Acousto-optic sensing for near-field acoustic holography

Samuel A. Verburg 1

Acoustic Technology group, Dept. of Electrical and Photonics Engineering, Technical University of Denmark Ørsteds Plads, building 352, 2800 Kongens Lyngby, Denmark

Efren Fernandez-Grande 2

Acoustic Technology group, Dept. of Electrical and Photonics Engineering, Technical University of Denmark Ørsteds Plads, building 352, 2800 Kongens Lyngby, Denmark

Earl G. Williams United States Naval Research Laboratory, Code 7106 Washington, DC 20375, USA

ABSTRACT Near-field acoustic holography (NAH) is a very powerful and widely used technique for the study of complex acoustic radiators. NAH enables to quickly understand how a complex source radiates into the medium. The technique is particularly suitable at low frequencies. At high frequencies, a dense transducer interspacing is required, and the measurement microphones can disturb the studied sound field when their size is comparable to the acoustic wavelength. In this study we examine the use of acousto-optic sensing in NAH. Acousto-optic sensing uses light beams as the sensing element, making it possible to acquire remote and non-invasive measurements without introducing extraneous objects in the vicinity of the source. The pressure, particle velocity and intensity fields, as well as the sound power radiated by a complex source, are determined from measurements in the near-field with an optical interferometer. The presented results demonstrate the potential of optical sensing to non-intrusively characterize sound fields, particularly at high frequencies.

1. INTRODUCTION

Near-field acoustic holography (NAH) is a powerful technique for studying and characterizing acoustic radiation from complex sources. Acoustic holography is of great relevance in many areas of acoustic research such as noise source identification, the study of sound radiation, or the characterization of vibrating structures and musical instruments. [1–4] In NAH, the sound pressure near the studied source is sampled over a surface using an array of transducers. The wavenumber spectrum is then calculated from the measured pressure, which makes

1 saveri@elektro.dtu.dk

2 efg@elektro.dtu.dk

a slaty. inter.noise 21-24 AUGUST SCOTTISH EVENT CAMPUS O ¥, ? GLASGOW

it possible to estimate the acoustic field over the entire 3D domain (not only on the measured surface) by propagating and back-propagating the wave components that conform the sound field. In order to estimate the wavenumber spectrum, the spatial sampling performed by the array must be dense enough to capture the highest spatial frequency present in the sound field. Central to NAH is the measurement of evanescent waves. Capturing evanescent components enables to reconstruct sources with a very fine spatial resolution and resolve features that are smaller than the acoustic wavelength (e.g., two point sources separated by a small distance, or circulatory energy flows below coincidence). Evanescent waves correspond to high spatial frequencies—with trace wavelengths that are shorter than the acoustic wavelength in the fluid medium. Therefore, a spatial sampling finer than the common ‘sensor spacing of half a wavelength’ rule is required to sample evanescent components while avoiding spatial aliasing. [5,6] In general, sampling sound fields at mid and high frequencies requires the use of dense arrays, which can be problematic. Dense arrays with a small transducer inter-spacing introduce scattering and di ff raction e ff ects that cannot be neglected (i.e., they are not acoustically transparent), e ff ectively biasing the measurements. The disturbance introduced by the array elements is particularly visible when the wavelength or the distance between transducers is comparable to the transducer size. Acousto-optic sensing is an alternative to conventional microphone arrays. The acousto-optic interaction describes the e ff ects that light experiences as it travels through an acoustic field. In acousto-optic sensing light is the sensing element, thus the measurements are non-invasive —in contrast to typical electro-mechanical transducers. Non-invasiveness is particularly relevant for problems where a dense spatial sampling is required, such as NAH at high frequencies. Unlike conventional transducers, acousto-optic sensing does not provide the pressure at specific locations but values of the pressure integrated over light beams (as in x-ray tomography). Hence, a tomographic reconstruction of the measured data is required in order to recover the pressure field. [7] Acousto-optic tomography has shown potential as a method for capturing the sound pressure in the near field of acoustic sources. [8, 9] In this study, we investigate sampling requirements for the tomographic acquisition and reconstruction of pressure fields.

2. BACKGROUND ON ACOUSTO-OPTIC SENSING

Acousto-optic sensing methods based on laser interferometry have drawn significant attention in recent years. [7–13] A laser beam traversing the harmonic pressure field p ( x , y , z ) e i ω t experiences the phase shift [7]

Z L

∆ φ = ω ℓ

∂ n ∂ p

0 p ( x , y , z ) dl , (1)

c 0

where l is an integration variable along the path of the laser beam, and L is the length of the path. The constants ω ℓ , c 0 , and ∂ n

∂ p are the frequency of the optical wave, the speed of light in vacuum, and the piezo-optic coe ffi cient in air, respectively. The time dependent term e i ω t is omitted for convenience. The phase shift described in Eq. 1 can be determined using optical interferometry. Such optical measurements are proportional to the line integral of p ( x , y , z ) along the path of a laser beam.

3. BACKGROUND ON ACOUSTIC HOLOGRAPHY

Acoustic holography makes it possible to fully characterize an acoustic field over an entire 3D domain by measuring the pressure on a single plane. [5] Let us consider the pressure field p ( x , y , z ) in the half space z > 0 where no sources are present. The wavenumber spectrum P ( k x , k y , z ) is given by the 2D Fourier transform of the pressure field at a plane z ,

P ( k x , k y , z ) = Z ∞

Z ∞

−∞ p ( x , y , z ) e − i ( k x x + k y y + k z z ) dxdy . (2)

−∞

The wavenumber spectrum at z can then be extrapolated to any other plane z ′ (from z ′ = 0 to infinity) via P ( k x , k y , z ′ ) = P ( k x , k y , z ) e ik z ( z ′ − z ) . (3)

Lastly, the pressure at the extrapolated plane is determined by the 2D inverse Fourier transform of P ( k x , k y , z ′ ),

Z ∞

Z ∞

p ( x , y , z ′ ) = 1 4 π 2

−∞ P ( k x , k y , z ′ ) e i ( k x x + k y y + k z z ′ ) dk x dk y . (4)

−∞

In NAH the pressure is measured clo se to the stud ied source in order to capture evanescent waves,

i.e. waves components with k z = + i q

k 2 x + k 2 y − k 2 . This enables to overcome the resolution limit imposed by the acoustic wavelength and resolve very fine spatial features. [1] Very dense microphone arrays are nonetheless required to sample evanescent waves, as such components correspond to high spatial frequencies ( k x or k y > k ). An alternative to dense microphone arrays is the use of optical measurements, as described in Eq. 1.

4. ACOUSTO-OPTIC TOMOGRAPHY SAMPLING

The spatial sampling of acoustic fields using point-wise measurements (i.e., microphone arrays) is well understood. [5] As measurements based on the acousto-optic interaction are line integrals of the pressure (as indicated by Eq. 1), di ff erent sampling requirements apply. In this study we derive sampling requirements for a uniform scan of the sound field using parallel laser beams. The projection of the pressure field in one plane at an angle ϕ is defined as

s ϕ (ˆ x ) ≡ Z ∞

−∞ p (ˆ x cos ϕ − ˆ y sin ϕ, ˆ x sin ϕ + ˆ y cos ϕ ) d ˆ y , (5)

where ˆ x , ˆ y represent the x , y coordinate system rotated by ϕ , as indicated in Fig. 1(a). The transformation in Eq. 5 is known as the Radon transform. [14] The uniform sampling of p ( x , y ) with parallel beams provides the value of s ϕ (ˆ x ) at a number of equally spaced points separated by ∆ ˆ x for a given angle ϕ . The view angle is then increased by ∆ ϕ and the process is repeated. While the spatial sampling along the radial direction ˆ x can be seen as a conventional sampling problem, the sampling along the angular direction is not as straightforward. In what follows we examine the number of view angles that are necessary to recover a pressure field. The 1D Fourier transform of the projection s ϕ (ˆ x ) is given by

S ϕ ( ˆ k x ) = Z ∞

−∞ s ϕ (ˆ x ) e − i ˆ k x ˆ x d ˆ x , (6)

where −∞ < ˆ k x < ∞ . Note that ˆ k x , ϕ and − ˆ k x , ϕ + π represent the same line, therefore S ϕ ( ˆ k x ) = S ϕ + π ( − ˆ k x ). The 2D Fourier transform of p ( x , y ) in polar coordinates is denoted as P pol ( ρ, ϕ ), so that

P pol ( ρ, ϕ ) = P ( ρ cos ϕ, ρ sin ϕ ) , (7)

where ρ ≥ 0 and 0 ≤ ϕ < 2 π . The Fourier slice theorem states that the 1D Fourier transform of each projection is a central cross-section of the 2D Fourier transform of p ( x , y ), [14] i.e.,

S ϕ ( ρ ) = P pol ( ρ, ϕ ) , S ϕ ( − ρ ) = P pol ( ρ, ϕ + π ) . (8)

Equation 8 is the basis of transform-based tomographic reconstruction methods since it enables to recover the 2D Fourier transform of a function from its measured projections. It follows from Eqs. 7 and 8 that if S ϕ ( ρ ) is known everywhere, one can simply convert this function to Cartesian coordinates

(a)

(b)

s ϕ (ˆ x )

ˆ x

k y

y

ˆ y

FT

∆ ϕ

∆ ϕ

ˆ x

ˆ k x ϕ

p ( x, y )

ϕ

x

k x

S ϕ ( ˆ k x )

∆ˆ x

Figure 1: Sketch of the sampling scheme. (a) Projections of the pressure field, s ϕ (ˆ x ), are sampled using parallel beams spaced by ∆ ˆ x . The view angle increment is denoted by ∆ ϕ . (b) The 1D Fourier transform of each projection, S ϕ ( ˆ k x ), gives a slice of the 2D Fourier transform of p ( x , y ) along a radial line. Values of the wavenumber spectrum are obtained on a polar grid.

and obtain P ( k x , k y )—and p ( x , y , z ) via Eq. 4. However, when measuring the projections of p ( x , y ), the function S ϕ ( ρ ) is only known at a discrete set of points along a number of radial lines, as illustrated in Fig. 1(b). It is observed that as the frequency increases (i.e., moving further away from the center of the k x , k y plane) the sampled data on the polar grid becomes sparser. Consequently, one cannot inverse Fourier transform the measured data directly. Several approaches to circumvent this problem exist. The most common is to filter the projections using | ˆ k x | , to then back-project each filtered projection independently. The filter derives from the Jacobian of a Cartesian-to-polar transformation, and leads to the filtered back-projection algorithm. [14,15] A di ff erent approach is the so-called direct Fourier inversion. In this case, the known values on the polar grid are interpolated to a rectangular grid. This study focuses on the direct Fourier formulation as it is possible to derive sampling conditions for which the interpolation is exact. [15,16]

4.1. Angular sampling The circular sampling theorem [16] enables to calculate the minimum number of samples necessary to recover a band limited periodic signal. The function P pol ( ρ, ϕ ) is periodic in ϕ with a period of 2 π . Let us assume that P pol ( ρ, ϕ ) is angularly band limited so that the highest angular frequency is w max = K / 2 π , where K is a positive integer. Then P pol ( ρ, ϕ ) can be recovered exactly from N ≥ 2 K + 1 of its samples via [16]

N − 1 X

n = 0 P pol ( ρ, n ∆ ϕ ) σ ( ϕ − n ∆ ϕ ) , (9)

P pol ( ρ, ϕ ) =

where ∆ ϕ = 2 π/ N , and the interpolation function σ ( ϕ ) is defined in Appendix A. The minimum even number of samples is 2 K + 2, and each measured projection S ϕ ( ˆ k x ) gives two samples of P pol ( ρ, ϕ ), as indicated by Eq. 8. Therefore the minimum number of view angles required to perfectly recover P pol ( ρ, ϕ ) is K + 1. An illustrative example is a simple axi-symmetric sound field for which the projections measured

at all view angles are the same. The highest angular frequency in such case is w max = 0 and just one view angle is necessary to recover P pol ( ρ, ϕ ). For most sound fields of interest 0 < w max < ∞ . One can conclude that if the number of available views is limited, an adequate low-pass filter of the measured projections in the angular frequency domain must be applied to avoid aliasing. An expression for such filter is given in Ref. [16].

4.2. Radial sampling Sampling the wavenumber space along the radial direction is a conventional sampling problem. If p ( x , y ) is zero outside the region defined by − A / 2 < x < A / 2 and − A / 2 < y < A / 2, the projections s ϕ (ˆ x ) are also band limited, and S ϕ ( ˆ k x ) can be recovered exactly from samples uniformly spaced by ∆ k ≤ 2 π/ A via the Whittaker-Shannon interpolation formula,

∞ X

m = −∞ S ϕ ( m ∆ k )sinc h ( ˆ k x − m ∆ k ) / ∆ k i . (10)

S ϕ ( ˆ k x ) =

Typically, p ( x , y ) is not exactly zero outside of the measurement region (although it decays significantly for large enough apertures) and a tapered window in the space domain should be applied. [5] The sum in Eq. 10 must be truncated to a finite number of terms M , which is given by the sampling resolution ∆ ˆ x ( M = A / ∆ ˆ x ). One must ensure that ∆ ˆ x is fine enough to capture the highest wavenumber present in the studied sound field, i.e., ∆ ˆ x ≤ π/ k max . For sound fields where evanescent wave components are present k max > k . In order to avoid aliasing it is necessary to apply a filter in the k -space with a cuto ff determined by k max , as typically done in NAH. [5] A general expression that allows to exactly interpolate from a polar grid to any point in the wavenumber domain is obtained by combining Eqs. 8, 9 and 10:

M / 2 X

N − 1 X

n = 0 P pol ( m ∆ k , n ∆ ϕ ) σ ( ϕ − n ∆ ϕ ) sinc  ( ρ − m ∆ k ) / ∆ k  . (11)

P pol ( ρ, ϕ ) =

m = − M / 2

The expression in Eq. 11 enables to calculate samples on a rectangular grid via Eq. 7, and ultimately recover the pressure in the 3D domain using Eqs. 2, 3 and 4. Equation 11 relies on the assumption that p ( x , y ) is band limited in the space domain (with a bandwidth A ), band limited in the angularly frequency domain (with a bandwidth 2 w max ) and band limited in the wavenumber domain (with a bandwidth of 2 k max ). Otherwise the measured data must be appropriately filtered prior to applying Eq. 11.

4.3. Number of view angles The minimum number of view angles necessary to recover P ( k x , k y ) is K + 1, where K = 2 π w max (as discussed in Section 4.1). Normally, the maximum angular frequency, w max , is not known a priori for an arbitrary sound field. Yet the approximate number of view angles necessary to sample p ( x , y ) can be derived if k max is known. The maximum spatial frequency that can be resolved is given by π/ ∆ ˆ x , hence, the maximum distance between two adjacent samples in the polar grid is ∆ ϕπ/ ∆ ˆ x . This distance should be similar to the distance between points along the radial direction to achieve an accurate interpolation, thus ∆ ϕπ/ ∆ ˆ x ≈ ∆ k . Noticing that ∆ ˆ x ≤ π/ k max and ∆ k ≤ 2 π/ A , one can write

∆ ϕ ≈ ∆ k ∆ ˆ x

π ≤ 2 π Ak max . (12)

The view angle increment ∆ ϕ is related to the minimum number of samples by

∆ ϕ = π K + 1 . (13)

Figure 2: (a) Pressure field p ( x , y ) over the aperture. The white dashed line indicates the position of the vibrating plate and the white X indicates the excitation point. (b) Wavenumber spectrum P ( k x , k y ) in a rectangular grid.

Combining Eqs. 12 and 13 we obtain

K + 1 ⪆ Ak max

2 . (14)

Equation 14 shows that the minimum number of view angles necessary to sample p ( x , y ) can be approximated if the highest spatial frequency present in the sound field k max and the aperture size A are known.

5. NUMERICAL EXPERIMENTS

The sound pressure radiated by a finite aluminum plate mounted on an infinite, rigid ba ffl e is computed numerically using a discrete implementation of the Rayleigh’s integral. [17] The plate has dimensions 30 × 40 × 0 . 3 cm, and it is excited at a frequency of 1600 Hz ( k ≈ 29). The pressure p ( x , y ) [shown in Fig. 2(a)] is calculated on a rectangular grid with a resolution of 1 cm over an aperture of A = 1 m and at a stand o ff distance of 5 cm from the plate. A Tukey window (with cosine fraction 0.5) is applied in the space domain and the array is zero-padded before computing P ( k x , k y ). Figure 2(b) shows the wavenumber spectrum, P ( k x , k y ), where it can be observed that the maximum spatial frequency present in this sound field is k max ≈ 44 m − 1 . The projections s ϕ (ˆ x ) are then calculated with a spacing ∆ ˆ x = 1 cm and for view angles from 0 ◦ to 179 ◦ with an increment of ∆ ϕ = 1 ◦ . The wavenumber spectrum P pol ( ρ, ϕ ) in the polar grid is obtained via Eqs. 6 and 8. Figure 3 shows the Fourier transform of P pol ( ρ, ϕ ) along the angular dimension. It can be observed that the angular spectrum is band limited—it drops significantly outside of the band − w max < w < w max , with w max ≈ 2 . 2 rad − 1 . In accordance to the circular sampling theorem, K = 2 π w max , thus the minimum number of view angles to recover this sound field is K + 1 = 16. Equation 14 gives an approximate minimum number of view angles Ak max / 2 = 22. Figure 4 shows the tomographic reconstruction of the pressure field using 22 , 16 and 10 view angles. When the sound field is over-sampled [ K + 1 = 22, Fig. 4(a)] a very accurate reconstruction is obtained when compared with the reference field [Fig. 2(a)]. At the sampling limit [ K + 1 = 16, Fig. 4(b)] the obtained reconstruction is accurate, with small artifacts towards the edges of the

Figure 3: Angular spectrum of P pol ( ρ, ϕ ).

Angular specturm of Pror(o,4) (absolute value) oe loos q w (tad-*)

Figure 4: Reconstruction of p ( x , y ) from tomographic measurements. (a) Using 22 view angles. (b) Using 16 view angles. (c) Using 10 view angles.

(a) K +1=22 (b) K+1 0.06 0.04 0.02 0.5 -0.5 x (m)

domain [notice the small ripples in the outer dark red areas in Fig. 4(b)]. When the sound field is under-sampled [ K + 1 = 10, Fig. 4(c)] large angular aliasing errors appear throughout the domain, particularly toward the edges. The results show that Eq. 14 provides a good indication for the approximate minimum number of view angles required to recover the sound field. In addition, Fig. 4(c) manifest the aliasing errors that appear when the measured data is not appropriately filtered before reconstruction, as discussed in Section 4.1.

6. CONCLUSIONS

In this study we have examined the sampling requirements for the tomographic acquisition and reconstruction of a sound field. It has been shown that a sound field can be reconstructed from a limited number of view angles provided that its wavenumber spectrum is angularly band limited. In order to avoid aliasing, a filter must be applied to band limit the angular spectrum prior to reconstruction. The su ffi cient sampling requirements, in terms of number of view angles and discretization along the radial direction, are linked to the maximum spatial frequency present in the sound field as well as to the dimensions of the aperture. This is analogous to conventional sampling for NAH. The sampling requirements developed in this study provide guidance in the design of acousto-optic tomography sampling schemes. The presented results are relevant for sound field reconstruction problems that require a dense spatial sampling, such as NAH at high frequencies, since acousto-optic tomography enables non-invasive measurements.

7. ACKNOWLEDGMENTS

Samuel A. Verburg and Efren Fernandez-Grande were supported by the VILLUM foundation (grant number 19179). Earl G. Williams was supported by the O ffi ce of Naval Research.

APPENDIX A: CIRCULAR SAMPLING THEOREM

The circular sampling theorem, as it is described in Ref. [16], provides the number of samples necessary to perfectly recover a band limited periodic function. For a periodic function g ( t ) with a period of T = 2 π and a highest frequency of w max , the product Tw max is a positive integer denoted by K . If the sampling period ∆ t = 2 π

N satisfies 2 w max < 1 / ∆ t , then g ( t ) can be perfectly recovered from N ≥ 2 K + 1 uniformly spaced samples via

N − 1 X

n = 0 g ( n ∆ t ) σ ( t − n ∆ t ) , (15)

g ( t ) =

where the function σ ( t ) is defined as

σ ( t ) ≡ sin( tN / 2)

N sin( t / 2) . (16)

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