Design of a Michelson interferometer for the measurement of electrostrictive strains

Oprim & Laser Technology,

Vol. 28, No. 6, pp. 4X1-484.

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Design of a Michelson interferometer the measurement of electrostrictive

for strains

G. KLOOS In the field of quadratic electrostriction of non-ferroelectric materials, efforts to optimize interferometers for the accurate measurement of small strains are being undertaken because the displacements that have to be measured are very small (1 0-13m). Here, a design for a Michelson interferometer is proposed that combines a double-face detection scheme with a null method. Copyright @ 1996 Elsevier Science Ltd. KEYWORDS: interferometers, Michelson interferometer, small-strain quadratic electrostriction, null method, double-face detection

measurement,

In this article, a modular measurement system is proposed that combines the compensation method presented by Bohaty with features of the instrument proposed by Van Sterkenburg.

introduction Quadratic electrostriction is the strain response of a dielectric material that is proportional to the square of an applied electric field and does not originate from electrostatic stresses (Maxwell stresses). The accurate measurement of electrostrictive coefficients of nonferroelectric materials is a challenging task from the point of view of optical engineering because the strains that have to be resolved are very small. Different solutions to this problem have been proposed in the literature: Zhang ef al,‘>2implemented a kind of automatic compensation for spurious vibrations in their Mach-Zehnder interferometer by using one laser beam to measure two opposite surfaces of the dielectric sample. The interferometer employed by Van Sterkenburg et 01.~ consists of two Michelson interferometers that measure the dilatations of both surfaces.

Interferometer

design

Employing a sophisticated polarization-optical scheme, Zhang et al.’ designed an instrument where compensation for spurious vibrations of the back face was ‘built in’. The main idea that underlies Van Sterkenburg’s design3 is the use of two independent Michelson interferometers to measure the strain at the top and at the bottom of the sample. The signal recorded at the top is the superposition of a signal originating from the sample itself and a contribution caused by vibrations of the sample holder. These vibrations are induced by the high voltage. They gain importance if the strains to be measured are very small. It is then assumed that the information on this spurious effect is contained in the strain signal detected at the bottom. Under this condition, the unwanted effect of these vibrations can be eliminated by subtracting both signals, which is done electronically in Van Sterkenburg’s scheme.

The authors’-3 point out that the back-face motion affects the measurement of small electrostriction coefficients and has to be compensated for by doubleface detection. A significant improvement in the performance of a Michelson interferometer was achieved by Bohaty4. The design concept, which is mainly based on a null technique5, has also been applied in recent work on the quadratic electro-optic effec@. A comparable compensation method for spherical Fabry-Perot interferometers was put forward by Bruins ef aL7

It is difficult to decide which of these design concepts is superior, because both schemes have their advantages. The core of the optical configuration proposed in this article (see Fig. 1) is formed by two Michelson interferometers, as in Van Sterkenburg’s device. For double-face detection the sample can either be mounted on a transparent support or on a sample holder, with a spacing to allow the laser beam to pass. Using a gold filament for acoustic insulation, an alternating voltage in the kilohertz range with a typical amplitude of 1 kV rms can then be applied to the sample. The system is based

The author is in the Department of Electrical Engineering, Eindhoven University of Technology, Den Do&h 2, 5600 MB Eindhoven, The Netherlands. Received 16 October 1995. Revised 14 December 1995.

481

Michelson

482

interferometer

for electrostrictive

strain measurement:

G. Kloos

Inverter

B D

Lock-in Amplifier

Preamplifier

Piezoelectric

He-Ne Laser I

I

-u

reference

Oscillator

Qusrterwave

He-Ne Laser

i i i

i i II

P D

Fig.1 Optical configuration and associated electronics

Lock-in

Amplifier

Michelson

interferometer

for electrostrictive

on the concept of modulation interferometry. The homodyne method applied to dilatometry can be described making use of the equation of state that links the induced strain E and the applied electric field E E=dE+y’E’

(1)

where d is the piezoelectric coefficient and y’ is the apparent coefficient of quadratic electrostriction. In the general case, a bias voltage EO and distortions can be superimposed on the input signal E,cosot E = E. + E, cos ot + E2w cos 2mt + E3w cos 3mt Combining

both equations

(2)

leads to

i=dE++E;+;y’(E;+E;,+E&,) + [dE, + Zy’EoE,

+ y’EwE2,

+ [dEz, + 2y’EoEz,,, + y’E,Ej, f...

+ -y’Ezi,Ejw]cosmt + ;71E;]cos2rut

...

(3)

The interferometer is kept at the point of maximum sensitivity of the transfer function linking intensity and optical path-length difference and the changes of optical path length that have to be detected have such a small amplitude that the transfer function can be considered linear in a good approximation. The effective voltage Vex measured using the photodiodes can then directly be related to the effective dilatation

of the sample where x, is the thickness of the sample. For a typical instrument3 the corresponding relationship is Veff = Ax,ff x lo8 V m-’ If the relationship U = XEE is used for the applied voltage U and the distance of the electrodes xE , and the input voltage is assumed to be free of distortion ( UO = 0, U,, = 0, U30 = 0), Equation (3) can be rewritten in terms of effective values Ax,ff = $

+‘:;,,,

+ dXEUw,eK cos ot

( + +$u;,+,

(4) cos 2O.u >

The small electrostrictive strains are measured by the lock-in technique. The frequency of the sinusoidal signal from the function generator is doubled to measure the quadratic effect. If the frequency doubler is not used, the instrument can also serve to determine piezoelectric constants. A separate read-out of both interferometers is preferred in order to avoid the noise caused by the two lasers from being superimposed. Another reason is that a null method can then be applied to each interferometer. The intensity is sinusoidally dependent on the optical path length difference. To keep the interferometer at the point of highest sensitivity and to reduce the influence of

strain

measurement:

G. Kloos

483

thermal fluctuations and acoustic noise, a feedback loop is used in interferometers for small-strain measurements8-9. It should operate at a frequency far below the modulation frequency in order not to affect the measurement. An additional beam-splitter can be used for electrical insulation of the measurement signal from the feedback circuit. Concave lenses are placed in front of the photodiodes. Van Sterkenburg et aL3 gave an account of the noise limitations of a double Michelson interferometer. While the above-mentioned electronic feedback can effectively be employed to suppress acoustic noise and noise caused by low-frequency changes of temperature, mode beating of the laser is found to be a major factor deteriorating the stability of an electrostriction measurement. Careful choice of the light source is therefore strongly recommended. A formula has been derived”,” to decide the question of whether the measurement can be affected by thermoelastic interaction in the dielectric sample. In some materials, like alkali halides for example, this effect can be neglected”. The polarization-optical components introduced in the optical path3 are not essential for the operation of the instrument but improve the signal-to-noise ratio by providing an additional read-out and by minimizing the DC component in the signal. (The author of this article failed to verify experimentally the elimination of laser noise that has been reported.) The quarter-wave plate serves to polarize elliptically the beam coming from the linearly polarized laser. Reflection changes the sign of the helix describing elliptical polarization. After the light beam has passed the quarter-wave plate again, it is linearly polarized in a direction perpendicular to the original one, so that its intensity is fully reflected by the polarizing beam-splitter. In the electronic signal path, an inverter is used because the AC components of the signals from both photodiodes have a phase difference of 180”. If both signals are added then the DC part originating from noise is reduced and the AC component is doubled. In experimental practice the amplitudes of both signals are not equal. Therefore, it is necessary to determine appropriate amplification factors before measurement, by scanning the intensity curve (manually or automatically) with the piezoelectric transducer. Kuwata et al. and Bohat$ have presented optical set-ups that allow one to apply a null technique, which is wellknown from electrical measurement bridges (Wheatstone bridge) to Michelson interferometers. A crystalline or ceramic sample with known piezoelectric or electrostrictive properties is introduced in the reference arm of the interferometer. In the case of a piezoelectric reference it has to vibrate at the double frequency. The amplitude and phase of these vibrations are then tuned in such a way that the read-out of the interferometer is brought to zero. The unknown electrostrictive coefficient can now be calculated from the applied voltages, the thicknesses and the known material constant of the reference sample. Besides the advantage of a null method, where disturbances which affect both arms in the same way are eliminated, this

484

Michelson

interferometer

for electrostrictive

procedure has the advantage that the sign of the observed dilatation can be concluded from the phase of the voltage that is needed for compensation. Contrary to the experimental arrangement proposed by Van Sterkenburg3, it does not seem advantageous to mount the sample and reference close to each other on one sample holder if the reference sample is vibrating. It is possible that these vibrations might deteriorate the signal of the sample itself. Typical dimensions of the optical set-up proposed in this article are 80 cm x 80 cm. A more compact design could be realized using a fibreoptic approach. This question is now being investigated. Conclusions

Besides the polarization-optical scheme, no additional complications are introduced into the optical set-up so that there are reasons to expect improved operation characteristics.

G. Kloos

References 1

2

3

4

5

6

The Michelson interferometer that has been described is a combination of a double-face detection scheme with a compensation method. A modular design was considered advantageous because it allows step-by-step testing. The use of two independent and symmetric interferometers makes a multitude of consistency checks possible that help to ensure the reliability of the values measured.

strain measurement:

Zhang, Q.M., Jang, S.J., Cross, L.E. High-frequency strain response in ferroelectrics and its measurement using a modified Mach-Zehnder interferometer, J Appl Phys, 65 (1989) 2807-2813 Pan, W.Y., Cross, L.E. A sensitive double beam laser interferometer for studying high-frequency piezoelectric and electrostrictive strains, Rev Sci Instrum. 60 (1989) 2701-2705 Van Sterkenburg, S.W.P., Kwaaitaal, Tl, Van den Eijnden, W.M.M.M. A double Michelson interferometer for accurate measurements of electrostrictive constants, Rev Sci Instrum, 61 (1990) 2318-2322 Bohati, L. Dynamisches Verfahren zur Messung von elektrostriktiven und elektrooptischen Effekten. Beispiel: Tinalkonit Na2B40s(OH)_, .3HzO, Z Krisfallogr, 158 (1982) 2333239 Kuwata, J., Uchino, K., Nomura, S. Electrostrictive coefficients of Pb(Mgt,sNbz,j)Os ceramics, Jpn J Appl Phys, 19 (1980) 2099-2103 Kucharczyk, W., Gunning, M.J., Raab, R.E., Graham, C. Interferometric investigation of the quadratic electro-optic effect in KDP, Physica B, 212 (1995) 5-9 Bruins, D.E., Garland, C.W., Greytak, T.J. New interferometric method for piezo-electric measurements, Rev Sci Instrum, 46 (1975) 1167-1170 Uchino, K., Nishida, S., Nomura, S. A highly-sensitive ac interferometric dilatometer, Jpn J Appl Phys, 21 (1982) 596-600 White, R.G., Emmony, D.C. Active feedback stabilisation of a Michelson interferometer using a flexure element, J Phys E Sci Instrum, 18 (1985) 658-663 Preu, P., Haussiihl, S. Quadratic electrostrictive effect in NaCl and KAI(S04). 12HzO derived from stress dependence of dielectric constants, Solid State Commun, 45 (1983) 619 Kloos, G. The correction of interferometric measurements of quadratic electrostriction for cross effects, J Phys D: Appl Phys, 28 (1995) 939-944

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