Sandwich, double-reference-wave,holographic, phase-shift interferometry

James D. Trolinger and Neal J. Brock

Phase-shift interferometry has provided a straightforward method for converting interferograms tophase maps. Unfortunately, some of the most powerful pulsed holographic interferometry techniquesare generally not compatible with phase-shift interferometry. One solution is to employ two referencewaves, one for each of the two object waves to be interfered, that can be phase stepped duringreconstruction. Practical aberration and alignment problems render this approach difficult. A simplemethod that employs a second hologram, sandwiched with the first, which produces the required tworeference waves during reconstruction, is presented. The process compensates for both chromatic andgeometric aberrations that otherwise render the phase-shift method unusable.Key words: Holographic interferometry, phase shift. r 1995 Optical Society of America

1. Introduction

A. Phase-Shift Holographic Interferometry with PulsedHolography—the Problem

The interferogram equation,

Ii 5 A 1 B cos1u 1 fi2, 112

can be solved by shifting the arbitrary phase term fithree ormore times to produce three linearly indepen-dent equations. Various algorithms have been devel-oped to improve the accuracy and usefulness of themethod,1 resulting in what is today a highly refinedmeasurement tool. Methods must be devised forshifting the phase term in an experiment. Phaseshifting in cw laser interferometry is easily imple-mented.Double-pulsed holographic interferometry is one of

the more difficult cases because both holograms aresuperimposed with little time between the two expo-sures. One way to achieve phase-shift interferom-etry with a double-exposed hologram is to record theholograms so that the two wave fronts can be recon-structed independently. For example, if the twowave fronts are recorded with two angularly spacedreference waves,2 fi can be shifted by shifting the

The authors are with MetroLaser, 18006 Skypark Circle, Suite108, Irvine, California 92714.Received 21 February 1995; revised manuscript received 25 May

1995.0003-6935@95@286354-07$06.00@0.

r 1995 Optical Society of America.

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phase of either reference wave during reconstruction.This requires high-speed switching from one refer-ence beam to the other during recording. This can beachieved, for example, by using Pockels cells to switchthe reference beam from one leg of the system toanother.3When two separate reference waves are employed,

the phase difference between the two reference wavesappears in the interferogram along with the phaseterms of interest in the object, losing some of thebenefits of holographic interferometry. Conse-quently, high-quality optics and precision alignmentmust then be used in the reference beams of therecording system. One way to resolve half of theproblem is to use the same optics during reconstruc-tion, but precise alignment remains critical. Also,this works only if the same wavelength is employedfor reconstruction.In resonant holographic interferometric spectros-

copy4–6 1RHIS2 the problem is compounded evenmore. In RHIS two waves of slightly different wave-length are passed simultaneously through the objectfield. One of the waves is tuned to a resonance of themedium, and the second is tuned slightly off-resonance. The two wave fronts experience the me-dium as well as the entire optical system almostidentically because the wavelengths are nearly thesame; however, the resonant wave is influencedstrongly by the resonant part of the medium. Thetwo waves are recorded holographically by mixingtheir respective reference waves with them at therecording. The two wave fronts are then recon-structed at a single wavelength and interfered. The

resulting interference pattern is due only to thepresence of the resonant component. All other opti-cal phase effects are subtracted from the hologram,resulting in a recording that is highly sensitive only tothe resonant material. Once again, relatively phaseshifting the two reconstructed waves requires record-ing with different reference waves.In RHIS the recording wavelength is sometimes in

the UV, although reconstruction must be done in thevisible or at least at a wavelength that can betransmitted through the hologram substrate.7 Re-constructing with the recording laser also can bedifficult with pulsed lasers because of alignment andstability. We found that the combinations of large-angle change and aberrations caused by the wave-length difference between recording and reconstruc-tion combined with the required optical quality andalignment requirement made two-reference-wave,phase-shift interferometry virtually impractical whenconventional methods are used. The resulting inter-ferograms contained many complex 1aberrational2 in-terference fringes, rendering them unusable. Amethod was needed to store and optically subtractsuch effects during the reconstruction process.The subfringe sensitivity of phase-shift interferom-

etry is extremely desirable because high sensitivity iscommonly a requirement in spectroscopy. In manyapplications of RHIS the resonant species underinvestigation will cause only a small fraction of awavelength shift in optical path length.The object of the study was to develop a holo-

graphic reconstruction technique that makes double-reference-wave holography and phase-shift interfer-ometry usable with RHIS even when a large wave-length difference exists between recording andreconstruction. The method employs holography tocompensate optically for the problems of reference-wave lens aberrations, misalignment, and chromaticaberrations.

B. Phase-Shift Sandwich Holographic Interferometry

The method developed in this study to achieve thedesired holographic correction for double-reference-wave holography employs sandwich holographic inter-ferometry.8,9 In sandwich holographic interferom-etry, two waves derived from two separate hologramsare mixed to produce the interferogram. The fringescan be shifted in phase as well as in number byadjusting the relative position of the two holograms.Consequently, infinite fringe and finite fringe interfero-grams with arbitrary phase can be produced at will.When the two holograms are sandwiched together,the interferogram can be phase shifted either byrotating the two plates together or by adjusting one ofthe two plates in its own plane. Sandwich holograminterferometry is a practical and straightforwardmethod to use and is easily compatible with therequirements of phase-shift interferometry. Conse-quently, we set out to adapt the method to double-reference-wave interferometry.

2. Summary

The result of this study is a technique that resolvesthe practical problems of both double-pulsed, phase-shift interferometry and double-reference-wave RHIS.The method uses phase-shift, sandwich, double-reference-wave holographic interferometry. Aholographic correcting plate is produced that containsthe necessary information to cancel out all the aberra-tions caused by differences in the optics used forrecording and reconstruction as well as the aberra-tions caused by wavelength changes between record-ing and reconstruction. An added 1and unexpected2benefit is the removal of tilt fringes in RHIS normallypresent because of recording at two wavelengths andreconstructing with a single wavelength.

3. Basic Principle

For double-reference-wave holographic interferom-etry, two data waves to be compared by interferenceare recorded on the same hologram with two indepen-dent reference waves and then reconstructed with thesame two reference waves. Ideally, the phase frontof the reconstructing reference waves must be identi-cal to the two reference waves used during recording.Any phase change in either of the reference waves1from recording to reconstruction2 that is not made inthe other appears as phase information in the inter-ferogram of the two reconstructed data waves. Suchchanges occur in practice because the optics in thereconstruction system are not identical to the record-ing system, the optical alignment is not identical inthe two systems, or large wavelength changes fromrecording to reconstruction result in aberrations.The principle used herein to avoid this problem

compensates for changes in the reference beams byproducing a hologram of the recording system refer-ence beams with a single, third reference wave. Thishologram can be used later to reconstruct referencebeams that are identical to those used in recordingthe two data waves. Because both reference wavesare recorded by use of a single third reference wave,they can both be reconstructed with a single referencewave. Then, any aberrations that are present in thisthird reference wave 1geometrical or chromatic2 arethe same for both reconstructed reference waves andtherefore do not affect the final interferogram of thereconstructed data waves. Changes in wavelengthfor the reconstruction process are automatically com-pensated by the holographic process. For example, ifa longer wavelength is used during reconstruction,the two reconstructed reference waves will emerge atwider angles from the hologram. The wider angle isprecisely what is needed to reconstruct two datawaves at a longer wavelength that are properlyaligned for interference 1from a hologram that con-tains two aligned data waves originally recorded at ashorter wavelength2.Sandwich holographic interferometry is a practical

method for implementing this principle. The datawave is recorded in one holographic plate, and thereference waves are recorded on a second holographic

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plate at precisely the same position as the datahologram. The two are then sandwiched togetherand used simultaneously to construct all the desiredwaves. A number of waves that are not neededappear in the process but are easily removed byspatial filtering.

4. Theory

A. Fundamental Equations

A hologram is formed by mixing a data wave and areference wave,

Udi 1 Uri, 122

where Udi is the ith data wave and Uri is the ithreference wave, and recording the intensity of thetwo,

Hi 5 1Udi 1 Uri21Udi 1 Uri2*

5 Udi2 1 Uri

2 1 Uri*Udi 1 Udi*Uri. 132

Modeling the hologram as a transmission gratingwith amplitude transmission proportional to the inten-sity of the exposure during recording, when thehologram is illuminated by a wave,Uri8, the transmit-ted complex amplitude is

Ut 5 Uri81o Hi2. 142

The hologram that is used to store and reconstructthe two reference waves can be described by

Hr 5 Ur32 1 Ur1

2 1 Ur1*Ur3 1 Ur3*Ur1,

1 Ur32 1 Ur2

2 1 Ur2*Ur3 1 Ur3*Ur2. 152

The various waves are identified in Fig. 1.When this hologram is played back with reference

wave Ur4, the two waves that are of interest to us are

Fig. 1. Definition of terms in the equations. The left-hand sideshows the recording of the two data-wave holograms. The topright shows a recording of the two reference waves used inrecording the first two waves. The bottom right shows thedouble-plate reconstruction of the original two waves.

6356 APPLIED OPTICS @ Vol. 34, No. 28 @ 1 October 1995

the fourth and the eighth terms in Hr, producing thereference waves that we require to reconstruct datawaves from the data hologram:

Ur18 5 Ur4Ur3*Ur1, 162

Ur28 5 Ur4Ur3*Ur2. 172

When these two waves strike the data hologram,they produce a series of waves, but the only ones thatinterest us are the two that emulate the original datawaves, Ud1 and Ud2. The hologram that is used tostore and reconstruct the two data waves can bedescribed as

Hd 5 Ur12 1 Ud1

2 1 Ud1*Ur1 1 Ur1*Ud1

1 Ur22 1 Ud2

2 1 *Ud2Ur2 1 Ur2*Ud2. 182

The waves reconstructed from Hd by referencewaves Ur18 and Ur2 are the final data waves that wewish to interfere. The desired waves are producedfrom the fourth and the eighth terms inHdmultipliedby the correct reference waves, Ur18 and Ur28. Theresulting waves are

Ud185 Ur4Ur3*Ur1Ur1*Ud1 5 Ur4Ur3*Ur12Ud1, 192

Ud285 Ur4Ur3*Ur2Ur2*Ud2 5 Ur4Ur3*Ur22Ud2. 1102

An interferogram formed bymixing these twowavescan be described as

I 5 2 Re1Ud18Ud282

5 2Ur12Ur2

2 Re1Ur4Ur3*Ud1Ur4*Ur3Ud2*2

5 2Ur12Ur2

2Ur32Ur4

2 Re1Ud1Ud2*2,

5 K Re1Ud1Ud2*2. 1112

Equation 1112 shows that this procedure cancels outall the phase errors in all the reference waves, leavingonly the interference between the two data waves.The process even corrects for errors otherwise causedby a wavelength difference between the recording andreconstruction waves. In ordinary RHIS where thetwo reference beams overlap, a tilt fringe caused bythe presence of the two wavelengths is unavoidable.This problem usually requires the two wavelengths tobe positioned close to each other. The method de-scribed here eliminates the tilt fringe problem so thatthe two wavelengths employed could be arbitrarily farapart.In our experiment Ud1 and Ud2 can be represented

as two waves traveling at angles a and b relative tothe normal of the recording. The waves have twodifferent wavelengths, l1 and l2, and carry phases, f1and f2, that represent the data.

Ud1 5 U1 exp i312px@l12sin1a2 1 f14, 1122

Ud2 5 U1 exp i312px@l22sin1b2 1 f24. 1132

Therefore

I 5 K cos512px23sin1a2@l1 2 sin1b2@l24 1 Df6. 1142

In our experiment a and b are nearly equal to zero.So

I 5 K cos1Df2. 1152

Therefore, regardless of reference-wave quality andthe wavelength selected, an infinite fringe interfero-gram containing only the desired phase data is pro-duced. This striking result allows us to concludethat the two-reference-wave hologram can be re-corded at two wavelengths with rather poor optics,then reconstructed at a different wavelength with asecond set of rather poor optics without sufferingaberration effects in the resulting interferogram!If no unforeseen practical problems exist with theresult, this can simplify an otherwise extremely com-plex type of interferometry. The experimental re-sults presented below confirm the conclusions of Eq.1152.

B. Phase Shifting

Phase shifting can be achieved by one of two methodsduring reconstruction:

112 Sliding one of the plates over the other.122 Separating the two holograms by a small dis-

tance 1in sandwich holography, the distance is theplate thickness2 and then rotating the plate pairtogether about an axis in the plane of the two plates.

Consider the phase shift caused by rotating thereconstruction angle by an amount De 1Fig. 22. Thenthe path lengths of the reference waves,Ur18 andUr28,change by different amounts before they reach Hd,producing a relative phase shift in Ud18 and Ud28.If the two reference waves emerge at equal angles, h,to the normal, the path-length difference 1betweenthe two2 changes approximately by amount Dp:

Dp 5 13t2De tan h@cos2 h. 1162

A2p phase shift occurs if

De 5 l cos2 h@13t2tan h. 1172

Fig. 2. Path-length difference change between the two referencewaves with a change in angle of reconstruction.

For typical cases such a phase shift is achieved witha De of ,1 mrad, when the adjustment is placed in agood range for typical rotating stages.Some of the practical considerations not clearly

shown in the above analysis are the following:

112 Reference wave Ur3 should not be along thesame axis as the data waves, because they would behard to remove from the data waves during theirreconstruction.

122 During reconstruction the fringe contrast canbe adjusted by rotating the hologram sandwich byangles large enough to change the relative diffractionefficiency for reconstructing the two reference waves.

132 Unlike ordinary sandwich holography, bothholograms can be bleached to improve the efficiency ofthe process because the first hologram produces thereference waves for the second.

5. Experiment

A simplified experiment was chosen to validate theprediction that the method eliminates geometricaland chromatic aberrations originating from the opti-cal recording and reconstruction system differences.Atwo-reference-beam, conventional, holographic inter-ferometer was set up as shown in Fig. 3. Both thedata hologram and the corrector holographic opticalelement 1CHOE2 were recorded with this optical sys-tem by use of a He–Ne laser. Optical alignmenterrors were intentionallymade to increase the aberra-tions in the interferometer. 1The lenses were tilted.2The data hologram was recorded in two steps with

an ordinary glass plate as a test object. In the firststep the object is present and one of the referencebeams is blocked. In the second step the object isremoved and the beam block is positioned to block theopposite reference beam. This method will ulti-mately produce a holographic interferogram, reveal-ing the optical quality of the plate. The CHOEhologram is recorded with the same two-step proce-dure except that the glass plate is never inserted intothe beam.We then sandwiched together the data and CHOE

holograms during reconstruction, utilizing a specialdouble-plate holder that was set up in a completelydifferent optical system. A HeCd laser, operating ata wavelength of 442 nm, was used to illuminate theholograms during reconstruction. The HeCd laserprovided a much shorter wavelength for reconstruc-tion than was used during recording to test the abilityof the CHOE to correct for the resulting chromaticaberration. The reconstruction system configura-tion is shown in Fig. 4.

6. Alignment and Phase Shifting in SandwichHolographic Interferometry

To perform sandwich holographic interferometry,two holograms 1test and reference2 must be sand-wiched together precisely and one of the hologramsmust be rotated relative to the other to obtain theoptimum fringe spacing. To achieve this, we devel-

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Fig. 3. Configuration for making corrector and data holograms.

oped a plate holder that accommodated two suchholograms and allowed the precise translational androtational movements required for the resulting inter-ferogram to be optimized. The plate holder has beenused in various types of interferometric recordings.10A front view of the double-plate holder is shown in

Fig. 5. The double-plate holder consists of two plates,each of which holds a 4 in.3 5 in. 110.16 cm 3 12.7 cm2holographic glass plate. Three clamps are designedin the front holder to press the two plates togetherlightly. The plate holder is designed to positioneither vertical or horizontal plates and to allow pre-cise positioning of the sandwiched holographic plates.The functions of the plate holder are as follows:

c To adjust the relative in-plane position of thetwo holographic plates. The X-X and Y-Y linearstage mounted on the front holder provides a spatialtranslation of the front plate with respect to the backplate. The back-plate holder ismounted on a goniom-eter 1Z-R2 that provides rotational adjustment withrespect to the front holder at the plate center 12.

Fig. 4. Reconstruction of the data hologram with the CHOE.

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The X-X and Y-Y translation and Z-R in-plane rota-tion provide the adjustment necessary to position thetwo plates together as they are constructed.

c To adjust finite fringe spacing. Another func-tion of the goniometer 1Z-R2 is to rotate one plate withrespect to the other plate in the X-Y plane to introducethe angular difference of the two holograms andtherefore modulate the finite fringe spacing of thereconstructed interferogram. Two plates will be offfrom the positions when they are constructed; how-ever, the rotation in Z-R is very small 1,1 arcmin orless in most cases2 to provide enough fringe spacingmodulation.

c To provide phase-shifting capability. Rotatingthe sandwiched pair about a vertical 1Y 2 axis by therotation state continuously changes the optical-path-

Fig. 5. Frontal view of the double-plate holder.

length difference between the two holograms. High-resolution rotation within a fraction of 1 deg for a 2pphase change is essential to the success of phase-shifting interferometry 1PSI2. Within a 2p phasechange, four or five interferograms are registered by aCCD camera at an equal rotation angle betweeninterferograms. The mechanical stability of thedouble-plate holder and optical setup for reconstruc-tion should be stable enough to maintain fringestability. Because the reconstruction geometry is acommon-path interferometer, this requirement is con-siderably relaxed. The mechanical actuator on therotation stage provides 0.3-arcsec accuracy.

7. Hologram Bleach Processing

The reconstructed beams from the CHOE are used toilluminate the data hologram. It is important thatthe diffraction efficiency of each hologram be suffi-ciently high so that the interferogram is bright enoughto be photographed. A conventional hologram-pro-cessing procedure produces holograms with a diffrac-tion efficiency of,6%. If this procedure were used toprocess both the data and the CHOE holograms, thetotal diffraction efficiency would be only 0.36%.Therefore a bleaching procedure was used to increasethe total diffraction efficiency. This procedure isdescribed below:

112 Develop the exposed plate in D-19 developeruntil an optical density of at least three is achieved.

122 Rinse with distilled or deionized water for 30 s.132 Soak in a rehalogenating bleach solution until

the plate becomes transparent again.142 Rinse with distilled or deionized water for 5

min.152 Dry for 10 min.

Fig. 6. Four phase-shifted interferograms of the test plate.

Total diffraction efficiencies of greater than 10%were achieved with this method.

8. Results and Conclusions

The interferograms that were made with the holo-graphic correction technique show none of the re-sidual aberrations. The double-plate holder simpli-fies adjustment of holograms for either finite orinfinite fringe conditions. The incremental phaseshifts required for data reduction by PSI were pro-duced by use of the rotation adjustment of the plateholder. Four interferograms of the glass test plate,each shifted by p@4, are shown in Fig. 6. Thestraightness of the fringes not passing through theglass plate indicate that the systematic aberrationshave been removed. The holographic correctionmethod using the CHOE appears to eliminate all theproblems encountered when different optical systemsand different lasers between hologram constructionand reconstruction are used. This method clears theway for incorporation of PSI to RHIS at any wave-length and with virtually any optical system.The analysis has predicted and the experiments

have confirmed a way to correct critical problems inthe implementation of two-color holographic interfer-ometry, especially when different wavelengths arerequired for reconstruction. Therefore hologramsmade at any two wavelengths with two separatereference waves can be reconstructed at other wave-lengths, nomatter how different, to produce interfero-grams that are chromatically corrected. This makespossible the use of highly sensitive methods such asPSI with pulsed holography.

The authors acknowledge the helpful suggestions oftheir colleagues, Peter DeBarber, Michael Brown,David Rosenthal, and Fred Unterseher. The re-search was supported by the U.S. Air Force WrightLaboratories and the U.S. Army Research Office.

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