Pulsed holographic interferometry has proven to be avaluable tool in the study of transient events, such aspropagating flexural waves in plates, acoustic near-field radiating from impacted structures, and shockwaves in liquids and air.1–4 Unlike many othermeasuring techniques, holographic interferometry isa whole-field technique: An entire object volume isevaluated at a specific time set by the time separationin between the two laser pulses. With CCD camerasinstead of photographic film, quantitative phase dataare quickly obtained in electronic form. Digital im-age plane holograms are directly recorded onto aCCD detector, stored, and processed in a computer.5,6
This interferometric technique is applicable whencomparatively small deformations are present.However, often in technical applications the object issubjected to a bulk motion that may be substantiallylarger than the quantity to be measured over the timeinterval of interest. A typical example is the mea-surements of out-of-plane vibrations for an in-planerotating object. Vibrations on brake disks have beenmeasured by use of double-pulsed holographic inter-
The author [email protected]! is with the Division of Experimen-tal Mechanics, Luleå University of Technology, S-971 87 Luleå,Sweden.
Received 12 July 2000; revised manuscript received 22 January2001.
2304 APPLIED OPTICS y Vol. 40, No. 14 y 10 May 2001
ferometry with holographic film. Only short timesbetween the exposures could be applied to obtainfringes with high visibility because of large in-planerotations. The lateral shift in the speckle patternmust be less than the speckle size in the image planeof the observation system to obtain high-contrastfringes. In situations when this requirement is notfulfilled, methods either to rotate the holographicsetup synchronously with the object or to use imagederotation with a rotating prism have been devel-oped.8,9 Blade vibrations on high-speed rotating im-pellers have been successfully recorded. However,these techniques need a quite complicated and expen-sive setup and can be used only when the bulk motionis a rotation. By use of CCD cameras instead ofphotographic film, the images ~corresponding to theundisturbed and disturbed condition of the object!can be stored on separate frames, which allows thepossibility of calculating the speckle shift and com-pensating for this in the interference phase evalua-tion. Digital speckle photography ~DSP! is aechnique that measures the displacement of apeckle pattern over several tens of speckle diame-ers and is described in detail in Ref. 10. In DSP theulk motion of a speckle pattern is determined fromhe peak position of the digital cross correlation be-ween subimages, typically of 32 3 32 pixels, from theeference and the deformed images. This techniqueas been combined with ordinary TV holography byse of a continuous wave Nd:YAG laser as the lightource to measure three-dimensional deformationelds.11 The speckle modulation was extracted from
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the phase-stepped TV holography images and pro-cessed with the DSP algorithm ~digital image corre-lation! to obtain the in-plane displacements. Theout-of-plane deformation was determined by compen-sating for the speckle shift in the phase calculation.This technique has also been used to retrieve highfringe visibilities when large in-plane displacementsare present.12
In this paper pulsed TV holography and DSP arecombined to measure out-of-plane deformations onmoving objects. The pulsed TV holography systemrecords two separate digital image-plane hologramswith a double-pulsed ruby laser as the light source.The technique was first tested out with a continuouswave Nd:YAG laser. Next, deformation that is dueto impact on a moving plate is demonstrated.
2. Experimental Setup for Pulsed TV Holography
Light pulses emitted by a ruby laser ~RL! are ex-panded by a negative lens ~NL! for object illumination~see Fig. 1!. A plane-convex lens ~L1! of focal length500 mm is positioned in front of the object ~OBJ! toachieve collimated light. The object illumination isreflected back from the object toward the beam split-ter ~BS!. A positive lens ~L2! of focal length 80 mmreproduces the object on the CCD detector. TheCCD camera ~PCO Computer Optics Sensicam dou-ble shutter, 1280 3 1024 pixels! is controlled from acomputer and can be triggered externally to be syn-chronized with the laser to capture two pulsed imageswithin 1 ms. The sensor is cooled to 212 °C and thedigital images have 12-bits depth, which gives a highdynamic range of the camera. Close to the frontfocal point of L2 a rectangular aperture is placed ~1.9mm 3 7.0 mm! to reduce the spatial frequencies atthe detector. A small portion of light is reflected atthe plane surface of NL to form the reference beam~R!. This beam is adjusted so that seen from thedetector the virtual image ~a bright spot! is locatedone slit width from the edge of the aperture. This isan important condition because it separates the in-terference term ~between the object and the reference
Fig. 1. Experimental setup. RL, ruby laser; NL, negative lens;BS, beam splitter; L1, lens to collimate the light; A, aperture; L2,imaging lens; O, object beam; R, reference beam; CCD, CCD cam-era; OBJ, object; AT, air track; P, pendulum ~impactor!.
eam! and the self-interference of the light passinghe aperture in the Fourier domain.
To measure small out-of-plane deformations whenhe object is subjected to comparatively large in-planeisplacements ~roughly a factor of 100! requires some
special demands on the optical setup. Unwantedstraight fringes caused by the translation may easilyoccur when there is a variation in the sensitivityvector over the field of view, see Ref. 1. The sensi-tivity vector lies along the bisector of the angle be-tween the illumination and observation directions.Assume a slight variation in the sensitivity vector of2° over the field of view and a translation of 0.2 mm.This results in ;20 unwanted fringes superimposedon the deformation of interest. The variation in thesensitivity vector over the field of view is minimizedby using collimated light and by making the obser-vation and the illumination direction coincide. Thesystem is adjusted by mounting a mirror on the objectto retroreflect the incident collimated light along thesame path. The aperture of the imaging system ~A!is placed at the position at which the reflected light isfocused by L1. Only light scattered in the same di-rection from different parts of the object will now passthe aperture and finally form an image.
3. Interference Phase Restoration
The interference phase difference between two re-corded holograms in pulsed TV holography ~beforeand after an event! is of vital interest. It is normallyevaluated with the Fourier transform method.13
However, large speckle translations on the CCD de-tector owing to an in-plane motion will degrade thespeckle correlation to the extent that the interferencephase is lost. With the use of DSP to compute thespeckle displacement from the recorded images, thephase can be restored.
Fig. 2. Fourier spectrum of the recorded hologram that is due toa single-slit aperture and an off-axis reference beam. The outerbright rectangular bands represent the spatial frequencies of theinterference pattern. The central part is the frequency content ofthe reference beam and light scattered from the object ~specklefield!.
10 May 2001 y Vol. 40, No. 14 y APPLIED OPTICS 2305
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2306 APPLIED OPTICS y Vol. 40, No. 14 y 10 May 2001
Let O and R ~Fig. 1! represent the object and ref-rence waves, respectively. The intensity I at a
pixel ~X, Y! on the detector can be written as @co-ordinates ~X, Y! are omitted for simplicity#
I 5 uO 1 Ru2 5 uOu2 1 uRu2 1 OR* 1 O*R. (1)
he first term is the image speckle pattern resultingrom the light reflected from the diffusely scatteringbject. If the object is subjected to an in-plane mo-ion, this speckle pattern will move in the detectorlane. If the reference beam is blocked, only thepeckle pattern is recorded and the setup is identicalo DSP. By recording two images with different in-lane positions of the object, the speckle translationan be calculated with the DSP algorithm ~digitalmage correlation!. It is, however, also possible to
easure the speckle translation when the referenceeam is present in the recordings. This may be nec-ssary for measuring nonrepeatable events. Theourier transform of Eq. 1 is
F$I% 5 F$uOu2% 1 F$uRu2% 1 F$OR*% 1 F$O*R%. (2)
he first and second terms are the Fourier trans-orms of the object speckle field and the referenceeld, respectively. The third and fourth terms rep-esent the frequency content of the interference pat-ern. They are conjugates to each other and containhe same information. In Fig. 2 the two-imensional Fourier spectrum of a recorded holo-ram is shown. The bright spot in the centeronsists mainly of the reference beam @the seconderm in Eq. ~2!# and is surrounded by the first termF$uOu2%!, horizontally between 2300 and 300 in spa-
tial frequency units. Owing to the angular offset ofthe reference beam, the third and fourth terms ~thetwo bright rectangular areas in Fig. 2! are spatiallyeparated from F~uOu2! 1 F~uRu2!. In the case that
uRu2 is constant, F~uRu2! will be a delta function locatedat the center of the Fourier plane and can easily beremoved. Thus the first term can be filtered out bycutting off the interference terms. After inverseFourier transformation, the object speckle field is re-calculated. However, if the reference beam is noisybecause of diffraction, the speckle field will not becorrectly recalculated, resulting in problems whenthe DSP algorithm is used to find the right correla-tion peak. One way to reduce the influence of anoisy reference would be to subtract F$uRu2% from F$I%before filtering and inverse Fourier transformation@see Eq. ~2!#. However, uRu2 must be recorded sepa-rately with the object beam blocked. This necessi-tates that the reference beam is repeatable from shotto shot.
The interference terms ~one of the rectangular
Fig. 3. In-plane translation and out-of-plane deformation of aplate. ~a! Phase map for different pixel shifts of 0 and 8–12 pixels,from the bottom to the top of the image. ~b! In plane displace-ment, each arrow corresponds to 10 pixels in displacement. ~c!Restored phase map of the out-of-plane deformation.
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bright areas in Fig. 2! for each of the two hologramsare filtered out by a high-pass filter in the usual way.6Inverse Fourier transformation yields two maps ofcomplex numbers from which the wanted interfer-ence phase difference is calculated. By shifting oneof these the same amount as the calculated speckledisplacement before the phase-difference calculation,the phase change caused by the out-of-plane defor-mation is restored.
4. Experimental Results
A. In-Plane Translation
The technique to restore the interference phase inpulsed TV holography was first tested with a contin-uous Nd:YAG laser as the light source. The opticalsetup was similar to that in Fig. 1 except for theoptical components ~now coated for 532 nm instead offor the ruby laser wavelength of 694 nm! and theaperture size. The experiments were static; that is,one image was recorded before deformation, transla-tion, or rotation. Successive images were recordedin between changes in the object state. The objectwas a white-painted plate mounted on a rotation andtranslation stage. This permitted a controlled in-plane displacement of the plate. A small lever wasmounted on the backside in the middle of the platewhere a small weight could be hinged, thus introduc-ing a couple of forces in the plate center. Recordingsof the plate before and after translation and loadingwere made. The translation was 400 mm. Figure3~a! shows a montage of phase maps for differentshifts ~in pixels! of the image of size 30 mm 3 30 mmcorresponding to the deformed state of the object.From 0 to 5 mm in the vertical direction, no imageshift has been applied and, as expected, the phaseinformation is lost; no fringes are seen. Largershifts are from 8 to 12 pixels. Highest fringe visibil-ity is obtained when the shift is ;10 pixels. Observethe phase jump between adjacent pixel shifts. Thisis due to the angle between the object and referencewaves, resulting in a phase difference between pixelswith different shifts. This can be compensated forby multiplying the maps of complex numbers with alinear phase factor before the interference phase res-toration. Using the procedure described above, fil-tering out the two speckle fields and correlating themby the DSP technique, we obtain results of a calcu-lated speckle shift of ;10 pixels @see Fig. 3~b!#. In
ig. 3~c! the whole image corresponding to the de-ormed state of the object has been shifted 10 pixels.
B. In-Plane Rotation
The interference phase can also be restored when theobject is subjected to an in-plane rotation. The ap-
Fig. 4. In-plane rotation of a flat plate. ~a! In-plane displace-ment field calculated with DSP. ~b! Phase map without compen-sation for the rotation. ~c! Phase map, compensated for therotation, showing that there also is a small out-of-plane compo-nent.
10 May 2001 y Vol. 40, No. 14 y APPLIED OPTICS 2307
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erture is now reduced to the same size in the verticaldirection as in the horizontal so that the speckleshave the same size in both directions. A rotation of809 ~1.33°! of the object plate results in a displace-ment field calculated with DSP, as shown in Fig. 4~a!.
t 20 mm from the rotation center, the in-plane dis-lacement of the plate is ;460 mm, resulting in apeckle shift in the image plane of ;12 pixels. Thehase map without compensation for the rotation ishown in Fig. 4~b!. The fringe contrast has disap-eared over the whole image except at a small arearound the rotation center. In contrast to the linearranslation case, compensations of the speckle move-ents must now be performed in both X- and-directions. The restored phase map is shown inig. 4~c!. The appearance of fringes in the imagehow that the large in-plane rotation was not perfect;t also had an out-of-plane component. This compo-ent, however, is small, ;0.35% of the maximal in-lane displacement.
C. Transient Wave Propagation—Impact on a MovingPlate
For the study of out-of-plate vibrations caused by animpact on a moving plate, OBJ in Fig. 1, the trans-lation device must not introduce unwanted vibrationsin the object. For this purpose an air track ~AT!~normally used for demonstrations in dynamicscourses! was showed to be suitable. A 2-mm-thickaluminum plate ~150 mm 3 200 mm! was mountedon a sledge. The track was inclined a small amount~;1.1°!, allowing the sledge to accelerate because ofgravity. Before the sledge reached the field of view,a trigger signal is sent to an electromagnet via a delayunit to release the pendulum ~P!. Immediately be-fore the impact of the pendulum, a laser beam aimedat an optical fiber connected to a photodiode is inter-rupted, and a trigger pulse is optionally delayed andsent to the ruby laser ~;1 ms before the first laserpulse!. The first laser pulse is recording the objectin an undisturbed condition. A short time thereaf-ter ~300 ms!, the second laser pulse is fired. Theimpact can be triggered to occur a preset time beforethe second laser pulse. The second exposure gives adifferent interference pattern owing to changes in theoptical path caused by the deformation and the trans-lation. The impactor of the pendulum is an acceler-ometer ~Bruel & Kjær 4393! with a small roundingcap to act as the impact surface. The time from theimpact start ~accelerometer signal! to the second la-er pulse is monitored on a digital oscilloscope.The velocity of the plate was 0.67 mys at the impact
instant and the computed speckle shift was ;5 pixels,corresponding to an in-plane translation of 200 mmbetween the laser pulses. Figure ~5a! shows the re-stored phase map representing the deformation of theplate 20 ms after the start of the impact. The phasehas been restored by shifting the second image 5pixels. By use of an unwrapping procedure,14 thecontinuous phase and hence the deformation field canbe determined; it is presented here as a three-dimensional plot @see Fig. ~5b!#. Here the maximal
308 APPLIED OPTICS y Vol. 40, No. 14 y 10 May 2001
deformation that is due to the impact is ;2% of thebulk in-plane motion.
5. Conclusions
Vibration measurements on in-plane moving objectsby use of holographic interferometry often requirespecial and expensive recording and evaluation tech-niques. Otherwise, the fringe patterns will be lostbecause of large speckle motions. Also, unwantedspurious fringes owing to variation in the sensitivityvector over the field of view will appear. Pulsed TVholography can be used in combination with DSP toretrieve fringes when in-plane motions of severalspeckle diameters are present. By filtering out thespeckle fields from the pulsed TV holography record-ings and correlating them with the DSP technique,the in-plane speckle shift can be determined and com-pensated for in the interference phase calculation.
Fig. 5. Transient vibrations on a moving plate recorded by pulsedTV holography. ~a! Restored phase map of the deformation on a2-mm-thick moving aluminum plate caused by impact of a pendu-lum. ~b! Three-dimensional plot of the deformation.
7. R. G. Hughes, “The determination of vibration patterns using
Bulk linear translations as well as rotations of anobject can be compensated for. This technique maybecome very attractive in the study of vibrations~preferable transient! on moving or rotating objects.
This project is supported by the Swedish ResearchCouncil for Engineering Sciences. Thanks are dueto Mikael Sjodahl and Per Synnergren for help withthe DSP technique.
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