallel planeswithin which all the points on a surface must lie.

1 July 2002 � Vol. 41, No. 19 � APPLIED OPTICS 3853

Dimensions such as the parallelism and size are oftenreferenced to a part datum Q, a virtual construct inimaginary contact with a designated surface, herethe lower surface A of the part. For example, par-allelism is often defined as the minimum separationof two planes parallel to the datum Q that contain theentire upper surface B of the part.

We propose here that the ideal optical GDT instru-ment provides high data density, absolute surfacetopography maps of both sides of a test part withrespect to a common coordinate system. The proposalto simultaneously profile both the front and the backsurfaces optically does not come without some room fordebate. Well-established multiple-wavelength tech-niques for the calibration of gauge blocks do not re-quire imaging both sides of the test object.3 One endof the gauge block is wrung to a polished surface plateand the interferometer views the other end of theblock and the surface plate in the same field of view,from the same side. However, the reality is thatmost ground and lapped industrial parts are insuffi-ciently smooth and clean for wringing to a surface

plate, and many others are too small or have surfacefeatures that make it impractical to mechanicallysimulate the datum surface. Although one canimagine a variety of kinematic mounts for specificpart shapes, we conclude that the most flexible andreliable system is entirely optical.

Figure 3 therefore shows our generic configurationfor simultaneous measurement of flatness, size, andparallelism by use of two optical profilers viewingfrom opposite sides of the part. Each optical profilermeasures heights hopt in the z coordinate directionwithin a limited measurement range for any givenpoint x, y on a surface with respect to an opticalprofiler datum H. Knowing how two optical profilerdatums HA,B relate to each other, one can then mapthe complete geometry of the part within a commoncoordinate system. Once this is done, we can ana-

Fig. 1. Two-sided three-dimensional image generated from the optical profiler data of a precision-engineered plane-parallel part.Diameter, 32 mm; size �thickness�, 5 mm; flatness, 5 �m.

Fig. 2. Common GDT definitions including flatness, size, andparallelism.

Fig. 3. Conceptual diagram illustrating the combination of twoprofilers with a displacement-measuring interferometer �DMI� forthe measurement of geometric dimensions.

3854 APPLIED OPTICS � Vol. 41, No. 19 � 1 July 2002

lyze the data using any GDT standards that wechose, including a purely software simulation of thepart datum Q shown in Fig. 2.

3. Locating the Profiler Optical Datum Planes

In some cases, we can imagine an interferometer de-sign in which there is a clear physical incarnation ofthe optical datum H. This is the case, for example,of the reference flat of a laser Fizeau interferometer,although such an instrument would not satisfy thepresent need for a profiler free of interference fringeuncertainty. In the most general case, the opticalprofiler datum H is not necessarily a physical object;rather it is an imaginary surface representing pointsin space for which the instrument would report a zeroheight h. Therefore the position, orientation, andeven the shape of H are initially unknown.

To find and characterize the optical profiler datum,we introduce a physical surface Z such as a precisionoptical flat, measure its position, and define the arrayof measured heights hsys as the system height offsetto the true zero position for the interferometer. Thesystem height offset effectively locates the opticalprofiler datum H with respect to the artifact surfaceZ so that we can relate subsequent measurements tothe position of the physical artifact. In a GDT ver-ification system for plane-parallel parts, the pre-ferred artifact has two tightly toleranced, parallel flatsurfaces of known separation Dz to which we canrelate two nominally parallel datums HA,B simulta-neously. We call measuring the artifact to calculatehsys the zero-gauge procedure.

It is generally necessary to move the datums HA,Bfrom their zero-gauge positions by mechanical dis-placement or equivalent adjustment of the profilers,so as to accommodate various part sizes and shapes.As shown in Fig. 3, displacement-measuring inter-ferometers �DMIs� monitor these adjustments, aswell as potential drifts and other relative motionsbetween the profiler datums HA,B. Treating the op-tical datums as rigid bodies, we need at most threevalues from the DMI system: the change DDMI inseparation and the changes �, � in the two anglesthat describe the relative orientation of the profilerdatums. In the two-dimensional representation ofFig. 3, for example, the change in separation is theaverage of two measurements D1

DMI, D2DMI:

DDMI � �D1DMI � D2

DMI��2, (1)

whereas the relative angle change is

tan�� D1DMI � D2

DMI��L, (2)

where L is the separation of the DMI beams. Theorthogonal angle � is in this example assumed stable,e.g., equal to zero. We then relate heights hA,B forboth the front and the back surfaces A, B of the objectto a common datum HA,B using

hA� x, y� � hoptA� x, y� � hsys

A� x, y�, (3)

hB� x, y� � hoptB� x, y� � hsys

B� x, y� � DZ � DDMI

� x tan��. (4)

Here we assumed that the field coordinates x, y al-ready have the same sign and the same point of originin both profiler images relative to a common coordi-nate system, achieved by straightforward coordinatetransforms of the profiler images.

Once we have the heights hA,B representing pointson the two surfaces A, B of the object, we can calcu-late a number of geometric parameters. Calculationprocedures depend on the choice of definition for theflatness, parallelism, and size. Table 1 includes ex-ample processing. Alternative definitions, includingthose involving software simulations of a datum ref-erence Q, are also available depending on the func-tion of the part or the specific GDT standard to whichthe result is to be compared. This flexibility is in-herent to the measurement principle, which gener-ates all the data necessary to completely describe theform of the relevant part surfaces in the software.

4. Profiler Technology

Several innovative interferometric profiling tech-niques accommodate the nonoptical surface finish�e.g., 0.1–0.5-�m average surface roughness� charac-teristic of machining, grinding, and lapping. Thesetechniques include grazing incidence,4 moire,5 multi-ple wavelengths,6 geometrically desensitized inter-ferometry,7 and coherence radar.8 One can adaptany of these technologies to the measurement prin-ciple shown in Fig. 3.

In selecting the best sensor technology, we takeinto account the stringent requirements for single-surface characterization. If a part’s flatness toler-ance is 1 �m, then the 1� repeatability of themeasured flatness, including loading and unloadingthe part with random orientation, has to be betterthan 0.02 �m. This provides a 99% certainty thatany single measurement is within �5% of the aver-age value, assuming no systematic errors. It turnsout that few optical techniques are capable of thislevel of performance with acceptable reliability,speed, and fault tolerance when rough surfaces aremeasured. Fringe projection, moire, holographic,and speckle contouring techniques as well as visible-

Table 1. Example Data Analysis

Step 1 Characterize the flatness of each of the surface profiles A and B by calculation of the difference between the highest andlowest height values after subtraction of a best-fit plane.

Step 2 Characterize the parallelism of the surface profiles A and B by fitting planes to each of the surfaces by use of least-squares techniques, then calculate the difference between the highest and lowest separations of these best-fit planes.

Step 3 Characterize the size of the part by calculation of the average difference of the two surface profiles A and B.


wavelength low-coherence interferometry suffer fromspeckle phenomena that limit their repeatability andcomplicate alignment and light-level control, partic-ularly over large surface areas �e.g., �10 mm�.9

We therefore elected to try something entirely newto achieve the optimum precision and flexibility onindustrial surfaces. Because flatness is a form pa-rameter, there is no need to resolve �or be affected by�the high-frequency roughness components that cre-ate speckles. A solution therefore consists in use oflonger wavelengths, which are not affected by thenow unresolved roughness. Our experiments andthose of other researchers show that most machinedparts become mirrorlike at wavelengths longer than5 �m.10,11 We combine this observation with mod-ern techniques for low-coherence interferometry andconstruct an infrared �IR� height profiler having asimple glowing filament as a light source.12

As shown in Fig. 4, clear interference fringes ap-pear in the IR even on machined surfaces that gen-erate fully developed speckle patterns at visiblewavelengths. Metals, ceramics, and ground glasssurfaces that have nonuniform reflectivity at visiblewavelengths appear uniform in the IR. An addi-tional benefit is the speed of data acquisition, whichexceeds 75 �m of surface height per second with a60-Hz camera without resorting to sub-Nyquist sam-pling of the interference carrier.

The broad IR spectrum from the filament localizesthe interference fringes and provides a wide range ofwavelengths for use in frequency-domain analysis, aprofiling technique that we have previously appliedto visible-wavelength microscopy.13 A computercaptures a sequence of camera frames of intensitydata during a mechanical scan of the reference mir-ror. These data are Fourier transformed on a pixel-

by-pixel basis to extract the interference phaseversus the spatial frequency. A linear fit to thesedata provides a slope that is closely related to thefringe localization in the interference pattern and amean phase equivalent to what would be obtainedfrom a laser-based IR system. The slope data pro-vide the correct fringe order for the interferencephase. In this way, frequency-domain analysis usesboth the coherence information and the underlyinginterference phase data to achieve a higher level ofprecision than is possible with coherence data alone,but without the fringe-order ambiguity characteristicof laser-based IR interferometry. For most indus-trial surfaces, the benefits derived from having usefulphase information without speckle noise outweighthe scaling penalty of longer wavelengths.

5. Instrument Design

Because this is a low-coherence interferometer hav-ing a spectral bandwidth of 2 �m, we need to balancethe optical paths from reference to object surface, andthis naturally leads to the Twyman–Green geometryof Fig. 4. To facilitate data-acquisition synchroniza-tion, we combine the two profilers required for plane-parallel parts into the single interferometer shown inFig. 5 having two measurement paths but a commonreference mirror. The reference mirror mounts on acapacitive feedback, closed-loop piezoelectrically ac-tuated mechanical transducer calibrated off line to20 nm by He–Ne laser interferometry. A singlemicrobolomer camera captures data frames encom-passing images from both sides of the object simul-taneously. Each image covers 150 110 camerapixels. Mechanical focusing stages carrying roofmirrors adjust the location of the optical profiler da-tums HA,B to bring the object surfaces within range ofthe profilers. A data acquisition consists of a200-�m reference mirror scan while data frames arecollected at 1.25-�m intervals, which correspond toone fourth fringe increments at the 10-�m meanwavelength of the system. In the actual instrument,a large-scale casting maintains stable positions forthe zinc selenide and germanium IR optics of a foldedTwyman–Green geometry. Figure 6 is an exampledata profile display, showing the two nominally par-allel surfaces of a machined metal part.

The optomechanical design requires only two DMImeasurement axes to monitor the unconstrained de-grees of freedom in the optical system. A third axisfor the angle � is not required thanks to the extensiveuse of highly stable roof mirror structures that act asretroreflectors in one dimension, for example, on thefocus stages and the two fold mirrors near the partarea. Two DMI beams based on the Zygo model ZMI510 heterodyne laser system projected symmetricallyto either side of the plane of Fig. 5 by use of a beamdoubler account for rotations about a horizontal axisof the roof mirrors and the overall optical path of thesystem. Figure 7 is a detailed drawing of the DMIbeam paths.

The zero-gauge artifact is integrated into the opto-mechanical design as a partially transparent window

Fig. 4. Low-coherence IR scanning interferometer. The fila-ment provides broadband radiation to which the microbolometercamera is sensitive. When the equal-path condition is satisfied,high-quality localized fringes appear even on rough-surface ob-jects.


having sufficient reflectivity �17%� on one of its sur-faces for data collection �Fig. 5�. In the zero-gaugefocus position, profilers A and B measure the frontand back of this artifact surface to initialize the sys-tem for subsequent measurements. Table 2 summa-rizes the zero-gauge procedure. Because the artifactis always in the instrument path, the zero-gauge pro-cedure is fully automated and does not require theoperator to do anything other than initiate the pro-cedure through a computer command. Frequentzero-gauge commands, for example once an hour,compensate for drifts in optical component thicknessand shape that are not detected by the DMI.

The part itself mounts in a cartridge that slidesinto a motorized tip and tilt stage. After adjustmentof the focus stages according to stored position pa-rameters, a rapid sub-Nyquist scan of the part sur-face informs the instrument of the initial orientationof the part, which is then corrected for minimuminterference fringe density across the image by use ofthe stage motors.

Table 3 summarizes the measurement procedure,leading to data for processing as summarized, e.g., inTable 1. Automated part alignment, data acquisi-tion, and data processing typically total less than 10s�part.

6. Calibration and Performance

The most important performance parameter forproduction testing is the gauge repeatability andreproducibility, or GRR. This specification relatesessentially to the consistency of measurement, a

critical and fundamental characteristic of the mea-surement tool. The 1� estimated by a detailed un-certainty analysis is 0.02 �m for measurements ofpeak-to-valley �PV� flatness, including part removaland replacement. For relational measurements, wemust add the noise in the DMI measurements as wellas other factors that come into play when measuringseparated surfaces, bringing the estimate to 0.03 and0.06 �m for parallelism and part size, respectively.The expectation based on the uncertainty analysis isfor 10% GRR results for part tolerances of 1.0, 1.5,and 3.0 �m for flatness, parallelism, and size, respec-tively. These results are generally achieved in prac-tice for a clean, nominally plane-parallel part lessthan 1 cm thick in an environmentally controlledroom. Extensive GRR studies validate these esti-mates on the instrument itself. Table 4, for exam-ple, shows an average 1� deviation of 0.037 �m forpart size �i.e., thickness� for ten different part sam-ples similar to that of Fig. 6 by use of two operators �1and 2� and three trials each �A, B, C�.

The 2� standard uncertainty of measurement in-cluding bias, sometimes termed informally the sys-tem accuracy, is 0.10, 0.15, and 0.30 �m for flatness,parallelism, and size, respectively. Parallelism un-certainty is validated experimentally by a reversalstest, whereas flatness uncertainty is verified by ahigh-quality optical flat.

The size uncertainty depends critically on calibra-tion with respect to gauge blocks traceable to Na-tional Institute of Standards and Technology �NIST�certified standards to provide a first-order correction

Fig. 5. Optical layout for a dual-profiler version of the IR scanning interferometer, including DMI instrumentation to monitor optical pathlengths within the system. The camera receives two images, A and B, corresponding to nominally parallel part surfaces. PZT,piezoelectric transducer.


for the environmental conditions of a shop floor in-stallation. After measuring several blocks over arange of sizes, we perform a least-squares linear fit ofmeasured values versus traceable calibration values.The intercept is the apparent thickness of the zero-gauge artifact DZ. The slope determines the scaling

factor for the DMI measurements, which provide thedistance DDMI and the angle �.

The 0.3-�m 2� size uncertainty corresponds to adetailed uncertainty analysis for a part 10 mm inlength, in a room having a �0.5 °C temperaturestability. NIST-traceable gauge blocks also verifysystem accuracy after setup. Table 5 shows, forexample, verification results for gauge blocks 3 hafter a complete system calibration. These resultsshow values generally consistent with the predicteduncertainty of measurement, given that the truelength of the 30-mm gauge block varies by 360nm�°C because of thermal expansion. Smaller de-viations are expected and indeed observed forsmaller parts and also immediately after calibra-tion. However, we have observed larger deviationsof as much as 1.5 �m on long gauge blocks in a

Fig. 6. Example data showing the A and B surface profiles of a metal part.

Fig. 7. Detail of the optics for one of the two DMI paths. Aftertraversing the system in double pass to compensate small-angleerrors, the DMI beams recombine and generate a 4-MHz beatsignal at the detector, the phase of which is monitored for displace-ment measurement. PBS, polarizing beam splitter.

Table 2. Zero-Gauge Sequence

Step 1 Remove any part that may be in the instrument andmove the focus stages to zero-gauge initializationposition. Images A and B are now of the frontand rear sides of the uncoated surface of the zero-gauge artifact �see Fig. 5� and there will be visibleinterference fringes.

Step 2 Measure zero-gauge artifact by means of a referencemirror scan. Generate reference profiles that willbe subtracted from all subsequent profiles.

Step 3 Zero DMI measurements.


production environment and with infrequent cali-bration, i.e., once every few months. A full systemcalibration therefore should be regularly scheduled,consistent with common shop practice, for best ac-curacy in unstable environments.

7. Conclusion

From Fig. 5 and the discussions above, we can seethat the measurement of part geometry is consid-erably more complex than the measurement of sur-face form alone. Our solution to simultaneousflatness, parallelism, and size measurement re-quires nothing less than the combination of a newprofiler technology, IR scanning, with an advancedheterodyne laser DMI in a large, highly stable op-tomechanical system.

There are examples of specific measurement tasksfor which the optical design can be simplified, and thealert reader will no doubt discover several of these.For example, repeated, dedicated measurements ofthe same part type allow for specialized in situ ref-erence artifacts placed directly in the field of view,next to the part itself. This approach in some cases

obviates the need for focus stages and the DMI mon-itoring system. Our experimental systems beganthis way but evolved to the more flexible design toaccommodate a larger variety of part types and sizes.One can also imagine specific part types for whichalternative profiling techniques such as grazing inci-dence or coherence radar would be adequate and per-haps more cost effective.

We find nonetheless that the combination of a dual-sided, IR optical profiler with a He–Ne laser DMIprovides the most flexible and robust approach togeneral-purpose gauging of the geometric dimensionsof high-precision industrial parts. The underlyingprinciples of the instrument can also be applied toother surface geometry problems, including orthogo-nality, step height, and nonplanar surface shapessuch as cylinders and spheres, areas of industrialmetrology in which there is still considerable re-search to be done to further advance interferometricmetrology for precision manufacturing.

The technology described in this paper has beenincorporated into a commercial product for the shopfloor.14 An advanced machine of this complexity re-quires the contributions of many skilled scientistsand engineers that we acknowledge here, includingJim Kramer, Tony Lynn Miles, and Kharta Khalsafor software; Dave Bourque for electronics; AlfredRatajczak and Kurt Stechman for mechanical design;and Randy Young and Mike Majlak for product linemanagement.

Table 3. Measurement Sequence

Step 1 Insert part and move focus stages so that the partsurfaces are sharply in focus and there are in-terference fringes. In production testing, thisis done when the stored part parameters arerecalled. Images A and B are now of the frontand rear sides of the part �see Fig. 5�.

Step 2 Adjust part position as needed by use of stage mo-tors to minimize tip and tilt. This is normallyan automatic function by use of informationgathered from a rapid sub-Nyquist scan of thepart surface.

Step 3 Measure part surfaces by means of a referencemirror scan. Generate surface profiles.

Step 4 Correct for the relative orientation of the opticaldatum planes by use of the DMI data, e.g., bymeans of Eq. �4�.

Step 5 Subtract from surface profiles A and B the corre-sponding profiles of the artifact generated dur-ing the zero-gauge initialization procedure.

Table 4. Example Repeatability Data �in Micrometers� for Size of a Machined Metal Part

Sample

Trial

1A 1B 1C 2A 2B 2CStandardDeviation

1 1244.85 1244.90 1244.87 1244.89 1244.92 1244.85 0.0282 1244.28 1244.28 1244.28 1244.26 1244.24 1244.27 0.0163 1245.20 1245.20 1245.16 1245.15 1245.17 1245.20 0.0234 1244.67 1244.66 1244.60 1244.60 1244.67 1244.68 0.0355 1244.14 1244.13 1244.19 1244.10 1244.16 1244.10 0.0356 1244.75 1244.87 1244.75 1244.77 1244.75 1244.74 0.0487 1244.71 1244.64 1244.68 1244.61 1244.64 1244.60 0.0448 1243.92 1243.89 1243.79 1243.82 1243.85 1243.85 0.0469 1244.15 1244.13 1244.07 1244.16 1244.17 1244.13 0.03710 1244.41 1244.53 1244.50 1244.45 1244.55 1244.56 0.058Average 0.037

Table 5. Example Accuracy Data for Measurement of NIST CertifiedGauge Block Size

SampleNIST�mm�

Simetra�mm�

Difference��m�

1 2.000052 2.000121 �0.0692 4.999978 4.999857 0.1213 10.00001 10.000033 �0.0234 29.999912 29.999539 0.373Average

difference0.147


References and Note1. X. Colonna de Lega, P. de Groot, and D. Grigg, “Dimensional

measurement of engineered parts by combining surface profil-ing with displacement measuring interferometry,” in Proceed-ings of Fringe 2001: the Fourth International Workshop onAutomated Processing of Fringe Patterns, W. Osten and W.Juptner, eds. �Elsevier, Paris, 2001�, pp. 47–55.

2. American Society of Mechanical Engineers, ASME Y14.5M-1994: Dimensioning and Tolerancing �ASME, New York,1994�.

3. D. C. Barnes and M. J. Puttock, “National Physics Laboratoryinterferometer,” Engineer 196, 763–769 �1953�.

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7. P. de Groot, X. Colonna de Lega, and D. Stephenson, “Geomet-rically desensitized interferometry for shape measurement offlat surfaces and 3D structures,” Opt. Eng. 39, 86–90 �2000�.

8. T. Dresel, G. Haeusler, and H. Venzke, “Three-dimensional

sensing of rough surfaces by coherence radar,” Appl. Opt. 31,919–925 �1992�.

9. G. Hausler, “About the scaling behavior of optical range sen-sors” in Fringe ’97: Automatic Processing of Fringe Patterns,W. Juptner and W. Osten, eds., Akademic Verlag Series inOptical Metrology �Akademic Verlag, Bremen, Germany,1997�, pp. 147–155.

10. C. Ai, “IR interferometers using modern cameras,” in OpticalManufacturing and Testing II, H. Stahl, ed., Proc. SPIE 3134,461–464 �1997�.

11. C. R. Munnerlyn and M. Latta, “Rough surface interferometryusing a CO2 laser source,” Appl. Opt. 7, 1858–1859 �1968�.

12. X. Colonna de Lega, L. Deck, and P. de Groot, “Method andapparatus for the metrology of precision engineered compo-nents,” U.S. patent 6,195,168 B1 �27 February 2001�.

13. P. de Groot and L. Deck, “Surface profiling by analysis ofwhite-light interferograms in the spatial frequency domain,” J.Mod. Opt. 42, 389–401 �1995�.

14. The Zygo Simetra, including scanning infrared interferometryand techniques for geometric measurements by combining pro-files from multiple viewing angles of a part and by use of DMIand profilometry together, is the topic of multiple U.S. andforeign patents pending assigned to the Zygo Corporation.


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