Optical imaging of sound fields is gaining attention in acoustic engineering for transducer characterization [1–3], elucidation of physical principles underlying musical instruments [4–6], and investigations of aeroacoustic sound generation [7–9], to name a few. Over the past decades, several optical methods have been applied to sound-field imaging both in water and air, including shadowgraph and schlieren methods [10–12], electric speckle pattern interferometry [1,13,14], off-axis digital holography [15–18], and parallel phase-shifting interferometry and digital holography [3,19]. Using high-speed cameras allows us to observe sound fields at frame rates higher than the sampling frequency of sound (typically 40,000 frames per second or higher for audio applications), resulting in direct observation of the spatiotemporal properties of acoustic phenomena.

A unique characteristic of the optical measurement of sound is the spatiotemporally harmonic nature of the measurand. When the measurement is based on the acousto-optic effect, the optical phase is modulated by sound. Since the sound is governed by the wave equation, so is the phase variation of light. This characteristic has been used for filtering [12,20,21] and tomographic reconstruction of a sound field [22,23]. In these applications, static noise and slow fluctuations can be distinguished from sound in temporal or spatiotemporal spectra.

We hypothesize that a sound field can be retrieved even from heavily distorted or noisy interferograms by applying appropriate post-processing based on the physical characteristics of sound. Examples of such low-quality interference fringes are those observed with an optical system composed of Fresnel lenses. The Fresnel lenses have several desired properties for sound-field imaging, including thinness, lightweight, low cost, and ease of making a large aperture. In particular, a large-aperture imaging system is desired for audio applications because the acoustic wavelength of audible sound ranges from 17 mm at 20 kHz to 17 m at 20 Hz. Therefore, the benefits of using Fresnel lenses in a sound-field imaging system are significant. However, due to their low imaging quality, they have never been used for sound-field imaging.

In this Letter, we propose to use off-the-shelf Fresnel lenses for interferometric sound-field imaging. An overview of the proposed method is shown in Fig. 1. Two Fresnel lenses are positioned in the object path of a speckle holographic interferometer. The main difficulty in using Fresnel lenses is their low-imaging quality due to jagged shapes, which are not suitable for common imaging and interferometric applications. We addressed this difficulty by temporal and spatial filtering to extract harmonic oscillations of the acoustic field. We conducted a proof-of-concept experiment comparing the imaging results by the proposed method with those by parallel phase-shifting interferometry (PPSI) [3]. We confirmed that the sound-field information can be extracted from the specklegrams obtained by the interferometer composed of two Fresnel lenses.

 

Fig. 1. Overview of proposed method. (a) Sound field generated by a loudspeaker between the two Fresnel lenses is measured. (b) Schematic of the optical system. M, mirror; HM, half mirror; L, lens; FL, Fresnel lens. (c) Procedure in the proposed method.

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The proposed optical system comprising Fresnel lenses is shown in Fig. 1(b). The two Fresnel lenses, FL1 and FL2, were used to magnify and demagnify the illumination beam emitted from a single longitudinal mode laser at 532 nm. The Fresnel lenses have a focal length of 350 mm, a numerical aperture of 0.58, and size of 280 mm $\times$ 280 mm (43-018, Edmund Optics). The 4$f$ imaging system consisting of L1 and FL1 (L2 and FL2) has a magnification (demagnification) of 11.7 (0.0857), where numerical apertures of L1 and L2 are 0.42. The light field on the back focal plane of L2 was projected onto a high-speed camera (HX-3, nac Image Technology Inc.) with the light passed through the reference path, and the interference between the object and reference light was observed.

Before measuring the sound field, we checked the intensity of the object light through the Fresnel lenses. Figure 2(a) shows an example of an intensity image recorded without the diffuser at the back focal plane of L2. Due to the low-quality of the Fresnel lenses, it was unable to create a uniform and collimated beam. Therefore, the diffuser plate was placed at the back focal plane of L2 to ensure that the object light uniformly illuminates the image plane, as shown in Fig. 2(b).

 

Fig. 2. Raw intensity image (a) without a diffuser and (b) with a diffuser. (c) Specklegram with diffuser and (d) spatial Fourier power spectrum of panel (c) on the linear scale. (e) Extracted wrapped phase map by the two-dimensional Fourier transform method.

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The speckle interferometry is employed to extract optical phase modulation caused by the sound from the diffused light. The measurement principle of sound fields by speckle interferometry is explained as follows. The intensity of the interferogram at an initial state, where no sound exists, can be written as

For the proof-of-concept experiment, we measured the sound fields generated by a loudspeaker (FT 48D, Fostex) by the proposed method and parallel phase-shifting interferometry [3], and compared the results to verify the feasibility of the sound-field imaging using Fresnel lenses. The PPSI system comprises a high-speed polarization camera and an ordinary projection lens with a diameter of 200 mm, which enables the measurement of sound fields quantitatively; therefore, it was used for reference data in this experiment. The hyper ellipse fitting in the subspace method was used for phase calculation of the PPSI data [26]. For both optical systems, the cameras were operated at the frame rate of 50,000 frames per second and the exposure time of 10 $\mathrm{\mu}$s. We recorded 1000 frames for each measurement. The field of view (FOV) was a 100 $\times$ 100 mm square. The PPSI observed the FOV with $160 \times 160$ pixels. The proposed system provided a FOV with $220 \times 220$ pixels, which is depicted by the white dashed box in Fig. 2(e). The integral lengths of the measured volumes were 2.5 m for PPSI and 0.7 m for the proposed method. It should be note that the difference of the integral lengths does not significantly affect the measurement results because the sound pressure apart from the central acoustic axis is small. The loudspeaker was placed just below the FOV in the center of the integral path, as shown in Fig. 1(a). For the acoustic signal, we used the sinusoidal wave of 5, 10, 15, and 20 kHz. A monitor microphone was placed 10 cm above the loudspeaker [not shown in Fig. 1(a)], and the amplitudes of each sinusoidal sound wave were adjusted to 110 dB in the sound pressure level relative to 20 $\mathrm{\mu}$Pa at the monitor microphone position. The acquisition timing of each system relative to the generated sound fields was aligned using an external trigger input of the high-speed cameras.

The obtained data of both systems has the unit of optical radian that is proportional to the line-integral of sound pressure along the laser path. To obtain a point-wise sound pressure value from the line integral, tomographic reconstruction from multiple angle measurements or single-shot reconstruction of the axisymmetric field have been proposed [22,23,27]. Here, we do not apply any reconstruction method and display the projected sound field with the unit of an optical radian because comparing the projected field without reconstruction is the most straightforward way they evaluate any acousto-optic imaging method and is independent of any parameters and algorithms of reconstruction processes.

Figure 3 shows the imaging results. As shown in Fig. 3(a), the PPSI captures the smooth spherical wavefronts propagating upward for the frequencies 5, 10, and 15 kHz. The sound field of 20 kHz shows a unique spatial characteristic, i.e., the radiated sound is divided into two spatial components. This should be due to a non-uniform vibration mode of the loudspeaker diaphragm, which often occurs in a high-frequency signal whose acoustic wavelength is smaller than the diameter of the loudspeaker diaphragm. The imaging results of the same sound fields by the proposed method are shown in Fig. 3(b). It can be seen that similar sound fields with the PPSI are obtained for 5, 10, and 15 kHz, although there are significant wavefront distortions. These results suggest that the sound field information is preserved in the distorted phase maps even when the object light passes through Fresnel lenses. For 20 kHz, however, the spatial distortion is dominant, and the sound field is difficult to recognize.

 

Fig. 3. Imaging result of (a) PPSI and (b) the proposed method. From left to right: sound field of 5, 10, 15, and 20 kHz, respectively.

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To verify whether the captured spatial wavefronts are the generated sound waves, the power spectra are calculated from the temporal averaging of 10 by 10 pixels at the center of the images for both PPSI and the proposed method. Figure 4 plots the power spectra of the monitor microphone, PPSI, and the proposed method. The microphone data are shown by the sound pressure level, whereas the optical data are shown by the decibel relative to 1 radian. The sound pressure levels of all frequencies are 110 dB as adjusted before the experiment. The signal frequency components are indicated by square markers. Both PPSI and the proposed method have peaks at the same frequencies as the microphone in their power spectra. The signal power of the optical data varies with acoustic frequencies because the line integral of sound pressure depends on the sound field along the laser path. Since the PPSI detects the optical phase quantitatively, the results indicate that the proposed method enables detection of the acoustic signal quantitatively both in frequency and power. Note that the peak components around 3 kHz observed in all PPSI spectra were due to the vibration caused by the fan of the polarization high-speed camera. The power spectra also show that the noise floor of the proposed method is approximately 20 dB larger than that of the PPSI.

 

Fig. 4. Power spectra of (a) the microphone and (b) PPSI and the proposed method. From left to right: sound field of 5, 10, 15, and 20 kHz, respectively. The squares indicate the signal frequencies.

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The imaging results of the proposed method shown in Fig. 3(b) indicate that a significant amount of spatial noise is contained in the sound-field images. Figures 5(a) and 5(b) show the spatial frequency spectra of the sound-field images of PPSI and the proposed method, respectively. The two-dimensional (2D) Fourier transform was calculated after the images were zero-padded to be $512 \times 512$ pixels. As shown in Fig. 5(a), the spatial frequency spectra of 2D sound fields are localized within a circle of radius $k$, where $k$ is the magnitude of the 2D acoustic wavenumber vector [12,21]. In our experimental setup, since the loudspeaker is located below the imaging area and the sound wave propagates upward, the spatial frequency spectrum is localized near the vertical axis. Figure 5(b) indicates that the spatial frequency spectrum of the proposed method consists of the same peaks with the PPSI and the other components that stand for noise. Since the noise components are separated in this spatial frequency domain, we designed spatial masks as depicted in Fig. 5(c). The masks have value 1 where $k-\Delta \leq \sqrt {k_x^2+k_y^2} \leq k -\Delta$ and $|k_x| < |k_y|$ are holds, and otherwise 0. Note that $k_x$ and $k_y$ are the horizontal and vertical variables in the spatial frequency domain, and $\Delta$ is the width of the spatial bandpass filter. Here, $\Delta = 8.6$ m$^{-1}$ for all frequencies, corresponding to two pixels in the spatial frequency domain. The first inequality determines the ring shape, whereas the second eliminates horizontal components. The spatial frequency mask of each frequency was multiplied by the amplitude of the spatial frequency spectrum of each frame of the temporally filtered sound-field image, followed by the inverse 2D Fourier transform of the masked spectra.

 

Fig. 5. Spatial frequency spectra of the (a) PPSI and (b) proposed method on a linear scale. The horizontal and vertical axes are enlarged around zero. (c) Spatial frequency masks used for spatial filtering. From left to right: sound field of 5, 10, 15, and 20 kHz, respectively.

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The spatially filtered sound-field images are shown in Fig. 6. For the PPSI images, the diagonal stripe patterns in Fig. 3(a) are eliminated, and the filtered images show a smooth sound field. The diagonal patterns are nonlinear errors associated with the phase-shift error of the four-step phase-shifting method, and the spatial filter removed the patterns. The 5-kHz results in Figs. 3 and 6 show differences on the left and right sides because the spatial mask unintentionally removed a part of the spatial frequency spectra leaking outside of the 90 degrees annular sector. This change may be avoided by carefully designing a spatial frequency mask. Finally, by comparing Figs. 3(b) and 6(b), we can confirm that the spatial noise is significantly reduced. Especially at 5, 10, and 15 kHz, the sound fields after spatial filtering agree with those of PPSI, indicating the effectiveness of spatial frequency filtering based on the spatiotemporally harmonic nature of sound. At 20 kHz, however, their intensity and spatial characteristics do not match those of PPSI, although some wave-like patterns are extracted. As can be seen from Fig. 5(b), the spatial noise components that existed within the spatial frequency mask can be considered to change the appearance of the filtered sound field. The reason why the spatial frequency noise of 20 kHz is larger than the other frequencies needs further investigation in the future.

 

Fig. 6. Spatially filtered sound fields. (a) PPSI and (b) proposed method. From left to right: sound field of 5, 10, 15, and 20 kHz, respectively.

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In summary, this Letter investigated the feasibility of sound-field imaging using Fresnel lenses. The experimental results verified that sound-field imaging via Fresnel lenses is possible with spatial frequency filtering. The advantages of using Fresnel lenses over conventional lenses include low cost, thinness, and lightweight. A Fresnel lens of the size used in this experiment may cost approximately an order of magnitude less and is much lighter compared to an optical grade lens of the same size; these benefits increase rapidly as the size increases. For more complicated sound fields rather than the pure tones used in this Letter, one may use a spatiotemporal filter bank [12,21] or sparse optimization-based sound-field reconstruction method [28]. Further investigations, including investigation of differences in sound field visualization performance depending on the lens and optical systems used, and optimization of the imaging system and signal processing method are the subjects of future work. In addition, since the proposed method has a higher noise floor than the PPSI, investigating the causes of the measurement noise and the applicability of the proposed method to weak sound is also necessary. Pushing forward the proposed concept may lead to a low-cost and large-aperture optical imaging solution for audio applications. Especially by combining the proposed method with a tomographic reconstruction scheme, large-volume 3D sound-field reconstruction will be feasible in the future. For example, a commercially available 1-m class Fresnel lens could expand the imaging area by a factor of 100 and the tomographic volume by 1000 compared to a typical sound-field imaging system. The proposed method will be effective for many situations generating loud sounds, such as characterizations of public address loudspeakers, musical instruments, and impact sounds.