Contour Generation of Vibrating Object, by Weighted Subtraction of Holograms Takuso Sato, Hiroshi Ogawa, and Mitsuhiro Ueda

Research Laboratory of Precision Machinery & Electron­ics, Tokyo Institute of Technology, 2-12-1 0-okayama, Meguro-ku, Tokyo 152, Japan. Received 23 November 1973. Time-average hologram interferometry is now being

used widely to measure vibration amplitude.1-3 To get high sensitivity for very small vibration amplitude, Hari-haran has reported a simple method of holographic vibra­tion analysis which uses the simple subtraction of time-average holograms.4 In this method, dark lines appear at the stationary points of the object, so that the portion of the object of small vibration amplitude can be seen against the dark background.

In this letter, the weighted subtraction of time-average holograms is adopted to estimate the small vibration am­plitude quantitatively, that is, by introducing this tech­nique it becomes possible to generate the contour lines of

Fig. 1. Optical system to obtain double-exposed hologram. L, laser; S, shutter; BS, beam splitter; M, mirror; 0, vibrating

object; H, hologram; A, slit; G, trapezoidal glass plate; P, photodetector.

1280 APPLIED OPTICS / Vol. 13. No. 6 / June 1974

equal vibration amplitude at any given small level. How­ever, the smallest level of the amplitude that can be contoured is limited by the practical accuracy of the holo­graphic subtraction, and in our experiments using a crystal vibrator as the vibrating object the sensitivity is improved by a ratio of 2.5 over the conventional time-average meth­od.

The configuration for recording the time-average holo­grams is shown in Fig. 1. The first exposure is made with the object stationary for T1 sec, and the second exposure is made for T2 sec with the object vibrating and the phase of the reference beam shifted by Π rad. This double-exposed hologram may be considered as a hologram ob­tained as the result of weighted subtraction.

The intensity of the reconstructed image from the holo­gram is given Fig. 3. Theoretical variation of the visibility as a function of

phase error.

where a one-dimensional model is assumed for simplicity and the variation is assumed only in the x direction, mix) is the amplitude of the sinusoidal vibration, J o is the Bes-sel function of order zero, λ is the wavelength of light, I1(x) is the intensity of the reconstructed image when the object is stationary, and θ1, θ2 are the angles between the displacement vector of the vibration and the directions of the incident and scattered light, respectively. From this result, it is seen that the dark lines in the reconstructed image, that is the point where I{x) = 0, correspond to the vibration amplitude mo that satisfies the following rela­tion:

This equation could possibly have a large number of solu­tions for mo, but it is assumed that mo represents only the lowest order solution. Consequently, by adjusting the ratio T1/T2 properly, the contour lines of any small vibra­tion amplitude can be generated, and the theoretical vari­ation of the intensity of the reconstructed image of a vi­brating object as a function of the vibration amplitude for several values of T 1 /T 2 is given in Fig. 2.

So far, it is assumed that the phase of the reference beam changes exactly π rad between the two expo­sures. To study the effect of phase error, let the phase change between the two exposures he π + α rad. Then, Eqs. (1) and (2) are rewritten as follows:

In this case the intensity of the dark lines does not drop to zero but is given by

Since the value of α can be assumed to be small, the shift of the vibration amplitude corresponding to the dark lines given by Eq. (4) can be considered negligible. However, the rise of the minimum level given by Eq. (5) may affect the measurements of the vibration amplitude seriously. As seen from Fig. 2, if we want to contour the smaller vibration amplitude, the intensity of the stationary points [m(x) = 0] becomes smaller. Let Io be the intensity of the stationary points; then, it is given by

In the case of Imin ≃ I 0 , we cannot determine the location of the dark lines and the measurement of the vibration amplitude becomes impossible.

It may be appropriate to introduce the concept of visi­bility to estimate the observability of dark lines, and the visibility is defined by

Fig. 2. Theoretical variation of the intensity of the reconstructed image of a vibrating object as a function of the parameter bm(x).

In Fig. 3, the variation of the visibility as a function of the phase error for several values of (T1 /T2) is shown. It can be seen from Fig. 3 that the phase error affects the mea­surements more seriously when attempting to obtain a contour of smaller vibration amplitude.

A 5° X-cut crystal vibrator of which the resonance fre­quency was 83.7 kHz was used as the vibrating object, and the following experimental parameters were adopted, that is, θ1 = 30°, θ2 = 90°, and λ = 0.6328 μm. To detect the phase difference between the object and reference beams, a Mach-Zehnder interferometer is constructed in the opti­cal system as shown in Fig. 1 and the fringe signal is de­tected by a photodetector. To assure the phase difference of π rad between two exposures, the output voltages of the photodetector is maximized at the first exposure and is minimized at the second exposure by moving a trape­zoidal glass plate.

June 1974 / Vol . 13, No. 6 / APPLIED OPTICS 1281

Fig. 4. Reconstructed images of a crystal vibrator: (a) station­ary object, (b) time-average hologram (T1/T2 = 0.0), (c) weighted subtraction (T1/T2 = 0.6), (d) weighted subtraction (T1/T2 =

0.8), and (e) simple subtraction (T1/T2 = 1.0).

Some results are shown in Fig. 4. In Fig. 4(a) the re­constructed image of the stationary object is shown, and in Figs. 4(b)-(e) the crystal was excited at the same level and only the ratio (T1 /T2) was changed. The ratio in Figs. 4(b)-(e) are 0, 0.6, 0.8, and 1.0, respectively, and the corresponding vibration amplitude of the dark lines are 0.25 μm, 0.15 μm, 0.10 μm, and 0.0 μm, respectively. Figure 4 (b) shows a result by the conventional time-av­erage hologram interferometry, and Fig. 4(e) shows a result using Harihanan's method. Figures 4(c) and (d) indicate that it becomes possible to generate the contour lines of vibration amplitudes smaller than those generated by the conventional method, and more quantitative analysis of the vibration amplitude becomes possible by introducing the technique of weighted subtraction. That is. Fig. 4(d) shows about 2.5 improvement of the sensitivity compared with the conventional method.

As for the phase error in the subtraction operation, we could not eliminate the slow fluctuation of the output voltage of the photodetector, which corresponded to a phase fluctuation of ±6°. Though the source of the phase fluctuation could not be determined exactly, it might be caused by low-frequency vibration of the optical table. Consequently, it was rather difficult to generate a con­tour of the vibration amplitude smaller than indicated in Fig. 4(d) due to this phase fluctuation. In these experi­ments the double-exposed holograms were produced with­out biasing. The prebiasing would contribute to accu­rately produce the proper weighted subtraction.5

The authors wish to thank H. Fukuyo and N. Ohura of their Laboratory and the referees of this journal for their ad­vice and suggestions.

References 1. R. L. Powell and K. A. Stetson, J. Opt. Soc. Am. 55, 1593

(1965). 2. A. Macovski, S. D. Ramsey, and L. F. Schaefer, Appl. Opt. 10,

2722(1971). 3. C. C. Aleksoff, Appl. Opt. 10, 1329 (1971). 4. P. Hariharan, Appl. Opt. 12, 143 (1973). 5. H. J. Caulfield, Sun Lu, and J. L. Harris, J. Opt. Soc. Am. 58,

1003(1968).

1282 APPLIED OPTICS / Vol. 13, No. 6 / June 1974