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    Sound field reconstruction using acousto-optic tomography

    Torras Rosell, Antoni; Barrera Figueroa, Salvador; Jacobsen, Finn

    Published in: Acoustical Society of America. Journal

    Link to article, DOI: 10.1121/1.3695394

    Publication date: 2012

    Document Version Publisher's PDF, also known as Version of record

    Link back to DTU Orbit

    Citation (APA): Torras Rosell, A., Barrera Figueroa, S., & Jacobsen, F. (2012). Sound field reconstruction using acousto-optic tomography. Acoustical Society of America. Journal, 131(5), 3786-3793. https://doi.org/10.1121/1.3695394

    https://doi.org/10.1121/1.3695394 https://orbit.dtu.dk/en/publications/2f9943d7-6e74-44a5-b1c1-7913f11b6795 https://doi.org/10.1121/1.3695394

  • Sound field reconstruction using acousto-optic tomographya)

    Antoni Torras-Rosellb) and Salvador Barrera-Figueroa Danish Fundamental Metrology A/S, Matematiktorvet 307, 2800 Kongens Lyngby, Denmark

    Finn Jacobsen Acoustic Technology, Department of Electrical Engineering, Technical University of Denmark, Ørsteds Plads 352, 2800 Kongens Lyngby, Denmark

    (Received 28 September 2011; revised 15 February 2012; accepted 19 February 2012)

    When sound propagates through a medium, it results in pressure fluctuations that change the instanta-

    neous density of the medium. Under such circumstances, the refractive index that characterizes the

    propagation of light is not constant, but influenced by the acoustic field. This kind of interaction is

    known as the acousto-optic effect. The formulation of this physical phenomenon into a mathematical

    problem can be described in terms of the Radon transform, which makes it possible to reconstruct an

    arbitrary sound field using tomography. The present work derives the fundamental equations govern-

    ing the acousto-optic effect in air, and demonstrates that it can be measured with a laser Doppler

    vibrometer in the audible frequency range. The tomographic reconstruction is tested by means of

    computer simulations and measurements. The main features observed in the simulations are also

    recognized in the experimental results. The effectiveness of the tomographic reconstruction is further

    confirmed with representations of the very same sound field measured with a traditional microphone

    array. VC 2012 Acoustical Society of America. [http://dx.doi.org/10.1121/1.3695394]

    PACS number(s): 43.35.Sx [JDM] Pages: 3786–3793

    I. INTRODUCTION

    The acousto-optic effect has been extensively used to

    characterize ultrasonic waves in underwater acoustics.1–7 In

    such measurements, the acoustic properties of sound are

    determined by measuring the small changes of the refrac-

    tive index that are induced by the pressure fluctuations of

    the acoustic field. These variations of the refractive index

    will cause diffraction and changes of the speed of light that

    influence the propagation of light in amplitude and phase.

    In practice, in the low ultrasonic frequency range, diffrac-

    tion effects can be neglected when the acoustic field has

    small amplitudes.2,6,7

    Ultrasonic measurements in air are not commonly

    based on the acousto-optic effect (although some examples

    can be found in Refs. 8 and 9). This is perhaps because

    ultrasound is highly attenuated in air and because there

    normally is a significant impedance mismatch between the

    air and the ultrasonic transducer.8 These constraints are

    less severe when using conventional loudspeakers and

    microphones in the audible frequency range. However, only

    a few investigations have been reported for audible

    sound,10–12 and they have been concerned with visualiza-

    tion purposes rather than quantification.

    The aim of this work is to demonstrate that light can be

    used as a means to characterize airborne sound in the audible

    frequency range. First, the physical principles governing the

    acousto-optic effect are presented. This will show that sound

    pressure fluctuations are captured in the phase of a light

    beam that travels through the medium. Next, we describe the

    measurement procedure used to retrieve the phase of the

    light as an apparent velocity measured with a laser Doppler

    vibrometer (LDV). This apparent velocity is interpreted as a

    projection of the sound field. Several projections in different

    directions can be used to reconstruct the acoustic field using

    tomography. The quality of the tomographic reconstruction

    is finally assessed by means of simulations and experimental

    results.

    II. ACOUSTO-OPTIC TOMOGRAPHY

    A. Acousto-optic effect

    The phenomenon of sound inherently involves pressure

    fluctuations that change the density of the medium. Assum-

    ing adiabatic conditions, the total pressure pt and the density q of the medium are related by means of the following expression:13

    pt p0 ¼ q

    q0

    � �c ; (1)

    where p0 and q0 are the pressure and the density under static conditions, and c is the ratio of specific heats. Note that, when sound propagates, pt corresponds to the superposition of the static and the acoustic pressures, that is, pt¼ p0þ p. The influence of the density variations on the propagation of

    light can be determined by combining the mechanical and

    optical properties of the medium. In 1863, Gladstone and

    Dale established an empirical relation between the refractive

    index n and the density q of various liquids14

    n� 1 ¼ Gq; (2) b)Author to whom correspondence should be addressed. Electronic mail:

    atr@dfm.dtu.dk

    a)Portions of this work were presented in “Sound field reconstruction based

    on the acousto-optic effect,” Proceedings of Inter-Noise 2011, Osaka,

    Japan, September 2011.

    3786 J. Acoust. Soc. Am. 131 (5), May 2012 0001-4966/2012/131(5)/3786/8/$30.00 VC 2012 Acoustical Society of America

    Downloaded 21 May 2012 to 192.38.67.112. Redistribution subject to ASA license or copyright; see http://asadl.org/journals/doc/ASALIB-home/info/terms.jsp

  • where the Gladstone–Dale constant G is an intrinsic feature of the liquid. This relation also holds for air.15 The most im-

    portant property of the latter expression is not the exact

    value of G, but the fact that density and refractive index ex- hibit a linear relationship. The relation between the refrac-

    tive index and the acoustic field can be established by

    combining Eqs. (1) and (2)

    n ¼ ðn0 � 1Þ pt p0

    � �1=c þ 1 ¼ ðn0 � 1Þ 1þ

    p

    p0

    � �1=c þ 1;

    (3)

    where n0 is the index of refraction under standard atmos- pheric conditions. As shown in Appendix A, this expression

    can be approximated by a first order Taylor expansion when

    the acoustic pressure is much smaller than the static pres-

    sure, p� p0;

    n ffi n0 þ n0 � 1 cp0

    p: (4)

    Thus, under weak acousto-optic interaction, the variations of

    the refractive index are proportional to the sound pressure.

    For ease of reference, a sound pressure of 1 Pa yields an

    increase of the refractive index of air of about 2� 10� 9 of its value under static conditions.

    B. Measurement principle

    The understanding of the physical phenomenon govern-

    ing the acousto-optic effect opens up the possibility of char-

    acterizing sound by measuring light that travels through an

    acoustic field. In this context, light can be regarded as an

    electromagnetic wave E that satisfies the electromagnetic wave equation

    r2E� n c0

    � �2 @2E @t2 ¼ 0; (5)

    where c0 corresponds to the speed of light in vacuum. Although this fundamental equation is normally derived for

    waves propagating through a homogeneous quiescent me-

    dium (which is not completely true in the presence of

    sound), correct solutions can still be obtained when the fol-

    lowing condition is fulfilled:16

    1

    n

    @n

    @t T

    ���� ����� 1; (6)

    where T corresponds to the oscillation period of the electric field. This is indeed the case for weak acousto-optic interac-

    tion (see Appendix B for further details). Under such condi-

    tions, the acousto-optic effect modifies the phase of light

    rather than its amplitude. One can think of it as an electro-

    magnetic wave that travels faster or slower depending on the

    pressure fluctuations caused by the acoustic field. The light

    travels slower when the medi