Effect of phase mismatch on pseudo-phase-stepanalysis of time-average hologram recordings ofvibration modes

Karl A. Stetson

I present the results of an experiment to demonstrate the effect of phase mismatch between an objectvibration and a bias vibration in pseudo-phase-step analysis of time-average holographic interferogramsof vibration modes. Pseudo-phase-stepping applies conventional phase-step equations to zero-orderBessel function fringes and during phase unwrapping corrects for the errors incurred. A circular diskvibrating in a quadrature combination of its two one-diameter modes was used as a test object andprovided a 360° phase distribution. The results indicate that the process has considerable tolerance tophase mismatch. © 2006 Optical Society of America

OCIS codes: 090.1680, 120.2650, 120.3180, 120.3040.

1. Introduction

Two methods have been developed for extracting nu-merical data from time-average holographic inter-ferograms of vibrating objects.1,2 Both introducephase modulation at the frequency of the object vi-bration into either the reference beam or the objectbeam to shift the zero-order fringe of the J0 fringefunction. Such modulation has been shown to add abias constant to the argument of the Bessel function,which may be described as a phase vector.3 Themethod due to Ellingsrud and Rosvold2 is suitableonly for measurement of small vibrations of an objectand involves introducing a bias modulation thatshifts the zero-order fringe of the J0

2 function to thenearly linear region that lies between its peak at zeroand its first zero at 2.4048. By recording four inter-ferograms, with four values of phase separated by 90°and with proper calibration, they were able to solvefor both the amplitude and the phase of vibrationpatterns. The object vibrations must be small enough,however, that the J0 function does not depart signif-icantly from its linear approximation.

Four years earlier, Stetson and Brohinsky1 pre-sented a method for measuring larger vibration am-

plitudes based on what may be called pseudo-phase-stepping. This method recognized that the J0 functionhas a periodicity very close to that of a cosine functionfor most of its range, departing most significantly atsmall values of its argument. It also recognized thatwhen an object vibrates in one vibration mode, itspoints move either in-phase or out-of-phase with oneanother, which makes phase measurement unneces-sary. When the phase of the bias modulation matchesthat of the object vibration, the phase vector shift ofthe argument of J0 becomes a simple additive con-stant and may be used to shift the J0 fringes in amanner equivalent to phase shifting of cosine fringes.Analysis of recordings of shifted J0 fringes by equa-tions based on cosine fringes gives errors that arepredictable a priori, and a lookup table may be usedto correct this error after a calculation is made. Ac-curate implementation of this method depends ontwo calibrations: setting the phase of the bias modu-lation equal to the object vibration and calibratingthe amplitude of the bias vibration so that knownamounts of shift can be introduced. As is describedbelow, the amplitude calibration can be performedaccurately, but the phase setting is more difficult.The purpose in this paper is to present an experimen-tal demonstration of the effect of mismatch betweenthe bias modulation and the object vibration.

A. Pseudo-Phase-Step Analysis

The author’s company manufactures an electronicholography system (the K�100 system plus theHG7000 interferometric computer), which was used

Karl Stetson (kastetson@snet.net) is with Karl Stetson Associ-ates, LLC, 2060 South Street, Coventry, Connecticut 06238.

Received 21 February 2006; accepted 18 April 2006; posted 24April 2006 (Doc. ID 68286).

0003-6935/06/256473-04$15.00/0© 2006 Optical Society of America

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for this experiment. A description of the steps pro-vided by its PCHolo32v2.05 operating program forvibration data recording will help the reader under-stand the process. This system has two coupled fre-quency generator boards in the computer system, onethat supplies excitation to the object and the otherthat provides bias modulation to the reference beam.The program first records a speckle averaged imageof the object, without vibration, to serve as a maskduring phase unwrapping. It is usually assumed thatimage points below a certain threshold lie outside theobject and that those above the threshold lie withinthe object, but the mask can be edited to eliminatepoints that are ambiguous or troublesome. The pro-gram then displays a standard time-average holo-gram in real time that the user can observe to locatethe vibration mode and identify its frequency of max-imum response. The user is instructed to adjust theamplitude until no more than the fifth zero of the J0

function is displayed, because beyond that level theamplitude of J0

2 may be too low to provide accuratedata.

Next the program switches control to the bias vi-bration generator, and the user is instructed to in-crease its amplitude until the zero-order fringedisappears and then to adjust its phase until thezero-order fringe returns with maximum brightness.The range over which the zero-order fringe can beseen is actually quite broad, and accuracy is in-creased by finding phase values below and above thepoint where the zero-order fringe appears equallydegraded and then taking the average of those val-ues. The phase is set to that average value, and theprogram proceeds to the next step, where the ampli-tude of the bias vibration is calibrated. Here the ex-citation to the object is turned off, and the user isinstructed to increase the bias amplitude until theimage appears as black as possible. This correspondsto the first zero of the J0 function at 2.4048. Theprogram then proceeds to capture an image of theobject with excitation and asks the user to mark apoint on the zero-order fringe. This point is used totell the unwrapping program where the unwrappedphase function should be approximately zero. (Theentire unwrapped phase function will be shifted by amultiple of � to accomplish this, and then the resultwill be read through a lookup table.) After this pointis marked, the program then captures four interfero-grams at bias levels corresponding to an equivalentphase of 135°, 45°, �45°, and �135° relative to theperiodicity of the J0

2 function. After data capture, theinterferograms are processed by the standard four-step algorithm to give the unwrapped phase. Thesedata, and auxiliary data, are stored for unwrappingby a program based on calculated unwrap regions.4(This program actually unwraps a phase map wherethe wrapping has been doubled to a range of � fromits original range of 2� in order to create a map ofunwrap regions, and this is why the shifting of theunwrapped phase function is by a multiple of �rather than 2�.)

Calibration of the amplitude of the bias modulationis required for each excitation frequency, because res-onances of the piezoelectric mirror affect the responsewith frequency, and this can be done quite accuratelybecause it is quite easy to see the when the entireimage goes black. This can usually be done to one partin 50 or better. By comparison, the phase calibrationis less accurate, and, although accuracy of this cali-bration increases with the amplitude of the bias vi-bration, it usually cannot be set to better than a fewdegrees. Phase calibration is also required for eachdata recording, because the response of a vibrationmode relative to an exciting force changes rapidly inphase from nearly zero below resonance to nearly180° above it. There is no way to determine what thisphase will be beforehand, so no permanent calibra-tion can be applied. It is of interest to know, therefore,what the effect of phase mismatch will be and how toidentify whether it has occurred.

B. Experiment

The object chosen for this experiment is a thin circu-lar disk cemented to a heavy circular frame. It wasexcited so that it responded in a quadrature combi-nation of its two one-diameter vibration modes.5 Thisgave a traveling wave that circulated around the cen-ter of the disk, making one circuit for each vibrationcycle. The vibration phase therefore varies through360° around the center of the disk. Figure 1 shows atime-average hologram image of this vibration pat-tern as recorded by the electronic holography systemfrom its real-time display. There is a circular spot inthe center of the disk at the point around which thewave rotates. The argument of the J0 function is zeroat this point and on the boundary where the disk iscemented to the frame.

The procedure for capturing vibration data wasapplied to this vibration pattern and the phase cho-sen so that the zero-order fringe was translated ver-tically. The resulting wrapped phase is shown inFig. 2.

Fig. 1. Time-average hologram recording of a circular disk vibrat-ing with a quadrature combination of its two one-diameter vibra-tion modes.

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It is interesting to note that the fringes are quitecircular, which indicates that the effect of the phasevariation is not severe. The fringes become ratherindistinct, however, at the sides where the bias vi-bration is plus or minus 90° to the bias modulation.An attempt to unwrap these fringes failed because ofthe loss of fringe quality in these regions. The maskwas edited, therefore, to remove these regions fromthe unwrap computation, and the result is shown inFig. 3.

These results show that the tolerance of this pro-cess to phase mismatch is quite good, even up to�75°. Detailed study, however, shows that when thephase mismatch gets beyond about �45° there is abanding to the phase results that follows the fringes.Figure 4 shows plots of data scans at 0° to verticaland at 15° to vertical. The plot of data for 15° liesslightly inside that for 0°, and this is an artifact of theway the program reads the data. Both plots are ver-tical lanes, and, although the one at 15° is slanted atan angle, the data are still averaged across sets of fivehorizontal pixels, and the results are plotted as a

function of the corresponding vertical pixel. Accord-ingly, this foreshortens the pattern slightly. Bothplots are quite smooth and offer reasonable measure-ments.

Figures 5 and 6 show two more scans from Fig. 4taken at 40° and 63°. The data taken at 40° are moreirregular than the data taken at 0° and 15°, and thedata at 63° are much rougher still. This shows thatthe effect of phase mismatch is to create irregularitiesin the calculated data that depend on where the datalie in the phase cycle and that increase with theamount of mismatch.

2. Conclusion

The results of this experiment show that the effect ofmismatch between an object vibration and a bias vi-bration in pseudo-phase-stepping is not severe andthat mismatch up to �15° can be tolerated. The re-sults also suggest that, if the phase mismatch is ac-tually known, the errors could be precalculated andcorrected in the same way in which we correct forerrors associated with the difference between cosineand J0 fringes.

Fig. 2. Wrapped phase resulting from the application of the vi-bration data capture procedure in the PCHolo32v2.05 program.

Fig. 3. Unwrapped phase of the data shown in Fig. 2 with trian-gular segments at the left and right of the center removed.

Fig. 4. Plot of data taken at 0° to vertical and at 15°. Each plot isthe average across a lane of data 5 pixels wide.

Fig. 5. Plot of data at 40°.

Fig. 6. Plot of data at 63°.

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References1. K. A. Stetson and W. R. Brohinsky, “Fringe-shifting technique

for numerical analysis of time-average holograms of vibratingobjects,” J. Opt. Soc. Am. A 5, 1472–1476 (1998).

2. S. Ellingsrud and G. O. Rosvold, “Analysis of TV-holographysystem used to measure small vibration amplitudes,” J. Opt.Soc. Am. A 9, 237–251 (1992).

3. K. A. Stetson, “The effects of beam modulation on fringe loci and

localization in time-average hologram interferometry,” J. Opt.Soc. Am. 60, 1378–1388 (1970).

4. K. A. Stetson, J. Wahid, and P. Gauthier, “Noise-immune phaseunwrapping by use of calculated wrap regions,” Appl. Opt. 36,4830–4838 (1997).

5. N.-E. Molin and K. A. Stetson, “Measuring combination modevibration patterns by hologram interferometry,” J. Phys. E 2,609–612 (1969).

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