Extended unambiguous range interferometry

Extended unambiguous range interferometry

Timothy C. Strand and Yigal Katzir

Phase measuring interferometers generally measure phase modulo 2ir. We present a system which usesfringe contrast to help determine the absolute phase in the interference image within the limits of thecoherence length of the illumination. This approach obviates the need for phase unwrapping and isunaffected by surface discontinuities or by data dropout. Since the phase is determined on a point-by-pointbasis, the processing could be pipelined. The system is set up on a microscope interferometer and producessurface profiles over an array of 512 X 512 points. The measurement range is related to the coherence lengthof the source and can easily be varied from 0.5 to 2.5,um. The resolution is limited by the 8-bit quantization ofthe output.

1. Introduction

Optical interferometry is an old and well-estab-lished tool, widely used by the optical industry fortesting and quality assurance.',2 The ever shrinkingtolerances on such surfaces as magnetic and opticalstorage media, solid-state wafers, and thin-film de-vices, together with increasing production volumes,have created a strong demand for surface-profilometrytechniques for manufacturing line inspection. Thedesirable profilometer should be noncontacting, havehigh spatial resolution, and combine high accuracywith high data rate. In addition, manufacturing lineoperation dictates robustness, reliability, and simpleautomated operation. Optical interferometry, beingsimple and sensitive, was a natural candidate for theseapplications, and consequently several microscope in-terference devices have been developed to answer vari-ous industrial needs.

Whereas interference devices have indeed foundwidespread applications in the electronics industry,their use has mostly been limited to visual semiquanti-tative evaluation. Recently, several automated opti-cal profilometers have become commercially available.Using the principles of phase shifting interferometry,originally developed for testing optical elements, they

have potential repeatability of better than 1 nm.However, their actual accuracy, i.e., the degree towhich the measurement corresponds to the actual sur-face, is limited by the inherent ambiguity of the inter-ference pattern. Consequently, they depend on someform of phase-unwrapping algorithm.3 While compu-tation of the fractional phase can be made very fast,phase-unwrapping to retrieve the fringe order is itera-tive and recursive and, therefore, adds considerably tothe processing time, especially in situations where sig-nal dropout is a problem. Furthermore, phase track-ing is impossible whenever the phase difference be-tween adjacent resolution elements becomes greaterthan 7r.

We present an interferometer in which the unambig-uous range is extended beyond the 2-r limit. Using abroadband light source, the resulting variation of in-terference modulation depth, or contrast, is measuredand serves to identify the fringe order. This obviatesthe need for phase-unwrapping and makes the inter-ferometer both potentially faster and capable of han-dling surfaces containing sharp gradients.

11. Ambiguity of the Phase Shifting Algorithm

The most general expression for the light irradianceat the output plane of a two-beam interferometer witha quasimonochromatic light source is usually given as

I(x,y) = B(x,y)[1 + M(x,y) coso(xy)].

When this work was done both authors were with IBM AlmadenResearch Center, 650 Harry Road, San Jose, California 95120-6099;Y. Katzir is now with Orbot Systems, Ltd., Industrial Zone, P.O. Box215, 70650 Yavne, Israel.

Received 20 April 1987.0003-6935/87/194274-08$02.00/0.© 1987 Optical Society of America.

(1)

The above expression describes the interference pat-tern at every point as the sum of a bias term and aninterference term. For the purpose of profilometry,the relevant information, namely, the height of the testsurface compared with some reference surface, is con-tained in X, the relative phase between the interferingbeams. For an object observed in reflection, the phasedifference is related to profile height z by

4274 APPLIED OPTICS / Vol. 26, No. 19 / 1 October 1987

0(xy)= 2 2z(xy), (2)

where X is the wavelength of the light and 2z is theoptical path difference (OPD). In the following we usethe terms phase and optical path difference inter-changeably. Here we have neglected the phase factorin the Fresnel reflection coefficient. We will comeback to that later. M in Eq. (1) is the modulationdepth, or contrast, of the interference pattern. Formonochromatic light, M is approximately constant at agiven point, independent of the OPD, and satisfies IMI< 1.

Attempting to use Eq. (1) to solve for m and assumingwe can somehow account for B and M, we note that atbest we can obtain 0 mod(7r). This constraint arisesfrom the periodicity of the interference term, whichlimits the unambiguous range of our hypothetical pro-filometer to X/4.

The phase shifting interferometer provides an ele-gant way of accounting for the bias and modulationterms while doubling the unambiguous range.66 Gen-erally speaking, the idea is to apply a controlled spa-tially uniform phase shift between the two arms of theinterferometer. Equation (1) can now be modified toinclude such a phase term explicitly:

I(x,y) = B(xy)[1 + M(x,y) cos(4 + )]. (3)

The irradiance given by Eq. (3) is now measured withseveral known values of the phase shift . Since thereare three unknowns in Eq. (3), a minimum of threelinearly independent measurements has to be made.More measurements may be taken to improve the SNRprovided other error sources such as vibration anddrift are well controlled.

Rather than go into the mathematical detail of thephase shifting algorithm, we choose to illustrate itgraphically. As before, we assume that we have asufficient number of linearly independent measure-ments to account for B and M, so that we can computethe net interference term cos( + ). Although anytwo values of 6 can be used, as long as they do not differby an exact multiple of r, we assume that we havecomputed two such terms with = 0 and = 7r/2,respectively. We now have the values of both cosk andcos( + r/2) = sino for every point. This pair ofparameters defines 0 uniquely in the range [0,27r), aswe readily observe from Fig. 1 where the values of inthat range trace out a circle in the plane defined by thecos, sink axes. We refer to this technique of takingtwo measurements with a 1r/2 phase shift as quadra-ture detection.

For values of 0 beyond 27r, the circle is retraced,which illustrates the ambiguity problem. The variousimplementations of the phase shifting approach differprimarily in the particular combination of phase shiftsthat are linearly combined to yield the quadraturecomponents, namely, coso and sino. However, theyall share the same ambiguity problem.

Removal of the ambiguity can be done by a recursiveprocess known as phase unwrapping.3 7 This processassumes that the difference in phase between adjacent

0

cos f

Fig. 1. Graphic representation of the phase shifting algorithm:The points with Cartesian coordinates given by (coso,sinp) trace outa circle as 0 goes from 0 to 27r. For higher values of 0 the circle is

retraced, which illustrates the ambiguity of the algorithm.

sampling points in the interference image does notexceed -X and uses this assumption to track the phasefrom one pixel to the next. Thus the process is addingor subtracting whole multiples of 27r to the phase val-ues so as to force the differences between adjacentpixels into the interval [-7r,7r). In practice, some sys-tems limit the phase difference to less than 7r, so thatan optimized phase unwrapping path may be chosen tominimize errors.

From the point of view of surface profilometry, thedependence on phase unwrapping is problematic forthe following reasons:

The restriction on the allowed phase difference be-tween adjacent pixels places severe requirements onspatial resolution. These requirements are unlikely tobe met with surfaces containing such features as sharpsteps, scratches, dust, debris, or occlusions.

The process is sensitive to errors generated by elec-tronic and optical noise, and signal dropouts such ascaused by defective sites in the area detector or by lowreflectivity object points.

Errors that occur at certain pixels generally propa-gate through the image, the amount and form of thepropagation depending on the specific algorithm.

Being iterative and recursive, phase unwrappingadds considerably to the processing time even when itcan be smoothly accomplished.

Ill. Extending the Unambiguous Range

Our approach to extending the unambiguous rangeof a two-beam interferometer is based on the fact thatthe fringes generated with wide-bandwidth sourceshave a different appearance according to order. Thisfeature is being used extensively, mainly by microsco-pists, for qualitative interpretation of white light inter-ferograms. The idea in white light interferometry is tomatch the observed fringe color with a color table.Automated interference-color analyzers exist for suchapplications as thin-film thickness measurements.These are generally point measurement systems. Forthe purpose of area profilometry, however, the high-

1 October 1987 / Vol. 26, No. 19 / APPLIED OPTICS 4275

pixel throughput rate required renders such apointwise spectral analysis impractical. Thus, ratherthan analyzing fringe spectral content, we decode thedegree of temporal coherence to identify fringe order.This information is combined with the quadraturedetection outlined above to extend the unambiguousrange.

To discuss the underlying principles of our ap-proach, we now rewrite Eq. (1), the general expressionfor the irradiance of a quasimonochromatic two-beaminterferogram, so as to include explicitly the physicalparameters involved:

I(x,y) = R(xy) + S(x,y) + 2j0m(x,y;p) cosp. (4)

In Eq. (4) R and S are the separate irradiances of thereference and object beams, respectively, and it is as-sumed that the two beams are in the same state ofpolarization. The parameter m is the modulus of thecomplex degree of coherence. 8 It satisfies 0 < m < 1and describes a slowly varying envelope of the interfer-ence term. Like 0, m is a function of the optical pathdifference. If we further assume that the interferome-ter introduces no shear between the two images of thesource, m contains only temporal effects. In the ab-sence of dispersion, m is simply proportional to thedegree of temporal coherence of the source. The latteris equal to the Fourier transform of the normalizedpower spectral density of the source. This impliesthat, for a source of finite bandwidth, m drops from avalue close to unity for very small path differences toessentially zero at a path difference of the order of thecoherence length of the source. If the source spectraldistribution is reasonably smooth, this drop is essen-tially monotonic with little or no sidelobes. In thesecircumstances, there is a one-to-one correspondencebetween the value of m and the optical path difference,provided we limit ourselves to either positive or nega-tive path differences.

A typical interferometer output as a function ofoptical path difference, for a source with a widebandGaussian spectral distribution, is illustrated in Fig. 2.In this case the function m is Gaussian and, in theabsence of dispersion, symmetric about the origin ofzero path difference.

For a wideband source the function m is no longerslowly varying. Nevertheless, we can still talk aboutan effective wavelength in terms of the well-definedzero crossings of the interference term. The effectivephase shift is now given by Eq. (2) with X denoting theeffective wavelength. If we now draw the equivalentof Fig. 1 for a wideband source, with coordinates givenby m(0) coso and m( + 7r/2) sin5, the resulting curvewill be a spiral rather than a circle. This is illustratedin Fig. 3. As is clearly evident from Fig. 3, the quadra-ture detection with a wideband source is potentiallyunambiguous over an OPD range of several phaseturns or fringe orders. This range is basically deter-mined by the slope of the temporal coherence functionm and the capability of the data acquisition and pro-cessing system to measure accurately the quadratureinterference signals. As a crude rule-of-thumb for the

O.P.D.

Fig. 2. Interferometer output with a wideband source: Fringevisibility gradually drops from a maximum value of -1 for smallODP to 0 for an OPD approximately equal to the coherence length ofthe source. The temporal coherence m is the slowly varying enve-

lope of the rapidly oscillating output function.

Fig. 3. Graphic representation of the extended range algorithm:for a wideband light source, the points with Cartesian coordinatesgiven by [m(¢) coso,m(o + r/2) sink trace out a spiral as 0 goes from0 in either a positive or negative direction. The radius of the spiraldecreases in accordance with m, the temporal coherence of thesource. The spiral shown is for positive OPD; negative OPD wouldgive an opposite handed spiral which intersects the positive spiral.The spiral is unambiguous over a range determined by the source

temporal coherence length.

useful range in terms of fringe orders, we assume alinear slope of the coherence function. If we denotethe coherence time and length by i-r and c, respective-ly, we have

c \2I= ci> = = ,AP ct AX (5)

where AP is the source bandwidth and c is the speed oflight. Bearing in mind the ± ambiguity, only half ofthe coherence length is useful. Therefore, the corre-sponding number of fringe orders is

1 2c X 2=2X A=- (6)

Given that the quadrature interference terms areproperly measured and computed, the value of thecorresponding OPD can be read from a calibrationlookup table, which is essentially a digital representa-tion of the spiral curve of Fig. 3.


IV. Algorithm

Our purpose now is to show how the desired netquadrature interference signals can be measured andcomputed. To this end, we rewrite Eq. (1) for thethird time to include the possible presence of opticalnoise. This is primarily stray light reflected fromvarious surfaces, which constitutes additive noise, i.e.,is independent of the presence of the interferometer.Since many surfaces of interest are poorly reflective(-5% or less), this noise can be significant. The inter-ferometer output is now given by

I(x,y) = N(x,y) + R(x,y) + S(x,y) + 2Rm(o) cosp(xy). (7)

A look at Eq. (7) reveals a total of five unknowns.Hence at least five light measurements are needed atevery point. We acquire the following five images:

(1) The noise light distribution N(ij), where (ij)are pixel coordinates.

(2) The reference irradiance, i.e., light reflectedfrom the reference mirror. This measurement corre-sponds to

N(ij) + R(ij).

(3) The total intensity. This is the combined irra-diance of the reference and object surfaces with theinterference term set to zero. This image is given by

N(ij) + R(ij) + S(ij).

(4) The cosine interference image, as given by Eq.(7):

N(ij) + R(ij) + S(ij) + 2XRiSm(q5) coso(ij).

(5) The sine interference image:

N(ij) + R(ij) + S(ij) + 2RSm(0 + ir/2) sino(ij).

The five light measurements are now manipulatedby simple algebraic steps to yield the normalized quad-rature interference terms:

m(¢) cos(i,j);

m(o + r/2) sinp(ij).

These normalized interference terms are now usedas coordinates to read the corresponding value of theOPD from a 2-D lookup table. The use of a lookuptable saves computation time and automatically com-pensates many system parameters. The procedure forgenerating this lookup table is discussed in the nextsection. It is important to note at this point, however,that the lookup table is purely empirical and does notrely on the functional form of the interference terms.This means that the exact phase relation between im-ages 4 and 5 is not critical. As long as it is consistent, itcan take on any value other than a multiple of r,although a value around 7r/2 is best. This distin-guishes the current approach from other phase shiftinginterferometers which assume a specific phase shift.

V. Lookup Table Generation

As implied in the previous section, a key element tothe EXTRA interferometer algorithm is a suitable cali-

bration lookup table. This table is essentially a digitalrepresentation of the spiral curve of Fig. 3. It is a dual-input table that accepts rescaled and shifted represen-tations of the normalized interference terms definedabove as inputs. The table output is a number propor-tional to the corresponding OPD.

Our approach to generating the lookup table is toperform a spatial scanning of the phase range. Theidea is to use the interferometer in the biased mode,whereby a test surface of very good flatness is tilted atan angle to produce a set of straight parallel fringes.We now have a phase distribution that is linear inimage coordinates. The desired calibration spiral isnow obtained by computing the normalized quadra-ture interference terms for a line of points that runsperpendicular to the fringes. The experimentalpoints, with coordinates computed as described above,are assigned z values proportional to the correspond-ing OPD in accordance with the depth of the lookuptable. Thus, for an 8-bit deep table, numbers rangingfrom 0 to 255 are assigned. The next step is to proper-ly interpolate and smooth the table, so that table loca-tions that fall between the experimental calibrationpoints are assigned appropriate values. Points not onthe spiral are assigned z values equal to that of theclosest point on the spiral. Thus, during subsequentmeasurements, data points which do not fall exactly onthe spiral due to noise or other errors are assignedappropriate z values.

The limitation of this calibration technique is itsdependence on the availability of a high-quality opti-cal flat that has the same spectral reflectivity proper-ties as the test surface. This may potentially be aproblem with some surfaces, particularly ones whichare translucent or textured. Fortunately, however,our experience shows that very often it is possible touse a generic lookup table prepared from a metallicmirror for work with nonmetallic surfaces. The effectof such surfaces is usually to introduce an overall re-duction in fringe visibility. This can be compensatedfor by properly scaling the normalized interferenceterms before using them as lookup table coordinates.In case a suitable calibration surface cannot be found, atemporal scanning technique may be considered,whereby an appropriate calibration sample is translat-ed along the optical axis in known steps. The spiralcalibration data are obtained by taking measurementsat each step until a full spiral has been obtained. Thisassumes the ability to produce well-quantified submi-cron steps and that system stability can be maintainedfor the duration of the calibration process.

VI. Design and Practical Considerations

A. Spatial Coherence Considerations

Since our interferometer uses the same calibrationlookup table for all pixels, the fringe visibility envelopemust be uniform across the entire field of view. Suchconditions usually do not prevail in interferometrywith extended sources, and the degree of coherence min Eq. (4) in general also includes space-variant effects.


These effects stem from the finite source size. Sourcesize and shape determine the spatial coherence in theinterferometer output plane. A limited spatial coher-ence due to finite source size brings about a reductionof fringe visibility as a function of the shear or lateralseparation between the two images of the source asviewed from the output plane of the interferometer.In general, the shear varies linearly with the amount ofrelative tilt between the reference and object surfaces.

Shear is eliminated in practical interferometers byimaging both the reference and object surfaces ontothe output plane. In a reflected light microscope thefield stop is imaged by the objective lens onto the testsurface and so constitutes an effective source. By thevan Cittert-Zernike theorem, the spatial coherencerange of a reflected light microscope is identical to thediffraction-limited resolution element of the objectivelens. In the Michelson and Mirau interferometersboth the test and reference surfaces are imaged by thesingle objective. In these interferometers, the twoimages of the field stop remain in perfect registrationwith no shear between them as long as both surfaces liewithin the depth of field of the objective lens. In theLinnik-type interferometer, the reference and objectsurfaces are imaged by separate objectives. Althoughthese are carefully selected and supplied as matchedpairs, residual geometrical distortions (e.g., a smalldifference in magnification) between the two opticalchannels create enough spatial contrast variation topotentially preclude an implementation of our systemwith a Linnik interferometer. Our demonstrationswere all done with a commercial Michelson interfer-ometer attachment. We have also established the fea-sibility of a Mirau implementation.

With existing detector technology and 8-bit digitiz-ers, we expect that practical implementations of thiswill be limited to a useful height range of up to -3 ,um.This range is well within the depth of field of all but thehighest magnification (40X) attainable with a Mirauinterferometer. Therefore, no practical problems as-sociated with spatial coherence are expected with theMichelson and Mirau attachments up to an objectivemagnification of 20X or so, provided care is taken tofocus carefully the reference mirror during the setupprocedure.

B. Environmental Stability Requirements

This interferometer system depends for its practicaloperation on temporal repeatability with respect toboth fringe visibility and phase shift between thequadrature interference patterns. Both of these pa-rameters may be adversely affected by evironmentaleffects such as vibrations and drift. In an industrialenvironment, in particular, it is usually impractical toeliminate totally these effects. Therefore, in additionto a good shielding of the system, such as mounting iton a vibration isolated table, the data acquisition pro-cedure should be optimized to reduce sensitivity toenvironmental effects. Thus the acquisition of thephase shifted interference patterns should be made ina rapid sequence to reduce the sensitivity to drift. We

have also experimented successfully with software vi-bration monitoring. In that mode, the data acquisi-tion program checks the stability of the interferencepattern by sampling a few pixels and comparing theirvalues at different times against an empirically deter-mined window. This allows the system to pick rela-tively quiet time periods for doing the acquisition.

C. Phase Shift Uniformity and Repeatability

The shape of the spiral curve in Fig. 3 is extremelysensitive to the phase shift between the two quadra-ture components. Computer simulations have shownthat in extreme ranges (i.e., around ten phase turns)even a variation of 20 between the phase shift assumedin the lookup table and that of the actual data may beintolerable. In our evaluation system the phase shiftis effected by a piezoelectric translator on which thetest surface is mounted. The PZT is only specified tohave a uniformity of better than 5%, which for a shift of90° would translate to an error of 4° across the fullwidth of the PZT. To ensure the necessary phase shiftuniformity, the PZT is chosen to be considerably widerthan the field of view of the microscope. Any PZT tilt isdemagnified by the ratio of the width of the field ofview to the width of the PZT base. In our experimentsthat ratio was of the order of 1:50.

To ensure temporal repeatability of the phase shift,the PZT is always cycled between the same voltages toavoid hysteresis effects. The phase shift is still subjectto errors due to vibrations, and ways to combat thesehave already been discussed.

D. Detector Requirements

A successful operation of the interferometer systemdepends on the following detector requirements: goodgeometrical repeatability; linear response; high SNR;and high spatial resolution.

Generally speaking, this system shares the same de-tector requirements with conventional phase shiftinginterferometers. These requirements are readily metby several modern high-resolution solid-state imagersof the CCD or MOS varieties. While good geometricalrepeatability and linear response are inherent to thesolid-state design, the SNR of these imagers (around45-50 dB) generally matches that of the 8-bit videodigitizers which are in common use. Just like all otherphase shifting interferometers, the algorithm de-scribed here is insensitive to the cosmetic quality of theimager, since the effects of fixed pattern and responsenonuniformity are automatically canceled. The effectof limited detector resolution on the performance ofthis system, however, is somewhat different. The rea-son lies in the finite area of the detector elements. Ifthe area of a detector element is larger than the resolu-tion element of the image, the detector performs anincoherent weighted integration of several interfer-ence terms. In our system such errors are limited tohighly textured areas or sharp steps and have no effecton neighboring pixels. However, it is best to matchthe detector element area to the resolution of the imag-ing system. By limiting the field of view of the system


to typically two-thirds of the total field, the imageresolution can be matched to that of the 512 X 512sampling of most image digitizers. Assuming the cam-era has at least that much resolution and the micro-scope has a sufficiently high magnification to resolveall surface features, the computed wavefront may beconsidered a faithful replica of the test surface.

E. Calibration Lookup Table Design

The calibration lookup table described earlier in thispaper defines the useful optical path difference rangefor the interferometer. The lookup table determinesboth the maximum profile depth that can be accom-modated and the depth resolution, since the table has afinite number of bits. By controlling the bandwidth ofthe illumination using filters, the range covered by thetable can be tailored to suit the application.

Ideally, the fringe envelope should have a constantslope, i.e., equal separation between the turns of thespiral in Fig. 3. We note that the useful range shouldbe truncated at the high visibility end when the coher-ence function starts leveling off. At the other end, thetable is truncated when fringe visibility becomes toolow to provide an acceptably accurate phase measure-ment. For 8-bit resolution, table values beyond theseextremes are clipped at 0 and 255.

F. Focusing and Biasing

Focusing and biasing the interferometer into itsproper range of operation present special problems.The useful domain of the interferometer, as deter-mined by the lookup table, constitutes less than half ofthe coherence region where fringes are visible. Thatregion in turn is usually much narrower than the depthof field of the microscope objective. Commercial au-tofocus systems that use a geometrical optics sharp-ness criterion are inadequate for automatic setup ofthe interferometer.

A special interferometric video autofocus techniquehas been developed for this system interferometer.The technique uses the contrast of the interferencefringes as a focusing criterion. The algorithm employsFourier analysis to decode the fringe envelope and findits offset. That offset in turn is used to compute theerror signal applied to the focusing fine translatormechanism. Computing the focusing offset at severalpoints (at least three) allows determination of the sur-face tilt. Thus alignment of the system can be auto-mated.

VIll. Extra Interferometer Demonstration System

A. System Configuration

This interferometer concept has been demonstratedwith a system consisting of commercially availablecomponents. The optical front end is based on a com-mercial Michelson interference attachment mountedon a standard reflected light microscope (see Fig. 4).Illumination is provided by a quartz halogen lamp fedby a regulated dc power supply. The test object is laidon top of a tilt table, which in turn is mountd on a PZT,

PiezoelectricCrystal

Fig. 4. Interferometer system: the interferometer consists of acommercial microscope and an interferometer attachment. A tung-sten lamp with appropriate spectral filters is used for illumination.A piezoelectric transducer is used to shift the object one-fourth of a

fringe. The images are acquired via a CCD array camera.

hooked up to a programmable power supply. In thecurrent configuration, the output image is sensed witha high-resolution frame transfer CCD camera having604 horizontal and 485 vertical elements. The videodata are transferred to a personal computer via a com-mercial image acquisition board that digitizes andstores the video images. The personal computer pro-cesses the data and also controls the PZT power sup-ply. The acquisition and processing of the data re-quired to produce a 512 X 512 profile image is -35 s.

B. Experimental Procedure

Assuming a suitable illumination spectrum and amatching lookup table has already been established forthe test surface under consideration, the measurementprocess starts with the acquisition of the noise or straylight image. This is done by removing the object andinterference attachment, so that the microscope islooking into empty space. In practice, this measure-ment is only required with poorly reflecting, usuallynonmetallic, surfaces. It needs to be performed onlywhen either the optical configuration or the light levelis changed, usually when setting up for a new type ofsurface.

The next step is installing the interference attach-ment, taking care to focus properly the reference mir-ror and aligning it perpendicular to the optical axis.The reference image can now be digitized and stored.Like the noise image, this image can be used for all


Intensity(height)

Fringe Profile Surface Profile

I ( > ia xFig. 5. Fringes from a tilted plane mirror: The experimentallyobtained fringes from a tilted plane mirror object show decreasingcontrast across the image due to the finite coherence length of thesource. The superimposed line is the resultant profile as computed

by the system.

subsequent measurements using the same configura-tion. Now the test surface is brought into view, andthe interferometer is proposed focused and adjustedfor minimum tilt. The focusing is then detuned slightlyso that no interference fringes are visible but the objectis still within the geometric depth of focus. This al-lows the acquisition of the total intensity image.

The interferometer is now tuned back, and the focusis adjusted for proper biasing of the interference pat-tern. The purpose here is to have all the fringe orderswithin the object's region of interest lie inside theuseful range as determined by the lookup table. Thisprocess can be included in the autofocusing procedurediscussed above. The quadrature interference patternsare now acquired in rapid succession. Processing thedata to compute the surface profile can now be done.

C. Experimental Results

An initial indication of the system performance canbe seen from Fig. 5, which shows an interference imageof a tilted plane mirror object. The fringe contrast canbe seen to decrease across the object due to the limitedcoherence length of the source. Superimposed on thefringe image is a profile along a horizontal line in themiddle of the image. The profile was generated by thesystem using the algorithm described above withoutphase unwrapping or fringe tracking.

To calibrate the system, a commercial step heightstandard was measured. The standard consists of achrome covered silicon dioxide relief structure. Thestructure includes several patterns such as a single barand surface relief gratings of various pitches. Therelief of all the structures is 0.99 ym for the target used.

To demonstrate the profiling capability we showsome profiles of a magnetic tape head mock-up. Fig-ure 6 shows a schematic drawing of the tape head. Thefigure also shows a cross section of the head with anindication of how the tape lies when moving across thehead. The tape is drawn down into close contact withthe head in the regions of the active read/write ele-ments. The edges of the tape fall into the slots ateither end of the tape. This assures the correct posi-tion of the tape with respect to the active elements.

A

(a)

Tapev

A'- , e Ia 7A-I Profiled

Area

Ib)

Fig. 6. Magnetic tape head: (a) The magnetic tape head consists oftwo cylindrical components, a read head and a write head. The slotsalong the leading and trailing edges are aligned with the tracks on thetape. The slots along either end of the head define the tape positionon the head. (b) A cross section through the head shows how the

tape lies as it moves over the head.

Height

100 nm

450 um

(a)

(b)

Fig. 7. Profiles of worn tape head mock-up: (a) profile along thegap at a track; (b) profile at the slot along the edge showing the wearfrom the tape edge. The flat region is the outside edge of the head

behind the tape so there is no wear.

The regions of close contact and particularly the re-gions where the tape hangs over the edge of the outsideslots experience wear. Figure 7 shows profiles of theseareas on a glass mock-up of the head. The first profileis along the gap in the region of the track. This showshow the glass mock-up has been worn in this region.The second profile shows the outside edge region of thehead. The flat region is the outermost portion beyondthe tape; thus it experiences no wear. The taperedregion is the result of the edge of the tape wearing thehead. Note that with this interferometer, the absolute


I

amount of wear of the inner region with respect tounworn outer region can be determined even thoughthere is no continuous profile to trace between the tworegions. This would not be possible with conventionalphase measuring interferometers which have to rely onphase-unwrapping or fringe tracking.

Vil. Conclusions

We have demonstrated a variation on phase measur-ing microscopes that provides absolute phase measure-ment over a range of several wavelengths on a point-by-point basis. It relies on illumination with a limitedcoherence length and uses fringe contrast informationto essentially determine fringe order. The system iscapable of measuring surface profiles over a 512 X 512matrix. Currently the processing time required to dothis is 35 s. The output resolution of the currentsystem is limited by the 8-bit output. Thus the resolu-tion is the measurement range/256. The measure-ment range can be anywhere from one wavelength, asin conventional systems, up to -10 times that, depend-ing on application requirements.

The authors would like to thank G. Sincerbox for hissupport and many useful discussions and B. Lee for hiswork in developing the personal computer version ofthis system.

References1. W. H. Steel, Interferometry (Cambridge U. P., London, 1967).2. D. Malacara, Ed., Optical Shop Testing (Wiley, New York, 1978).3. L. N. Mertz, "Phase Estimation with Few Photons," Appl. Opt.

23, 1638 (1984).4. J. H. Bruning, J. E. Gallagher, D. P. Rosenfeld, A. D. White, D. J.

Brangaccio, and D. R. Herriott, "Digital Wavefront MeasuringInterferometer for Testing Optical Surfaces and Lenses," Appl.Opt. 13, 2693 (1974).

5. J. H. Bruning, "Fringe Scanning Interferometers," in Ref. 2.6. J. C. Wyant, C. L. Koliopolis, B. Bhushan, and 0. E. George, Am.

Soc. Lubr. Eng. Trans. 27, 101 (1984).7. K. Itoh, "Analysis of the Phase Unwrapping Algorithm," Appl.

Opt. 21, 2470 (1982).8. J. W. Goodman, Statistical Optics (Wiley-Interscience, New

York, 1985).

Meetings Calendar continued from page 4255

1987November

17 Lasers in Industry-What Can They Do For You, Sem.,Mississauga H. McAdie, Ontario Res. Foundation,Sheridan Park Res. Community, Mississauga, Ontar-io L5K 1B3

17-20 Reliability of Repairable Systems: Analysis & Applica-tions course, Los Angeles UCLA Ext., P.O. Box24901, Los Angeles, CA 90024

30-5 Dec. Materials Research Soc. Mtg., Boston MRS, 9800McKnight Rd., Ste. 327, Pittsburgh, PA 15237

December

1-3 Optical Information Systems Conf. & Exhibition, NewYork J. Emard, 11 Ferry Lane West, Westport, CT06880

9-11 Fiber Optic Communications course, Tempe Ctr. forProfessional Development, Coll. of Eng. & Appl. Sci.,Arizona State U., Tempe, AZ 85287

9-15 Optics & Glass Expo/China '87, Beijing China Promo-tion, Ltd., Rm. 1810 Shum Tak Centre, Office Tower,200 Connaught Rd., Central, Hong Kong

9-15 Fiber Optic Communications course, Tempe ArizonaState U., Elect. & Computer Eng. Dept., Ctr. forProfessional Dev., Tempe, AZ 85287

15-17 Solid State Lasers course, Williamsburg C. Keen, Sci-ence & Tech. Inst., 101 Research Drive, Hampton,Va. 23666

1988January

6-9 Electron Device Int. Mtg., Wash., DC M. Widerkehr,Courtesy Assocs. Inc., 655 15th St., N.W., Wash., DC20005

7-11 Fundamentals & Applications of Lasers course, Albu-querque Laser Inst. of Am., 5151 Monroe St., Tole-do, OH 43623

7-11 Lasers '87, Lake Tahoe Lasers '87, P.O. Box 245,McLean, VA 22101

7-11 Laser Beam Propagation & Interaction Effects course,Denver Eng. Tech. Inst., P.O. Box 8859, Waco, TX76714

7-11 NAVSTAR Global Positioning System: Operation, Im-plementation & Applications course, Los AngelesUCLA Ext., P.O. Box 24901, Los Angeles, CA 90024

4-5 Low Noise Amplifier Design course, Lake Buena VistaV. Amico, U. Central Florida, Orlando, FL 32816

4-5 Laser Principles Sem., Lake Buena Vista V. Amico, U.Central Florida, Orlando, FL 32816

5-16 Optical Science & Engineering course, TucsonSlater, P.O. Box 18667, Tucson, AZ 85731

P.

6-8 Infrared Detectors & Systems course, Lake Buena VistaV. Amico, U. Central Florida, Orlando, FL 32816

6-8 Fiber Optics Workshop & Laboratory, Lake Buena VistaV. Amico, U. Central Florida, Orlando, FL 32816

continued on page 4292


Extended unambiguous range interferometry

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