May 1982 / Vol. 7, No. 5 / OPTICS LETTERS 233

Method of vibration measurements in heterodyneinterferometry

Karl A. Stetson

Optics and Acoustics Group (86), United Technologies Research Center, East Hartford, Connecticut 06108

Received January 8, 1982

A novel method is presented for determining the vibration amplitude of a vibrating fringe pattern in the presenceof a low-frequency heterodyne phase shift. It consists of counting the average number of positive-going zero cross-ings, per vibration cycle, in the signal from a photodetector on which the moving fringes are imaged.

IntroductionThe purpose of this Letter is to present and describe anovel method for determining the vibration amplitudeof a vibrating interference fringe pattern. The methodhas several inherent advantages: it is a digital method,it is insensitive to vibration phase, it is highly accurate,it requires no calibration, and it is easily adapted tosignal averaging.

MethodThe method consists of (a) generating a linearly in-creasing phase change between two interfering lightbeams by means of a heterodyne phase shifter, (b) ex-tracting an electrical signal proportional to the lightintensity of the resultant fringe pattern at the point ofinterest, and (c) counting the number of positive-goingzero crossings in the electrical signal per vibration cycle.If the frequency of the signal generated by the hetero-dyne phase shifter is considerably lower than, and ir-rational to, the vibration frequency, then as the sam-pling time increases, the number of positive-going (ornegative-going) zero crossings per vibration cycle willasymptotically converge to the peak-to-peak vibrationamplitude in cycles of an interference fringe. If thevibrating fringe pattern derives from light reflected atnormal incidence from a vibrating surface, then eachcycle of an interference fringe corresponds to a dis-placement of A/2 of the surface, and the result is thepeak-to-peak vibration amplitude of the surface inhalf-wavelengths of light. For reflection at other thannormal incidence, the displacement units are X/2 timesan obliquity factor.

It is not immediately obvious that the foregoing as-sertion is true. Let us consider, therefore, the time-varying output of the detector, which will be of theform

f(t) = sin ir[A cos ct + 0(t)], (1)

where f(t) is the detector output, A is the vibrationamplitude, and 0(t) is the slowly varying bias phaseprovided by the heterodyne phase shifter. We willconsider 0(t) to vary so slowly that it is effectivelyconstant over any vibration cycle. The zeros will occurwhen the argument of the sine function is N-7r, where N

is any integer-positive, negative, or zero. By sym-metry, there are an equal number of positive- and neg-ative-going zero crossings so that half the total numberof zeros will equal either the positive or the negativecrossings. The condition for zeros may be written as

A cos wt + 0(t) = N. (2)

As the argument of the cosine function varies from 0 toir to 27r (i.e., through one complete cycle), A cos ct willvary from A to -A to A. If 0(t) = 0, the number ofzeros will equal twice the number of integers betweenthe values of -A and +A. A numerical example facil-itates explanation of the process. Consider a vibrationamplitude of A = 1.2; in this case, there would be sixzeros, three of which would be positive-going crossings.If k = 0.21, however, the number of zeros would be twicethe number of integers between -0.99 and +1.41. Inthis case, there are only four, again, two of which wouldbe positive-going crossings. As s increases, therefore,the number of positive-going zero crossings per vibra-tion cycle alternates between two and three. If 0 in-creases linearly with time, it takes but little imaginationto perceive that there will be two crossings 60%o of thetime and three crossings 40% of the time so that, on theaverage, there will be 2.4 positive-going zero crossingsper vibration cycle. The value of 2.4 is, in this case, thepeak-to-peak vibration amplitude.

Implementation

This method is quite simple to implement. A photo-detector is required whose frequency response is a factorof 10-100 times that of the vibration frequency, de-pending on the magnitude of vibration amplitude to beencountered. The signal processing may be done witha frequency counter that has provision for an externalclock input. The vibration signal (which is generallyavailable) is provided to the external clock input of thecounter, and the detector output is provided to thecounter input. Most counters are designed to count thenumber of positive crossings of a threshold voltage in100, 1000, 10000, etc. cycles of the clock frequency asselected by a scaling switch. The counter thereby canprovide the average vibration amplitude over the cor-responding number of vibration cycles.

0146-9592/82/050233-02$1.00/0 © 1982, Optical Society of America

234 OPTICS LETTERS / Vol. 7, No. 5 / May 1982

-POSITIVE-GOING ZERO CROSSINGS

Fig. 1. Detector output of a heterodyne interferometer witha low-vibration amplitude.

(3 0 3

0

O 01

° 01He = 0.0661

H/ =0.0441

0

Z0 0 03 -H/-0=S025

003 01 0.3

PEAK TO PEAK VIBRATION AMPLITUDE(FRACTION OF T RADIANS)

0.6

Fig. 2. Zero crossings versus vibration amplitude. Thisfigure shows computer simulation of the vibration detectorat low-vibration amplitudes. Note that the detection schememakes a sharp transition between detecting the heterodynesignal and detecting the vibration amplitude.

Minimum-Detectable Vibration

Questions immediately arise with this detection schemeabout how it performs with small-amplitude vibrationsand, further, what lower limit exists for vibration de-tection. Figure 1 shows a plot of the detector output,where the vibration amplitude is so small that zerocrossings occur for only a portion of the heterodynecycle. Clearly, if the vibration amplitude is reduced tothe point at which it generates no local maxima orminima, zero crossings will be caused only by the het-erodyne signal, and the vibration will no longer be de-tectable. Examine the slope of the argument of the sinefunction in Eq. (1), and let 0(t) = Ht. When the max-imum slope of A cos ct is less than or equal to that ofHt, the vibration will no longer generate local maximaor minima. Thus the condition for a detectable vibra-tion amplitude is that

A > Hiw,

and it is the ratio of heterodyne to vibration frequencythat sets the minimum-detectable vibration ampli-tude.

Below the minimum-detectable vibration amplitude,the vibration readout system will count zero crossingscreated by the heterodyne signal and will yield the valueH/c. It may be questioned, therefore, whether the zerocrossings generated by the heterodyne signal do not, infact, represent a constant error to the system that

should be subtracted. This question was answered bya computer model of the process wherein the numberof positive-going zero crossings of f(t) were counted in10,000 cycles of vibration, for different values of A andratios of H/lc. The results are shown in Fig. 2. Thevibration-detection scheme makes a sharp transitionfrom the value Hiw to the peak-to-peak vibration am-plitude, and there is no residual error from the hetero-dyne signal itself.

Accuracy

In principle, the accuracy of the scheme is 11 count andis independent of the vibration amplitude. For apeak-to-peak vibration amplitude of X/2, averaged over1000 cycles, this would yield a resolution of ±A/2000.

In practice, however, there may exist noise in thedetector output because of ambient vibrations of theapparatus sufficient to cause local maxima and minima.The magnitude of this noise will simply add to themagnitude of the vibration signal to yield an error. Ifthe noise is of sufficient low level to yield no localmaxima or minima, it will not affect the measurement.This consideration leads to the idea that this systemcould also give the sum of the magnitudes of two inde-pendent vibrations if they were simultaneously presentin the motions of the fringes.

Differential Vibrations

If, for example, the vibrating fringe pattern is that foundon a vibrating object in a concomitant (real-time) ho-logram interferometer, it may be of interest to know thedifference in vibration amplitude between adjacentpoints on the object. The object may be imaged ontoa pair of small photodetectors and the two resultingsignals fed to a reversible counter that is gated by thevibration signal. Such counters often provide a switchselector for A + B or A - B, etc. so that either the sumor the difference of the two vibration amplitudes maybe obtained. Care must be taken in such applicationsthat the two vibrations are not at some arbitrary phaseto each other, for if this is so, the sum or the differenceof their magnitudes will not equal their true sum ordifference, which can only be found by vectorial addi-tion or subtraction.

Applications

This detection scheme has been set up and tested at thislaboratory, and its operating characteristics have beenconfirmed. In particular, it has been applied to thevibratory readout of an optical strain sensor' withconsiderable success. The combination of convenience,simplicity, and accuracy inherent to this method suggestits suitability to a wide variety of applications.

Reference

1. K. A. Stetson and R. K. Erf, Optical Heterodyne StrainSensor, Air Force Wright Aeronautical Laboratories Rep.AFWAL-TR-80-2088 under contract F33615-80-C-2067,September 1981. (United Technologies Research CenterRep. R81-995310, September 1981.)