Acousto-Optical Volumetric Sensing of Acoustic Fields

The characterization of sound fields over space is fundamental in acoustics. Yet, the volumetric acquisition of sound fields using conventional electromechanical transducers is challenging due to the high sampling requirements needed in a three-dimensional domain. In this study, we introduce an acousto-optic sensing method to remotely sample acoustic fields in a volume using a sparse set of noninvasive optical measurements. Optical methods that use light as a sensing element have drawn significant attention recently, since they allow for remote, noninvasive sampling of acoustic fields with fine spatial resolution. The practical deployment of optical sensing in acoustics, however, has been hindered by the limited acquisition of data available, and the unsuitability of existing methods to reconstruct sound fields in scarce-data regimes. The proposed methodology relies on projecting the measured data on a basis that satisfies the wave equation, thus alleviating the measurement requirements associated with traditional reconstruction methods. The proposed approach leads to accuracy improvements of 20 dB relative to current methods, and a 90% reduction of measured data. We demonstrate the approach experimentally by reconstructing in situ the three-dimensional sound field inside a room from remote measurements of the acousto-optic interaction. These results are key to advancing the use of acousto-optical sensing methods, which are currently limited to simplified domains. The work presented in this paper brings about the possibility to solve so far intractable problems, such as the volumetric and remote acquisition of complex three-dimensional acoustic fields.

Figure 1

(a) Optical interferometer used to measure acoustically induced phase shifts of light. A laser beam is generated and split into a sensing branch $s$ and a reference branch $r$. The sensing beam is sent through the sound field $p(\mathbf{r})$, where it is backscattered off the surface $\mathrm{\partial }\mathrm{\Omega }$, and collected back at BS2. After passing through BS3, both branches interfere on a photodiode detector, which generates an electrical current proportional to the acoustically induced phase shift of the sensing beam. (b) Tomographic scan of $p(\mathbf{r})$ with parallel beams. The projections $S(x^{\prime },\theta )$ are sampled with beams of equal spacing $\mathrm{\Delta }{x}^{\prime }$ and rotation increment $\mathrm{\Delta }\theta$. A sample of the projection measured by a single beam $m$ is $s_m$.

Figure 2

(a) Sampling scheme with $M=1682$ measurements distributed inside two square-pyramidal volumes of base $2.1\times 2.1{\mathrm{m}}^2$ and height 1.15 m. (b) Vector $\mathbf{n}=(\sum _{i=1}^K|{\mathbf{v}}_i^{\mathrm{null}}{|}^{\odot 2}{)}^{\odot 1/2}$ for the null-space orthonormal basis of the measurement set shown in (a). The $x$ and $y$ axes represent the azimuth and elevation angles of the wave coefficients over the radiation sphere. The number of waves included in the expansion is $N=2500$, and the acoustic frequency is set to 3000 Hz.

Figure 3

Comparison of the proposed wave-based reconstruction method with traditional tomographic FBP and ART algorithms. (a) Reference sound field (real part), generated by a monopole radiating in free field with a frequency of 2.5 kHz on a $1\times 1{\mathrm{m}}^2$ plane. The pressure is calculated at $N={260}^2$ points on the plane. Projections are sampled along $M_{x^{\prime }}$ parallel beams (uniformly spaced by $1/\sqrt{N}\mathrm{m}$) at $M_{\theta }$ uniform rotation angles, resulting in a total of $M=M_{x^{\prime }}\times M_{\theta }$ measurements [see Fig. 1]. (b)–(e) Reconstructions achieved by the FBP and ART algorithms, using $M={260}^2$ and $M={52}^2$ measurements, respectively (reconstructions computed with the ASTRA toolbox [31]). (f),(g) Reconstructions achieved by the proposed method (PW), based on a plane-wave expansion of the sound field, using $M={260}^2$ and $M={52}^2$ measurements, respectively.

Figure 4

Experimental setup. (a) Picture of the room, interferometer, and loudspeaker. The room has dimensions $3.29\times 4.38\times 3.28{\mathrm{m}}^3$. The room walls, floor, and ceiling are made of concrete, and no special retroreflecting covering is applied to the surfaces (just regular white paint). Two of the walls (the $y$-$z$ wall closer to the loudspeaker and the $x$-$z$ wall opposite to the scanned surface) are treated with acoustically absorptive material, which attenuates the reflections from such surfaces. The average reverberation time of the room in the studied frequency range (1 to 4 kHz) is 1.1 s. (b) Schematic of the measurement setup. The sound field is generated by a loudspeaker (in black) placed sideways with respect to the interferometer (in gray). The sound field is sampled with $M=814$ beams (in red), and reconstructed in a rectangular volume (dashed line). A null-space analysis of this sampling scheme is included in the Appendix.

Figure 5

Time-domain sound field in a room at four time frames: (a) 5.16 ms, (b) 6.76 ms, (c) 8.71 ms, and (d) 9.65 ms. The colored surfaces represent the pressure on three of the faces of the rectangular volume in which the sound field is reconstructed. The sound field is band limited between 1.5 and 3 kHz. Left column: reconstruction from $M=814$ optical measurements (see Fig. 4). Right column: numerical simulation of the sound field. The simulation is based on the image source method [33], computed using the implementation in Ref. [34].

Figure 6

(a) Normalized mean squared error and (b) normalized correlation achieved by the proposed sensing method and by the simulation for the 27 positions measured with a microphone inside the reconstructed volume. The central red mark indicates the median, the top and bottom of the blue box indicates the 75th and 25th percentiles, and the whiskers extend to the most extreme datapoints not considered outliers.

Figure 7

Acoustic pressure at position $x=0.572$, $y=0.378$, $z=1.370$ inside the sampled volume. Reference pressure measured with a microphone (in black) and (a) reconstructed from optical measurements (in blue) and (b) simulated with the image source method (in red).

Figure 8

(a) Sampling scheme with $M=814$ measurements distributed inside a square-pyramidal volume of base $0.84\times 0.84{\mathrm{m}}^2$ and height 1.15 m. (b) Vector $\mathbf{n}=(\sum _{i=1}^K|{\mathbf{v}}_i^{\mathrm{null}}{|}^{\odot 2}{)}^{\odot 1/2}$ for the null-space orthonormal basis of the measurement set shown in (a). The $x$ and $y$ axes represent the azimuth and elevation angles of the wave coefficients over the radiation sphere. The number of waves included in the expansion is $N=2500$, and the acoustic frequency is set to 3000 Hz.