Vibration phase mapping using electronicspeckle pattern interferometry

Ole J. L0kberg and KAre H0gmoen

The contours of constant phases across sinusoidally vibrating objects have been mapped using referencewave phase modulation in time-average electronic speckle pattern interferometry.

Introduction

To describe the motion of a sinusoidally vibratingobject completely we need to know both its vibrationamplitude and phase. Sometimes, the phase infor-mation is not essential as it can be deduced from theamplitude information, e.g., in pure modes with onlytwo opposite phases. However, if the object vibratesin a complex mode, the phase information is necessaryfor a correct understanding of the object's motion.

Time-average holography and electronic specklepattern interferometry (ESPI) both provide a fringepattern that represents only the vibration amplitudedistribution across the object. The phase informationmay be revealed if we phase modulate either the objectillumination or the reference wave.' The resultingfringe pattern is then determined by the relative motionbetween the object and the phase modulating device.

Several modulation techniques have been proposedfor use by holography to determine the vibrationphase.1-4 However, the holographic methods are timeconsuming and possess the usual interferometric sen-sitivity to instabilities of object and setup. Time-av-erage ESPI is normally less sensitive to these distur-bances due to the high sampling rate and the corre-sponding short exposure time of its TV system. TheESPI fringe pattern is presented in real-time, whichmakes the technique well suited for applications wherewe need to adjust parameters and rapidly observe theeffect of the adjustments.

In this paper we describe how phase modulationcombined with time-average ESPI can be used to plotcontours of constant phase in a fast and simple way.

Reference Wave Phase Modulation

We suppose a point (x,y) on the object to vibratesinusoidally with frequency f, amplitude ao(x,y), and

The authors are with The Norwegian Institute of Technology,Physics Department, N-7034 Trondheim-NTH, Norway.

Received 27 May 1976.

phase (po(x,y). The reference wave is reflected at nor-mal incidence from a mirror vibrating at the same fre-quency f, but with amplitude a, and phase °r. As-suming normal illumination and observation, the in-tensity I(x,y) of the object as reconstructed on the TVmonitor can be written

I(x,y) = Io(xy)J 02 ( ao2(xy) + ar2

- 2ao(x,y)ar cos[oo(x,y) -V 1r1/2), (1)

where Io(x,y) is the reconstructed intensity with theobject and the reference mirror at rest, Jo is the zeroorder Bessel function of first kind, and X is the wave-length of the laser light.

Equation (1) gives that the argument of the Besselfunction is determined by the vectorial difference be-tween the movement vectors of the object and the ref-erence mirror. The information about the amplitudeand phase of the object is coded against the amplitudeand phase of the reference mirror. The modulatedfringe pattern is not readily interpreted, especially notif the object movement is complicated. As shown byLevitt and Stetson,3 the phase of the object can be ob-tained if we compare the unmodulated and modulatedfringe patterns at different phase settings of the refer-ence mirror. However, this method is not suited forESPI recordings since the number of resolvable fringesis too low to give sufficient accuracy in the phase cal-culations. Instead we use the fact that the zero orderfringe, i.e., J0

2(0), will be observed wherever the phaseand amplitude of the object are equal to the phase andamplitude of the reference mirror, that is,

ao(x,y) = ar and o(x,y) = (P.r (2)

These conditions must be simultaneously fulfilled if thezero order fringe shall attain its maximum brightness.The zero order fringe is easily detected as its brightnessis six times greater than the first order fringe.

November 1976 / Vol. 15, No. 1 / APPLIED OPTICS 2701

Phase shift[O]

100

10

100%

50%

20%

0.10 4 8

Vibration amplitude V M]

Fig. 1. Phase shift required to reduce the intensity of the zero orderfringe by 20%, 50%, and 100% as function of vibration amplitude

(assuming equal object and reference mirror amplitude).

Phase Contour Mapping

In a previous paper5 we have used the phase modu-lation technique and the conditions given by Eq. (2) toincrease the amplitude range by ESPI recordings. Herewe describe how these conditions can be used to map thephase distribution across the object, independent ofamplitude variations.

The phase of the reference mirror, i.e., the phasedifference between the mirror drive signal and theobject drive signal, is set at a value so,. We then vary theamplitude of the reference mirror between zero and amaximum value determined by the object amplitudedistribution. This variation of the reference mirroramplitude makes the fringe pattern change, but thecenter of the zero order fringe will always be positionedwhere the object phase ip(x,y) equals the referencephase Spr Therefore, the shifting of the zero orderfringe traces out the line of constant phase iop(x,y) = ;o,across the object. To obtain the complete phase dis-tribution of the object vibration we vary the referencephase in appropriate steps Ago, and repeat the proce-dure with variation of the reference mirror amplitude.In this way, a satisfactory phase mapping can usuallybe performed in 5 min, although this time is largelydependent on the complexity of the vibration.

The accuracy of the phase mapping by this techniqueis mainly determined by the vibration amplitude.Larger vibration amplitudes give better accuracy. Thisis illustrated in Fig. 1, where we have plotted the nec-essary phase shift to reduce the intensity of the zeroorder fringe by 20%, 50%, and 100% as a function of vi-bration amplitude. The curves are based on Eq. (1).At large object and reference vibration amplitudes, onlya small change in the cosine term in Eq. (1) is needed tomake the argument of the Bessel function differ con-siderably from zero. Therefore, at large amplitudes thezero order fringe appear as a small spot or a narrow line.

At smaller amplitudes, e.g., close to nodal regions, thefringe will broaden, and we have to determine the centerof the fringe to find the position of the phase contour.

By using this technique we are able to separate thephase information from the amplitude information.Thus we have a complete description of the objectsmotion.

Essentially, the same information could be obtainedby stroboscopic techniques. Stroboscopic ESPI 6 showshow an object deforms during a vibration cycle and givesa good impression of its dynamical motion and thepresence of any vibration anomalies. However, at aparticular strobing instant we observe a fringe patternthat is determined by the combined amplitude andphase distributions. We must compare fringe patternsfrom different parts of the vibration cycle if we want toseparate the amplitude and phase information. Theexact analysis of a complex vibration mode may there-fore be very time consuming. The stroboscopic ESPIalso demands a more complicated setup, and thestrobing process gives a poor light economy.

Experimental Apparatus and Results

The ESPI setup used for the experiments is shownin Fig. 2. The object is imaged by two lenses onto thetarget of the TV camera. The reference wave is re-flected by two mirrors, filtered and expanded and su-perimposed in line with the object wave. The videosignal from the TV camera is rectified and filtered in theusual way, and the processed image interferogram ispresented on the TV monitor.

The modulating mirror, which is mounted on a highfrequency loudspeaker, provides phase modulation ofthe reference wave. The object is excited by a sine wavegenerator. From the same generator we obtain a co-herent reference signal that makes the modulatingmirror vibrate sinusoidally. The reference signal iscontrolled by an amplifier and a phase shifter. Thephase shift is measured by a digital phasemeter.

In the first example we consider a 12.7-cm (5-in.)circular brass plate, sprayed with retroreflective paintto improve the light economy. Figure 3(a) shows anordinary time-average ESPI recording of the plate vi-brating at 1755 Hz. Since this is a pure mode we expect

Modulatingmirror

Fig. 2. Experimental setup for ESPI with phase modulated referencewave.

2702 APPLIED OPTICS / Vol. 15, No. 11 / November 1976

isfactory picture quality, we kept a low object excitationlevel. The pictures show clearly how the trace of thezero order fringe approaches a line when the amplitudeincreases. In practical applications of the technique,we plot the path of the zero order fringe on a transparentpaper that is fastened to the front of the monitor. Thisplotting procedure has been used to obtain the morecomplete phase map shown in Fig. 5; here we also havedrawn lines of constant amplitude.

The last example is a 10-cm loudspeaker membranevibrating at 1000 Hz. The membrane was not coatedwith retroreflective paint as this would alter its vibra-tion pattern. The low reflectivity and the surfacestructure of the membrane therefore gave fringe pat-terns of poor quality, not suited for reproduction. Theamplitude distribution could be deduced from the un-

(b)

(a)

(c)

(b)

Fig. 3. (a) Ordinary time-average ESPI recording of plate vibratingin resonance at 1755 Hz. (b) Areas of constant phase po0. (c) Areas

of constant phase po + 7r.

the phase to be constant in each antinode and a phaseshift of ir between adjacent antinodes. To show this,we adjusted the phase of the modulating mirror tocoincide with the phase (po of one-set antinodes. Thenwe varied the amplitude of the mirror and photo-graphed the averaged light distribution traced out bythe zero order fringe. Figures 3(b) and 3(c) show theresulting light distributions, giving the areas of constantphase soo and soo + r, respectively.

Figure 4(a) shows the plate excited at a higher fre-quency-1805 Hz. This is a composite mode with anodal point in the center. In this case, the phase variescontinuously around the object with radial lines ofconstant phase. Figures 4(b) and 4(c) show phase linesat the phase values o and <,o + 7r. In order to get sat-

id~~~~~~~~~~~~~~c

Fig. 4. (a) Time-average recording of plate vibrating in compositemode at 1805 Hz. (b) Areas of phase po. (c) Areas of phase ~o + r.

November 1976 / Vol. 15, No. 11 / APPLIED OPTICS 2703

(a)

go

Fig. 5. Map showing phase and amplitude distribution across theplate. The lines of constant phase are fully drawn, and iso-amplitude

lines are shown dotted.

Fig. 6. Map showing phase (fully drawn) and amplitude (dotted)distribution across a loudspeaker vibrating at 1000 Hz. N.P.-Nodal

point located at the voice coil.

modulated fringe pattern, but the more detailed patternshown with dotted lines in Fig. 6 was obtained by use ofreference wave modulation to extend the amplituderange.5 The phase distribution obtained by phasecontour mapping is shown with fully drawn lines in thesame figure. The lines of constant phase have the nodalpoint as common center, and the phase value changescontinuously around the periphery of the membrane.One might visualize this motion by comparing it witha coin, flapping after falling on the floor.

Conclusion

Compared with other methods for vibration phasemeasurements, ESPI with phase modulation of thereference wave has proven to be a useful method, wellsuited for routine measurements. It is a fast workingmethod, and the experimental setup is relatively un-complicated. A phase contour map of the vibratingobject is readily obtained independent of vibrationamplitude, although the accuracy of the phase mea-surements depends upon the amplitude.

Thanks are due to H. M. Pedersen for valuable sug-gestions and helpful discussions and to 0. M. Holje forassistance during the experiments. The work has beenpartly supported by the Royal Norwegian Council forScientific and Industrial Research.

References1. C. C. Aleksoff, in Holographic Nondestructive Testing, R. K. Erf,

Ed. (Academic, New York, 1974), Chap. 8.3, p. 247.2. D. B. Neumann, C. F. Jacobson, and G. M. Brown, Appl. Opt. 9,

1357 (1970).3. J. A. Levitt and K. A. Stetson, Appl. Opt. 15, 195 (1976).4. N. Takai, M. Yamada, and T. Idogawa, Opt. Laser Technol. 8, 21

(1976).5. 0. J. Lokberg and K. Hogmoen, J. Phys. E 9, 847 (1976).6. H. M. Pedersen, 0. J. Lokberg, and B. M. Forre, Opt. Commun.

12, 421 (1974).

Engineering Research Initiation Program

NSF is seeking proposals for its EngineeringResearch Initiation program designed to encouragethe development of research programs by engineeringfaculty who have had no previous substantial researchsupport. The program is for full-time engineeringteaching faculty of U.S. undergraduate and graduateinstitutions that award degrees in engineering.Application deadline is Dec. 1, 1976: awards will beannounced about April 1, 1977. For further informa-tion, request NSF 76-32 on the Publications OrderForm or call 632-5790.

2704 APPLIED OPTICS / Vol. 15, No. 11 / November 1976