heterodyne fizeau interferometer for testing flat surfaces

6
Heterodyne Fizeau interferometer for testing flat surfaces T. H. Barnes A heterodyne Fizeau interferometer, which uses a rotating radial grating to achieve the required optical frequency shift, is described. An analysis of the effects of grating ruling errors shows that they may be nearly eliminated by averaging the interferometer phase readings over integral numbers of grating revolutions. Experimental tests indicate that the interferometer is capable of measuring with a reproducibility of X/200 (X = 632.8 nm), limited by temperature effects. 1. Introduction The Fizeau interferometer has for many years been one of the most popular instruments for comparing optical flats. While simple versions are to be found in almost every optical workshop, more complex systems have been used to measure optical surfaces to extreme- ly high accuracy'" 2 and to derive standards of optical flatness based on liquid surfaces or the intercompari- son of three optical flats. 3 For a detailed review of the use and applications of this type of interferometer, see the article by Murty. 4 Modern developments include the application of lasers, TV systems, and image pro- cessing techniques to Fizeau-type interferometers, 5 and several different types of instrument using these techniques are commercially available. 6 - 8 The determination of surface profile using the Fi- zeau interferometer was conventionally achieved by measurement of the positions of the fringes of equal thickness produced by the instrument. Interpolation procedures were used to obtain fringe phase as a func- tion of position and hence the surface profile. In more modern instruments the phase map is obtained by digitally processing a TV image of the fringes. Heterodyne techniques have also been used in inter- ferometers designed to measure the surface profile to high accuracy (X/100). Crane 9 described a heterodyne Twyman-Green-type interferometer incorporating a frequency shifter, which used polarization effects, in one arm of the interferometer. Sommargren' 0 de- scribed several similar systems but with the frequency T. H. Barnes is with DSIR Physics and Engineering Laboratory, Private Bag, Lower Hutt, New Zealand. Received 30 January 1987. 0003-6935/87/142804-06$02.00/0. ©1987 Optical Society of America. shifter outside the interferometer. Other systems us- ing Bragg cells to obtain the required frequency shift have also been constructed. 11 ' 12 Dandliker et al.' 3 de- scribed a double-exposure holographic heterodyne in- terferometer for measuring the deformation of diffuse- ly reflecting surfaces with a resolution better than X/ 1000. However, the application of heterodyne techniques to Fizeau-type interferometers presents certain prob- lems. The two interfering beams in the Fizeau travel very nearly the same path with the exception of the region between the two test plates. At first sight, any optical frequency shifter used to achieve heterodyne operation should, therefore, be placed in this region, but the practical difficulties of manufacturing and in- stalling a suitable device make this approach virtually impossible. The fact that the instrument is very near- ly a common-path interferometer makes it very stable and easy to use but at the same time very difficult to convert to heterodyne operation. This paper describes a heterodyne Fizeau-type in- terferometer where the two frequency-shifted beams needed for heterodyne operation are generated outside the interferometer by a rotating radial grating. This approach gives much greater flexibility in the optical modulator arrangement, while maintaining the advan- tages of simplicity and stability which make the Fizeau such a useful instrument. The instrument is designed and operated so that it is insensitive to grating errors. Despite grating phase errors of up to 60°, reproducibil- ity tests indicate that measurements made with the interferometer have a standard deviation of <X/200(X = 632.8 nm). 11. Interferometer Optical System The optical system of the heterodyne Fizeau inter- ferometer is shown in Fig. 1. The light source is a 1.5- mW He-Ne laser with a 1-nm diam output beam. Light from the laser passes through a 1100-line radial grating of 45-mm diameter, and the resultant diffract- 2804 APPLIED OPTICS / Vol. 26, No. 14 / 15 July 1987

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Heterodyne Fizeau interferometer for testing flat surfaces

T. H. Barnes

A heterodyne Fizeau interferometer, which uses a rotating radial grating to achieve the required opticalfrequency shift, is described. An analysis of the effects of grating ruling errors shows that they may be nearlyeliminated by averaging the interferometer phase readings over integral numbers of grating revolutions.Experimental tests indicate that the interferometer is capable of measuring with a reproducibility of X/200 (X= 632.8 nm), limited by temperature effects.

1. Introduction

The Fizeau interferometer has for many years beenone of the most popular instruments for comparingoptical flats. While simple versions are to be found inalmost every optical workshop, more complex systemshave been used to measure optical surfaces to extreme-ly high accuracy'"2 and to derive standards of opticalflatness based on liquid surfaces or the intercompari-son of three optical flats.3 For a detailed review of theuse and applications of this type of interferometer, seethe article by Murty.4 Modern developments includethe application of lasers, TV systems, and image pro-cessing techniques to Fizeau-type interferometers,5and several different types of instrument using thesetechniques are commercially available.6 -8

The determination of surface profile using the Fi-zeau interferometer was conventionally achieved bymeasurement of the positions of the fringes of equalthickness produced by the instrument. Interpolationprocedures were used to obtain fringe phase as a func-tion of position and hence the surface profile. In moremodern instruments the phase map is obtained bydigitally processing a TV image of the fringes.

Heterodyne techniques have also been used in inter-ferometers designed to measure the surface profile tohigh accuracy (X/100). Crane9 described a heterodyneTwyman-Green-type interferometer incorporating afrequency shifter, which used polarization effects, inone arm of the interferometer. Sommargren'0 de-scribed several similar systems but with the frequency

T. H. Barnes is with DSIR Physics and Engineering Laboratory,Private Bag, Lower Hutt, New Zealand.

Received 30 January 1987.0003-6935/87/142804-06$02.00/0.© 1987 Optical Society of America.

shifter outside the interferometer. Other systems us-ing Bragg cells to obtain the required frequency shifthave also been constructed.1 1' 12 Dandliker et al.'3 de-scribed a double-exposure holographic heterodyne in-terferometer for measuring the deformation of diffuse-ly reflecting surfaces with a resolution better than X/1000.

However, the application of heterodyne techniquesto Fizeau-type interferometers presents certain prob-lems. The two interfering beams in the Fizeau travelvery nearly the same path with the exception of theregion between the two test plates. At first sight, anyoptical frequency shifter used to achieve heterodyneoperation should, therefore, be placed in this region,but the practical difficulties of manufacturing and in-stalling a suitable device make this approach virtuallyimpossible. The fact that the instrument is very near-ly a common-path interferometer makes it very stableand easy to use but at the same time very difficult toconvert to heterodyne operation.

This paper describes a heterodyne Fizeau-type in-terferometer where the two frequency-shifted beamsneeded for heterodyne operation are generated outsidethe interferometer by a rotating radial grating. Thisapproach gives much greater flexibility in the opticalmodulator arrangement, while maintaining the advan-tages of simplicity and stability which make the Fizeausuch a useful instrument. The instrument is designedand operated so that it is insensitive to grating errors.Despite grating phase errors of up to 60°, reproducibil-ity tests indicate that measurements made with theinterferometer have a standard deviation of <X/200 (X= 632.8 nm).

11. Interferometer Optical System

The optical system of the heterodyne Fizeau inter-ferometer is shown in Fig. 1. The light source is a 1.5-mW He-Ne laser with a 1-nm diam output beam.Light from the laser passes through a 1100-line radialgrating of 45-mm diameter, and the resultant diffract-

2804 APPLIED OPTICS / Vol. 26, No. 14 / 15 July 1987

L3

Fig. 1. Optical system of the interferometer.

ed beams are focused in the plane of spatial filter F1 bylens L1. Filter Fl blocks all but the +1 and-1 orders,these passing unimpeded into the interferometer.When the grating rotates, a frequency difference of 2f(where f is the line-passing frequency of the grating) isintroduced between the 2 orders.'4 This light thenpasses through the beam splitter B and is collected bylens L2. This lens is an f/No. matching lens whichexpands the two beams to fill the collimator lens L3(750-mm focal length, operating at f/15). L3 colli-mates the light into two (nearly coaxial) 50-mm diambeams, which then pass on to the test flats T1 and T2which are separated by <1 mm. It is not necessary forthese flats to be aluminized. The lens combinationL1, L2, L3 is arranged so that it images the plane of thegrating into the space between Ti and T2 to minimizeunwanted shearing effects.

Light reflected from the test surfaces travels backthrough the same system until it meets the beam split-ter B. Here a portion is reflected off to the secondspatial filter F2. F2 is placed in a focal plane of thesystem, i.e., F1 and F2 occupy the same relative posi-tion as viewed through the beam splitter from lens L2.The test surfaces are adjusted so that, in the plane ofF2, the focus of the -1 order reflected from T1 issuperimposed on the focus of the +1 order reflectedfrom T2, as shown in Fig. 2. Spatial filter F2 has asingle hole which selects the light from these superim-posed orders but at the same time is big enough toavoid unwanted diffraction effects. The rear surfacesof T1 and T2 must be at a slight wedge angle to thefront surfaces so that unwanted rear surface reflec-tions may also be filtered out by F2. The filtered lightcarries on through lenses L4 and L5 and finally reachesthe detector plane D containing detectors D1 and D2.D1 is fixed, while D2 is mounted on a stage driven by astepper motor which enables it to be scanned over theoutput plane. Lenses L4 and L5 in conjunction withlens L2 image the region between the flats T1 and T2into the detector plane, again to minimize unwantedshear effects.

With the grating stationary, an image of the test flatscrossed by fringes is formed in the detector plane. Thefringe spacing may be changed by fine adjustment ofthe test flats. When the grating rotates, the fringesmove across the field of view, the fringe-passing fre-quency being twice the grating line-passing frequency.

ORDERS REFLECTED BY T2

+1 -1

I I

SPATIAL FILTER HOLE

Fig. 2. Superimposition of orders in the plane of the spatial filter F2.

Fig. 3. Block diagram of the interferometer data collection system.

The fringe phase map may be obtained from the elec-trical phase difference measured between fixed andmoving detectors as a function of the position of themoving detector. The mapping resolution of the in-terferometer, as determined by detector size, is 1 mm.

The interferometer is constructed on a vibrationisolated optical table and is insulated and shieldedfrom air currents by a polystyrene cover with an alumi-num liner.

111. Data Collection and Processing System

Figure 3 is a block diagram of the electronic datacollection and processing system used with the inter-ferometer. The electrical signals from detectors D1and D2 are amplified and filtered by matched band-pass filters (Q = 3) to give clean signals for input to thephasemeter. The grating rotates at 6 rpm giving afringe-passing frequency of 220 Hz. The phase differ-ence between the filtered signals is then measured by aKrohn-Hite 6500A phasemeter, whose analog outputis fed to a DEC ADV1lA A-D converter controlled by aHeath LSI 11 computer. The phase resolution of thephasemeter-A-D converter combination is better than0.10.

The bandpass-filtered signal from the fixed detectoris also used to generate pulses which interrupt thecomputer each time a fringe crosses the detector. An

15 July 1987 / Vol. 26, No. 14 / APPLIED OPTICS 2805

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interrupt service routine, written in machine code,counts these pulses and keeps track of the position ofthe rotating grating. Every 7.2° of rotation the com-puter triggers the A-D converter to log the phasemeteroutput and keeps a continuously updated table ofphasemeter reading as a function of angle of rotation ofthe grating. The computer also generates stepper mo-tor control pulses to control the position of the movingdetector.

The procedure used to collect the data is as follows.The operator first enters into the computer the diame-ter over which measurements are to be made and thedistance between measurement points. The movingdetector is then set to the center of the measurementarea, and control is handed to the computer whichscans the detector over the measurement area, loggingthe data as follows:

On arrival at each measurement point, the stage isstopped, and the computer (which keeps track of thegrating position via the interrupt routine) waits untilthe grating next passes through the zero degrees orstart position so that data logging begins at exactly thesame grating rotational position for each measure-ment. The computer then accepts fifty phasemeterreadings logged at 7.2° intervals during the next com-plete grating rotation. These readings are examinedfor any phase ambiguities (phase jumps over the 0-360° boundary), corrected if necessary, and their aver-age is stored as the electrical phase reading for thatmeasurement point.

When data have been logged for all measurementpoints, the computer processes them to produce a pro-file plot. First, 360° phase ambiguities from point topoint are removed by comparing the phase reading ateach point with a value obtained by interpolation fromadjacent points. The readings in the resultant unam-biguous phase map are then converted to path differ-ence and a least-squares procedure used to determinethe best-fit plane. This plane is then subtracted fromthe actual readings to remove any residual displace-ment and tilt that may be present. The surface profileis then plotted on a VT125 graphics terminal, and ahard copy is produced if required.

IV. Effects of Grating Errors

The grating used in the interferometer was manu-factured by first drawing it on paper using a KantoDenki Corp. FPL 2000 digital plotter, then photore-ducing it and etching into chrome-on-glass using litho-graphic techniques. Grating phase errors are quitelarge (up to 600 or one-sixth of the spacing of the lines)and cause fluctuations in the phase difference mea-sured between the detectors, which can give rise tosignificant errors in the measurements made by theinterferometer. An analysis of the effects of gratingruling errors is given below.

The grating and detector planes are conjugate viathe lenses and reflections in the interferometer system.An image of the grating, spatially filtered by the aper-tures at F1 and F2, and formed from light diffractedinto the +1 and - grating orders is formed in the

detector plane. Thus at any point in the detectorplane the two beams of light forming the grating imagemay be written as

cos(wt + PI + g) (1)(light from +1 grating order),

cos(Wt + - gI) (2)(light from -1 grating order),

where the amplitudes are assumed to be the same andwhere 'kID45Ib are phase shifts introduced by the inter-ferometer optics, including errors in the test flats Tiand T2, and q5g is the phase shift introduced on diffrac-tion at the grating (grating phase).

Now the grating phase term may be written as

og = 2rN + E(O), (3)

where N is the number of grating lines,0 is the rotation angle of the grating,

E(O) represents ruling errors which repeat foreach rotation of the grating.

Provided that E(O) is averaged over integral numbersof grating revolutions, its fluctuations average to zero.When the grating rotates 0 may be written as At + a,where fi is the grating angular velocity, t is time, and ais a constant angle representing the position of themeasurement point in the output plane (or its conju-gate in the grating plane). The detector signal may bederived by substituting for 0 in Eq. (3) and dig in Eqs.(1) and (2) followed by the addition of Eqs. (1) and (2)and squaring of the resultant. The ac component of thedetector signal may then be written as

cos[4rN(Ot + a) + E(#t + a) + d)-b . (4)

Now the term Oja - 0Ib represents errors in the ancil-lary optical components of the interferometer plusphase changes caused by differences in the surfaceprofiles of the test flats T1 and T2. Let this term beS(x,y), where x and y are coordinates in the outputplane. The phase difference between signals from thetwo detectors in the output plane may, therefore, bewritten as

4irN(al - a2 ) + [S(x,y) - S(X2 ,Y2 )]

+ [E,(#t + a,) - E2(flt + a2)], (5)

where the suffixes 1 and 2 are associated with thepositions of the two detectors.

The first term in this expression represents a patternof fringes whose mean spacing may be altered by suit-able adjustment of the test plates. When the testplates are adjusted for perfect superimposition of theorders as shown in Fig. 2, this term reduces to zero.The second term represents the profile of the air spacebetween the test surfaces plus any residual interferom-eter errors.

The final term represents phase fluctuations causedby grating ruling inaccuracies. The nature of the grat-ing is such that this fluctuation function repeats foreach grating rotation, and, therefore, averaging themeasured phase difference over integral numbers ofgrating rotations reduces the effect of grating errors on

2806 APPLIED OPTICS / Vol. 26, No. 14 / 15 July 1987

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ANGLE OF ROTATION OF GRATING (DEGREES)

Fig. 4. Phase fluctuation as a function of grating rotation anglebetween two fixed detectors in the output plane.

Table 1. Average Phase Readings for Successive Grating Rotations

Rotation no. Average phase reading

1 330.592 330.533 330.3754 330.43

the interferometer output to zero (assuming that thegrating rotates in a reproducible fashion). If the grat-ing lines are not straight, this averaging may give rise toa constant term which could appear as a systematicprofile error. This may be incorporated into the termS(xl,yl) - S(X2,y2).

Figure 4 shows the phase fluctuations measured be-tween two fixed detectors in the output plane as afunction of grating rotation angle. Data were loggedover four grating revolutions. The cyclic nature of thefluctuations introduced by grating ruling errors canclearly be seen, and for the particular positions of thedetectors for this run the phase fluctuated over a rangeof 240. Table I gives the averaged phase values ob-tained for each of the four revolutions and shows that,despite large grating-induced phase fluctuations, aver-aging over integral grating revolutions gave a phaserepeatability from revolution to revolution of betterthan 0.30

V. Instrument Reproducibility

The reproducibility of the complete interferometersystem was checked by measuring the same two testsurfaces 5 times in succession. For this experiment,the test flats were two uncoated Fabry-Perot platesmade from Schott BaK50 glass. Each had a diameterof 50 mm, a thickness of 10 mm, and a wedge anglebetween first and second surfaces of 12 min of arc.Measurements were taken over the central 40 mm ofthe flats at a spacing of 2 mm. Each measurement runtook -10 min. The temperature of the laboratory(outside the insulated interferometer enclosure) var-ied by <0.10C during the experiment.

-1/20

I 1/200 WAVELENGTH

DISTANCE ACROSS SURFACE

Fig. 5. Reproducibility of the interferometer.

The results are plotted in Fig. 5, which shows theaverage of the five measured profiles together witherror bars representing the standard deviation of thevalue at each measurement point. These standarddeviations are all <X/400.

VI. Determination of Interferometer Errors

The term S(xyl) - S(x2,y2) in Eq. (5) representsthe profile of the airspace between the test flats plusany residual interferometer errors. Provided that theinterferometer is only used for the comparison of flats,direct determination of the residual errors is not neces-sary since the flats may be mounted in turn in theposition of T1, and the comparison is made by sub-tracting the profiles so obtained-the interferometererrors substracting out. However, if the interferome-ter is to be used to establish an absolute flatness stan-dard based on the intercomparison of three flats, de-termination and subtraction of the interferometererrors are necessary before profiles measured by theinterferometer are processed to produce the standard.

To measure these errors, both test flats are mountedon a translation stage which allows them to be movedsideways across the field of view of the instrument.Two sets of measurements are made, the flats beingtranslated by a known distance in between. Subtrac-tion of the measurement sets gives a shear functionfrom which the profile of the airspace between the flatsmay be calculated, independent of residual interfer-ometer errors, by methods used in shearing interfer-ometry, (see, for example, Rimmer15. Knowing this,the interferometer error function may also be calculat-ed.

This rather indirect method of obtaining the inter-ferometer error function is somewhat unsatisfactorysince noise in the interferometer during the two mea-surement runs can accumulate in the calculation of theprofile to appear as quite large errors in the interfer-ometer calibration. A far more satisfactory method ofcalculating the interferometer is to use a liquid surface,but this proved very difficult in the author's laboratorybecause of continuous slow changes in the tilt of thefloor.

15 July 1987 / Vol. 26, No. 14 / APPLIED OPTICS 2807

\w # l

+1/20

40-11

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N

11

8

Ed

EI

2;

-1/20 -

�i

8N

�s

111.C.I

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Fig. 6. Interferometer error function.

I 1/200 WAVELENGTH

' -_ - - 40mm -

DISTANCE ACROSS SURFACE-1/20 1

Fig. 7. Profile of the airgap between the test flats corrected forinterferometer errors.

The reproducibility of the determination of interfer-ometer error function was investigated by repeatingthe measurement sequence above 5 times. For thesetests data were logged over the central 40 mm of theflats at 2-mm spacing, and the shear was 2 mm. Figure6 shows the error function obtained, the error barsrepresenting the standard deviation of each measure-ment point. The standard deviations are all <X/200.Figure 7 shows the profile obtained, corrected for re-sidual errors.

VIl. Discussion

The interferometer described here is capable ofmeasuring an optical surface profile with a repeatabili-ty and resolution of better than X/200. At this level, amajor limitation is the thermal stability of the compo-nents and their mountings. This problem is com-pounded by the rather long time to complete a mea-surement run (typically 10 min). Since the systemscans the measurement points in sequence, slow driftsin the orientation of the test flats or (to a lesser extent)other ancillary optical components cause errors in themeasured profiles. The performance of the interfer-ometer might be significantly improved if the mea-surement time could be reduced.

Use of a slowly rotating grating as an optical fre-quency shifter is one main reason for the slowness ofthe instrument. An easy way to improve matters is toturn the grating faster, but there is a limit to thegrating speed set by the speed of the computer. Inaddition to this, the method by which the system ismade immune to grating errors demands that the grat-ing rotates at least one complete revolution at eachmeasurement point, and this further slows the loggingprocedure. The immunity to grating errors meansthat the grating may be made relatively easily andcheaply, but the penalty for this is a long measurementtime.

These problems could easily be overcome by using aBragg cell as the frequency shifter, provided that thepath lengths through it are sufficiently stable. Unfor-tunately such a device was not available in the author'slaboratory, but the experience of others (see, for exam-ple, Massie et al.1

2) indicates that a heterodyne Fizeau

interferometer with this method of frequency shiftingcould be a very useful and convenient device.

Vill. Summary

A Fizeau interferometer, modified to allow hetero-dyne fringe phase measurement, has been described.The two frequency-shifted beams required are gener-ated outside the interferometer by a rotating radialgrating. Errors in the grating give rise to fluctuationsin the measured fringe phase and normally introduceerrors into surface profile measurements made withthe instrument. However, because of the cyclic natureof the grating errors, averaging each phase measure-ment over an integral number of grating revolutionsmakes the interferometer essentially immune to grat-ing errors.

A shearing technique was used to determine theresidual systematic errors introduced by the auxiliaryoptical components of the interferometer. This is im-portant if the instrument is to be used for establishinga flatness standard based on the intercomparison ofthree flats.

Experimental tests indicate that simple compari-sons may be made with the interferometer with stan-dard deviations of <X/400, and the interferometer er-ror function may be determined with standarddeviations of <X/200.

The author wishes to acknowledge the help of M. K.Andrews and B. Moore in the manufacture of the grat-ing, D. Cochrane for supplying the optical flats used forthe tests, and C. M. Sutton and R. B. Hurst for helpwith the manuscript.

References1. G. D. Dew, "Optical Flatness Measurement-the Construction

and Use of the Fizeau Interferometer," NPL Report on OpticalMetrology 1 (Apr. 1967).

2. R. Bunnagel, H.-A. Oehring, and K. Steiner "Fizeau Interferom-eter for Measuring the Flatness of Optical Surfaces," Appl. Opt.7, 331 (1968).

3. G. D. Dew, "The Measurement of Optical Flatness," J. Sci.Instrum. 43, 409 (1966).

2808 APPLIED OPTICS / Vol. 26, No. 14 / 15 July 1987

4. M.V.R.K. Murty, "Newton, Fizeau, and Haidinger Interferome-ters," in Optical Shop Testing, D. Malacara, Ed. (Wiley, NewYork, 1978).

5. W. H. Augustyn, A. H. Rosenfeld, and C. A. Zanoni "An Auto-matic Interference Pattern Processor with Interactive Capabili-ty," Proc. Soc. Photo-Opt. Instrum. Eng. 153, 146 (1978).

6. Zygo Interferometer, Zygo Corp., Middlefield, CT.7. Specac FOTI-100 Interferometer, Analytical Accessories Ltd.

Orpington, Kent, England.8. Wyko IR3-10 Interferometer, Wyko Corp., 1955 East Sixth St.,

Tucson, AZ 58119.9. R. Crane, "Interference Phase Measurement," Appl. Opt. 8, 538

(1969).10. G. E. Sommargren "Up-down Frequency Shifter for Optical

Heterodyne Interferometry," J. Opt. Soc. Am. 65, 960 (1975).11. F. M. Mottier, "Microprocessor-Based Automatic Heterodyne

Interferometer," Proc. Soc. Photo-Opt. Instrum. Eng. 153, 146(1978).

12. N. A. Massie, R. D. Nelson, and S. Holly, "High-PerformanceReal-Time Heterodyne Interferometry," Appl. Opt. 18, 1797(1979).

13. R. Dandliker, B. Ineichen, and F. M. Mottier, "High ResolutionHologram Interferometry by Electronic Phase Measurement,"Opt. Commun. 9, 412 (1973).

14. W. H. Stevenson, "Optical Frequency Shifting by Means of aRotating Diffraction Grating," Appl. Opt. 9, 649 (1970).

15. M. P. Rimmer, "Method for Evaluating Lateral Shear Interfero-grams," Appl. Opt. 13, 623 (1974).

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15 July 1987 / Vol. 26, No. 14 / APPLIED OPTICS 2809