high-amplitude vibrations detection on rough surfaces using a photorefractive velocimeter

$: High-amplitude vibrations detection on rough surfaces using a photorefractive velocimeter$
*Corresponding author. Tel.: #33-1-6935-8750; fax: #33-1-6935-8700.E-mail address: [email protected]'fr (Ph. Delaye).

Optics and Lasers in Engineering 33 (2000) 335}347

High-amplitude vibrations detection on roughsurfaces using a photorefractive velocimeter

Ph. Delaye*, S. de Rossi, G. RoosenLaboratoire Charles Fabry de l+Institut d+Optique, Unite& mixte de recherche 8501 du CNRS, Bat. 503,

Centre Scientixque d+Orsay, BP.147, 91403 Orsay Cedex, France

Received 8 May 2000; accepted 7 August 2000

Abstract

We present and characterize theoretically and experimentally a photorefractive velocimeter.This device, based on two-wave mixing in a rapid photorefractive crystal, measures theinstantaneous velocity of a vibrating target. It is particularly adapted to the measurement ofhigh-amplitude (as high as some mm) low-frequency (until some kHz) vibrations. Instantaneousvelocity as high as 25 mms~1 are expected to be measured with common photorefractivesemiconductors and CW lasers. ( 2000 Elsevier Science Ltd. All rights reserved.

Keywords: Vibration measurement; Photorefractive e!ect; Two-beam coupling; GaAs

1. Introduction

Laser vibrometry is now widely spread in the "eld of non-destructive testing, for thedetermination of the vibration spectra of objects, as well as for ultrasonic testing ofmechanical parts [1]. Vibration sensors are commercially available. Most of thesesensors are based on coherent detection schemes (homodyne or heterodyne detec-tions). Although very sensitive, these sensors lose their e$ciency when used withscattering surfaces that often prevents the measurements, unless the object is "rstpolished which is not possible most of the time. We have recently studied anddeveloped an ultrasonic sensor based on two-wave mixing in a dynamic holographicmedia [2,3]. This sensor, allows to make sensitive ultrasound vibration measurementson scattering objects. Nevertheless, this device perfectly adapted to the measurement

0143-8166/00/$ - see front matter ( 2000 Elsevier Science Ltd. All rights reserved.PII: S 0 1 4 3 - 8 1 6 6 ( 0 0 ) 0 0 0 5 6 - 7

of the sub-nanometer ultrasonic vibrations, does not work anymore as soon as thevibration amplitude becomes greater than a fraction of the wavelength used, becauseof the erasing of the hologram.

In this paper, we will describe a photorefractive interferometer derived from thephotorefractive ultrasonic sensor, that keeps its main advantage, i.e. the possibility towork with speckled beams, and extends its measurement ability to the high amplitudevibrations (micrometric vibrations). In this new working regime, the device measuresthe instantaneous velocity of the vibrations and thus works as a velocimeter. We willdescribe theoretically the operation of the photorefractive velocimeter in the followingsection, showing its performances. Then we will present its experimental implementa-tion and "rst results that clearly demonstrate its di!erentiating behavior. The experi-mental results will also be compared to the theoretical predictions.

2. Theoretical model

In its principle, the device is based on a two-wave mixing experiment in anholographic material. This setup has already been used in a great number of experi-ments, to develop a photorefractive ultrasonic sensor [2,3], a double-exposure holo-graphic camera [4], or for the characterization of materials [5]. The tested point onthe object is illuminated by a laser source. The signal wave re#ected and scattered bythe object is sent on the dynamic holographic material together with a coherent pumpbeam. These beams write in the holographic material, a hologram of the wavefrontstructure of the signal beam. This hologram is read by the pump beam that creates inthe direction of the transmitted signal beam, a local oscillator that has exactly thesame wavefront structure as the signal beam, whatever be the structure. These twobeams will then interfere in a wide "eld of view homodyne detection scheme, what willtransform the phase modulation imprinted on the signal beam (and caused by thevibration of the object) into an intensity modulation. The phase shift between thesignal and the local oscillator can be chosen experimentally to be equal to p/2 (beamsin quadrature) in order to optimize the operation of the sensor.

We consider here, an object vibrating at low frequency compared to the cut-o!frequency of the dynamic holographic material (i.e. the inverse of the response time ofthe material). In this regime, the hologram follows the vibrating interference patternwith, nevertheless, a delay. This means that the local oscillator will carry someinformation on the vibration, as it comes from the di!raction of a stationary beam ona mobile hologram.

From a general point of view, we will show that in this low-frequency regime thetwo-wave mixing setup works as a velocimeter. If the phase modulation imprinted onthe signal beam is given by a vibration of amplitude d(t), we measure a signalproportional to the instantaneous velocity v(t) of the surface. We will also show thatthe saturation of the signal at high displacement, observed at high frequency andlinked to the partial erasure of the hologram, still exists but is less stringent comparedto the photorefractive ultrasonic sensor. This saturation is linked to the response time

336 Ph. Delaye et al. / Optics and Lasers in Engineering 33 (2000) 335}347

Fig. 1. Scheme of a photorefractive-beam combiner based on the anisotropic di!raction con"guration. PBSis a polarizing beam splitter, PC is the photorefractive crystal, j/2 and j/4 are half and quarter-wave plates,respectively.

of the dynamic holographic material, and at a "xed frequency it is all the more highthan this short response time.

2.1. Beam coupling in the anisotropic diwraction conxguration

For presenting the theoretical model, we will consider a photorefractive crystal usedin the anisotropic di!raction con"guration [3] (Fig. 1). In this con"guration, twos-polarized signal and pump beams, interfere to create the grating in a photorefractivecrystal in the di!usion regime. The pump beam di!racts into a p-polarized localoscillator. Then the two beams (s-polarized signal and p-polarized local oscillator) aresent on a quarter-wave plate with axes oriented along the polarization axes. This waveplate induces a p/2 phase shift between the two beams that are either in or out of phase(depending on the sign of the index grating) at the output of the crystal. The two crosspolarized beams are then sent on a 453 polarizing beamsplitter. The beams issuedfrom the beamsplitter are sent on two detectors whose signals are subtracted ina di!erential ampli"er. We choose this con"guration because it is easy to implementand it does not require an external applied electric "eld. It is also di!erential what canbe interesting to eliminate an eventual intensity noise of the laser.

For simplicity of the study, we neglect the absorption of the crystal. We also placeourselves in the undepleted pump approximation, which means that the pump beamamplitude stays constant in the thickness of the crystal (condition usually ful"lledwhen the pump beam intensity is much higher than the signal beam intensity). In sucha case, the equation that governs the di!raction of the pump beam E

14on the grating

to form the p-polarized signal beam E41

, is:

RE41Rx "

cG

*n(t)E14

, (1)

where c is the photorefractive gain in amplitude (real in the di!usion regime weconsider), *n(t) is the index grating variation whose temporal evolution is given by

Ph. Delaye et al. / Optics and Lasers in Engineering 33 (2000) 335}347 337

"rst-order kinetics, with a real time constant q, and a steady state that is proportionalto the fringe modulation with a proportionality constant G:

R*n

Rt #

*n

q"

G

qm(t)

2. (2)

The fringe modulation m, is given by the interference of the s-polarized components ofthe pump beam E

14and of the signal beam E

44(with the corresponding intensities

I14

and I44):

m(t)"2E

44(t)EH

14I44(t)#I

14

. (3)

If the target vibrates with an amplitude d(t), it induces on the signal beam, a phasemodulation u(t)"4p d(t)/j, leading to an amplitude of the s-polarized component ofthe signal beam (constant on the whole crystal thickness as absorption is neglected)equal to E

44(t)"E

44e*r(t). Thus, the vibration generates a displacement of the fringe

pattern according to a law:

m(t)"m0e*r(t). (4)

Eq. (1) is solved very easily for a crystal of thickness x, using the boundary conditionE41

(x"0)"0,

E41

(t)"cxG

*n(t)E14

. (5)

To this equation we have to add the value of the other polarization component of thesignal beam E

44(t).

After the crystal, the signal beam passes a quarter-wave plate where the p-polariza-tion component is p/2 phase shifted with regard to the s-polarization. Then bothpolarization components are sent on a 453-polarizing beamsplitter where they createtwo beams having the amplitudes:

E1(t)"

1

J2CE44e*r(t)#e*p@2

cxG

*n(t)E14D,

E2(t)"

1

J2CE44e*r(t)!e*p@2

cxG

*n(t)E14D.

(6)

Both beams having intensities

I1(t)"DE

1(t)D2"

1

2CKicxG

*n(t)E14 K

2#DE

44e* r(t)D2#2ReCi

cxG

E14

EH44

*n(t) e~* r(t)DDI2(t)"DE

2(t)D2"

1

2CKicxG

*n(t)E14 K

2#DE

44e* r(t)D2!2ReCi

cxG

E14

EH44

*n(t) e~* r(t)DD(7)


Both beams are sent on two detectors whose signals are subtracted to give anelectrical signal S(t). The experiments are performed in the general condition of use ofa two beam-coupling setup, i.e. the intensity of the pump beam is much larger than theone of the signal beam (I

44;I

14). Using relations (3) and (4), the signal S(t) can thus be

written as (c is real):

S(t)"2cxI44Re[S

C(t)], (8)

with

SC(t)"2i

*n(t)

m0G

mH(t)mH

0

. (9)

Using Eq. (2) together with relation (4), we easily show that Sc(t) is governed by the

following di!erential equation:

RSCRt "!S

CAiRuRt #

1

qB#i

q(10)

whose solution is

SC(t)"

i

qe~* r(t) e~t@qAP

t

0

e* r(t{ ) e~t{@qdt@#CB. (11)

Constant C is given by the initial condition. At t"0, SC(t)"S

C0, which gives

C"!i qSC0

e* r(0). In the case of a vibration measurement, we usually write thegrating until steady-state is reached (i.e. *n(0)"*n

45"(Gm

0e*r(0))/2) and at time

t"0 we apply the vibration with a phase origin u(0). Thus, according to relation (9)we have S

C0"i.

Making the variable change t@"t!u in the integral, we "nally arrive at

SC(t)"

i

q Pt

0

cos(u(t!u)!u(t)) e~u@qdu!1

q Pt

0

sin(u(t!u)!u(t)) e~u@qdu

#SC0

e~* (r(t)~r(0)) evt@q (12)

2.2. Case of a low-frequency signal

Until now, no supposition has been made concerning the phase-modulation char-acteristics. The above expressions are thus valid whatever be the nature of the phasemodulation (high or low amplitude, or high or low frequency). We now placeourselves in the case we are interested in, i.e. we consider a modulation with a lowfrequency compared to the inverse of the response time of the dynamic-holographicmaterial. This means that the signal varies with a time period much greater than theresponse time of the material. We thus have t<q. As e~u@q is non-negligible for valuesof u close to q only, that means that we can consider that u;t and thus t!u+t, in


the integrals of relation (12). This allows to make a Taylor development of u(t), thatgives

u(t!u)!u(t)+!uRuRt . (13)

The two integrals of relation (12) can then be explicitly calculated

Pt

0

sinA!uRuRt Be~u@qdu"

!q2 Ru/Rt#q e~t@q[q Ru/Rt cos(t Ru/Rt)#sin(t Ru/Rt)](1#q2 (Ru/Rt)2)

(14a)

and

Pt

0

cos(!u Ru/Rt) e~u@qdu"q#q e~t@q[!cos(t Ru/Rt)#q Ru/Rt sin(t Ru/Rt)]

(1#q2 (Ru/Rt)2) .

(14b)

As we consider that t<q, all the terms in e~t@q in relations (12) and (14) can beneglected. The signal S(t) then becomes:

S(t)"2 cx I44

q Ru/Rt(1#q2 (Ru/Rt)2) . (15)

This expression shows that the signal of the two-wave mixing interferometer in thelow-frequency domain depends on the derivative of the phase modulation signal only.If now, the amplitude of the phase-modulation derivative is small enough, such thatq2 (Ru/Rt)2;1, the expression simpli"es and becomes

S(t)"2cxq Iss

RuRt . (16)

The signal is directly proportional to the instantaneous velocity of the target. Thetwo-wave mixing interferometer thus works as a velocimeter in the low-frequencydomain.

2.3. Limit of the linear regime

The above point was already established earlier but only in the case of a low-amplitude phase variation (u(t);p/2) [2,5]. We see that the previous results applymore largely to large-amplitude modulation, as soon as q2 (Ru/Rt)2;1. For a phasemodulation induced by a surface displacement, we have u(t)"4p d(t)/j andRu/Rt"4p v(t)/j. The condition becomes

A4pqv(t)

j B2;1 (17)


A good limit for such a condition is

A4pqv(t)

j B(1

3(18)

which gives

v(t)(v-*.

"

1

12p

jq. (19)

The performances of the velocimeter, i.e. the maximum velocity measurable in thelinear regime, will be controlled by the response time of the crystal only. The shortestresponse time will be the best. Considering usual photorefractive crystals used withCW laser, response time of the order of a microsecond should be reachable with themore rapid crystals such as the semiconductors. For a wavelength of 1.06lm anda crystal having a response time of 1 ls, the limit velocity is v

-*."28mms~1.

In the case of a sinusoidal displacement d(t)"d0

sin (2p ft), we havev(t)"2pfd

0cos(2p ft), and condition (19) becomes

f d0(

1

24p2

jq

. (20)

For the same experimental parameters (j"1.06lm and q"1 ls) this leads toa maximum measurable displacement of 4.5lm at the frequency of 1 kHz, or 4.5mmat a 1Hz frequency.

2.4. Numerical simulation

In order to have an idea of the characteristics of the photorefractive velocimeter, wecalculate the response of the system to di!erent incident signals. For this calculationwe use expression (8) with S

C(t) given by the solution of the di!erential equation (10)

for a given displacement d (t). The parameters of the crystal are a response timeq"1 ls, a photorefractive gain in amplitude c"0.3 cm~1, a thickness of 1 cm. Thecalculated values are normalized to the incident signal beam intensity. The "rstcalculation allows to show the derivative behavior of the velocimeter. We see in Fig. 2,the response to a sinusoidal and a sawtooth periodic vibration. For the sinusoidalvibration, the response is phase shifted by a quarter of a period, i.e. the p/2 phase shiftof a cosinusoidal response. Moreover, no deformation of the signal appears, despitethe high amplitude of the displacement (1mm). For the sawtooth vibration theresponse is a square waveform, corresponding to the derivative too.

In Fig. 3 we have plotted the response for a "xed amplitude (1mm) as a function ofthe frequency of the signal. As expected, the signal amplitude increases with thefrequency until a saturation of the signal that begins to appear at a frequency of 10Hz,and that becomes clearer at higher frequencies, with a signal that is no more thederivative of the initial signal (curves at 100 and 1000 Hz). Then, we calculate theresponse as a function of the amplitude of the vibration at a "xed frequency of 1 kHz


Fig. 2. Numerical simulation of the velocimeter response, for a sinusoidal (plain lines) and a sawtoothdisplacement (dashed lines) of 1mm amplitude at 1Hz. Bold line represents the calculated signal and thinlines the initial displacement.

Fig. 3. Numerical simulation of the velocimeter response, for a sinusoidal displacement of 1mmamplitude at di!erent frequencies (the abscissa are normalized to the frequency for comparison of thedi!erent curves).

(Fig. 4). The possibility to linearly measure high-amplitude vibrations is clearly seen(curves for a displacement amplitude up to 1lm). A saturation at amplitude between1 and 10lm corresponding to the value given by relation (20) appears, with a defor-mation of the signal that increases with higher vibration amplitudes due to theapparition of higher harmonics.

We also calculate the RMS value of the signal observed for a sinusoidal vibration ofincreasing amplitude (from 1 nm to 10mm), as a function of the frequency (Fig. 5). Atlow amplitude of the vibration (below 100nm), the signal increases linearly asa function of the frequency, until a cut-o! frequency of 105Hz (corresponding to the1ls response time). Above this cut-o! frequency the signal is constant as the ampli-tude of the vibration is not su$cient to erase the grating. This regime is the one that


Fig. 4. Numerical simulation of the velocimeter response, for a sinusoidal displacement at 1 kHz. Thecurves at the lowest amplitudes (0.01}1lm) are compensated for their amplitude. They superpose exactlyshowing the linearity of the response of the velocimeter.

Fig. 5. Frequency response of the photorefractive velocimeter, for di!erent amplitude of the sinusoidaldisplacement between 1 nm and 10 mm. The presented signal is the RMS value of the calculated temporalresponse of the velocimeter.

corresponds to the photorefractive ultrasonic sensor we previously studied [2,3].With higher amplitude of vibrations (above 100 nm), the signal increases linearly withthe frequency (corresponding to a signal proportional to the velocity) until a max-imum value. When the frequency continues to increase, the RMS signal decreases asthe displacement is su$cient to partially erase the grating (see Fig. 3) and themeasured signal is no more sinusoidal. We can note that the maximum RMS signal isindependent of the displacement amplitude (except at very low-displacement ampli-tude), it is just the frequency at which this maximum occurs that depends on thedisplacement.


Fig. 6. Photo of the experimental setup used in the study, with the beams indicated. PBS : polarizing beamsplitter, PM piezomirror, j/4 : quarter wave plate, j/2: half-wave plate, Det : di!erential detector.

3. Experimental characterization

We have experimentally implemented the photorefractive velocimeter, in order toperform a "rst characterization of its performances, mainly focused here to a veri"ca-tion of its temporal response and its ability to measure high-amplitude vibrations.

3.1. Experimental set-up

The set-up is shown in Fig. 6. A CW Nd :YAG laser emitting at 1.06lm is split, toform a high-power pump beam, and a signal beam that is sent to a piezomirror. Theretrore#ected beam is mixed with the pump beam into the photorefractive crystal. Thephotorefractive crystal is a semi-insulating undoped GaAs crystal, used in the anisot-ropic di!raction geometry, i.e. a grating vector along the (1 1 0) direction and theincident beams vertically polarized along (0 0 1) axis. The crystal is antire#ectioncoated and has a thickness of 1 cm. It is used with a grating spacing of the order of2lm, the photorefractive gain is 0.13 cm~1. After the crystal, the signal beam passesthrough a quarter-wave plate with axes oriented along the direction of the transmittedbeam polarization and of the cross-polarized di!racted beam. The beam is then sent


Fig. 7. Experimental response of the photorefractive velocimeter to a sinusoidal (black curves) anda sawtooth (grey curves) phase modulation of 0.6Hz frequency. The dashed lines represent the displacementof the piezomirror that creates the phase modulation.

on a 453 polarizing beamsplitter. The two beams issued from the beamsplitter are senton two detectors connected to a simple low-frequency di!erential ampli"er.

The piezomirror is calibrated and its displacement is measured by an internalsensor which gives a reference signal. With this piezomirror, displacement amplitudesof 10lm are obtained until frequencies of about 60Hz. These performances are clearlyinsu$cient to characterize the ability of the setup to measure very high-amplitudevibrations, and see its ultimate performances. Nevertheless, this piezomirror is su$-cient to realize a "rst demonstration of the working principle of the photorefractivevelocimeter. The characteristics of the crystal (especially a relatively long responsetime) are chosen in order to see the di!erent operating regimes of the velocimeter(linear regime and saturation regime), with this piezomirror.

3.2. Experimental results

The "rst experiment (Fig. 7) allows to illustrate the derivative behavior of thephotorefractive velocimeter. As expected, the sawtooth signal is transformed intoa square-shaped periodic signal, whereas the sinusoidal signal is p/2 phase shiftedcompared to the initial displacement. We also see that there is no deformation of thesinusoidal response despite the high amplitude of the displacement (2lm), showingthe ability for the velocimeter to measure high-amplitude signals without the wrap-ping that is seen in an homodyne detection. For the frequency response of thevelocimeter, we see (Fig. 8) that the signal increases with the frequency and saturatesfor high frequencies (6Hz curve), before being distorted (62Hz curve). The theoreticalcurves calculated with relations (8) and (10), correspond to the experimental oneswhen the response time of the crystal (i.e. the only adjustable parameter) that governsthe shape of the curves is about 200 ls. This gives an idea of the response time of thecrystal in this speci"c experimental arrangement. The saturation begins to occuraround 0.6Hz for the 11.4lm displacement amplitude used. This corresponds to whatcan be found using Eq. (20).


Fig. 8. Experimental response (A) of the photorefractive velocimeter to a sinusoidal phase modulation of"xed amplitude (11.4lm, except for the 62Hz signal corresponding to a 9.9lm displacement amplitude)having an increasing frequency. In (B) are shown the calculated responses of the photorefractivevelocimeter.

Fig. 9. Experimental response (A) of the photorefractive velocimeter to a sinusoidal phase modulation of"xed frequency (60Hz) and of increasing amplitude. In (B) are shown the calculated responses of thephotorefractive velocimeter.

For the response as a function of the amplitude, there is also a good accordancebetween calculated and experimental curves (Fig. 9). For the "xed frequency of 62 Hzused, the saturation occurs around 144nm which still corresponds to what is given byrelation (20). Here again the signal is distorted at higher displacement.

Finally, we perform a comparison of the vibration signal obtained with plane waveswith the signal obtained with the speckled waves re#ected by a rough surface (Fig. 10).The surface is imaged in the photorefractive crystals and imaged again on thedetectors. No di!erence in the signal is observed when changing the structure of thewavefront, proving that our system works with rough surfaces. On the curves we alsosee that when the response time of the photorefractive crystal varies (by changing thepump beam illumination), we change the form of the response signal, going froma linear regime (for q"15ls) to highly non-linear regime (for q'200ls). Comparedto the curves of Fig. 9, we see that for a similar frequency the saturation occurs at anamplitude one order of magnitude higher, corresponding to a response time reducedby one order of magnitude.


Fig. 10. Experimental responses of the photorefractive velocimeter to a sinusoidal phase modulation of"xed frequency (50Hz) and "xed amplitude (4.5lm) for di!erent response time of the photorefractivecrystal. In (A) the object is a mirror that re#ects a plane wave, in (B) the target is a rough surface that re#ectsa speckled wave. The di!erent amplitude of the signals is due to the di!erent re#ected power. Both curvesare in good accordance with theoretical curves not presented here.

4. Conclusion

We present the "rst characterization of a new holographic vibration sensor, thephotorefractive velocimeter. This sensor measures linearly the instantaneous velocityof high-amplitude (several lm), low-frequency (0}1 kHz) vibrations on scatteringsurfaces. We develop a simple theoretical model that perfectly explains the behavior ofthe velocimeter. The performances of this velocimeter are controlled by the responsetime of the photorefractive crystal. Using rapid photorefractive crystals like GaAs orCdTe will allow to measure vibration velocities as high as 25 mms~1. The proof ofprinciple of this photorefractive velocimeter is successfully demonstrated. We will nowoptimize its performances, to be able to measure such high vibration velocities, witha compact and low cost system.

References

[1] Monchalin JP. IEEE Trans Ultrason Ferroelectrics Frequency Control 1986;UFFC-33:485.[2] Delaye Ph, de Montmorillon LA, Roosen G. Opt Commun 1995;118:1549.[3] Delaye Ph, Blouin A, Drolet D, de Montmorillon LA, Roosen G, Monchalin JP. J Opt Soc Am

1997;14:1723.[4] Labrunie L, Pauliat G, Launay JC, Leidenbach S, Roosen G. Opt Commun 1997;140:119.[5] Sugg B, Shcherbin KV, Frejlich J. Appl Phys Lett 1995;66:3257.


high-amplitude vibrations detection on rough surfaces using a photorefractive velocimeter

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