phase-shifting birefringent scatterplate interferometer · phase-shifting birefringent scatterplate...

Phase-shifting birefringent scatterplateinterferometer

Michael B. North-Morris, Jay VanDelden, and James C. Wyant

We realized what we believe is a new phase-shifting scatterplate interferometer by exploiting thepolarization characteristics of a birefringent scatterplate. The common-path design of the interferom-eter reduces its sensitivity to environmental effects, and phase shifting allows quick and accuratequantitative measurements of the test surface. A major feature of the birefringent scatterplate ap-proach for phase shifting is that no high-quality optical components are required in the test setup. Thetheory of the interferometer is presented, the procedure for the fabrication of the birefringent scatterplateis described, and experimental results are shown. © 2002 Optical Society of America

OCIS codes: 120.3180, 120.3940, 120.5050, 120.6650.

1. Introduction

The most significant advancement in interferometryand other types of instrumentation over the past 30years has been the development of the computer.The prime manifestation of this occurrence in opticaltesting has been the evolution of phase-shifting inter-ferometry. Phase-shifting interferometry records aseries of interferograms while the reference phase ischanged. The wave front is encoded in the irradiancedistribution of the fringe pattern, and a simple point-by-point calculation in the computer recovers thephase. Some of the advantages of phase-shifting in-terferometry over other measurement techniques arehigh measurement accuracy, rapid measurement ca-pability, and good results with low-contrast fringes.Accuracies of the order of 1�1000 of a wave areachievable. Phase-shifting interferometry was firstconceived by Carre1 in 1966 and was more fully de-veloped later by Crane,2 Bruning et al.,3 and Wyant4

to name a few. Unfortunately, there is a limit im-posed on every measurement technique. For phase-shifting interferometry the limiting error is usually

When this research was performed, the authors were with theOptical Sciences Center, University of Arizona, Tucson, Arizona85721. M. B. North-Morris is now with Agilent Technologies,3910 Brickway Boulevard, Santa Rosa, California 95403. J. C.Wyant’s e-mail address is [email protected].

Received 14 June 2001; revised manuscript received 5 October2001.

0003-6935�02�040668-10$15.00�0© 2002 Optical Society of America

668 APPLIED OPTICS � Vol. 41, No. 4 � 1 February 2002

fringe movement caused by vibrations and indexvariations in the optical path.

Common-path interferometers are one possible so-lution to the environmental limitations. Theircommon-path design makes them largely insensitiveto vibrations and index variation in the optical path.Unfortunately, because the test and reference beamstraverse the same optical path, it is difficult to sepa-rate the test and reference beams for phase shifting.

A few clever groups have phase shifted common-path interferometers with varying success. Kwon,who fabricated a point-diffraction interferometeronto a sinusoidal grating, was the first to phase shiftthe point-diffraction interferometer.5 The diffrac-tion orders contained the desired phase shift. LaterMercer et al.6 produced a phase-shifted point-diffraction interferometer by embedding a micro-sphere into a thin liquid-crystal layer. Themicrosphere created the reference beam, and the liq-uid crystals produced a variable phase shift. Mostrecently the point-diffraction interferometer wasphase shifted by Medecki et al.7 In this setup thebeam coming from the test optic is divided into twobeams with a small angular separation by use of agrating. The zero-order beam focuses onto a smallpinhole creating the reference, and the first-orderbeam passes directly through a much larger hole thatis slightly offset from the pinhole. The phase shift iscreated when the grating is shifted laterally.

The scatterplate interferometer has also been suc-cessfully phase shifted by two research groups:Huang et al.8 and Su and Shyu.9 Both methods ex-ploit polarization to separate the test and referencebeams when an auxiliary optic is placed near the test

ter fo

mirror. Huang et al. placed a small quarter-waveplate near the test surface that rotated only the in-cident linear polarization of the reference beam by90°. With orthogonal polarizations in the test andreference beams, the interferometer is phase shiftedwith an electro-optic light modulator. Su and Shyu,playing on the same theme, placed a large polarizerwith a small hole in the center near the test surface.In combination with additional polarization manipu-lating optics, the polarizer rotates the polarization ofthe test beam 90°.

The phase-shifting scatterplate interferometer pre-sented in this paper also uses polarization to separatethe test and reference beams. However, it does notrequire optics to be placed near the test surface. Thesecret to this more practical approach is use of abirefringent scatterplate to separate the test and ref-erence beams.

2. Conventional Scatterplate Interferometer

A. Common-Path Interferometer

Because of its common-path design and unique abil-ity to average many measurements at one time, thescatterplate interferometer is the ideal choice for aphase-shifting common-path interferometer. Thescatterplate interferometer belongs to a subset of in-terferometers known as common path. As the nameimplies, a common-path interferometer is one inwhich the test and reference beams traverse nearlythe same optical path. The advantages of common-path interferometers are well documented10–14: �1�reduced sensitivity to vibrations and air turbulence,�2� no precision auxiliary optics required, and �3� awhite-light source can be used.

Because both the test and the reference beamstravel the same path, it is easy to imagine that, forthe most part, both beams experience the same phasevariations that are due to optical path perturbationsbrought on by vibrations and air turbulence. Lessobvious is the absence of piston error; relative longi-tudinal motion between the interferometer and thetest optic introduces defocus only, which is usually

Fig. 1. Scatterplate interferome

undetectable if the vibration movements are small.13

Another advantage of the common-path design is theability to use moderate-quality optics in the inter-ferometer. The aberrations present in the illumina-tion and imaging optics will primarily affect the testand reference beams equivalently. In addition, com-mon path also implies equal path, accordingly white-light sources can be used. There are, however,limitations imposed on the source size and spectraldistribution.13

In addition to being common path, the scatterplateinterferometer has the unique property of averagingmany measurements at one time. Because eachscatter point on the scatterplate scatters the light ina unique manner, each scatter point makes a statis-tically independent measurement of the surface un-der test, and the resulting interferogram is anaverage of all the measurements. Any small spatialphase variations present in the beam incident on thescatterplate will be averaged, significantly reducingthe errors associated with them.

B. Qualitative Description of Scatterplate Interferometer

The scatterplate interferometer is schematically asimple design. The configuration for the testing ofspherical mirrors is shown in Fig. 1. A small circu-lar aperture is illuminated with a laser or broadbandsource producing a source of limited extent for theinterferometer. A focusing lens is used to image thesource onto the test mirror by way of a beam splitterthat removes the source from the path of the returnbeam and a scatterplate. The scatterplate is placedat the center of curvature of the test mirror. Part ofthe light incident on the scatterplate scatters, illumi-nating the entire mirror surface; part of it passesdirectly through, imaging to a point on the test mir-ror. After reflecting off the test surface the lightagain propagates through the scatterplate, scatteringa portion of the beam. Finally a focusing lens is usedto image the interference fringes onto a screen ordetector.

The question now arises, if all the light travels thesame path, what is interfering to produce fringes?

r the testing of concave mirrors.

1 February 2002 � Vol. 41, No. 4 � APPLIED OPTICS 669

6

To answer the question it is necessary to look moreclosely at the effect of the scatterplate. Each timethe light encounters the scatterplate, some of it isscattered and a portion passes directly through theplate. Because the scatterplate is traversed twice,there are four permutations of the beam that arrivein the image plane: �1� scattered–scattered, �2�scattered–direct, �3� direct–scattered, and �4� direct–direct. An examination of each of these combina-tions will uncover their role in producing fringes.The direct–direct beam passes directly through thescatterplate both times it is encountered, forming animage of the source called the hot spot in the imageplane. Because the hot spot is never scattered, itdoes not contribute to the production of interferencefringes. Similarly, the scattered–scattered beamdoes not play a role in the formation of interferencefringes. The light is scattered both times it passesthrough the scatterplate, producing background irra-diance in the image plane. If a laser source is used,the background irradiance, as well as the interfer-ence fringes, will contain a speckle pattern. Thedirect–scattered beam is the reference beam of theinterferometer. The light passes directly throughthe scatterplate on the first pass and forms an imageof the source on the test surface. If the image of thesource is small enough, the phase variations intro-duced into the beam on reflection are negligible. Onthe return leg the light is scattered. The scattered–direct beam serves as the test beam of the interferom-eter. The light is scattered on its initial passthrough the scatterplate, illuminating the entire testmirror. Any departures from a sphere will intro-duce phase variations in the beam. The light thenpasses directly through the scatterplate, producingfringes when it interferes with the reference beam.

It seems unlikely that two beams of light that havebeen scattered at different positions in the inter-ferometer could possibly produce interferencefringes. In general, the process described abovewould not produce interference fringes; however,Burch10 came up with the solution: The scatterplatemust have inversion symmetry. Inversion symme-try means that each scatter point has an exact twinlocated directly opposite the center point of the scat-terplate. Figure 2 shows an example of inversionsymmetry designed into a binary photomask used toexpose scatterplates. The reason inversion symme-try works is revealed when we examine individualscatter points and their effect on the reference andtest beams. Keep in mind that an examination ofindividual points is not the complete story. Sum-ming the wave fronts from all the scatter points formsthe contour fringes. Figure 3�a� summarizes the ef-fect of inversion symmetry for a perfectly alignedsystem. When the incident light interacts with thescatterplate, the light scattered at scatter point S�x,y� acts like a point source. Because the scatterplateis located at the center of curvature of the test mirror,point S�x, y� is imaged back into the plane of thescatterplate at the conjugate point S��x, y�. The ref-erence beam passes directly through the scatterplate,


reflects off the test mirror, and scatters at point S��x,�y�. With inversion symmetry and proper align-ment, point S�x, y� and S��x, �y� are scattered inexactly the same manner, and the test and referencebeams both appear as point sources located at pointS��x, �y�. As a result, the phase change that is dueto scattering is the same for both beams.

The above discussion of inversion symmetry is for asystem with perfect alignment, but this is not alwaysthe case. The effect of misalignment of the scatter-plate is now discussed. Lateral movement of thescatterplate produces tilt in the contour fringes.Figure 3�b� demonstrates the consequence of whenthe center point of the scatterplate is placed above thecenter of curvature of the test mirror. The image ofS�x, y�, although still in the plane of the scatterplate,no longer coincides with the symmetric point S��x,�y�. The result is tilt in the contour fringes pro-duced by the interference of two laterally shiftedpoint sources. Similarly, longitudinal misalignmentof the scatterplate adds defocus to the contour fringepattern. Figure 3�c� shows the effect of when thescatterplate is placed inside the center of curvature ofthe test mirror. The image of point S�x, y� is still atthe same lateral position as the symmetric pointS��x, �y�; however, it no longer lies in the plane ofthe scatterplate. The interference of the two longi-tudinally shifted point sources produces defocus inthe contour fringes. Adjusting the scatterplate po-sition adds an important flexibility that enables us tominimize the number of contour fringes across theimage plane. Figure 4 contains typical fringe pat-terns obtained with the scatterplate interferometer.The fringe patterns contain varying amounts of tiltand defocus. Note that the fringe patterns containspeckle as is consistent with use of a coherent source.

3. Phase-Shifting Scatterplate Interferometer

A. Birefringent Scatterplate

Before the operation of the interferometer as a wholecan be understood, it is necessary to independently

Fig. 2. Inversion symmetry designed into a binary scatterplate.

examine the characteristics of the birefringent scat-terplate. In the phase-shifting scatterplate inter-ferometer, the goal is to control when the test andreference beams are scattered. Here the birefrin-gent scatterplate shown in Fig. 5 provides the desiredcontrol. The appropriate aperiodic pattern with in-version symmetry is etched into a calcite retarderwith a chemical etching process. An index-matching oil chosen to match the ordinary index ofthe crystal is then pressed between the calcite and aglass slide. The end result is that, for light polarizedalong the ordinary axis of the crystal, the index of theoil and the index of the crystal appear the same and

Fig. 3. Point-to-point analysis of inversion symmetry: �a� perfemisalignment of scatterplate.

the light passes directly through the scatterplate,whereas light polarized along the extraordinary axisof the crystal sees an index difference and is scatteredby the rough surface. This is the property that sup-plies the control over scattering that is necessary tophase shift the scatterplate interferometer.

Matching the index of the oil with the ordinaryindex of the calcite is fairly straightforward. How-ever, it is important to consider the dispersion of bothmaterials. Literature usually quotes the indices ofrefraction at the sodium D line. Unfortunately,matching the indices at 589 nm does not guaranteethat the indices will match at a different wavelength.

gnment, �b� lateral misalignment of scatterplate, �c� longitudinal
ct ali

6

The dispersion of calcite is described by the followingequations15:

no��2 � 1 �0.8559�2

�2 � �0.0588�2 �0.8391�2

�2 � �0.141�2

�0.0009�2

�2 � �0.197�2 �0.6845�2

�2 � �7.005�2 , (1)

ne��2 � 1 �1.0856�2

�2 � �0.07897�2 �0.0988�2

�2 � �0.142�2

�0.317�2

�2 � �11.468�2 , (2)

where � is the wavelength of the source in microme-ters. The manufacturer usually supplies the wave-length dependence of the index-matching oil. Thedispersion equation for Cargille’s nd � 1.662 oil is

n1.662 � 1.621298 �1192615��104�2 �

7.658881 � 1012

��104�4 .

(3)

At the operating wavelength of 633 nm, the indicesdiffer by only 0.00005.

When the index of the oil is matched with the or-dinary index of the calcite rather than the extraordi-nary index, the ordinary index does not change foroblique illumination of the birefringent scatterplate.This is an important advantage because the scatter-plate will be placed in a converging beam, and indexmatching is crucial to the operation of the phase-shifting birefringent scatterplate interferometer.

B. Qualitative Description of Phase-Shifting ScatterplateInterferometer

It is instructive to examine the operation of thephase-shifting scatterplate interferometer qualita-tively. The schematic diagram of the interferometeris shown in Fig. 6. If the polarization elements suchas the wave plates and polarizers are removed, it is aconventional scatterplate interferometer. The po-larizer passes linearly polarized light oriented at 45°with respect to the optic axis of the calcite scatter-plate, providing equal amplitudes for the componentof the beam polarized along the optic axis and thecomponent polarized orthogonal to the optic axis.The component parallel to the optic axis will see theextraordinary index of the crystal, and the perpen-dicular component will see the ordinary index. Aliquid-crystal retarder produces a variable phase

Fig. 4. Typical scatterplate interferograms.


shift between the two orthogonal components of po-larization, and the rotating ground glass plate re-duces the speckle in the interferogram because iteffectively enlarges the apparent source size. Therotating ground glass plate is offset slightly from theintermediate image of the source such that the diam-eter of the incident beam is larger than the originalsource. At each point of the rotating ground glassplate the phase of the scattered light is changingrapidly compared to the integration time of the CCDarray. The result is an extended source, made up ofincoherent point sources, that is, the same size as thebeam incident on the ground glass plate. We canadjust the size of the source and consequently thecoherence by longitudinally shifting the rotatingground glass plate. The source created at theground glass plate is imaged onto the test mirror.The scatterplate is located at the center of curvatureof the test mirror and scatters only the component ofthe beam that is polarized parallel to the optic axis ofthe crystal. The perpendicular component passesthrough the scatterplate and forms an image of thesource on the test mirror. Both the scattered andthe direct beams pass through a quarter-wave platetwice producing a 90° rotation in the direction ofpolarization of each. As a result, on the second passthrough the scatterplate the beams change roles:The one that traveled directly through on the firstpass is now scattered, and the one that was scatterednow passes directly through. The outcome is ascattered–direct and direct–scattered beam with or-thogonal polarization. Note that, if there is no un-wanted scattering when the beams pass directlythrough the scatterplate, there is no background ir-radiance in the interference pattern. Similarly, ifall the light is scattered by the scatterplate on thepasses when scattering is wanted, the hot spot willnot exist. In general, not all the light polarized par-allel to the optic axis is scattered, and the quarter-wave plate does not rotate the two polarizationsexactly 90°. Consequently, both orthogonal polar-ization components will contain a direct–direct and

Fig. 5. Birefringent scatterplate.

erfer

scattered–scattered element producing a hot spot andbackground irradiance in the interference pattern.Finally, the test mirror is imaged onto a CCD arraythrough an analyzer, which serves to combine thetest and reference beams for observation of interfer-ence fringes. The end result is that we can phaseshift the interference fringes by applying a voltage tothe liquid-crystal retarder. Figure 7 contains a se-ries of four shifted interferograms.

4. Scatterplate Manufacture

The birefringent scatterplate is the key component ofthe phase-shifting scatterplate interferometer pre-sented in this paper. In this section we discuss itsmanufacture. There are many issues to be consid-ered, including pattern generation, average featuresize, inversion symmetry, and etching rates. We ad-dress each of these issues in the context of describingthe entire manufacturing process. First, an over-view of the entire process is presented in generalterms. Then we provide a detailed presentation forexposing the scatterplate pattern using two differentmethods.

A. Manufacturing Process

Because the operation of the interferometer dependson the birefringent properties of the scatterplate, it isnecessary to etch the scatterplate pattern directlyinto the birefringent material. The process pre-sented here will work for most birefringent materials;however, the chemicals used to clean and etch thescatterplate may vary. Etching precise patternsinto birefringent materials is a new concept, and toour knowledge there is no literature to speak of thatoutlines a process. The recipes presented here arespecific to calcite and were obtained through trial anderror.

The birefringent scatterplate is made by a six-step

Fig. 6. Phase-shifting scatterplate int

etching process outlined in Fig. 8. First, a goodquality wave plate made with calcite is cleaned infour stages with acetone, isopropanol, deionized wa-ter, and a plasma chamber. The calcite is placed ineach of the chemicals in the order listed above for 30 sand agitated gently. Upon removal from the deion-ized water, the calcite is dried and placed in a plasmachamber for 30 s to remove any remaining impuritieson the surface. If the substrate is not clean, thephotoresist will not adhere to the surface. The nextstep in the manufacturing process is to spin coat thesample with photoresist. There are many photore-sists available with different viscosities and sensitiv-ities. Shipley’s 1813 photoresist was used becauseof its availability. We coated the scatterplate withphotoresist by spinning it at 500 rpm for 5 s andincreasing the speed to 5000 rpm for 30 s. The speedand duration of the spin coating, as well as the vis-cosity, determine the thickness of the deposited pho-toresist. For this application the thickness of thephotoresist is not critical provided that it stands up tothe chemical etching. After coating, the sample issoft baked on a hot plate for 1.5 min at 100 °C tosolidify the photoresist. The scatterplate pattern isthen exposed into the photoresist by use of either aspeckle pattern or a photomask. Developing re-moves the photoresist from the calcite only in theregions that are exposed with flux levels above 150mJ�cm2. We developed the pattern by placing it inShipley’s 352 developer for 45 s and rinsing in deion-ized water for 1 min. After developing, the sample ishard baked for another 1.5 min on the hot plate.The birefringent scatterplate is then chemicallyetched with an extremely weak solution of hydrochlo-ric acid. A 37% solution of HCl is diluted 5000 to 1in deionized water, and the scatterplate is gently ag-itated in the solution for 3–5 min depending on theetch depth desired. Figure 9 shows a plot of the etch

ometer. LQR, liquid-crystal retarder.


6

depth as a function of etching time for calcite. Rins-ing the calcite in deionized water for 1 min termi-nates the chemical etching. Finally, the photoresistis removed by the same cleaning process used to cleanthe blank substrate. Figure 10 shows a surface plotof an etched scatterplate.

B. Holographic Exposure

A common method to expose scatterplates is to doubleexpose a speckle pattern, in which the scatterplate isrotated 180° between each exposures.13 Figure 11shows the holographic setup used to expose scatter-plates. An argon-ion laser beam tuned to a wave-length of 458 nm is expanded with a beam expander.The beam is then scattered by a ground glass plate,and a speckle pattern is created in the film plane.The size of the smallest features in the speckle pat-tern is determined by the diameter of the beam inci-dent on the ground glass plate and the distance fromthe ground glass to the film plane. In turn, the sizeof the speckle pattern features determines the

Fig. 7. Phase-sh
ifted fringe patterns.

Fig. 8. Scatterplate manufacturing process.

f�number of the mirror that can be measured. Thephotoresist-coated sample is placed in the film planeand rotated 180° between two exposures. The rota-tion is controlled by a kinematic mount, shown in Fig.11, to �0.05°. The end result is a random patternwith inversion symmetry.

A simple geometric analysis will reveal the rela-tionship between the holographic setup and thef�number of the mirror that can be measured.There are two geometries to consider in this analysis.First, the exposing geometry will determine thesmallest feature size on the scatterplate, and then thegeometry of the scatterplate interferometer will de-termine the fastest mirror that can be tested. Fig-ure 12 shows both geometries. When the speckle

Fig. 9. Etch depth versus etch time for calcite in diluted HCl.

Fig. 10. Surface plot of birefringent scatterplate manufacturedwith a photomask.

Fig. 11. Holographic exposure of a scatterplate.

pattern is exposed, interference between the twomost widely separated points on the ground glassproduces the smallest features. We can determinethe radius of the smallest features by finding the firstinterference minimum off the optical axis. For thegeometry shown, the radius is given by

r �z�

D, (4)

where z is the distance from the ground glass plate tothe film plane, � is the wavelength of the source, andD is the diameter of the beam incident on the groundglass plate. The analysis for the scattered light issimilar except that the smallest features on the scat-terplate are used to determine the radius at whichthe irradiance at the mirror falls off to zero. Repeat-ing the analysis for the interferometer geometry andsubstituting for r, we obtain the desired relation:

f�number �z

2D, (5)

Fig. 12. f�number analysis for holographic exposure: �a� record-ing geometry, �b� interferometer geometry.


6

where

f�number �f

2�. (6)

Here � is the radius of the mirror aperture and f is thefocal length of the mirror. The ratio of z over Ddetermines the f�number of the mirrors that can bemeasured by use of the exposed scatterplate. In apractical measurement, it is not desirable to have theirradiance drop off significantly at the edge of themirror. If the minimum f�number is half of thef�number of the test mirror, the irradiance will dropby 50% at the edge of the test mirror.

The advantage of using the holographic exposure isthat we can easily create low f�number scatterplates.The disadvantage is the elaborate setup required foreach exposure and the possible errors in symmetrythat are due to the rotating of the scatterplate be-tween exposures.

C. Photomask

A better method to expose the scatterplates is to usea photomask. A photomask is a glass substrate onwhich the desired pattern is written into chrome withan electron beam. Once the pattern is written, thephotomask can be used repeatedly to manufacturemany scatterplates. The photomask is placed inclose contact with the photoresist-coated sample, andillumination of the sample with a UV source exposesthe pattern. The exposing is usually done with anapparatus known as a mask aligner and takes only afew minutes. With one photomask many scatter-plates can be rapidly exposed. Most of the work as-sociated with use of a photomask is in its design.

As with the holographic exposure, the feature sizesin the mask determine the f�number of the mirrorthat can be measured with that particular scatter-plate. It turns out that, for a randomly generatedpattern, the average feature size in one dimension istwo pixels and the minimum feature size is one pixel.Because the irradiance pattern at the mirror is thesquared modulus of the Fraunhofer diffraction pat-tern of the scatterplate and the shape of the smallestfeature is known, the f�number can be determinedwhen we take the Fourier transform of the smallestfeature in the mask. The fastest mirror that can becompletely illuminated is one for which the envelopeof the irradiance pattern drops to zero at the edge ofthe mirror. The envelope function is given by

Eenvelope � �FF�rect�xa�rect�y

a��2

� �a2 sinc�a�sinc�a��2

� a4 sinc2�a�sinc2�a�, (7)where

�x�f

, (8)

�y

. (9)
�f

Here f is the focal length of the test mirror, a is thewidth of the smallest feature, and � is the wavelengthof the source. The irradiance pattern drops to zeroat

x ��fa

, (10)

which gives the following relation for the minimumf�number:

� f�number�min �a

2�. (11)

In a practical measurement, it is not desirable to havethe irradiance drop off significantly at the edge of thetest mirror. As a rule of thumb, a scatterplateshould be designed such that the minimum f�numberis one half of the f�number of the test mirror.

The photomask patterns are generated when atwo-dimensional array is randomly filled with onesuntil approximately 25% of the array is full. Then acopy of the array is created, rotated 180°, andsummed with the original. Pixels with valuesgreater than one are set equal to one. The finalpattern is an array of ones and zeros that are pseu-dorandom with inversion symmetry. The size of thepixels can be specified when the pattern is sent to thephotomask manufacturer.

An f�6 mirror with a focal length of 90.93 cm wasmeasured for the results presented in Section 6. Wefabricated the scatterplate used for the measurementusing a photomask with 4-�m pixels. At 633 nm theminimum f�number was 3.16, approximately half ofthe f�number of the mirror. The overall dimensionsof the scatterplate were 8 mm � 8 mm.

5. Experimental Results

The best way to estimate the accuracy of an inter-ferometer is to measure a known surface or, equiva-lently, compare measurements with a calibratedinstrument. Figure 13�a� contains a surface plot ofthe test mirror obtained by the phase-shifting scat-terplate interferometer, and Fig. 13�b� shows a sur-face measurement of the same mirror acquired witha commercial phase-shifting Fizeau interferometer.Because of the long focal length of the test mirror andthe limited size of the isolation table, a folding flatwas used in the Fizeau measurement. In doublepass, the possible peak-to-valley error in the mea-surement created by the flat was 0.073 waves. De-spite the measurement uncertainty because of thefolding flat, the scatterplate and Fizeau measure-ments compare well. The peak-to-valley differenceis less than 0.035 waves.

A second method to determine the performance isto establish the repeatability of the interferometer bythe subtraction of two consecutive measurements.The rms difference between two consecutive mea-

surements was found to be of the order of 0.0025waves.

6. Discussion

In this paper we described and analyzed a new phase-shifting scatterplate interferometer. The inter-ferometer separates the test and reference beams byexploiting the polarization characteristics of a bire-fringent scatterplate. The birefringent scatterplatewas manufactured with the pattern etched into agood quality calcite wave plate. The scatterplatescatters only the component of polarization orientedalong the extraordinary axis of the crystal. To-gether with two polarizers and a quarter-wave plate,the birefringent scatterplate produces test and refer-ence beams with orthogonal polarizations. A vari-able phase shift is induced between the beams by useof a liquid-crystal retarder.

The performance of the interferometer was admi-

Fig. 13. �a� Surface measurement taken with a phase-shiftingscatterplate interferometer. �b� Surface measurement taken witha Wyko 6000 phase-shifting Fizeau interferometer.

rable. Subtracting two consecutive measurementsdemonstrated a repeatability of 0.0025 waves rms,and a comparison of the measurements with a com-mercial interferometer established a peak-to-valleyaccuracy of 0.035 waves, which was well within theuncertainty of the commercial interferometer mea-surement.

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