Key words: Pulsed holographic interferometry, television holography, transient wave propagation.
1. Introduction
For the measurements of transient and steady-stateevents, such as propagating sound waves in air, bend-ing waves in structures, and compressible flow fields,pulsed holographic interferometry is a widely usedtechnique.1–3 Usually, a photographic plate is usedas the storage medium for recording and a continuouslaser for reconstruction of a hologram. Quantitativephase data can be obtained by use of two referencebeams during recording and a temporal phase-stepping technique during reconstruction.4,5 A CCDcamera can eventually be used to view the recon-structed holograms. This method yields phase mapsof quite good spatial resolution, but it is relativelytime consuming.
Using solid-state detectors rather than photographicplates avoids the time-consuming wet processing ofholograms. Thus, practical measurements can besimplified and speeded up, something that should beattractive for industrial applications. One drawback,however, is the limited spatial resolution of CCD de-tectors and the need for computer capacity for process-ing. However, in recent years the resolution of CCDdetectors and computer capacity have been in constantincrease, and it is expected that this progress will con-tinue in the future.
The authors are with the Division of Experimental Mechanics,Luleå University of Technology, S-971 87 Luleå, Sweden.
Received 10 October 1996; revised manuscript received 28 Jan-uary 1997.
Electronic speckle pattern interferometry or TV ho-lography systems that use continuous laser light forthe measurement of harmonic waves in mechanicsand acoustics ~time-average techniques! are wellknown.6,7 Experiments with TV holography by useof a pulsed laser are reported by Spooren et al.8 Ti-ziani et al.9 demonstrated an electronic speckle pat-tern interferometry system using a spatial-carriershift method for quantitative evaluation of phasedata from double-exposed interferograms. Thismethod needs only a single interferogram for phaseevaluation. A direct digital recording method of aFresnel hologram has been presented by Schnars.10
The hologram is captured directly on the CCD detec-tor, and digital reconstruction of the wave fronts isperformed with mathematical methods. The phasedata are obtained from the complex amplitudes of thereconstructed wave fronts.
An alternative method, proposed by Saldner etal.,11 is to use Fourier transforms for evaluation ofspatially phase-shifted image-plane holograms.The phase shift is achieved by a small angular offsetbetween the object and reference beams. This ap-proach is adopted here, and the method for obtainingthe phase difference between two electronically re-corded image-plane holograms is demonstrated.Using a properly designed lens aperture that formsthe image on the CCD detector yields a Fourier spec-trum of the image pattern that contains spatiallyseparated parts. Those parts, which correspond tothe interference between the object and referencewaves, are separated from the rest of the pattern.After inverse transformation, they yield a complexfunction that contains the phase information. Re-
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cording two images ~corresponding to two objectstates! allows the phase difference to be calculated.From the phase difference, the deformation field ~nor-mally the out-of-plane displacement! of an object canbe determined.
In Ref. 11, an aperture in the form of a double slitis suggested. Such a double slit is tested in thisstudy, and the conditions for a pulsed system to workwith this slit are investigated. Also, a single-slit anda three-hole aperture are tested. One reason to usethe three-hole aperture is that, theoretically, the in-plane deformation of the object would also be possibleto measure.12 More than one slit yields more thanone interference term in the Fourier spectrum.Thus, for improving the quality of the phase maps,spatial speckle averaging between the filtered-imageterms is performed for the cases of double-slit andthree-hole apertures.
The newly developed pulsed TV holography tech-nique is demonstrated in experiments showing prop-agating transient bending waves in an aluminiumplate, as well as in a cymbal. The light source is adouble-pulsed ruby laser ~2 3 0.5 J!. The systemallows electronic recording of two events in the rangefrom 20 to 800 ms with successive double pulses fromthe ruby laser.
2. Theory
The optical unit in the pulsed TV holography systemfocuses the image of the object onto the CCD detector.This is usually called image-plane holography. Theorthoscopic and pseudoscopic images of an object arethus located at the image plane rather than at dif-ferent positions in space, as in the case of a Fresnelhologram. When the aperture of the optical unit is asingle slit, one orthoscopic and one correspondingpseudoscopic image are formed by the interferencebetween the object and the reference beams. Theyare shifted in the Fourier plane as a result of theangular offset of the reference beam. If the apertureconsists of two slits, then two orthoscopic and twopseudoscopic images are formed. When these im-ages are formed by the same angular offset of thereference beam but with opposite signs, the ortho-scopic and pseudoscopic images will overlap in theFourier plane. This situation occurs when the ref-erence beam is positioned symmetrically between thetwo slits.
Consider Fig. 1, which shows the double-slit aper-
Fig. 1. Double-slit aperture. Focusing of the object light field.
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ture and the light field focused on the detector. Theincident object field is approximated as plane wavesand the Fraunhofer approximation is applied ~see theright-hand side of Fig. 1!. Let U9 be the amplitudefield on the camera detector after deformation of theobject. Then
U9 5 R 1 S1 exp@i~kd1 sin u 1 DF!#
1 S2 exp@i~2kd2 sin u 1 DF!#, (1)
where R is the reference field, S1 and S2 are thecomplex image fields, and DF is the phase changeintroduced as a result of the change in optical pathafter deformation of the object. The intensity, whichis registered on the CCD detector, is, after deforma-tion, ~sin u ' u ' xyf !:
I9 5 U9U9* 5 RS1 exp@i~kd1u 1 DF!#
1 RS2* exp@i~kd2u 2 DF!# 1 RS1* exp@i~2kd1u
2 DF!# 1 RS2 exp@i~2kd2u 1 DF!# 1 S1S2*
3 exp@ik~d1 1 d2!u# 1 S1*S2 exp@2ik~d1 1 d2!u#
1 R2 1 S1S1* 1 S2S2*, (2)
where the asterisk indicates the complex conjugate.The first four terms represent the interference be-tween the reference and object fields. The fifth andsixth terms are the interference between the twoslits, i.e., the outer autocorrelation terms of the ap-erture. The last two terms represent the centralautocorrelation of the aperture. Before object defor-mation, the amplitude and intensity fields are de-noted by U and I, respectively. The expressions forthose are the same as in Eqs. ~1! and ~2! but with thephase change DF omitted.
Next, the intensity field, Eq. ~2!, is Fourier trans-formed. The first four terms are shifted in the Fou-rier plane. Filtering out the first two interferenceterms from the rest of the image pattern and inversetransforming yields the following complex terms thatremain:
after deformation of the object. For the case inwhich the reference beam is positioned symmetri-cally between the slits, we have d1 5 d2. Thus, thetwo terms in Eqs. ~3! and ~4! will overlap in the Fou-rier plane. Furthermore, if uS1u 5 uS2u 5 uS1*u 5 S,Eqs. ~3! and ~4! yield
s 5 2RS@cos~kdu! 1 i sin~kdu!#, (5)
s9 5 RS@cos~kdu 1 DF! 1 i sin~kdu 1 DF!#
1 RS@cos~kdu 2 DF! 1 i sin~kdu 2 DF!#. (6)
The phase change DF between the two object states~before and after deformation! can be calculated di-rectly by the formula11
DF 5 arctanRe~s!Im~s9! 2 Im~s!Re~s9!
Im~s!Im~s9! 1 Re~s!Re~s9!. (7)
In the case where d1 5 d2, the numerator in Eq. ~7!will be identical to zero when Eqs. ~5! and ~6! areinserted into Eq. ~7!. Obviously, if a symmetricalposition of the reference beam between the slits ischosen, it is not possible to determine the phasechange. This was not noted in Ref. 11.
Instead, positioning the reference beam unsym-metrically between the slits, such that d1 Þ d2,causes the numerator in Eq. ~7! to have values otherthan zero. With an appropriate choice of the aper-ture dimensions and a properly positioned referencebeam, nonoverlapping interference terms can be ob-tained in the Fourier plane. The interference termsmust be separated from the central autocorrelationterm of the slit. Hence, the minimum distance fromthe edge of one slit to the reference beam is equal tothe slit width W. Furthermore, the distance fromthe reference beam to the other slit must be at least2W for the two interference terms to be separated.Obviously, for obtaining completely nonoverlappinginterference terms, the distance between the slitsshould be at least 3W, and the reference beam shouldbe positioned at the distance W from one of the slits.
Using the double-slit aperture makes a speckle-averaging procedure possible. Each of the two slitsyields interference patterns on the detector with sta-tistically independent speckle distributions, sepa-rated in the Fourier plane. They could be used toevaluate two similar phase maps of quite poor qual-ity, each containing statistically independent noise.The quality of the phase map is improved if both theinterference terms in the Fourier plane are filteredout. The spatial resolution is increased since thetwo terms together cover up to twice as large an areain the Fourier plane as does one term alone. Inpractice, one carries this out by averaging the numer-ators and denominators of Eq. ~7!. First, the numer-ators and the denominators of Eq. ~7! are evaluatedfor each interference term. Then averaging is per-formed. Finally, the phase difference DF is com-puted. This yields significantly smaller phaseerrors than does averaging the phase-difference DFvalues directly.13
Speckle averaging can also be performed for thecase of an aperture with three holes. Figure 2~a!shows a symmetric three-hole aperture and Fig. 2~b!the resulting Fourier spectrum for the case in whichthe reference beam is positioned at the center of theaperture. The central and the six outer cones cor-respond to the autocorrelation spectrum of the aper-ture. Theoretically, the outer autocorrelation termscan be used to derive the in-plane displacement.12
Thus, using a three-hole aperture makes it possible tomeasure all three deformation components of an ob-ject simultaneously. However, in this experiment,
the dimension of this aperture is chosen such that thesix outer terms are beyond the resolution limit of theCCD camera. Aliasing effects are minimized sincethe six outer terms will be aliased back mainly inbetween the interference terms, i.e., the six cylindersin between the central and outer cones. Three of theinterference terms are filtered out and inverse trans-formed, and finally the phase difference is calculated.
3. Recording of Pulsed TV Holograms
A ruby laser ~wavelength l 5 694.3 nm! emitting twohigh-energy pulses with a short time separation ~1–800 ms! is used. Two images, corresponding to thetwo pulses, are recorded by a CCD camera ~PULNIX,Model TM-9700! run in a noninterlaced mode. Thecamera has 768 3 484 pixels, and each pixel has asize of 11.6 3 13.6 mm. The experiments are per-formed with the CCD camera operating in normalvideo mode, and triggering is controlled by the videosignal. Firing the laser pulses at a certain time de-lay relative to the vertical sync pulse of the video
Fig. 2. ~a! Symmetric three-hole aperture and ~b! the Fourierspectrum obtained for the case in which the reference beam ispositioned in the center.
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Fig. 3. ~a! Experimental setup: BS, beam splitter; EB, excitation beam; NL, negative lens; M, mirrors; O, object; OL, object light; OU,optical unit; R, reference light; RL, ruby laser. ~b! Optical unit: A, aperture; AR, aperture for the reference light; BS, beam splitter; FL,field lens; M, mirror, NL, negative lens; OL, object light; R, Reference light; REL, relay lens; VL, video lens.
signal achieves control of triggering. The time delayis chosen such that the first image is recorded at theend of one frame, and the second one at the beginningof the next frame. Thus, two successive images, cor-responding to the first and the second laser pulses,are recorded on separate frames.
Figure 3~a! shows the experimental setup. Thelight beam coming from the ruby laser is split intotwo beams of equal energy at the beam splitter ~BS!.The reflected beam is used for excitation of the object~O!, whereas the transmitted beam is used for holo-graphic recording. A diverging light field illuminat-ing the object is obtained with a plano-concave lens~NL! whose optical axis is tilted a few degrees relativeto the beam. Hence, the reflected portion of light~approximately 5%! at the planar lens surface is usedfor the reference beam ~R!. The object and referencefields interfere at the CCD detector at the optical unit~OU!.
Figure 3~b! shows the optical unit in detail. Theobject is imaged by a video zoom lens ~VL! onto a fieldlens ~FL! and relayed to the camera by a two-elementrelay lens ~REL!. The aperture ~A! is placed be-tween the two elements. Finally, the object light isreflected at the beam splitter and illuminates theCCD detector. The reference beam is filtered by anaperture ~AR! and passes a negative lens ~NL! tospread the light uniformly on the detector. The vir-tual image of the reference beam seen from the cam-era appears as a point located at a certain positionrelative to the aperture. For the case of the double-
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slit aperture, the negative lens is adjusted such thatthis position is at one third the distance between theslits, whereas for the case of the three-hole aperturethe position is in the center.
4. Experimental Results
Both a rectangular aluminium plate ~of dimensions220 mm 3 480 mm 3 0.7 mm! and a musical cymbal~having a diameter of 410 mm! are used as test ob-jects. The plate is clamped at its lower and upperedges, whereas the cymbal is fastened at the center toa heavy stand. Excitation is accomplished when thefirst laser pulse is focused toward the backs of theobjects @see Fig. 3~a!#. This is a convenient excita-tion because the impact is controlled by the laser andno external triggering is needed, unlike with an im-pact device such as a pendulum. Another advantageis that the focused laser pulse approximates a Diracdelta function in that it is small spatially and of shortduration. The results of the impact experiments arepresented as phase maps, wrapped between 2p and1p. Convolution with a 3 3 3 kernel is used tosmooth the data.
Figure 4~a! is a phase map showing the propagat-ing bending waves in the aluminium plate 30 ms afterimpact. An increase from dark to bright indicatesan out-of-plane displacement in the direction of theimpact. In this experiment, a single-slit aperture ofdimensions 1.2 mm 3 2.3 mm is used. This choiceyields an average speckle size on the detector of ds '42 mm ~ds 5 lfyD, where l 5 0.694 mm, f 5 73 mm,
Fig. 4. Phase maps showing propagating bending waves in the aluminium plate at 30 ms after impact: ~a! from a single-slit apertureand ~b! from a double-slit aperture with speckle averaging.
and D 5 1.2 mm! in the horizontal direction and ds '22 mm in the vertical direction, i.e., a speckle sizemore than twice the pixel size of the detector. Thisexperiment was repeated with a double-slit aperture.The width and height of each slit are 0.7 and 2.4 mm,respectively, and the distance between the slits is 2.4mm. The resulting phase map after the averagingprocedure is shown in Fig. 4~b!. The excitation en-ergy was not exactly equal in Figs. 4~a! and 4~b! be-cause of the difficulty of getting repeatable pulsesfrom the ruby laser. Thus, the fringe pattern looksdifferent for these recordings, especially close to theimpact point.
Figure 5~a! shows the resulting phase map at 60 msafter impact on the plate for the case of the three-holeaperture. The distance from the center of the aper-ture to the center of one hole is 1.6 mm, and thediameter of each hole is 1.0 mm @compare with Fig.2~a!#. A second difference in the setup is an ex-
change of the relay lens to achieve unit magnification.In the new setup each element has a focal length of 90mm, rather than 73 mm. Applying an unwrappingalgorithm14 to the phase map allows a three-dimensional plot representing the deformation fieldto be obtained @see Fig. 5~b!#. The laser pulse impactoccurred in the positive z direction. Just at the im-pact point, decay of the deformation can be seen.This is due to local heating of the back of the plate,which induces deformation in the direction oppositeto the impact force.15
Finally, recording of transient bending waves ofthe cymbal is performed. The three-hole aperture isalso used in this experiment. The impulse is appliedat a location 100 mm from the outer edge of thecymbal. Figure 6~a! displays the phase map 300 msafter impact. The field of view is 205 mm 3 152 mm,and the outer edge of the cymbal is seen at the right-hand side. Reflected waves from the central dome
Fig. 5. Bending waves at 60 ms after impact on the aluminium plate. The three-hole aperture is used with speckle averaging. ~a!Wrapped phase map. ~b! Three-dimensional plot showing the unwrapped phase map.
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Fig. 6. Bending waves at 300 ms after impact on the cymbal. The three-hole aperture is used with speckle averaging. ~a! Wrappedphase map. ~b! Three-dimensional plot showing the unwrapped phase map.
~near the upper left-hand side! and from the outeredge can be seen. In Fig. 6~b! the correspondingunwrapped phase map in form of a three-dimensionalplot of the deformation field is shown.
5. Conclusions
A double-pulsed TV holography system permittingfast quantitative evaluation of interferograms wastested. Recordings of TV holograms with a double-pulsed ruby laser as the light source were carried outin separate frames of a CCD camera. Two eventscan be recorded in the range from 20 ms up to 800 mswith this system. An important component of theoptical unit is an aperture of appropriate dimensions.This aperture introduces a spatial offset into the in-terference pattern between the reference and objectirradiance fields. A Fourier spectrum of the imageis computed that allows the phase information to beevaluated. The interesting quantity to be deter-mined is the phase change between two exposures,representing two object states ~for instance, beforeand after impact on a plate!. The phase change isevaluated directly from the fast Fourier transform-processed complex image fields before and after de-formation of the object. The computer time requiredfor processing an image of 768 3 484 pixels is ap-proximately a couple of minutes on a SUN worksta-tion.
Experiments were carried out with impact on analuminium plate and a musical cymbal as test ob-jects. Focusing the laser pulse toward the objectsachieves impact. Phase maps of good quality show-ing transient bending waves arising from the impactwere obtained. Imaging systems with a single-slit, adouble-slit, and a three-hole aperture were tested.A speckle-averaging procedure was performed for thecases of the double-slit and three-hole apertures.One reason to use a three-hole aperture is that, inaddition to the out-of-plane displacement normallymeasured, it would also be possible to measure the
3946 APPLIED OPTICS y Vol. 36, No. 17 y 10 June 1997
in-plane displacement. However, using a three-holeaperture requires that the total area of the aperturebe reduced, compared with the use of a single aper-ture, to avoid overlapping terms in the Fourier plane.Unfortunately, this results in lower irradiance expos-ing the CCD detector and lower spatial resolution.However, in the test experiments, an aperture withthree circular holes was used. This is advantageousin that the spatial resolution is equal in the horizon-tal and vertical directions of the image. Further-more, the random speckle noise is reduced by speckleaveraging. A larger area could be covered by theinterference term in the Fourier plane by use of anenlarged rectangular single slit.16 This choicewould allow more light to fall on the detector andwould increase spatial resolution. However, a rect-angular slit naturally gives a different spatial reso-lution in the vertical and horizontal directions.
This project is supported by the Swedish ResearchCouncil for Engineering Sciences ~TFR!. The au-thors thank Karl A. Stetson for valuable comments.
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