Acousto-optic sensing – spatial reconstruction of the sound field enclosed in aroom
Samuel A. VERBURG(1) , Efren FERNANDEZ-GRANDE(2)(1) Acoustic Technology, Technical University of Denmark, Denmark, saveri@elektro.dtu.dk(2) Acoustic Technology, Technical University of Denmark, Denmark, efg@elektro.dtu.dk

Abstract
Measuring the sound pressure field over medium and large volumes, such as rooms, can be cumbersome. Conventional sensing methods acquire the sound pressure (or other field quantity) at a single point. Therefore, toacquire the three-dimensional sound filed in a room, a large number of measurement points must be distributed
throughout. In this study, we propose a method to capture the sound field in a volume based on the acoustooptic effect. The acoustic pressure field is captured along beams of laser light. As the light (the sensing elementof this method) travels through the sound field, it captures information of the sound pressure along its travel
path. In this study, the spatial reconstruction of the sound field in a room from acousto-optic measurements ispresented and compared with conventional point-wise measurements. The results show a first approach to thespatial characterisation of the sound field in a room using the acousto-optic sensing principle.
Keywords: Acousto-optic, microphone arrays, sensing methods, signal processing

1

INTRODUCTION

Capturing the spatio-temporal properties of the sound field enclosed in a room is crucial in many applications;from architectural and room acoustic design (1, 2), to 3D audio recording and reproduction (3, 4). Pressure
measurements at a discrete set of positions (either using microphone arrays (5) or sequential measurements (6))
have conventionally been used to sample sound fields over space. However, sampling large three-dimensional
domains, such as rooms, using point-wise pressure measurements is impractical. Firstly, a large number ofmeasurements (that increases with frequency and room size) is required. Secondly, the measurements have to
be distributed across large regions of the room that are often inaccessible. Alternative sensing methods, better
suited for sampling the sound field over large volumes, are lacking.
In this study, acousto-optic tomography is examined as an alternative (or complementary) sensing method toconventional pressure measurements. The acousto-optic effect describes the interaction between sound and light.
Pressure fluctuations in a medium caused by a sound field produce changes on the medium local density. Suchdensity changes, in turn, change the speed of light, i.e. light travel faster or slower depending on the medium
density. The small changes of the speed of light can be captured along beams of laser light using a laser
Doppler vibrometer (LDV). In other words, laser light can be used to measure acoustic pressure fluctuations.
LDVs can capture the projected sound field along laser beams, and if enough projections are measured, tomographic images of the sound field can be obtained.
Acousto-optic tomography has been applied to the visualisation of sound fields, both in ultrasonics (7–13), andmore recently in the audible frequency range (14–30). The use of acousto-optic tomography has been limited tolaboratory conditions partly due to the reconstruction methods (inherited from classic computerised tomography(31)) that are usually employed. For example, the commonly used filtered back-projection algorithm requires
measuring projections of the sound field from every direction along a set of parallel beams. As a result, sensing methods based on the acousto-optic effect have not yet been employed to characterise large, complex soundfields such as those that are found inside rooms.

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In this study an acousto-optic tomography method, suitable for capturing complex sound fields, is introduced.
The proposed method formulates the problem as an algebraic reconstruction, therefore it does not require a
specific configuration of the measurements.
In a previous study (32) the authors examined the proposed tomographic reconstruction method from numerical
data. In this paper, we present a first experimental study that explores the acousto-optic effect as a sensing
principle for capturing the sound field in a real room.

2

THEORY

2.1 Acousto-optic effectLet us consider a pressure field p(r) enclosed in the domain Ω (such as r ∈ Ω | Ω ⊂ R3 ). We assume that thedomain boundary ∂ Ω is an impenetrable surface (not porous), and that points on the boundary r∂ Ω vibrate withvelocity u. The vibration velocity of surfaces can be measured by a LDV: a laser beam is sent through Ω, backscattered at r∂ Ω , and collected again. The phase of the beam is determined by its optical path (15)Z L

φ = k`

n(r; l)dl,(1)0where n (defined as the ratio between the speed of light in vacuum and in the medium) is the refractive
index within Ω , l is an integration variable along the beam path, L is the length of the path, and k` is thewavenumber of the laser light. Laser Doppler vibrometers determine the velocity of a surface by measuring
the phase shifts induced by changes in the optical path. Therefore, the velocity v measured by a LDV is
proportional to the rate of change of φv = jωφ

1
.
k` n0

(2)

Generally, it is assumed a constant n across the measurement domain, n(r) = n0 ∀ r ∈ Ω. In this case, Eq. 2
reduces to changes in the length of the path L, i.e. the velocity of the surface in the direction of the laser beamv = uΨ (r∂ Ω ). However, acoustic pressure fluctuations in a medium induce changes on n as
n = n0 +n0 − 1
p,
γ p0

(3)

where n0 is the refractive index of the unperturbed medium, p0 is the static atmospheric pressure, and γ is theheat capacity ratio. Equation 3 describes the relation between the refractive index and the acoustic pressure –i.e. the acousto-optic effect. The expression in Eq. 3 is obtained by combining the Gladstone-Dale relationn = ρK + 1 (33) (where K is a constant characteristic of the medium), and the assumption that the pressure
fluctuations p are an adiabatic processpρ − ρ0
=γ
,
p0
ρ0

(4)

where ρ0 and ρ are the densities of the static and perturbed medium. Equation 4 relies on a first order approximation, as generally made in linear acoustics.
The actual measurement of a LDV is obtained combining Eqs. 1, 2 and 3
v = jωL + jω

n0 − 1
γ p0 n0Z Lp(r; l)dl.

(5)

0

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Figure 1. a: laser beam of length 1 m on the x-axis and a single plane wave with direction defined by θ . b:
magnitude of the acousto-optic term along the 1 m laser beam for unit amplitude plane waves propagating in
directions θ = 0 : π and frequencies 10Hz to 10kHz.
Equation 5 shows that LDV measurements
are the combination of the surface vibration velocity uΨ = jωL, andRL
the acousto-optic effect jω γnp00−1
p(r;
l)dl.
We can identify three cases. If uΨ is much larger than the acouston0 0
optic effect term, the LDV only measures the velocity of the surface at which the laser is directed (which is
the original purpose of LDVs). If the surface is rigid, the acousto-optic effect term dominates, and we obtainmeasurements of the sound field projected along the beam path. In case that the two terms are of the sameorder, the LDV measures a combination of the two phenomena.
The magnitude of the acousto-optic effect term for simple sound fields is examined in Fig. 1. Figure 1(a)represents a laser beam of length 1 m placed in the x-axis. The sound field is a single unit-amplitude plane
wave propagating in the direction defined by θ . Figure 1(b) shows the magnitude of the acousto-optic effectthat would be measured for plane waves propagating in different directions (θ from 0 to π) and frequencies(from 10 Hz to 10 kHz). For θ = 0, the acousto-optic term is zero for frequencies with wavelengths that are
an integer of the laser beam length. At other directions, this zero term is shifted to higher frequencies. For
θ = π/2, the integral in Eq. 5 is done over a constant. At frequencies below 100 Hz the wavelength is muchlarger than the length of the laser beam, thus the measurement magnitude is constant regardless of the directionof propagation. The loss of sensitivity at low frequencies constitutes an inherent limitation of the acousto-optic
sensing method: sampling at low frequencies require the use of longer beams. Between 100 Hz and 1 kHz, themagnitude of the acousto-optic effect generated by unitary amplitude plane waves is of the order of ≈ 60 dB
with reference to 1 nm/s.

3

METHOD

3.1 Tomographic reconstruction of sound fields
The reconstruction method proposed in this paper is based on the plane wave expansion of the pressure fieldNp(r) ≈ ∑ Xn e− jkwn ·r ,

(6)

n=1

where wn indicates the direction of propagation of the nth plane wave, and the complex coefficient Xn contains
its amplitude and phase. Only propagating waves are considered, i.e. k = 2π f /c, where f is the frequency of4942

the sound field and c is the speed of sound. Neither sound sources nor absorption within Ω are considered.
An expression that links the plane wave complex coefficients Xn to LDV measurements is obtained combining
Eqs. 5 and 6Nv = ∑ Xn jωn=1

n0 − 1
γ p0 n0Z L

e− jkwn ·r dl.

(7)0

Equation 7 assumes that the surface at which
the LDV is directed is rigid, so the measurement v is mainly due
RL
to the acousto-optic term (uΨ  jω γnp00−1
p(r;l)dl). The validity of this assumption will be tested section 4.
n0 0
Measuring with M beams with various paths lm leads to a system of equations that can be written algebraicallyasv = Hx,(8)where v ∈ CM are the LDV measurements,x ∈ CN are the plane wave complex coefficient, and H ∈ CM×N is
n0 −1 R Lm − jkwn ·r
dlm . Estimating the coefficients x from the measurements v
a matrix with elements Hmn = jω γ p0 n0 0 e
poses an inverse problem that can be solved via common regularised inversion (34). The pressure field canthen be estimated at any point rs within Ω by multiplying the estimated coefficients x̃ with the corresponding
propagation terms e jkwn ·rs . In contrast to classic reconstruction methods (such as filter back projection) used inprevious studies (20, 23, 28), the method proposed in this paper does not require a specific regular sampling
scheme.

4

EXPERIMENTAL STUDY AND RESULTS

An experimental study was conducted in a room to examine the suitability of using the acousto-optic sensing
principle in a real scenario. Vibrometer measurements over one plane were acquired using a compact LDV
placed on a tripod (see Fig. 2a). The LDV was sequentially moved to 8 positions inside the room, and thelaser beams were directed to two of the room walls, covering the plane of height 1.25 m (the beam paths areshown in Fig. 4b). An accelerometer was used to monitor the vibration of the walls at the sampled positions(Fig. 2b), making it possible to asses the assumption of rigid boundaries. The vibration of the LDV head was aswell monitored with an accelerometer. Rubber pads were placed under the tripod feet to minimise the vibrationsof the LDV head. In order to assess the quality of the reconstruction, extensive pressure measurements were
performed in the room. A robotic arm (shown in Fig. 2c) was used to move a microphone and sequentiallymeasure the room frequency response on S = 1734 positions, covering one plane (1.6 × 2.4 m, height 1.5 m) in
the room. This plane constitutes the reference sound field.
The room, with bare concrete wall, has dimensions 3.29 × 4.38 × 2.97 m and an approximate reverberation timeof 2.5 s. The excitation was a logarithmic sine sweep played through a loudspeaker at one of the corners. The
overall sound pressure level was ≈ 95 dB.
Figure 3 shows the velocity measured by the LDV, by the accelerometer placed on the wall, and by the accelerometer placed on the LDV head. At low frequencies (below 300 Hz), the LDV mainly captures the vibration of the wall. Above 300 Hz, the velocity measured by the LDV and the wall vibration start to deviate.
At frequencies higher than 500 Hz, the LDV measures a velocity at least 10 dB larger than the accelerometer.
This discrepancy cannot be explained by vibrations of the LDV head, although the head does resonate at certain frequencies. The results in Fig. 3 and Fig. 1 are not directly comparable since the amplitude of the soundfields and the length of the beams are different. However, a qualitative assessment of Fig. 1 suggests that the
acousto-optic effect term becomes significant above approx. 300 Hz. This result indicates that the discrepancy
observed in Fig. 3 above 300 Hz can be due to the acousto-optic effect.

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Figure 2. Experimental setup. a: LDV mounted on the tripod. b: accelerometer used to monitor the wall
vibration. c: robotic arm used for the extensive pressure measurements that serve as reference.

Figure 3. Blue: Velocity measured by the LDV. Red: velocity measured by the accelerometer placed on thewall (same position at which the laser is pointing). Yellow: velocity measured by the accelerometer placed on
the laser head.
The sound field is reconstructed following the method presented in section 3. In this first experimental study,
the reconstruction was performed using the M = 8 LDV measurements, from which the velocity of the walls(measured by one of the accelerometers) was subtracted. The subtraction is done to be able to reconstruct thesound field at low frequencies (where the wall vibration dominates). Reconstructions at frequencies above 300
Hz (where the acousto-optic term is larger that the wall vibration) would require a larger number of measurements.
Figures 4a and b show the magnitude of the reference measured and reconstructed sound fields across thereference plane at 55 Hz. The general features of the sound field (corresponding to an axial mode in the xdirection) are correctly recovered. The way beams were distributed favours the reconstruction of the 55 Hz
axial mode: projections both perpendicular and parallel to the waves that compose the mode are measured. The
lower half of the reference plane was practically unsampled (see Fig. 4b), which might explain the inaccurate
reconstruction in that area.
Figure 4c shows the modal assurance criterion (MAC) (35) across frequency. The MAC, defined as MAC =2
pH p̃ /(pH p)(p̃H p̃), is a measure of the spatial correspondence between the reference field p and the reconstructed field p̃. The MAC takes values between one (when reference and reconstruction are consistent) and4944

a) reference sound field - 55 Hzb) reconstructed sound field04c) similarity

1

-10

0.8

-20

2

-30

1

MAC

3

0.6
0.4
0.2
0
50

0

0

1

2
m

3 0

1

2

3

[dB]

-40

60

70

80

90

100

frequency [Hz]m

Figure 4. a: normalised pressure level of the measured reference sound field at 55 Hz. b: reconstructed soundfiled from LDV measurements. Red dots: positions of the LDV. Dashed lines: laser beam paths. c: Modal
Assurance Criterion indicating the degree of similarity between reference and reconstructed sound fields acrossfrequencies.
zero. Figure 4c shows a good agreement at low frequencies, although above 60 Hz the reconstruction degrades.
At higher frequencies the sound field becomes more complex. The number of waves increases, and they propagate with different directions. The simple scheme used (with just 8 beams along the x and y directions) cannot
properly sample this complex sound field. Therefore, additional beams measuring projections from other directions would be required at higher frequencies.
This first experimental study indicates that it is possible to recover a simple mode in a room from measurementsof the acousto-optic effect. However, we have identified technological limitations (the vibration of the LDV
system) and case-specific limitations (the vibration of the room walls). In this study, the vibrations have beenmonitored using accelerometers, making it possible to obtain ideal acousto-optic effect measurements non-biased
by wall vibrations. Other compensation methods to separate the acousto-optic effect from unwanted vibrations
will be investigated in the future. Another possibility is the use of the measured wall velocity as additional
information to solve the reconstruction problem (instead of regarding it as unwanted noise), for which thecoupling between the wall vibration and the sound field inside the room has to be studied.
It is important to differentiate technological and case-specific constrains (which can eventually be overcome)from the loss of sensitivity at low frequencies. The loss of sensitivity is a fundamental limitation inherent
to the sensing principle, and it cannot be compensated for. Finally, the authors will study alternative sensing
principles for sampling sound fields over large volumes.

5

CONCLUSION

In this study we have proposed a method to capture the sound field over large volumes from projection measurements of the acousto-optic effect. The method is based on the algebraic formulation of the reconstruction
problem, which makes it possible the use of arbitrary measurement arrangements.
The experimental study has shown that the acousto-optic effect inside a room is detectable by means of vibrometry measurements. One axial mode of the room has been recovered by means of 8 acousto-optic measurements.
At the same time, we have identified the limitations to the applicability of the acousto-optic sensing principle.

4945

ACKNOWLEDGEMENTS
This work was supported by a research grant from the VILLUM foundation (grant nr. 19179).

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