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Analysis of asymmetrical membrane vibrations by holographic interferometry
Volume 25, number 3
OPTICS COMMUNICATIONS
June 1978
ANALYSIS OF ASYMMETRICAL MEMBRANE VIBRATIONS
BY HOLOGRAPHIC INTERFEROMETRY R. ROHLER and C. SIEGER lnstitut ftir medizinische Optik, D-8000 Miinchen 2, FRG
Received 18 May 1977 Revised manuscript received 16 March 1978
Asymmetrical membrane vibrations were examined by time-average holograms as well as by strobe-pulses illuminating the vibrating membrane at its oscillation maxima. It is shown, that only by the second technique asymmetrical oscillations can be detected and the two vibration amplitudes can be measured.
1. Introduction In all vibration analyses using holographic methods
[1-14] so far symmetrical vibrations of the investigated specimen were assumed. For one-side damped membranes this assumption certainly is not legitimated, since the oscillation amplitudes towards and away from the damping material are unequal. An asymmetrical damping of a membrane may occur, if in transfering the energy from the vibrating membrane to the damping medium, frictional forces work different in both directions of oscillation. The membrane, for example, may in one direction loosely adjoin a damping material, or the transmission of the oscillators takes place by levers with gliding bearings, whose friction is unequal for both directions. Such an asymmetrical vibrating membrane e.g. could be the tympanic membrane, damped in the oscillation phase toward the middle ear by the ossicular chain, which transmits the vibrations to the fluid system of the inner ear. Usually it is not known in advance whether a surface vibration is symmetrical or not. Therefore in using without care standard methods of analysis which are based on the assumption of symmetry, deviations from symmetry may be overlooked and errors of interpretation may be introduced. In this paper it is investigated, whether standard
methods of holographic vibration analysis can be used to detect asymmetries in vibrations. We use the timeaverage method and the stroboscopic double-pulse method. Our object is a one-side damped membrane. As may be anticipated, we find that the time-average method is not suited for detecting asymmetries, while the double pulse method is very well suited for this task.
2. Experimental setup and experiments The object under examination was a round drum ( 9 : 1 0 cm), covered with a skin-membrane and excited acoustically. The one-side damping is realized by glass wool in the drum corpus. Time-average records were taken from this membrane at various frequencies and with different intensities of the exciting signal. By means of double-pulse holograms the amplitudes of the membrane-vibrations were measured in dependence of the vibrations phase. For that purpose a first exposure of the membrane at rest is taken using an exposure time half of that normally employed to get good reconstruction efficiency. Then the membrane is set into vibration and is illuminated with synchronous pulses of laser-light at one of its oscillation maxima, until the remaining half of the total required exposure time is achieved. The laser light flashes are produced from a neon-helium laser by means of an 297
Volume 25, number 3
OPTICS COMMUNICATIONS
June 1978
electro-optic switch, a pockels cell. An appropriate driver varies frequency and phase of the flashes synchronously to the duration of the exciting sinusoidal signal. The position of the stobe pulses relative to the vibration phase of the membrane is controlled by the "real-time" technique. For this purpose the hologram plate is processed in situ in a liquid gate. As was shown in [3,13], the normalized reconstructed intensity distribution I for such a double-exposure stroboscopic recording in this case is given by: 1 = I B cos2(MA/2),
(1)
where I B describes the intensity of the stationary object, M is a geometrical factor, considering the geometry of the set-up and the wavelength of the laser light, and A is the amplitude of vibration. Eq. (1) is also valid for the images of double-pulse stroboscopic recordings, viewed in real-time through a hologram of the non-vibrating object. In time-average holography, the normalized hologram image light intensity function of a sinusiodal vibrating membrane is described by: I = I B J~2(MA),
(2)
where J0 is the zero order Bessel function. As shown in detail in [7,8], the number of generated fringes is hereby directly proportional to the maximum vibration amplitude. The amplitude of the one-side damped membrane, measured by means of time-average interferograms are now compared with the two amplitudes, obtained with stroboscopic pulse illumination. There are three questions to be answered: 1) Is it possible to detect a difference between the both extremes of oscillation by strobe holography? 2) Is it possible to discover the asymmtrical oscillations, i.e. the dissimilar amplitudes related to the position at rest, by means of time-average holography? 3) Which quantitative result is yielded by a timeaverage hologram for such an asymmetrically vibrating membrane?
3. Results
Fig. 1 shows the "time-average" vibration pattern of a one-side damped oscillation. It does not give any reference to the asymmetrical vibration. The interfer298
Fig. l. The time-averagevibration pattern of a one-side damped membrane. ence pattern is rather equal to the pattern of an undisturbed vibrating membrane, whose centre has an amplitude of 0.65/am in accordance with eq. (2). If no preinformation exists on the asymmetry, no indication thereon can be gained from the time-average figure. Furthermore for the amplitude of the vibration at the damped side too large a value is deduced from the figure, as will be seen from the results of the double pulse analysis. It may be argued then, that results of the time-average method should be used with caution whenever the possibility of asymmetrical vibrations exist. With the stroboscopic double-pulse method the maximal deviations from resting position of the membrane on both sides can be detected. Fig. 2 shows the fringe pattern of the double-pulse interferogram of the undisturbed extreme, i.e. the maximum amplitude away from the damping material. The magnitude of this amplitude, evaluated by the number of interference fringes by means ofeq. (1) is 0.65/am, as it was already measured by the timeaverage hologram. On the other hand, the maximum amplitude towards the damping material, determined from photograph 3, is only 0.14/am. It is evident that at least for small elongations the amplitude as measured by time average method has the same value as the undisturbed maximum elongation when measured by stroboscopic illumination. To our opinion, this is due to surface vibrations arising in the damping medium, while the membrane approaches
Volume 25, number 3
OPTICS COMMUNICATIONS
Fig. 2. The stroboscopic double-pulse record taken at the undisturbed oscillation maximum (Le. away from the damping substance) of the membrane, vibrating in the same manner as above.
June 1978
Fig. 4. Stroboscopic record of the oscillation maximum towards the damping substance.
the resting position at this side. These vibrations interfere with the resting position of the membrane causing incoherent variations of the position, so that this phase of membrane vibration does not contribute in a coherent way to the time average phase value. This value therefore is dominated by the maximum elongation at the undamped side. With increasing amplitudes, the fringe pattern is more and more deteriorated, a phenomenon which is also observed with stroboscopic il-
lumination at the damped resting position. This is demonstrated by figs. 4 and 5. The accuracy of measuring the amplitudes by both methods is limited by uncertainties in the measuring of the geometry of thc set-up and in the identification of the fringe order. If the holographic device remains unchanged during the single exposures, the errors in
l:ig. 3. The stroboscopic double-pulse record of the membrane, now at its oscillation maximum towards the damping material.
Fig. 5. Stroboscopic record taken at the same conditions as at fig. 4, but with increased sound pressure level of the exciting loudspeaker. 299
Volume 25, number 3
OPTICS COMMUNICATIONS
measuring the geometry enter into the time average and the strobe interferograms in the same way. From an accuracy o f 0.5 ° for the illumination and observation angle follows an uncertainty of-+2 nm for the amplitude. The exact interference fringe order for any point on the surface o f the membrane can be interpolated in our interferograms down to 1/6 o f a fringe. That means, that the amplitudes can be measured with an accuracy o f -+26 nm in both procedures. By applying the dual-frequency method [15] this error could be reduced at least by the factor 15. While using the strobe technique the fight position of the strobe-flashes in the vibration extremes can be exactly controlled by means of the real-time technique. From an estimated error o f 5 ° in the phase of the light pulses in relation to the vibration follows an additional uncertainty of -+2.5 nm. Thus the undisturbed amplitudes measured by the stroboscopic double pulse method equal the time average amplitudes within the experimental error. However the amplitude towards the damping material is significantly different from the former value and is not indicated by the time average method.
This work was partly supported by the Deutsche Forschungsgemeinschaft within the Sonderforschungsbereich 88/B3. Part o f this paper has been presented at the annual meeting of the DGaO 1975 in Bad Ischl.
References Ill [21 [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [ 13] [14]
Acknowledgements We wish to thank K. Piwernetz and R. Krusche for valuable discussions and criticisms.
300
June 1978
[15l
C.C. Aleksoff, Appl. Opt. l0 (1971) 1329. E. Archbold and E.A. Ennos, Nature 217 (1968) 942. K. Biedermann and N.E. Molin, J. Phys. E3 (1970) 669. P. Hariharan, Appl. Opt. 12 (1973) 143. J. Janta and M. Miler, Optik 36 (1972) 185. S.M. Khanna and J. Tonndotf, J. Acoust. Soc. Am. 51 (1971) 1904. M. Lurie and M. Zambuto, Appl. Opt. 7 (1968) 2323. M.A. Monahan and K. Bromley, J. Acoust. Soc. Am. 44 (1968) 1225. T. Nakajima, Jap. J. Appl. Phys. 13 (1974) 471. P. Smigielski, F. Albe, A. Dancer, H. Fagot and R. Franke, Nouv. Rev. Optique 6 (1975) 49. K. Stetson, J. Opt. Soc. Am. 61 (1971) 1359. N. Takai, M. Yamada and T. Idogawa, Opt. anti Laser Tech. 8 (1976) 21. C.S. Vikram, G. Bose and J.N. Maggo, Nouv. Rev. Optique 6 (1975) 55. B.M. Watrasiewicz and P. Spicet, Nature 217 (1968) 1142. R. D~ndliker et al., Nature 217 (1968) 1142; Opt. Commun. 9 (1973) 412.
OPTICS COMMUNICATIONS
June 1978
ANALYSIS OF ASYMMETRICAL MEMBRANE VIBRATIONS
BY HOLOGRAPHIC INTERFEROMETRY R. ROHLER and C. SIEGER lnstitut ftir medizinische Optik, D-8000 Miinchen 2, FRG
Received 18 May 1977 Revised manuscript received 16 March 1978
Asymmetrical membrane vibrations were examined by time-average holograms as well as by strobe-pulses illuminating the vibrating membrane at its oscillation maxima. It is shown, that only by the second technique asymmetrical oscillations can be detected and the two vibration amplitudes can be measured.
1. Introduction In all vibration analyses using holographic methods
[1-14] so far symmetrical vibrations of the investigated specimen were assumed. For one-side damped membranes this assumption certainly is not legitimated, since the oscillation amplitudes towards and away from the damping material are unequal. An asymmetrical damping of a membrane may occur, if in transfering the energy from the vibrating membrane to the damping medium, frictional forces work different in both directions of oscillation. The membrane, for example, may in one direction loosely adjoin a damping material, or the transmission of the oscillators takes place by levers with gliding bearings, whose friction is unequal for both directions. Such an asymmetrical vibrating membrane e.g. could be the tympanic membrane, damped in the oscillation phase toward the middle ear by the ossicular chain, which transmits the vibrations to the fluid system of the inner ear. Usually it is not known in advance whether a surface vibration is symmetrical or not. Therefore in using without care standard methods of analysis which are based on the assumption of symmetry, deviations from symmetry may be overlooked and errors of interpretation may be introduced. In this paper it is investigated, whether standard
methods of holographic vibration analysis can be used to detect asymmetries in vibrations. We use the timeaverage method and the stroboscopic double-pulse method. Our object is a one-side damped membrane. As may be anticipated, we find that the time-average method is not suited for detecting asymmetries, while the double pulse method is very well suited for this task.
2. Experimental setup and experiments The object under examination was a round drum ( 9 : 1 0 cm), covered with a skin-membrane and excited acoustically. The one-side damping is realized by glass wool in the drum corpus. Time-average records were taken from this membrane at various frequencies and with different intensities of the exciting signal. By means of double-pulse holograms the amplitudes of the membrane-vibrations were measured in dependence of the vibrations phase. For that purpose a first exposure of the membrane at rest is taken using an exposure time half of that normally employed to get good reconstruction efficiency. Then the membrane is set into vibration and is illuminated with synchronous pulses of laser-light at one of its oscillation maxima, until the remaining half of the total required exposure time is achieved. The laser light flashes are produced from a neon-helium laser by means of an 297
Volume 25, number 3
OPTICS COMMUNICATIONS
June 1978
electro-optic switch, a pockels cell. An appropriate driver varies frequency and phase of the flashes synchronously to the duration of the exciting sinusoidal signal. The position of the stobe pulses relative to the vibration phase of the membrane is controlled by the "real-time" technique. For this purpose the hologram plate is processed in situ in a liquid gate. As was shown in [3,13], the normalized reconstructed intensity distribution I for such a double-exposure stroboscopic recording in this case is given by: 1 = I B cos2(MA/2),
(1)
where I B describes the intensity of the stationary object, M is a geometrical factor, considering the geometry of the set-up and the wavelength of the laser light, and A is the amplitude of vibration. Eq. (1) is also valid for the images of double-pulse stroboscopic recordings, viewed in real-time through a hologram of the non-vibrating object. In time-average holography, the normalized hologram image light intensity function of a sinusiodal vibrating membrane is described by: I = I B J~2(MA),
(2)
where J0 is the zero order Bessel function. As shown in detail in [7,8], the number of generated fringes is hereby directly proportional to the maximum vibration amplitude. The amplitude of the one-side damped membrane, measured by means of time-average interferograms are now compared with the two amplitudes, obtained with stroboscopic pulse illumination. There are three questions to be answered: 1) Is it possible to detect a difference between the both extremes of oscillation by strobe holography? 2) Is it possible to discover the asymmtrical oscillations, i.e. the dissimilar amplitudes related to the position at rest, by means of time-average holography? 3) Which quantitative result is yielded by a timeaverage hologram for such an asymmetrically vibrating membrane?
3. Results
Fig. 1 shows the "time-average" vibration pattern of a one-side damped oscillation. It does not give any reference to the asymmetrical vibration. The interfer298
Fig. l. The time-averagevibration pattern of a one-side damped membrane. ence pattern is rather equal to the pattern of an undisturbed vibrating membrane, whose centre has an amplitude of 0.65/am in accordance with eq. (2). If no preinformation exists on the asymmetry, no indication thereon can be gained from the time-average figure. Furthermore for the amplitude of the vibration at the damped side too large a value is deduced from the figure, as will be seen from the results of the double pulse analysis. It may be argued then, that results of the time-average method should be used with caution whenever the possibility of asymmetrical vibrations exist. With the stroboscopic double-pulse method the maximal deviations from resting position of the membrane on both sides can be detected. Fig. 2 shows the fringe pattern of the double-pulse interferogram of the undisturbed extreme, i.e. the maximum amplitude away from the damping material. The magnitude of this amplitude, evaluated by the number of interference fringes by means ofeq. (1) is 0.65/am, as it was already measured by the timeaverage hologram. On the other hand, the maximum amplitude towards the damping material, determined from photograph 3, is only 0.14/am. It is evident that at least for small elongations the amplitude as measured by time average method has the same value as the undisturbed maximum elongation when measured by stroboscopic illumination. To our opinion, this is due to surface vibrations arising in the damping medium, while the membrane approaches
Volume 25, number 3
OPTICS COMMUNICATIONS
Fig. 2. The stroboscopic double-pulse record taken at the undisturbed oscillation maximum (Le. away from the damping substance) of the membrane, vibrating in the same manner as above.
June 1978
Fig. 4. Stroboscopic record of the oscillation maximum towards the damping substance.
the resting position at this side. These vibrations interfere with the resting position of the membrane causing incoherent variations of the position, so that this phase of membrane vibration does not contribute in a coherent way to the time average phase value. This value therefore is dominated by the maximum elongation at the undamped side. With increasing amplitudes, the fringe pattern is more and more deteriorated, a phenomenon which is also observed with stroboscopic il-
lumination at the damped resting position. This is demonstrated by figs. 4 and 5. The accuracy of measuring the amplitudes by both methods is limited by uncertainties in the measuring of the geometry of thc set-up and in the identification of the fringe order. If the holographic device remains unchanged during the single exposures, the errors in
l:ig. 3. The stroboscopic double-pulse record of the membrane, now at its oscillation maximum towards the damping material.
Fig. 5. Stroboscopic record taken at the same conditions as at fig. 4, but with increased sound pressure level of the exciting loudspeaker. 299
Volume 25, number 3
OPTICS COMMUNICATIONS
measuring the geometry enter into the time average and the strobe interferograms in the same way. From an accuracy o f 0.5 ° for the illumination and observation angle follows an uncertainty of-+2 nm for the amplitude. The exact interference fringe order for any point on the surface o f the membrane can be interpolated in our interferograms down to 1/6 o f a fringe. That means, that the amplitudes can be measured with an accuracy o f -+26 nm in both procedures. By applying the dual-frequency method [15] this error could be reduced at least by the factor 15. While using the strobe technique the fight position of the strobe-flashes in the vibration extremes can be exactly controlled by means of the real-time technique. From an estimated error o f 5 ° in the phase of the light pulses in relation to the vibration follows an additional uncertainty of -+2.5 nm. Thus the undisturbed amplitudes measured by the stroboscopic double pulse method equal the time average amplitudes within the experimental error. However the amplitude towards the damping material is significantly different from the former value and is not indicated by the time average method.
This work was partly supported by the Deutsche Forschungsgemeinschaft within the Sonderforschungsbereich 88/B3. Part o f this paper has been presented at the annual meeting of the DGaO 1975 in Bad Ischl.
References Ill [21 [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [ 13] [14]
Acknowledgements We wish to thank K. Piwernetz and R. Krusche for valuable discussions and criticisms.
300
June 1978
[15l
C.C. Aleksoff, Appl. Opt. l0 (1971) 1329. E. Archbold and E.A. Ennos, Nature 217 (1968) 942. K. Biedermann and N.E. Molin, J. Phys. E3 (1970) 669. P. Hariharan, Appl. Opt. 12 (1973) 143. J. Janta and M. Miler, Optik 36 (1972) 185. S.M. Khanna and J. Tonndotf, J. Acoust. Soc. Am. 51 (1971) 1904. M. Lurie and M. Zambuto, Appl. Opt. 7 (1968) 2323. M.A. Monahan and K. Bromley, J. Acoust. Soc. Am. 44 (1968) 1225. T. Nakajima, Jap. J. Appl. Phys. 13 (1974) 471. P. Smigielski, F. Albe, A. Dancer, H. Fagot and R. Franke, Nouv. Rev. Optique 6 (1975) 49. K. Stetson, J. Opt. Soc. Am. 61 (1971) 1359. N. Takai, M. Yamada and T. Idogawa, Opt. anti Laser Tech. 8 (1976) 21. C.S. Vikram, G. Bose and J.N. Maggo, Nouv. Rev. Optique 6 (1975) 55. B.M. Watrasiewicz and P. Spicet, Nature 217 (1968) 1142. R. D~ndliker et al., Nature 217 (1968) 1142; Opt. Commun. 9 (1973) 412.