Remote vibration measurement of rough surfacesby laser interferometry

Robert A. Bruce and Gerald L. Fitzpatrick

A He-Ne laser interferometer employing a focusing telescope is used remotely to measure transient longi-tudinal displacements of rough surfaces by monitoring the resulting Doppler shift of the backscatteredlight. The feasibility and accuracy of this technique are considered theoretically relative to laser output,range, telescope aperture, sensor efficiency, bandwidth, noise, surface characteristics, and coherence. Thetechnique was successfully demonstrated on various test surfaces at distances up to 100 m using an unsta-bilized 4-mW laser, a 15-cm reflector telescope, and a PIN photodiode.

1. IntroductionUnequal path interferometers are currently being

used to measure surface motion over considerabledistances.' These devices monitor the Dopplershifts of the backscattered light relative to a superim-posed reference beam. Motion is indicated by a lightsensor in the resultant interference zone. These in-terferometers are inherently capable of resolving dis-placements to within a fraction of the light wave-length used.

A disadvantage of most of these interferometers isthe requirement that the surface being monitoredmust have a mirrorlike finish normal to the directionof the sensing beam. The usual method of effectingthis is by attaching a corner cube retroreflector to thesurface.

This paper discusses an alternative technique inwhich transient longitudinal displacements are re-motely measured on rough, unprepared surfaces. 2 Inlieu of an attached retroreflector on the surface, amoderate-aperture focusing telescope is employed atthe interferometer. When it is focused near the dif-fraction limit on an optically smooth surface, the col-lected wavefronts are mainly in phase. On an opti-cally rough surface, the reflected wavefronts are moreor less random in phase, and the resultant Dopplersignal is attenuated accordingly. 3 It is theoreticallyinferred and experimentally demonstrated that theparameters involved in this application can be ad-justed to allow remote vibration measurement onmost surfaces at 100 m. Accuracy is shown to be de-

The authors are with the Denver Mining Research Center, U.S.Bureau of Mines, Denver, Colorado 80225.

Received 8 January 1975.

graded primarily by surface roughness with lateralmotion, an effect that often can b6 minimized.

II. Theory of OperationThe photocurrent of a light sensor located in two

photomixing fields is

i = [i,2

+ i2 +

2ijsr cosA'(t)]"'2 , (1)

where AC1(t) is the instantaneous phase differencebetween the two fields at the point of detection, andi, and i are the photocurrents resulting when thefields are sensed separately. This phase difference isthe sum of the Doppler shift in phase A'p(t) and aconstant sac determined by the nominal optical path-length difference between the two interferometerarms and the difference in the number of reflectorsbetween the two arms. Classically at normal inci-dence the Doppler term is

A0(t) = 2TAx(t)X/2

(2)

where the phase modulates through 27r rad for everyhalf-wavelength change in displacement Ax.

The signal to be monitored is the alternating cur-rent following the Doppler shift,

id = {2isir cos[ X (t) + c] (3)

where s denotes the sensing beam and r the referencebeam.

The above discussion applies to the ideal situationwhere the wavefronts of the reflected beam are per-fectly in phase. The instantaneous phase differencewould be exactly specified. However, in the subjectinterferometer the parallel wavefronts of the refer-ence beam are photomixed with light collected from

July 1975 / Vol. 14, No. 7 / APPLIED OPTICS 1621

an illuminated spot with both lateral and longitudi-nal extent. If the lateral spot extent exceeds the dif-fraction limit criterion 4

6 = 1.22 AR (4)-A'

where A is the collecting aperture located at distanceR from the spot, the phase difference in Eq. (1) is notwell defined. The focusing telescope utilized can ap-proach this criterion, implying validity of Eq. (3) foroptically smooth surfaces.

Longitudinal extent, present with rough and in-clined surfaces, could still severely attenuate thephotocurrent of Eq. (3). This attenuation can beconsidered by regarding the surface illuminated bythe beam to be composed of m randomly distributeduniphase scatterers. This model can be consideredin terms of a random walk problem where each of them scatterers contributes a small Doppler photocur-rent. The resultant photocurrent would tend tom-1/2 of the maximum possible if all the scattererswere in phase. This model is admittedly not rigoroussince variable scatterer size is not considered. How-ever, it will be sufficient for this analysis.

A. Inaccuracy Due to Lateral MotionLateral surface motion can degrade accuracy of vi-

bration measurements by two effects. On level butoptically rough surfaces, lateral motion continuouslyreorients the resultant phase of the Doppler photo-current arbitrarily as one illuminated set of m scat-terers is replaced by another. This random effectcan be estimated by the ratio of lateral displacement,Ay in units of diffraction limited spot sizes to longi-tudinal displacement, Ax in units of half-wave-lengths, or [A/(2.44R)](Ay/Ax) [roughly equal to 0.01(Ay/Ax)]. This effect is small for relatively low lat-eral motion. The other effect is the actual addedlongitudinal displacement caused by the beam mov-ing laterally on an inclined surface. This inclined ef-fect can be considerable, equaling tan (y/Ax),where 0 is the surface inclination of the spot. The in-clination for rough surfaces will vary considerablywith lateral motion. The displacement caused bythis effect could be hard to distinguish from thatcaused by actual longitudinal displacement of thesame spot. Both kinds of displacements representvalid translation of the surface relative to the inter-ferometer. However, signal amplitude fluctuationsdue to changes in m and reflectivity can suggest thepresence of lateral motion. Lateral motion should beminimized or avoided if possible to diminish thesetwo deleterious effects on the analysis. The inclinedeffect does suggest a possible alternative application,that of surface delineation by lateral scanning.

B. Reflecting Characteristics of SurfacesThe backscattered light from an illuminated spot

on a typical surface reflects back throughout 27rsr.If the angular distribution of the reflected light is iso-tropic, the signal attenuation due to test geometry is

A2/8R2. Such a surface is diffusely reflecting andgenerally consists of many small scatterers that mayor may not be arranged in uniphase elements. Sur-faces that reflect nonisotropically consist of fewer butlarger elements that reflect light in a more or lessspecular manner. Positioning the sensing beam onsuch a surface changes the attenuation due to test ge-ometry by a backscattering factor k, where k could beas high as 8R2/A2 for a normally oriented mirrorlikesurface or could be zero for one oriented off axis.For uniform diffusely reflecting surfaces this factor is1.0. All surfaces scatter some light directly back byeither diffuse or specular manner and usually both.However, optimal beam placement can result in ahigher value of k, where specular scatterers are in-volved, a lower number of scatters m if inclined or ex-cessively rough surfaces can be avoided, and a highervalue of reflectivity for surfaces that are not homoge-neous. This placement can be made remotely by atrial-and-error process at the interferometer

C. Signal-to-Noise ConsiderationsThe photocurrent i due to the collected light is5

k7PqpA2

= 8hcR2 '

where - =P =q =P =

=h =C =

(5)

quantum efficiency of sensor;laser output;charge of an electron;surface reflectivity;backscattering factor;Planck's constant; andspeed of light.

The Doppler photocurrent may be obscured bybackground noise, sensor noise, and that of the laserand associated electronics. A sufficiently intensereference beam increases the Doppler photocurrentwell above the sensor noise. This procedure alsodominates the background noise leaving the purity ofthe reference beam the important factor in consid-ering noise.6 The ultimate lower limit of noise there-fore results from the shot noise (statistical fluctua-tions of the light) from the laser7 8 given by

in = (2qiB)12 , (6)

where B is the bandwidth of the sensor and accompa-nying electronics.

The highest possible SNR is determined by

SNR = -n2 8Bhcm1 l2R2 - (7)

The feasibility of this technique of vibration mea-surement can be inferred by computing this SNR forthe instrument parameters and typical test condi-tions involved. For a 1-mW beam from a He-Nelaser, reflecting from a surface of 0.2 reflectivity at100 m, and analyzed by a 15-cm aperture with a sen-sor of 0.2 quantum efficiency and 100-kHz band-

1622 APPLIED OPTICS / Vol. 14, No. 7 / July 1975

width, this SNR is 372 k/(m)1/2.- This value is im-pressive since a usable signal could be sensed even ifthere were as many as 105 separate randomly posi-tioned uniphase scatterers illuminated on a diffuse,uniformly reflecting surface. This suggests the feasi-bility of the technique even with poor surface charac-teristics. This SNR improves with higher beam out-put and lower bandwidth. (The Doppler signal of dcto 100 kHz accommodates surface velocities up to ap-proximately 3 cm/sec.) Other parameters could beadjusted as well.

The reference noise can be reduced by subtractingthe signals of two matched sensors that are posi-tioned to be nearly 180° out of phase relative to thereference beam.6 This would drastically improve theSNR.

D. Coherence Considerations

The light along the sensing beam both to the sur-face and back is required to be spatially and tempo-rally coherent. Spatial coherence of laser light re-flected from a rough surface is maintained, althoughthe reflected wavefronts can be spatially very chaotic.Temporal coherence is also maintained as long as theoptical pathlength difference between the sensingand reference beams is within the coherence length ofthe laser used. The product of coherence length andmonochromaticity AX/X is usually specified to be X/4or less, where AX accounts for drift and instabilitiesin the laser. The monochromaticity of a free run-ning, unstabilized He-Ne laser has been demon-strated to be 3 X 10-8 for a 1-min interval with AXmainly due to temperature drift.9 Exceedingly smallvalues of AX/X could be expected for very short timeintervals implying extreme coherence length andhence operating range. A short-term coherencelength of at least 1.6 km was demonstrated experi-mentally, implying a monochromaticity of 10-10 orless. The level of coherency for transient vibrationmeasurement is surprisingly lax. Air turbulenceprobably degrades the maximum short term coher-ence lengths possible.

E. Alignment Considerations

Focusing and alignment of the optical componentsare in general inexact diminishing the possible signalstrength. However, an adjustable aperture over thesensor can improve the signal strength. Equation (4)suggests a greater field of view for a reduced sensingaperture thus accommodating larger projected spotsizes due to improper focusing and alignment. Theoptimum SNR predicted by Eq. (7) can no longer beobtained because all the available light is not used.Yet with potentially high SNR's, there still can beconsiderable leeway for less than optimum signal andhigher noise levels.

The depth of focus is the usable longitudinal rangeabout a focused image of an unresolved point, inwhich the circles of confusion of the focused lightcone are no larger than some acceptable multiple Nof the Airy disk; this can be formulated as 2.44

NXR 2/A2. Thus, the depth of focus in this interfer-ometer system can be large, particularly for long-dis-tance tests. This is significant since the interferome-ter can accommodate situations involving large dis-placements and inexact focusing.

Another difficulty is the reference beam feedbackinto the laser. This noise can be eliminated by a po-larizer and a quarter-wave plate oriented to extin-guish the returning beam. An alternate method is toslightly offset the returning beam so that no feedbackoccurs. Both methods are accompanied by a loss ofsignal strength that is preferable to the noise.

A possible limitation of the technique is that ofambient vibration. If the interferometer is not ade-quately isolated from the surrounding sources of vi-bration such as machinery noise, wind, or even thesource causing the surface to move, the analysis be-comes inexact. Differential or gross motion of thecomponents is sensed in the same way as motion ofthe surface of interest.

F. . Signal Interpretation

The Doppler photocurrent id can be directly inter-preted by Eq. (3). For every half-wavelength changein displacement, id undergoes a 29r change in phase.Actually id is an FM representation of the surface ve-locity. Figure 1 demonstrates this for a 1-kHz sinus-oidal motion of a piece of Scotchlite located 6 m fromthe interferometer. The upper trace is a 2-msecsweep of the Doppler signal. The lower trace indi-cates the motion of the Scotchlite. It is actually thesimultaneous oscillator output to the vibrator.When the vibrator stops while reversing direction,the Doppler frequency drops to zero. For sinusoidalmotion the interval between these reversals repre-sents a one-half period of oscillation. The fringecount between these reversals represents the maxi-mum change in displacement in units of half-wave-lengths. The fringe density indicates velocity inunits of half-wavelengths per unit time. Thus, Fig. 1indicates a 1-kHz motion with -3-,gm peak-to-peakdisplacement. The maximum Doppler frequency of-25 kHz indicates a maximum surface velocity of-0.75 cm/sec.

Fig. 1. Oscillogram indicating Doppler shift of backscatteredlight from Scotchlite (6 m remote) vibrating at 1 kHz. Uppertrace: 2-msec sweep of interferometer signal. Lower trace: si-multaneous oscillator output to vibrator. Doppler frequency

ranges up to -25 kHz.

July 1975 / Vol. 14, No. 7 / APPLIED OPTICS 1623

-Attenuators

Mirror

Telescope

Fig. 2. Laser interferometer configuration.

The Doppler signal as described thus far does notindicate the direction of the motion; or for complicat-ed motion does not adequately indicate changesin directions (reversals as opposed to hesitations).Another sensor can be used to extract direction. Ifthe two sensors are arranged to be out of phase in thephotomixing field (optimally 900), then direction canbe obtained unambiguously. There also existschemes that shift the frequency of the referencebeam which has the added advantage of avoiding thehigh sensor and exciter noise at low frequencies.' 0

Ill. Interferometer DesignFigure 2 shows the configuration used. The beam

splitter transmits part of the light into the referencearm and reflects a nearly equal part into a reflectortelescope. The telescope simultaneously focuses thelight to a small spot on the surface of interest andcollimates the backscattered light incident on the tel-escope aperture. The collected light transmittedthrough the beam splitter photomixes with the refer-ence beam. The light sensor monitors the Dopplersignal in the resulting interference field thereby indi-cating the surface motion. Another equivalent ar-rangement consists of transposing the laser and sen-sor in Fig. 2. These designs can be adapted to anykind of telescope.

A. ComponentsThe telescope used consisted of a 15-cm, f/4 pri-

mary and a 6-mm orthoscopic eyepiece and could befocused from 1.5 m to infinity. The light sensor usedwas a PIN photodiode that could adequately senseDoppler frequencies up to 50 kHz with the amplifierused. The photodiode was used in lieu of a photo-multiplier because of its small size and capacity totolerate high light levels. Other components usedconsisted of a 4-mW laser, a 1/10 wave front-surfacemirror, and a 1/20 wave beam splitter. These com-ponents were first arranged on an optical table to testfeasibility under ideal conditions. A field versionwas then constructed by mounting all components onthe telescope tubing.

The manufacturing specifications of the laser usedincluded an rms noise level •0.3% for a passband upto 100 kHz. This noise is based on the dc sensorlevel of the beam and not the Doppler photocurrentutilized in the present application. Besides the lasershot noise, there was a repeatable 60-Hz line compo-nent that could ideally be subtracted and a sweeping

sinusoid that ranged from -30 kHz to 200 kHz thatwas probably due to three or more modes interactingwith each other. This is why limited filter passbandswere used in the tests. These-noise sources can beobviated by modifying or changing the laser and ex-citer used.

IV. Experimental ResultsThe interferometer performance was evaluated by

signal-to-noise tests using six different surfaces at 5m and 100 m. These surfaces were oscillated by a vi-brator that could be adjusted in both frequency andamplitude. The Doppler signal was adjusted torange in frequency from dc to 1 kHz or to 10 kHz de-pending on the test. The filter passband for the1-kHz test was set at 0.5-5 kHz; for the 10-kHz test itwas set at 0.5-50 kHz. A very low frequency (lessthan a Hz) and high amplitude motion were used toobtain a nearly uniform Doppler frequency for ashort time interval. The peak-to-peak amplitude ofthe Doppler signal was measured at the specified fre-quency. Noise was determined by blocking the sens-ing beam at the interferometer and measuring thefluctuations of the reference beam. Table I summa-rizes the test results.

The signal-to-noise evaluation is demonstrated byFig. 3. It shows a 20-msec sweep of both signal andnoise for a test on a coal fragment oscillating at 30Hz, 100 m from the interferometer. Here the filterpassband was set at 0.8-2.5 kHz, and the trace indi-cated that the Doppler frequency does vary through-out this range. The SNR apparently varies from-20 at 0.8 kHz to -2 at 2.5 kHz. There was a repea-table component caused by the 60-Hz line noise thatcould be subtracted for improved SNR. (Note thetwo similar negative spikes near both ends of thenoise trace which are 1/60 see apart.)

The Scotchlite reflects an intense component di-rectly back and as such is not a typical diffuse sur-face. However, its retroreflecting character was use-ful in establishing a good focus; sufficient light can becollected and seen on a white card over the eyepiecefrom distances in excess of 100 m. Proper collima-tion occurs when the spot on the card is minimized.

Fig. 3. Signal-to-noise test: 30-Hz motion of coal fragment lo-cated 100 m from the interferometer. Upper trace: 20-msecsweep of interferometer signal. Lower trace: reference beam

noise.

1624 APPLIED OPTICS / Vol. 14, No. 7 / July 1975

Table I. Observed Signal-to-Noise Ratios

Distance from interferometer 5 meters 5 meters 100 meters

Doppler frequency 1 kHz 10 kHz I kHzFilter passband .5-5 kHz .5-50 kHz .5-5 kHz

Test surface:

1. Scotchlite - red, retroreflecting, smooth. 90 dB 68 dB 52 dB

2. Sandstone - light, dull lustre, uniform fine- 50 26 22grained.

3. Oil shale - dark, dull lustre, very fine-grained. 48 26 16

4. Granite - multicolored, variable lustre, 60 38 19

medium to coarse-grained.

5. Coal - black, resinous lustre, conchoidally 59 37 19

fractured.

6. Soot - black, dull, smooth. 42 20 14

A SNR of -26 dB, using a 0.5-2-kHz passband, wasobserved outdoors at night using Scotchlite at 800m. This implies an adequate coherence length fortransient tests at least up to this distance.

The surface characteristics of reflectivity, back-scattering character, and roughness are crudely de-scribed by the parameters of p, k, and m, respective-ly, in Eq. (7). Of the three, k seems to be the mostsignificant for the data in Table I. The three high-lustre surfaces gave noticeably higher signals. Coalranked unexpectedly well, nearly matching the gran-ite, probably because of its resinous lustre. TheSNR's of the three dull surfaces by contrast seem tocorrelate with reflectivity. However, the effect ofroughness may tend to make the observed SNR's of-the dull surfaces more similar. The light, fine-grained sandstone is only 8 dB better than the black,smooth soot. The dark, very fine-grained oil shale isintermediate.

The drop in signal from 5 m to 100 m for the threehigh-lustre surfaces was -40 dB and for the three.dull surfaces was -30 dB. The drop in signal notonly depends on the inverse square attenuation withdistance, but also an interaction of changes in back-scattering character, roughness, and aperture withdistance. In addition there is the aberration causedby using a parabolic mirror for close focusing. Thereis insufficient data to quantify specifically these ef-fects.

The average 22-dB drop in SNR noted between the1-kHz and 10-kHz tests at 5 m was due more to adrop in signal amplitude than to the increase in noise[suggested by Eq. (6)] from the elevenfold increase inbandwidth. The amplifier and PIN photodiode useddid not respond well to higher frequencies as is ap-parent in Fig. 3. Another type of PIN photodiode/amplifier combination and a photomultiplier were

tried." Both gave uniform wideband frequency re-sponse and were somewhat noisier as suggested byEq. (6). These two wideband sensors had the disad-vantage of saturating at reference beam levels thatwere too low to be as usable for long-range experi-ments where the collected backscattered light was ex-tremely feeble. However, the combination usedcould tolerate up to a -1-mW reference beam andhence gave stronger Doppler signals for such low col-lected light levels.

A. Example of ApplicationFigure 4 is an oscillogram showing a 1-msec sweep

of an impulse test made of a hammerblow through alimestone slab located 6 m from the interferometer.The oscilloscope was triggered by a P-wave transduc-er near the impact point. The interferometer and

Fig. 4. Response to hammerblow across limestone slab located 6m from interferometer. Upper trace: 1-msec sweep of interfer-ometer signal. Lower trace: simultaneous output of nearby P-

wave transducer.

July 1975 / Vol. 14, No. 7 / APPLIED OPTICS 1625

another P-wave transducer simultaneously detectedthe arrival of the signal train near the same spot onthe opposite side of the slab. Note that the dead-time for the impulse to cross the slab is the same inboth traces and that the beginning of the signal traincan be roughly compared assuming the interferome-ter signal is an FM representation of the slope of theother. The P-wave transducer is actually propor-tional to acceleration. The frequency content of thehammerblow is predominantly -3 kHz and is easilyanalyzed by the Doppler signal which ranges up to 20kHz. The passband for this test was set at 1-50 kHz.

V. ConclusionsThe feasibility of remotely measuring transient

displacements of rough, unprepared surfaces by laserinterferometry has been demonstrated. The config-uration used can function as a remote transducer forvibration and impulse studies on unprepared surfac-es. Other possible applications include surface de-lineation and evaluation of surface characteristics.Accuracy of this technique is primarily limited by theeffects of surface roughness with lateral motion andambient vibration levels, both of which can often bereduced.

Further equipment refinements, particularly laserand sensor improvements, should greatly enhance

the performance of this instrument. It is potentiallyan excellent instrument for many varied applications.Applications in mining geophysics are discussed bythe authors elsewhere.' 2

The authors gratefully acknowledge the assistanceof Richard Myers with the sensor circuit and helpfuldiscussions on other aspects of this work.

References1. L. M. Barker, Exp. Mech. 12, 209 (1972).2. G. A. Massey, Appl. Opt. 4, 781 (1965).3. This technique is similar to that used in laser Doppler veloci-

meters to measure flow velocities.4. R. D. Kroeger, Proc. IEEE 53, 211 (1965).5. M. Ross, Laser Applications (Academic, New York, 1971),

Vol. 1, Chap. 2, pp. 119-129.6. M. Ross, Laser Receivers (Wiley, New York, 1966), Chap, 3,

pp. 111-112.7. B. M. Oliver, Proc. IRE 49, 1960 (1961).8. J. R. Kerr, Proc. IEEE 55, 1686 (1967).9. K. D. Mielenz et al., J. Opt. Soc. Am. 56, 156 (1966).

10. D. A. Neish et al., U.S. Patent 3,482,436 (1969).11. Figures 1 and 4 are oscillograms using the wideband PIN

photodiode/amplifier combination; Fig. 3 is an oscillogramusing the limited bandwidth combination utilized in the sig-nal-to-noise evaluation.

12. G. L. Fitzpatrick et al., in Proc. Symp. of Modern Innovationsin Subsurface Exploration, Transportation Research Board,National Research Council, Washington, D.C. (in press).

Tutorial Seminar onOPTICS IN DIAGNOSTIC MEDICINE

January 9-11, 1974

ABSTRACTS OF TECHNICAL PAPERS

Abstracts of the papers presented at this meeting are now avail-able. The fifty-one papers cover such subjects as:

fiber optics;image recording, storage- and Drocessing;light microscopy;Medical holography:nuclear medicine;thermography;ultrasound.

The text is offset from author's copy: 60 pages, 6 x 8 inches.saddle stitched. Price ... Prepaid $3.00; Invoice $3.50Order from:

Optics in Diagnostic Medicine, Optical Society of America2100 Pennsylvania Avenue, N.W., Washington, D. C. 20037

1626 APPLIED OPTICS / Vol. 14, No. 7 / July 1975