Vibration measurement by vibrating-plate holograms

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OPTICS

VIBRATION

COMMUNICATIONS

MEASUREMENT

January

BY VIBRATING-PLATE

1973

HOLOGRAMS

Seiichiro KAWASE?. Toshio HONDA and Jumpei TSUJIUCHI Imaging Science

and Engineering Laboratory, Tokyo Institute 0-Okayama, Meguro-ku, Tokyo, Japan

Received

If a object, reduced method

7 November

1972

hologram of a vibrating object is recorded on a photographic plate which vibrates synchronously with the the reconstructed image produces equi-amplitude fringes of the object vibration, the amplitude of which is by a constant determined from the amplitude of the plate vibration. This phenomenon can be used as a to extend the measurable range of vibration amplitudes, and some experimental results are shown.

A time-averaged hologram [ 1] of a vibrating object produces equi-amplitude fringes superposed on the reconstructed image, and this can be used as a powerful means of vibration measurements. But, as the irradiance of these fringes is proportional to ]Ju(cm)l* (where Jo is the zeroth order Bessel function of the first kind, m the amplitude of vibration and c a constant given by the configuration of the apparatus), bright fringes become darker with increasing amplitude m and the useful measurement range is limited by the number of fringes which have sufficient visibility to be observed. Various methods for a temporal modulation hologram, such as stroboscopic illutnination [2] of the object and phase modulation [3] of the reference wave, are tried to extend the useful range of measurement in that case. The method proposed here has the same performance as the phase modulation of the reference wave by using a simple mechanism for vibrating a hologram plate. As shown in fig. 1, suppose a hologram plate H which vibrates synchronously with the vibration of an object 0 to be measured and has a rectangular coordinate system (x, y, z), in which the x axis is parallel to the in-plane component of plate vibration and the z axis is perpendicular to the plate at C. At a certain point Q on the hologram plate, let crl and

t Now with the National Japan, Tokyo.

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/3, be the incident and the azimuth angles, respectively of the object wave originating from a point P on the object, and similarly cy2 and /3* for the reference wave. If the effective amplitude of the object vibration at P to be observed from Q is m, the x and z components of the plate vibration at Q are d and f respectively, and these vibrations are in-phase if the movement of P and Q is made at a moment as shown by vectors RZ, d and f, then the instantaneous complex amplitude of the object wave U, and that of the reference wave U, at Q become U, = A exp [(4ni/X) m cos (2nt/to) ] X exp [(2ni/X)(d

sin a1 cos /3j

+ fcos “1) cos (2nt/t())], U, = B exp [--(2ni/h)(d + f cos 9)

(1)

sin cy2 cosp2

cos (2nr/r())],

of Fig. 1. Geometry

of the apparatus

(2)

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7, number

where A and B are constant, and f,, is the period of vibration. The intensity at P in the reconstructed image is given by the time-average of U,q, and if the exposure time T is much longer than co we have, apart from a constant factor, 7

I(P) = $i_

k

2

1 U,U; dt

=

0

= IJO [(4nlh)(m-mo)ll2>

(3)

where m. = id (sin a1 cos (3, - sin o2 cos f12) + $(cos

January

OPTICS COMMUNICATIONS

1

011- cos a2).

(4)

This means that the amplitude of the object vibration is reduced by a constant m. which can be arbitrarily chosen by the amplitude of the plate vibration and the configuration of the apparatus. In practical applications of this method, various arrangements can be considered to obtain a suitable value of mo, and because of the facility of instrumentation we have used an in-plane vibration of the plate (i.e., f= 0), which was realized in the following two cases. Linear vibration method. The hologram plate has a linear vibration in the x direction with a small amplitude d, which is kept at a constant value over the hologram. Any values of cyl, cl12,01, and /.I2 may be taken, except for czl = CQ = 0, 01 = n/2, 3n/2, and

p2 = n/2,3rr/2. If we take, for simplicity, 13, = 0, fi2 = z, we have

1973

o1 = 09 = ty,

m o = d sin CY

(5)

and equi-amplitude fringes for the reduced amplitude (m - mo) are produced on the reconstructed image. Rotationa! vibration method. The hologram plate has a small rotational vibration with a small angular amplitude B. about the z axis, using the mechanism shown in fig. 2, and the amplitude d and the azimuth fit and 13, are determined according to the position of Q. If the choice of Q is limited only along a radial line passing through C, the y axis for example, we have d = Boy, fll = 0 and f12 = 71,where y is the distance 6Q. Any values of (ul and ~2 may be taken except for QII = cr2 = 0 similarly as the former case. The reconstruction of this hologram is to be made by setting at Q a slit-like aperture parallel to the x axis or simply by the incidence of a non-expanded tine laser beam at Q, and the values of m. can be changed continuously from zero to a certain value moving the point Q along the y axis while reconstructing. In practical cases, there may be a phase difference r#~ between the object and plate vibrations even if they are driven by the same generator. The instantaneous complex amplitudes (l), (2) in such a case become CJL= A exp [(4rri/X) m cos (2n/to)] X exp [-(2ni/X) Ui = B exp [-(2ni/X)

d sin cr cos (2.rrt/to + #)I,

(1’)

t $)I,

(2’)

d sin a cos (2nt/to

respectively under the simplified conditionsmentioned in [ 11, and we have for (3) Z(P) = IJo [(477/X) {(m -d

sin QIcos @)2

+ d2 sin20 sin2$}1/2] 12.

Fig. 2. The plate holder

for the rotational

vibration

method.

(3’)

This shows that the phase difference q5makes an analysis of the resultant fringes difficult and is to be kept zero or n. To make sure that r#~= 0 or rr, a vibration meter using a non-contact probe is 7

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OPTICS

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January

1973

(d)

(b)

(e>

Fig. 3. A series of interferograms ment of the reconstructing point taken with the rotational vibration m. = 0.03 pm; (b) d = 0.37 wrn, (c) d = 0.71 ,um, m. = 0.39 Frn; mo = 0.53 pm;(e) d = 1.39 pm,

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obtained by the movefrom a single hologram method: (a) d = 0.06 w, m. = 0.20 pm; (d) d = 1.05 burn; m. = 0.74 I.rm.

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appropriate as a monitoring device, and two such vibration meters are used both for the object and for the plate. Two output signals of these vibration meters are simultaneously observed in a two-channel oscilloscope, and the phase shift can be compensated for by a suitable phase-shifting circuit. In fact, this vibration meter detects only the motion of a point in the object or in the plate, thus it is incapable of measuring at once the distribution of the vibration amplitude about the whole area of the object. But as the hologram plate vibrates with a constant amplitude d or BO, this vibration meter can be used for the determination of the amplitude of the plate vibration, and it is very advantageous for the calibration of the amplitude of the object vibration. Fig. 3 shows a series of interferograms obtained by the movement of the reconstructing point Q on the hologram which is recorded by the above-mentioned rotational vibration method, and a displacement of the brightest fringe is observed which corresponds to m - m. = 0, and the amplitude of the area where the brightest fringe appears can be estimated by the measurement of the amplitude of the plate vibration. From this result, it is easily understood that if the reconstructing point Q is slightly moved to increase mo, the brightest fringe moves to the direction where m - m. > 0, and this phenomenon can be used to distinguish the phase inversion domain from the complicated equi-amplitude map. It is concluded that this technique is useful not only to extend the measurable range of vibration amplitudes m, but also to distinguish the phase inver-

January

1973

sion of vibration by observing fringe movements due to slight changes of mo. In practical use of this method, however, some weak points can be pointed out. (1) The necessity to replace a plate in the plate holder makes the plate vibration sometimes unstable. This effect may be reduced by a mechanism which enables tight holding of the plate, but to avoid the effect completely it will be necessary to adopt a re-usable photosensitive material such as a photoconductor-thermoplastic film fixed in the holder. (2) There will be a limit of frequency for the plate vibration mechanism, and this is the limit of the practical use of this method. (3) In the case of the rotational vibration method, reconstructed fringes are suffered by the speckle pattern owing to the reconstruction with a small aperture. The authors wish to express their sincere thanks to Dr. S. Ueha and Mr. T. Kuwayama in our laboratory for useful discussions and experimental collaboration.

References [l]

K.A. Stetson and R.L. Powell, J. Opt. Sot. Am. 55 (1965) 1694. [2] E. Archbold and A.E. Ennos, Engineering uses of holography, ed. E.R. Robertson (Cambridge Univ. Press, London, 1970) 381. [3] C.C. Aleksoff, Appl. Opt. 10 (1971) 1329.

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