Double-exposure phase calculationmethod in electronicspeckle pattern interferometry based on

holographic object illumination

Richárd Séfel* and János KornisBudapest University of Technology and Economics, Department of Physics, Budafoki út 8, Budapest, Hungary, H-1111

*Corresponding author: sefel@phy.bme.hu

Received 13 April 2011; revised 22 June 2011; accepted 24 June 2011;posted 27 June 2011 (Doc. ID 145786); published 5 August 2011

Multiple-exposure phase calculation procedures are widely used in electronic speckle pattern inter-ferometry to calculate phase maps of displacements. We developed a double-exposure process basedon holographic illumination of the object and the idea of the spatial carrier phase-shifting method toexamine transient displacements. In our work, computer-generated holograms and a spatial light mod-ulator were used to generate proper coherent illuminating masks. In this adjustment all phase-shiftedstates were at our disposal from one recorded speckle image for phase calculation. This technique can beused in the large scale of transient measurements. In this paper we illustrate the principle throughseveral examples. © 2011 Optical Society of AmericaOCIS codes: 120.4290, 120.5050, 120.6160, 090.2890, 120.7280.

1. Introduction

Electronic speckle pattern interferometry (ESPI) is awell-established method for optical metrology [1,2].Over the past decades, with the development of op-tical elements, the phase maps calculated fromphase-shifted speckle images [3] have been success-fully used in a wide range of applications, such as de-formation, displacement, or shape measurement.Within the field of transient deformation analysisthe rapid variations of phase prevent the use of mul-tiple-exposure phase calculation methods, becauseall the information must be recorded in the samestate of the object.

In ESPI an interesting phase calculation methodwas published in recent years. Pedrini et al. [4,5]have experimentally demonstrated the applicationof the spatial carrier phase-shifting method for tran-sient analysis. In their experiment, an ESPI setupwas used with a reference wave, tilted by angle Θwith respect to the optical axis. This angle was cho-sen so that the phase difference changes by a con-

stant amount from one pixel to the next one. Thecalculation of phase was based on sets of three con-secutive pixels and a standard phase-shifting algo-rithm. Because of its experimental simplicity andlow computational time requirement, it is an attrac-tive method for transient analysis. However, it hasseveral disadvantages. It presumes that there is lit-tle variation in the irradiance of interfering wavesover any set of three consecutive pixels. In generalthis is not true for a diffusely reflecting object.Furthermore, it can be noticed that more steps phasecalculation methods produce better quality phasemaps. In the last decades this technique has under-gone several improvements.

Because of the rapid development of spatial lightmodulators (SLMs), the object illumination can beperformed using digital holograms [6]. There are sev-eral examples in the literature, where computer-calculated wavefronts were used for phase shifting[7]. In our work, we combined these two ideas andshowed that SLM is capable of generating adequateobject illumination for a double-exposure phasecalculation during the measurement time, usingcomputer-generated holograms.

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In this paper, we present this new type of phasecalculation method for transient displacement analy-sis and demonstrate through several examples.Among others, the heating of an archeological sam-ple was monitored to detect hidden inhomogeneities,which are needed before its cleaning procedure.

2. Overview and Details of the Method

The capability to carry out accurate deformation ordisplacement measurements by ESPI mostly relieson the application of phase-shifting algorithms [8].These methods have various implementations. Actu-ally the techniques, based on the use of phase shif-ters, like piezoelectric (PZT)-mounted mirrors, arevery accurate, reliable, and capable of automation.They usually use 2–5 appropriately phase-shiftedpatterns to calculate wrapped phase maps of inter-ferometric fringe systems. In experiments the four-steps method is the most commonly used technique[3], because of its simplicity and robustness.

In the case of transient displacement measure-ment, there is not enough time to record phase-shifted images from the same state of the object. Itis not a significant problem in vibration analysisbased on digital holographic interferometry [9–12],where the complex amplitude of the object wave isa known quantity after reconstruction. In ESPI thereare several interesting methods to eliminate the pro-blem of phase calculation. They usually use a double-pulsed system [13,14] and the quantitative analysisof fringes based on the spatial carrier phase-shiftmethod [4,5]. Besides these, there are proceduresbased on the high-speed synchronization of a PZT-mounted mirror [15].

Our purpose was to use adequate computer-gener-ated holographic object illumination with an SLM fordouble-exposure phase calculation in transient mea-surements. Commercially available liquid crystalSLMs are capable of optically reconstructing digitalholograms in satisfactory quality, so the recon-structed real image of an object can be used as a co-herent illuminating mask in optical measurementmethods [16]. The developed procedure is based onthe combination of the ideas of a four-step phase-shifting method with the spatial carrier phase-shifting technique. The novelty in our work was thatall phase-shifted states were encoded in one holo-gram used for object illumination; therefore, thephase calculation and phase unwrapping was feasi-ble from only two images (one in initial, one indisplaced state of the object).

In practice, we simulated a hologram of a virtual,rectangular-shaped diffusely reflecting object as afirst step. This was used as the input image forthe SLM. Functionally there are several kinds ofSLMs, which may modulate the amplitude, intensity,phase, or polarization of the illuminating beam. Forholographic purposes the best choice is a phase mod-ulating device, due to the theoretically high diffrac-tion efficiency of phase modulation. The digitalholograms of an object were initially Fresnel-type

amplitude holograms, which can serve as the inputsignal for a phase-shifting SLM, but it is clearlynot the optimal solution. Generally, phase hologramsproduce better results on a phase-shifting device,because they encode the complex information moredirectly. Therefore, we converted our holograms toFourier-type phase holograms numerically. Afterthe conversion, the resulting phase holograms werematched to our SLM, using themeasured phase mod-ulation curve of our device at 647:1nm, to optimizethe working parameters. On a matched hologramthe phase values are represented by gray levels.

The complex amplitude on each point of the recon-structed hologram [Hðp; qÞ, see Fig. 1(a)] is a knownquantity; therefore, artificial phase manipulationcan be used:

Hmodðp; qÞ ¼ Hðp; qÞ · e

�i2πλ ·Mðp;qÞ

�; ð1Þ

where Hðp; qÞ is the reconstructed hologram; p, q arepixel coordinates; λ is the wavelength of the light;Mðp; qÞ is a matrix that modifies the phase on eachpoint; and Hmodðp; qÞ is the resulting hologram afterphase manipulation. In our work, the Mðp; qÞ matrixconsists of blocks that represent the phase shifts (0,α, 2α, 3α). An enlarged section from this matrix (4 × 4pixels) is shown in Fig. 1(b), where the gray levelsdenote the four phase-shifted states. As object illumi-nation, Hðp; qÞ and Hmodðp; qÞ holograms were usedby turns.

Denote with a1ðp; qÞ the recorded speckle imagefrom the object in state 1, and a2ðp; qÞ in state 2,using Hðp; qÞ and Hmodðp; qÞ as the object illumina-tion, respectively. Using the basic equation of speckleinterferometry,

Iðp; qÞ ¼ a1ðp; qÞ − a2ðp; qÞ; ð2Þ

the difference image (Iðp; qÞ) contains all four phase-shifted states, so the next step is the separation ofthem pixelwise. Let I1ðp; qÞ, I2ðp; qÞ, I3ðp; qÞ,I4ðp; qÞ denote the four phase-shifted differencespeckle images whose half resolutions compare to

Fig. 1. The reconstructed image of a (a) computer-generated ho-logram used for object illumination and (b) the enlarged part ofMðp;qÞ used for pixelwise phase shifting (0, α, 2α, 3α).

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Iðp; qÞ. Now these difference images can be inter-preted as

I1ðp; qÞ ¼ Aðp; qÞ þ Bðp; qÞ · cosðΔφÞ; ð3Þ

I2ðp; qÞ ¼ Aðp; qÞ þ Bðp; qÞ · cosðΔφþ αÞ; ð4Þ

I3ðp; qÞ ¼ Aðp; qÞ þ Bðp; qÞ · cosðΔφþ 2αÞ; ð5Þ

I4ðp; qÞ ¼ Aðp; qÞ þ Bðp; qÞ · cosðΔφþ 3αÞ; ð6Þ

where α denotes the phase shift and Δφðp; qÞ is thephase difference change between the initial and thedisplaced state of the object at a given pixel. It iscalled the interference phase. The parametersAðp; qÞ and Bðp; qÞ depend on the coordinates inthe interferogram. In practice these parametersare not known due to several disturbing effects:the brightness of the interferogram varies (the ex-panded laser beam, used to illuminate the object,has a Gaussian profile), the surface of the object un-der investigation may have a varying reflectivity,speckle noise, and additional noise from electronicrecording. The difference images exhibit the well-known speckled interference fringe patterns. Thesespeckles are smoothed over by a low-pass filter,namely a convolution with a Gaussian kernel withthe size of 4 × 4 pixels. This implies that the specklesize on the half resolution images has to be consider-ably less than the fringe spacing so that it is possibleto get rid of speckles without reducing fringe con-trast. After that we have four intensity values for

each pixel in the images. The measurements pre-sented are processed by a method known as theCarré technique [17–20]; therefore, the wrappedphase map of the displacement can be calculated be-tween any two states of the transient process. It hasthe advantage that it also works if the phase shift isnot constant throughout the object, as long as it re-mains linear. The generation of a continuous phasedistribution is called phase unwrapping. Severalunwrapping algorithms have been developed. Inour experiments, Goldstein’s cut-line method wasused [21].

The most important requirement for the presentedmeasuring technique is that the size of the recordedspeckle images (a1ðp; qÞ, a2ðp; qÞ) must be integermultiples of the size of the holograms (Hðp; qÞ,Hmodðp; qÞ) used for input image on the SLM. If thiscondition is met, the difference speckle image can beseparated pixelwise into four parts that belong to thefour phase-shifted states. Because of this strict re-quirement, we need to pay attention for the opticalsettings.

3. Experimental Setup

The optical arrangement used is basically a Mach–Zehnder type interferometer (Fig. 2). Because ofthe ESPI approach, the object is imaged with a lens(f ¼ 55mm, f =2:8 aperture Nikon objective) andthere is a diffuser in the reference arm. Our lightsource is a LEXEL 95L krypton gas-ion laser witha wavelength of 647:1nm. The camera used in ourexperiment is a Model Marlin by ALLIED VisionTechnologies. It has a CCD with a resolution of

Fig. 2. The optical setup using holographic object illumination wave. BE1, BE2 are beam expanders, BS1, BS2 are beam splitters, andSLM is a spatial light modulator.

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1600 × 1200 pixels and 4:4 μm pixel size, and the dy-namic range of 8 bit. Its maximum speed at full reso-lution is 12:5 fps.

The optical reconstruction of the digital hologramtakes place right after beam expander 2 (BE2), wherethe plane wavefront passes the SLM device, whichgets its signal from a PC. The SLM used is ModelLC2002 SLM by HoloEye, with a resolution of800 × 600 pixels, a modulation depth of 8 bits, andmaximum frame rate of 60Hz.

To obtain a clearly focused image in a finite dis-tance, lens elements have to be used. The SLM is fol-lowed directly by the first long focus lens of atelescope. In our experiment, it was a 200mm focuslength, f =5 aperture symmetric Fourier lens. It pro-duces the far-field diffraction pattern of the SLM’spixels in its back focal plane. The second componentof the telescope is a short-focus lens, which imagesand magnifies the low-order diffraction patterns likea slide projector. In the experiment it was a 29mmfocus length, f =2:8 aperture photographic lens. Thisgeometry has two obvious advantages: the size of theprojected image can be changed relatively easily andthe change of the lateral position of the image is alsopossible. Because these parameters (size and posi-tion of the optically reconstructed holograms) havegreat importance in the developed phase calculationmethod, the lenses were placed on a special mountfor fine adjustments. Moreover, the common compu-ter control of the SLM and digital camera is alsohelpful for settings.

4. Measurements and Results

In our measurements three different test objectswere used. One of them was a conventional deform-able membrane plate. It was used to calibrate thetechnique and test it within the confines of a well-known problem. It was also good to compare the re-sults to the conventional phase-shifting technique;therefore, in this case the phase maps were also mea-sured with the application of a PZT-mounted mirroras phase shifter. Besides this, a Peltier device and ahistorical wall painting sample were used as test ob-jects. Their transient displacements, due to electriccurrent or thermal heating were examined. Thesemeasurements were based on the computer synchro-nization of the digital camera and SLM control. Inour experiments the displacements of the sampleswere measured between two arbitrary moments.We could take captures with the speed of maximumfour images per second because of the high comput-ing requirements, but we note that using a powerfulcomputer only, the speed of the camera and the SLMwould be the limit for the speed of the monitored phe-nomena. Now our goal was to demonstrate only thatthe above explained phase calculation techniqueworks in a wide range of experiments.

A. Deformation Measurement

At first our test object was a bronze diaphragm,painted white, with an external size of 40mm by

40mm and a thickness of 0:25mm. The diaphragmwas fixed to a holder with 16 screws and could beloaded at the center via micrometer screw. The appli-cation of this test object was useful in the early per-iod of our examinations, because it has a well-knownbehavior. Figure 3 shows the wrapped (a) and the (b)unwrapped phase map in case of a central deforma-tion with a magnitude of 2:63 μm. As seen, some ofthe membrane fixing screws were not tight.

We also calculated the phase map based on thefour phase-shifted image using a PZT-mounted mir-ror as phase shifter to examine the accuracy ofthe method and make a comparison. The double-exposure phase map (a) has half resolution comparedto the multiple one (c), as we described above, whichis the most significant limiting factor of the uppermeasuring range. If the same number of fringes ap-pears in a smaller area, the too-dense fringes couldinhibit the phase unwrapping. Besides this, thephase fringes have poorer quality than using thestandard multiple-exposure method, but they havesufficient visibility for evaluation and the qualityof unwrapped images is also suitable. The poorerquality of fringes can be accounted for by the influ-ence of the speckle image separation. It means thatthe upper measuring range is about half of the con-ventional method.

We also examined the difference between the ac-curacies of the conventional and the introducedphase calculation techniques. In our work, the calcu-lated unwrapped phase maps were compared. In thisexample, the maximal deviation between the defor-mation profiles was about 0:06 μm, which is only2.3% of the amplitude of deformation.

B. Measuring the Displacement of the Front Surface of aPeltier Device Using a DC Power Supply

As second test object, a square-shaped Peltier devicewas used with an external size of 30mm by 30mm. Itwas fixed to a cooler and its front surface could beheated up using a DC power supply. During this pro-cess we measured the transient displacement of theobserved front surface. This way a time-resolved mea-surement could be done andwe could calculate the dif-ference phasemap between two arbitrary states of theobject. For instance, Figs. 4(a) and 4(b) show the dis-placement between 0.5 and 1:5 s after turning (a) on

Fig. 3. (a) Wrapped and (b) unwrapped phase maps in the case ofdouble-exposure phase calculation measurement. (c) Phase map ofthe same deformation using the conventional four-step phase cal-culation algorithm. Themagnitude of the deformation was 2:63 μmand some of the membrane fixing screws were not tight.

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and (b) off the power supply. As can be seen, the mag-nitude of the displacement is smaller in the case of therelaxation process during the same time interval. Itmeets our requirement.

C. Monitoring of a Thermally Excited Roman WallPainting Sample

Our most particular test object was a Roman wallpainting sample [Fig. 5(a)]. Archeological materialsare a highly precious historical heritage and, there-fore, their examination should be noncontact,nondestructive. The interferometric techniqueimplemented for the structural observations on thestudied sample is able to locate inborn cracks, de-tachments, and inhomogeneities hidden inside thebulk of materials and structures thatmay potentiallyaffect the examined surface. The exploration of themis essential before the cleaning process.

ESPI is able to measure changes of surface topo-graphy caused by a transient thermal expansion.Our measuring technique is well suited to calculatedifference phase maps between two arbitrary levelsof heating. Using a time-resolved measurement, thedisplacement of the surface can be monitored withdifference phase maps. On these images a suddenchange in the fringe density corresponds to a hiddendiscontinuity in relation to the surface. The data al-low postprocessing to reveal the relation of the dis-continuity to the surface as regards hidden defects.

We noticed that the structural inhomogeneities aremore observable if the sample is heated to a highertemperature. It meant in our case that the differenceimages, corresponding to the total displacement, arenot processable, because the fringe system they con-tain is too dense. Because high temperature changecan help visualize hidden cracks, we decided topreheat the sample and measure its displacementbetween two heated states. During the heating pro-cess the displacement of the front surface of the sam-ple changes permanently; therefore, conventionalmultiple-exposure phase calculation techniques arenot proper. In this case, the introduced measuringtechnique provides a sufficient way to calculate thedifference phase map.

In our measurements, the sample [Fig. 5(a)] wasthermally excited using an IR lamp. Figures 5(b)and 5(c) show the front surface displacement afterless and more preheating, respectively. We assumethat the change in the fringe structure in Fig. 5(c)is caused by sudden inhomogenities. To locate thestructural detachment exactly, we calculated an un-wrapped, masked image that shows the deflectionbetween the less and more preheated sample [seeFig. 5(d)]. It is a mathematically scaled gray-scaleimage where the available intensity range is consid-ered to cover the range of deflection. In this case themaximal intensity change describes a deflectionwhose magnitude is about 0:4 μm.

We note that these are our first results used todemonstrate the functionality of the measuring tech-nique. In the near future we want to take measure-ments to examine the structure of archeologicalsamples in detail.

5. Conclusion

In this paper we have demonstrated through variousexamples that a conventional ESPI arrangementwith proper holographic object illumination is applic-able to calculate the phase maps of displacementsfrom only two speckle images, recorded one beforeand one after the displacement. The principle isbased on the idea that digital holograms can be usedas a coherent illuminating mask and with artificialphase manipulation all the necessary informationcan be encoded into the holograms for phase shiftingand calculation.

In our work, a relatively simple optical setup wasused, but this process requires precise adjustment.The speed of the phenomena that can be monitored

Fig. 4. Phase maps of displacements of a square-shaped Peltierdevice in the interval of 0.5 and 1:5 s after turning (a) on and (b) offthe power supply. The amplitude of the displacements are 4:7 μmand 3:5 μm, respectively.

Fig. 5. (Color online) (a) Roman wall painting sample. Phasemaps of the front surface displacement after (b) less and (c) morepreheating, respectively. (d) Visualization of the deflection whosemagnitude is about 0:4 μm.

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depends on the speed of the digital camera, SLM, theavailable computational performance, and the expo-sure time. If it is too long, the changing of displace-ment can spoil the recording of good quality speckledimages.

Beside the aforementioned transient analysismethods, it is a good solution for measurement ofslow transient displacements, where the applicationof an expensive high-speed technique would be unne-cessary, but the conventional techniques are notworking because of the continuous displacement ofthe examined sample.

Our experimental measurements testify that theintroduced procedure works well and we presentedan interesting field of applications, which is thetesting of an archeological material before itscleaning.

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