Fourier-transform evaluation of phasedata in spatially phase-biased TV holograms

Henrik O. Saldner, Nils-Erik Molin, and Karl A. Stetson

TV holograms for spatial phase stepping are formed with a small angular offset between the object andthe reference beams to give a spatial frequency bias to the pattern recorded by the TV camera. It iscommon to set the bias so that there is a 90° or 120° phase shift between adjacent pixels and to use theirradiance of three or more adjacent pixels to evaluate the phase of the interference. We report theFourier-transform evaluation of such recordings to obtain their phase data. We also demonstrate thedirect calculation of the phase difference between successive recordings without intermediate calcula-tion of the random phase of each hologram. This technique is proposed as an approach to pulsed TVholography. r 1996 Optical Society of America

1. Introduction

In recent years a method for interferogram analysiscalled spatial phase shifting has been developed,1–3which allows analysis of the phase function of asingle interferogram, and it has become an attrac-tive approach to pulsed-laser holography. Thismethod appears to have originated with Mertz4 andinvolves introducing a spatial frequency carrier tothe fringe pattern so that neighboring pixels haveapproximately 120° or 90° phase difference. Irradi-ance values for sets of three pixels are recorded, andinterference phase values are calculated from themin themanner of a temporally phase-shifted interfero-gram. This method presumes there is little varia-tion in the amplitudes of the interfering waves overany set of 3 pixels, and although this is a goodapproximation for interferograms of aerodynamicphenomena,1 it is not true in general for thoseinvolving diffusely reflecting objects.3An alternative method has also become popular

that uses Fourier transforms5 and appears to haveoriginated with Takeda et al.6 In this method aspatial offset is introduced into the interferencepattern; it is digitized into pixels, and a Fouriertransform is performed of the pixel pattern. If the

H. O. Saldner and N.-E. Molin are with the Division ofExperimental Mechanic’s, Lulea University of Technology, LuleaS-97187, Sweden. K. A. Stetson is with Karl Stetson Associates,2060 South Street, Coventry, Connecticut 06238.Received 24 May 1995; revised manuscript received 8 Septem-

ber 1995.0003-6935@96@020332-05$06.00@0r 1996 Optical Society of America

332 APPLIED OPTICS @ Vol. 35, No. 2 @ 10 January 1996

spatial frequency content of the amplitude varia-tions of the interfering waves is less than the spatialoffset, then the Fourier spectrum will exhibit threedistinct parts. One of the outer parts can be win-dowed off from the rest, shifted to zero frequency,and inverse transformed to yield a complex functionwhose phase angle is the interference phase functionof the original interferogram. This phase can beobtained by taking the arctangent of the ratio of theimaginary to real parts of the complex function.Fourier-transform methods have also been appliedto holographic interferometry and shearography bymeans of direct digital reconstruction of holograms.7The purpose of this paper is to show that the

Fourier-transform method can be used to evaluatespatially phase-shifted interferograms of diffuselyreflecting objects in what is called TV holography orelectronic speckle pattern interferometry. This pro-cess has antecedents in the temporal frequencyfiltering of TV signals for analysis of interferograms8and for holographic interferogramdisplay.9 In theseearlier works, the individual TV scan signals weresubjected to electronic processing, whereas in thiswork the TV images are digitized and processed withcomputer methods. We believe that for diffusefields3 this approach is less error prone than spatialphase shifting because diffuse fields are nonuniformand that it is a more practical method of performingholographic interferometry than direct digital recon-struction.7

2. Theory

Let the complex field of the image formed on thecamera detector be S1x, y2 and the reference field be

A exp1ikx2, where x and y are the image coordinates,A is a constant, i is the square root of 21, and k is aconstant defining the spatial frequency offset of thereference beam. The intensity I1x, y2 on the cameradetector is

I1x, y2 5 A2 1 AS1x, y2exp12ikx2

1 AS*1x, y2exp1ikx2 1 0S1x, y2 02, 112

where * denotes a complex conjugate.Equation 112 is Fourier transformed with respect to

the spatial frequency variables j and c. Let usidentify the following transform pairs:

I1x, y2 = I1j, c2, S1x, y2 = S1j, c2.

The Fourier transform of Eq. 112 becomes

I1j, c2 5 A2d1j, c2 1 AS1j 1 k, c2 1 AS*1k 2 j, c2

1 S1j, c2 p S*1j, c2, 122

where p denotes a convolution. The first term is adelta function at the origin that corresponds to thetransform of the average irradiance of the image.The second and the third terms are Fourier trans-forms of the image field and, as such, correspond toimages of the field passing through the aperture ofthe lens used to form the image. They appearshifted in the spatial frequency plane because of theangular offset of the reference beam. The fourthterm is the halo corresponding to the transform ofthe image speckle pattern and is the convolution ofthe aperture image with itself. At this point, one ofthe image terms, for example, S1j 1 k, c2, is sepa-rated from the rest of the pattern, shifted to theorigin, and subjected to an inverse Fourier transform.This yields the result

S1j, c2 = S1x, y2,

from which it is possible to extract the phase angleby the formula

f 5 arctan5Im3S1x, y24@Re3S1x, y246 13a2

if Re3S1x, y24 is positive and

f 5 arctan5Im3S1x, y24@Re3S1x, y246

1 p sgn5Im3S1x, y246 13b2

if Re3S1x, y24 is negative.Our goal is not to obtain the phase of the image

field but to obtain the phase change of the image fieldbetween two exposures. It is common to subtractthe results of two calculations via Eqs. 132 for thefields before and after the deformation; however,doing this by itself is highly undesirable because thephase of the speckled image field f is randomlydistributed between 6p. Because the change inphase Df that is due to the deformation is added tothe random phase f, the result may lie above, below,or within the limit of 6p at which it will be wrapped

because of the arctangent calculation. When thetwo phases are subtracted, the values range between62p, and Df is peppered with random-phase wrapsof 62p, which are most numerous around the transi-tions of 6p. The random-phase wraps may beeliminated by rewrapping the Df values to 6p, orthe phase difference may be calculated directly bythe formula published originally by Stetson,10 whicheliminates the random phase from the calculation.If we use primes to denote the field after the deforma-tion,

Df 5 arctan53Re1S2Im1S82 2 Im1S2Re1S824

@3Im1S2Im1S82 1 Re1S2Re1S8246. 142

Either method for calculating Df eliminates theneed for shifting the windowed spectrum to zerospatial frequency. The spatial frequency offset inEq. 122 corresponds to an additive linear phase,which, like the random phase, disappears when thephase difference is calculated.If we are to separate one of the image terms in Eq.

122, e.g., S1j 1 k, c2, from the other terms, the aper-ture must be sized relative to the angular offset ofthe reference beam, which, in turn, must be chosenrelative to the resolution of the camera. Let ph andpv be the horizontal and the vertical pixel spacings.The aperture that transmits the most light is a pairof vertical slots in the configuration used by Bieder-mann and Ek9 and is shown in Fig. 1. The height ofthe slot is chosen so that the fringe spacing of theinterference of rays from the top and the bottom istwice the vertical pixel spacing, which defines theNyquist sampling limit. If H is the slot height, l isthe wavelength, and z is the distance of the aperturefrom the camera, then

H 5 zl@2pv. 15a2

As can be seen in Fig. 1, the slots are separated bytwice the slot width, and the reference-beam sourceis located at the center. The slot width,W, is set sothat the interference of the reference beamwith rayscoming from the outer edges of the slot form fringesthat have a horizontal period twice the horizontalpixel spacing. Therefore

W 5 zl@4ph. 15b2

Fig. 1. Aperture with a pair of vertical slots with the referencebeam positioned symmetrically between them as proposed inRef. 9.

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In principle, the Fourier spectrum of the imagerecorded by this aperture should appear as in Fig. 2,which shows the central autocorrelation locatedbetween the images of the two apertures and twoadditional autocorrelation spectra beyond them.In practice, the outer autocorrelation terms areattenuated by the resolution limit of the TV camera.In the experiments performed in this paper, how-ever, the double-slot aperture was not available.

3. Experiment

Figure 3 shows the interferometer part of the TV-holography system11 used to provide an image-planeTV hologram with a small angular offset to thereference beam. The object is imaged by the opticalsystem 1the video lens, the mirror, the relay lens2 andreflected at the front surface of the beam splitteronto the photosensitive surface of the CCD camera1NEC TI-23A with pixel spacing of 12.8 µm horizon-tally and 9.9 µm vertically2. A smooth sphericalreference beam emanating from a single-mode opti-cal fiber passes through the beam splitter, where itcombines with the object beam. The end of the fiberis located so that it appears to be the same distancefrom the beam splitter as the aperture. The fiber iscut to a length that equalizes, approximately, theoptical path lengths of the object and the referencebeams, in order to ensure their mutual coherence.The fiber end is held by an adjustablemount, slightlyoff axis.

Fig. 2. Fourier spectrum of the hologram obtained by the two-slitaperture as in Fig. 1.

Fig. 3. Interferometer of the TV-holography system used torecord the spatially phase-biased TV holograms. The single-slotaperture is placed inside the video lens. BE, beam expander; M,mirror; R, relay lens; BS, beam splitter; CCD, charge-coupled-device camera.

334 APPLIED OPTICS @ Vol. 35, No. 2 @ 10 January 1996

The angle between the reference field and theobject field was adjusted to yield approximately 4 to5 pixels between each carrier fringe. The presenttechnique tolerates a varying fringe spacing over thedetector surface so that setting the angle is notcritical and calibration is quite simple. When theaperture of the video lens used in this experimentwas minimized, it turned out not to be circular but,fortunately, more like one of the slits in Fig. 1, and itssize was adjusted so that one object speckle covered,5 horizontal neighboring pixels. Because no spa-tial phase coding was necessary in the verticaldirection, the vertical-slot aperture could be used toprovide more light. The intensity of the referencebeam was adjusted to be approximately equal to theobject beam to maximize contrast of the interferencefringes.The video signals from the camera were captured

by a digital image processor, which saved them incomputer memory. Before image capture, the inputgain and the black level were set on the analog inputsection of the processor. Figure 4 shows a part ofthe image plane hologram consisting of 50 3 50pixels, with a fringe spacing of 4 to 5 pixels withineach speckle.Figure 5 shows the two-dimensional fast Fourier

transform 1FFT2 of the recorded hologram. It hasthree main areas in the spatial frequency plane asopposed to the five areas in Fig. 2, which would bethe Fourier spectrum in the double-slit case. Thephase-biased parts of the spectrum are clearly sepa-rated from the central self-interference spectrum.A rectangular cut-off filter was used to separate thebiased part of the Fourier spectrum, shown insidethe white rectangle onto the right in Fig. 5. Thispart of the spectrum was transformed back to the

Fig. 4. Part 150 3 50 pixels2 of a recorded hologram as seen by theCCD camera with approximately 4 to 5 pixels between the carrierfringes.

space domain by an inverse FFT, and its real andimaginary parts as described in the theory sectionwere stored.The object—a rectangular, aluminum plate, 100

mm 3 80 mm 3 0.7 mm in dimensions and paintedwhite—was clamped along the upper and the loweredges. The out-of-plane deformation field of theupper-right part of the plate was measured in thisexperiment. A screw was fastened to the center ofthe back of the plate as a cantilever pin, perpendicu-lar to the plate. By putting a small load on this pin,we introduced a bending moment in the center of the

Fig. 5. Fourier spectrum of the recorded hologram for a single-slot aperture and a smooth off-axis reference beam. The areawithin the white frame was used for the inverse FFT transforma-tion.

Fig. 6. Deformation field of the object displacement shown as awrapped phase map calculated by use of Eq. 142.

plate that forced the upper half of the plate to deforminward and the lower part outward. Two record-ings on separate TV frames were made, one beforeand one after deformation of the object. Theseimages were processed to produce the real and theimaginary parts of the two inverse-filtered trans-forms as described above.Figure 6 shows the wrapped phase map obtained

by Eq. 142. Convolution with a 33 3 kernel was usedto smooth the data in the real and the imaginaryimages. The wrapped phase map was unwrappedwith the noise-immune algorithm described byHunt-ley,12 and the resulting deformation field is shown inFig. 7. The out-of-plane deformation of the object iscoded in different gray levels, where white is 21.13and black is 0.81 µm. The round circle in thepicture is a mask that removes the part of the objectin which the fringe spacing is too small to bespatially resolved in the presence of the noise 1com-pare to Fig. 62.

4. Conclusion and Discussion

The work presented in this paper establishes thatthe technique of Fourier-transform processing canbe used to evaluate spatially phase-biased TV holo-grams and that the results yield successful phasemaps for object deformations. A major reason forthe development of this technology is its potentialapplication to electronic recording of holograms withpulsed lasers, and it is thought that this methodmayoffer advantages over the earlier method describedin Ref. 3. First, the alignment of the reference-beam offset is less critical than with the spatialphase-step method. Second, this method does not

Fig. 7. Unwrapped phasemap showing the out-of-plane deforma-tion field of the object. The circular area in the lower left is amask used in the phase unwrapping to remove areas with too higha fringe density and too much noise.

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assume constant object-beam irradiance over sets of3 pixels and should be less prone to error. Thisreduced sensitivity to error should require lesssmoothing of the final data, and this reduced smooth-ing should allow better object resolution. Increasedresolution over that presented in Ref. 3 should alsobe possible in both techniques by the use of aninterline transfer CCD TV camera with progressivescanning, where the rows are read in order ratherthan by interlaced scanning. The experiments per-formed here are, in fact, representative of whatwould be obtained with this type of camera becauseentire TV frames are captured and processed.At present, the FFT operations are carried out in

the host computer that operates the electronic holog-raphy system or within a compatible computer. Afaster alternative is available through the use of aFFT processing board that can be resident within theholographic processor itself. It is expected that,with a resident FFT processor board, the phase couldbe displayed within seconds after the recording ofthe hologram. This would allow the user to evalu-ate the data in the process of testing. In manypulsed-laser applications, expensive test setups areinvolved in the project such as spin rigs, windtunnels, tensile testing machines, etc., and the delayrequired for chemical processing of holograms oftenties up a costly test facility. The use of electronicholography with rapid FFT processing of the inter-ferograms may make it possible to reduce the cost ofsuch tests by a significant amount, and such eco-nomic improvement may make holographic testingin such rigs more cost effective.

This project was supported by the Swedish Re-search Council for Engineering Sciences, TFR.

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