Visualization of sound field with uniform phase distribution using laser beam microphone coupled with computerized tomography method

Transcription

1 Acoust. Sci. & Tech. 29, 4 (28) #28 The Acoustical Society of Japan Visualization of sound field with uniform phase distribution using laser beam microphone coupled with computerized tomography method Tatsuya Sakoda ; and Yoshito Sonoda 2 Department of Electrical and Electronic Engineering, University of Miyazaki, Gakuenkibanadai-Nishi, Miyazaki, Japan 2 Institute of Industrial Science and Technical Research, Kyushu Tokai University, 9 Toroku, Kumamoto, Japan ( Received 3 October 27, Accepted for publication 9 January 28 ) Keywords: Audio wave, Sound field, Tomography, Fraunhofer diffraction, Reconstruction PACS number: 43.6.Rw [doi:./ast.29.29]. Introduction Microphones are widely used for detecting sound waves, where the sound waves vibrate a diaphragm and then electrical signals in proportion to the degree of vibration of the diaphragm are obtained. A microphone has high sensitivity, but itself disturbs the sound field to be measured and averages it over the diaphragm area. The disturbance is not negligible, particularly in precise scientific measurements of complex fields, such as a near field (Fresnel s interference zone) close to a vibrating solid surface or the front surface of a panel array speaker composed of many small speakers. In contrast, light wave sensing can realize high accuracy measurement without disturbing the sound field []. In particular, for audio wave measurements in air, some useful studies have recently been reported [2 ]. The light diffraction technique [2,3], which we here call the laser beam microphone (LBM) technique, is an effective sensing method and is more flexible for practical uses than other methods [4,], as it involves only a simple optical lens system. The LBM can detect sound waves in air, which have much longer wavelength than those treated in acoustooptics [], with a small laser beam diameter of a few mm. In the LBM, the extremely weak diffraction light induced by the slight changes in the refractive index owing to sound can be detected by an optical Fourier transform technique. In this method, as the ultrasmall modulation by the sound field is integrated along the laser beam path, it has the possibility of being used for sound field visualization by computerized tomography (CT) [6 8]. Our final objective is to establish the LBM coupled with CT (LBM-CT method), by which the amplitude and phase distributions of a complex field can be reconstructed and visualized. In this feasibility study, in the first stage of research, the sound field with almost uniform phase distribution in the cross section including the probing laser beam was chosen as the subject of measurements, and we carried out the experiment to verify the feasibility of the LBM-CT method for the visualization of only the amplitude distribution of the sound field. Diffraction signals due to sound waves with a frequency of 2 khz were measured, and the reconstruction of the sound field was examined by the CT method. The results were then sakoda@cc.miyazaki-u.ac.jp compared with those measured using an electrostatic microphone (EM). 2. Experiments and methods When a probing laser beam crosses a sound wave, diffracted light waves are generated and propagate with and in the penetrating beam through the Fourier optical system, which contains a lens for optical Fourier transformation (OFT) of the diffraction light. The spatial diffraction pattern, which oscillates at the frequency of the sound wave, is obtained at the detection plane or the back focal plane of the OFT lens [2,9,]. Here, the multidiffraction effect and the high-order frequency components, as observed in the strong interaction by a high-frequency ultrasonic wave, are negligible in the present interaction between light and audio waves. Then, the diffracted wave of the first order is homodynedetected with the penetrating optical wave, which functions as the local oscillating power, by a photodiode at the back focal plane. As shown in Fig., the x and z directions are defined as the forward directions of the laser beam and the sound wave, respectively. The y direction is taken here to be perpendicular to both the laser beam and sound wave propagation directions. x f, y f and z f are defined as the location coordinates on a detection plane. The spatial intensity I (W/m 2 ) of the resultant optical field at the back focal plane is given by [9,] I ¼ I ðexpf ½u 2 þðu nþ 2 Šg þ expf ½u 2 þðu þ nþ 2 ŠgÞ sin! p t; I ¼ð2P =w 2 f Þ exp½ 2ðy f =w 2 f ÞŠ; ¼ k i ð Þdp= p; where is the time-dependent component of phase modulation by the sound wave; w f is the beam size at the detection plane; u ¼ x f =w f is the normalized x-coordinate at the detection plane; p is the wavelength of the sound wave; k p (¼ 2= p ) is the wave number of the sound wave; w is the spot size; n ¼ k p w =2 is the normalized wave number;! p is the angular frequency of the sound wave; P is the total incident power; i is the wavelength of the probing beam; k i (¼ 2= i ) is the wave number of the incident beam; is the ðþ 29

2 Acoust. Sci. & Tech. 29, 4 (28) Semiconductor Laser s L L2 Digital Oscilloscope PC scanning Laser beam z y refractive index of air; d is the width of the sound wave; p is the sound pressure; is the specific heat ratio; and p is the atmospheric pressure. In the case where p is much more and n is less than, Eq. () can be approximated by I ¼ 4I expð 2u 2 Þun sin! p t: Because n ¼ k p w =2 ¼ w! p =2v p ¼ w f p =v p, where v p is the sound velocity and f p is the frequency of the sound wave, Eq. (2) indicates that I is proportional to f p and pd. The diffraction pattern of the first order appears along the z direction, and it consists of two simultaneous waves on the þz and z axes with temporal phases opposite to each other. If the interaction length (L) between the laser beam and the sound field is long, Eq. (2) is rewritten as Z I ¼ 4I expð 2u 2 Þunk i ð Þ= p sin! p t ~pðsþds; ð2 Þ where ~pðsþ is the sound pressure distribution along the s-axis. Incidentally, CT method can reconstruct a cross-sectional image using projected data from many directions of at least 8 degrees. Here, the two-dimensional sound field Sðx; yþ [Pa], with the propagation direction perpendicular to the x-y plane, is the subject of our present analyses. If the output voltage signal of the LBM is proportional to I or pd is normalized by the maximum signal amplitude, the detected signal, i.e., the projected data Dðr;Þ [Pam] along the interaction length (L) between the laser beam and the sound field, can be written as r S(x,y) D(r, ) y x Sound Transmitter B. P. Filter & AMP x Detection Plane (a) Relation between the coordinates of (x,y) and (s,r) Fig. L3 L4 L (b) Experimental apparatus Schematic view of apparatus. Pre-AMP L z f photodiode yf ð2þ Z Dðr;Þ¼ Sðx; yþds; where r ¼ x cos þ y sin and s ¼ x sin þ y cos in the r-s orthogonal coordinates which are rotated from the x-y coordinates, as shown in Fig. (a). The reconstruction image could be obtained by inversely projecting the data to the x-y coordinates, in which the filtered back-projection method and the Shepp/Logan filter function were used as the reconstruction algorithm and the convolution function, respectively [7]. Figure (b) shows a schematic view of the entire apparatus. The laser source was a semiconductor laser operated at i of 67 nm and P of 6.3 mw. The vertical and the horizontal beam spot sizes were.33 and.32 mm, respectively; therefore the laser beam was considered to be a round shape of Gaussian type. The number of optical lenses was, which enables the control of the beam size at the audio sound field and the detection plane. L (focal length of 3 mm) and L2 (focal length of mm) were used for the adjustment of the spot size, and the beam diameter at the sound field was 4 mm. L3 (focal length of mm) was a lens for OFT; however, the beam size at the rear focal plane was too small for measurements with high spatial resolution. L4 and L were used to adjust the beam width w f, which was set to be 3 mm. The penetrating laser light (the th) and the diffraction light of the first order were mixed and detected by a silicon photodiode with the sectional area of : : mm 2 and the sensitivity of.4 A/W. The silicon photodiode was mounted on a y-z stage. When the intensity distribution of the diffraction wave was measured, the photodiode was moved from the center (y f ¼ z f ¼ mm) in the z f direction. The only alternating components of detected signals were amplified by a preamplifier of 2 db; thereafter, a frequency range of 8 khz to 22 khz, which passed through a band-pass filter with an amplifier of 3 db, was sent to a digital oscilloscope. A sound transmitter with a diameter of 22 mm was positioned between L2 and L3 and below the probing laser beam, which was mounted on a y- stage. The location of the center of the sound transmitter surface was taken to be x ¼ y ¼ mm. There was a distance of 3 mm between the transmitter surface and the laser beam axis, and then the location of the center of the transmitter surface was expressed as x ¼ y ¼ mm with z ¼ 3 mm. The output frequency and the sound pressure were f p ¼ 2 khz and db, respectively. Since v p in air is commonly 33. m/s, p was calculated to be 6.6 mm. Additionally, n ¼ :4 was also calculated; therefore, our experimental conditions followed Eq. (2). The sound pressure was measured using an EM with a diameter of 6.4 mm and a frequency range of 2 Hz to 2 khz. Figure 2 shows the EM signals detected at y ¼ 3, 2,,,, 2 and 3 mm with x ¼ z ¼ mm. On moving the EM gradually away from the central axis of the sound transmitter, the detected intensity becomes lower. However, the phases of the signals are almost uniform. Figure 3 shows the sound pressure distribution on the x-y plane with z ¼ mm, obtained using the EM. The full width at half-maximum (FWHM) of the sound pressure is about L ð3þ 296

3 T. SAKODA and Y. SONODA: VISUALIZATION OF SOUND FIELD USING LASER BEAM MICROPHONE (a) y = mm (a) z f = -.2 mm (b) y = mm (e) y = - mm (c) y = 2 mm (f) y = -2 mm (b) z f = mm (d) y = 3 mm (g) y = -3 mm Fig. 2 EM signals detected at y = (a), (b), (c) 2, (d) 3, (e), (f) 2 and (g) 3 mm with x ¼ z ¼ mm (Vertical axis: mv/div., Horizontal axis: 2 ms/div.). (c) z f =.2 mm Fig. 4 Examples of signals detected at z f (a) = :2, (b) and (c).2 mm with y f ¼ mm at the detection plane (Vertical axis: 3 mv/div., Horizontal axis: ms/div.). Sound Pressure (Pa) x (mm) 2 36 mm. A determining factor of the sound radiation field, calculated as R D 2 = p, where R D is the radius of the sound transmitter surface, was 22 mm for the case of f p ¼ 2 khz. The measurement plane 3 mm above the sound transmitter was near the limiting distance of the nearby sound field. In the theoretical model of diffraction, the sound wave is considered to be a plane wave. Thus, visualization of the sound field was examined at a plane where the phase was almost uniform and the nearby sound field could be assumed, i.e., the experimentally obtained Dðr;Þ is the electrical signal proportional only to the amplitude of the sound wave integrated along the laser beam path, and CT analyses were implemented for the Fig. 3 Spatial distribution of sound pressure on the x-y plane with z ¼ mm, obtained using EM. uniform phase plane. The numerically obtained Sðx; yþ also referred only to the amplitude or sound pressure distribution of the sound wave. To obtain projection data for CT analyses, the sound field was rotated the in direction and moved toward the y direction. The rotation step angle was set to degrees, and the driving range in the y direction was from y ¼ mm to mm with the step length of mm. 3. Results and discussion Examples of detected signals at z f ¼ :2, and.2 mm with y f ¼ mm are shown in Fig. 4. z f ¼ :2 and.2 mm were the theoretically predicted intensity peak loci of diffraction waves, which were calculated using Eq. (2) [3]. The phases between two signals at z f ¼ :2 and.2 mm are opposite, and each signal has a period of ms, i.e., the frequency is 2 khz. The spatial intensity distribution along the z direction is plotted in Fig.. The obvious signal peaks appear at z f ¼ :2 and.2 mm, and the distribution profile is similar to that calculated using Eq. (2). In contrast, no diffraction signals were detected when there were two lenses between the interaction region and the detection plane. The spatial distribution of the sound field with temporal variation is converted into the Fourier transform image on a spatial frequency axis with temporal variation. The Fourier transform and the inverse Fourier transform are alternately executed for every additional lens. Therefore, the diffraction signals were detected only when the Fourier transform of the diffraction image was executed in front of the detection plane. This is the most noteworthy and important characteristic of the present technique. In addition, the signal peak position is independent of sound pressure [3]. In the measurements for collecting data used for CT analyses, the location of the photodiode was kept at z f ¼ :2 mm with y f ¼ mm. 297

4 Acoust. Sci. & Tech. 29, 4 (28) 3 Intensity (mv) z f (mm) Fig. Diffraction image pattern along the z f direction at the detection plane. Intensity (arb. units) x (mm) 8 Fig. 7 Reconstructed distribution of audio wave in the x-y plane with z ¼ mm by CT Projection data by EM Fig Original projection data obtained using LBM. 3 Intensity (arb. units) Figure 6 shows the original projection data. The sound field was rotated in the direction and was moved toward the x direction while the probing laser beam was fixed. Although the sound field was rotated and moved, the measurement is essentially the same as that by scanning the laser beam and the detection device. From Fig. 6, it can be expected that the sound field has an obvious peak on the y axis. The sound field at the x-y plane with z ¼ mm was visualized by a CT method. Figure 7 shows a reconstructed distribution of the sound field on the x-y plane with z ¼ mm. The peak is located at x ¼ y ¼ mm, and the reconstructed distribution is similar to that shown in Fig. 3. The rotation step angle and the step width were rough relative to the spatial spread of the sound field, since our aim was to verify the feasibility of the LBM coupled with CT; however, the setting is considered to be sufficient to grasp the outline of the spatial distribution. Profiles of the sound pressure distribution along the y direction with x ¼ z ¼ mm, obtained with CT, the LBM and the EM, are shown in Fig. 8. Circles denote projection data Fig. 8 Profiles of sound pressure distribution along y direction with x ¼ z ¼ mm, obtained with CT, LBM and EM. obtained using the LBM, and squares denote values obtained using the EM. The solid line denotes the CT profile. Each profile was normalized by the value at y ¼ mm because the values obtained using the LBM were of electrical signals in proportion to the sound pressure of the sound wave, whereas the EM values directly indicate the sound pressure. The maximum value at y ¼ mm corresponds to the sound pressure of 2 Pa. The three profiles differ somewhat in the 298

5 T. SAKODA and Y. SONODA: VISUALIZATION OF SOUND FIELD USING LASER BEAM MICROPHONE distribution near the edges of the sound field, although the difference is less than % of the maximum value, on the whole. When the EM measurements (with flat frequency characteristics from 2 Hz to 2 khz) were carried out near the edges of the sound field, the disturbance of the field due to the existence of the EM and the integrated background noise of about 6 db(flat) were not negligible and influenced the results because the amplitude of the sound field near the edges was low. On the other hand, though the phase difference between the center and the edge of the sound field was set to be very small in the present analysis, LBM signals (with a band-pass filter from 8 khz to 22 khz) were be slightly affected by it. These reasons are reflected in the distributions at the edge of the sound field. From these results, it is concluded that the LBM-CT method is useful for visualizing the sound field with uniform phase distribution. 4. Conclusions The reconstructed distribution of the sound pressure agreed with that measured using an EM, and the LBM-CT method was confirmed to be useful for visualizing a uniform-phase-distribution acoustic field. As the next stage, we are proceeding with the development of a CT program with which the phase and amplitude distributions of the sound field can be treated, and we will construct a measurement system with a narrow beam scanning space. Then, we will evaluate the possibility of applying the LBM- CT method to a general complex field, and its accuracy, which governs its usefulness in various fields where the disturbance of the sound field by the conventional EM is not negligible. References [] A. Korpel, Acousto-Optics, 2nd ed. (Marcel Dekker, New York, 997). [2] Y. Sonoda and M. Akazaki, Measurement of low-frequency ultrasonic waves by Fraunhofer diffraction, Jpn. J. Appl. Phys., 33, 3 34 (994). [3] T. Sakoda and Y. Sonoda, Transmission and detection of light diffraction signal by low-frequency ultrasonic wave, Jpn. J. Appl. Phys., 42, 82 8 (23). [4] K. Nakamura, Sound field measurements based on the acousto-optic effect of air, Proc. 27th Meet. Light Wave Sensing Technology, pp (2). [] Y. Ikeda, M. Goto, N. Okamoto, T. Takizawa, Y. Oikawa and Y. Yamasaki, A measurement of reproducible sound field with laser computed tomography, J. Acoust. Soc. Jpn. (J), 62, (26) (in Japanese). [6] K. C. Kim, H. Fukuhara and H. Yamazaki, Ultrasonic imaging of welded metals using simplified ultrasonic computerized tomography, Jpn. J. Appl. Phys., 4, (22). [7] T. Mihara, K. Hagiwara and T. Furukawa, Three-dimensional sound pressure field measurement using photoelastic computer tomography method, Jpn. J. Appl. Phys., 37, (998). [8] O. J. Lokberg, M. Espeland and H. M. Pedersen, Tomographic reconstruction of sound fields using TV holography, Appl. Opt., 34, (99). [9] D. E. Evans, M. von Hellermann and E. Holzhauer, Fourier optics approach to far forward scattering and related refractive index phenomena in laboratory plasmas, Plasma Phys., 24, (982). [] Y. Sonoda, Y. Suetsugu, K. Muraoka and M. Akazaki, Applications of the Fraunhofer-diffraction method for plasma-wave measurements, Plasma Phys.,, 3 32 (983). 299