Scanning Camera for Continuous-Wave Acoustic Holography
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Scanning Camera for Continuous-Wave Acoustic Holography Hillary W. Gao, Kimberly I. Mishra, Annemarie Winters, Sidney Wolin, and David G. Griera) Department of Physics and Center for Soft Matter Research, New York University, New York, NY 10003, USA (Dated: 7 August 2018) 1 We present a system for measuring the amplitude and phase profiles of the pressure 2 field of a harmonic acoustic wave with the goal of reconstructing the volumetric sound 3 field. Unlike optical holograms that cannot be reconstructed exactly because of the 4 inverse problem, acoustic holograms are completely specified in the recording plane. 5 We demonstrate volumetric reconstructions of simple arrangements of objects using 6 the Rayleigh-Sommerfeld diffraction integral, and introduce a technique to analyze 7 the dynamic properties of insonated objects. a)[email protected] 1 Scanning Holographic Camera 8 Most technologies for acoustic imaging use the temporal and spectral characteristics of 1 9 acoustic pulses to map interfaces between distinct phases. This is the basis for sonar , and 2 10 medical and industrial ultrasonography . Imaging continuous-wave sound fields is useful for 3 11 industrial and environmental noise analysis, particularly for source localization . Substan- 12 tially less attention has been paid to visualizing the amplitude and phase profiles of sound 13 fields for their own sakes, with most effort being focused on visualizing the near-field acous- 14 tic radiation emitted by localized sources, a technique known as near-field acoustic holog- 4{6 15 raphy (NAH) . The advent of acoustic manipulation in holographically structured sound 7{11 16 fields creates a need for effective sound-field visualization. Here, we demonstrate a scan- 17 ning acoustic camera that combines lockin detection with a polargraph for flexible large-area 18 scanning to accurately record the wavefront structure of acoustic travelling waves. Borrow- 12 19 ing techniques from optical holography, we use Rayleigh-Sommerfeld back-propagation to 20 reconstruct the three-dimensional sound field associated with the complex pressure field in 21 the measurement plane. These reconstructions, in turn, provide insights into the dynamical 22 properties of objects immersed in the acoustic field. 23 I. HOLOGRAPHY 24 A harmonic traveling wave at frequency ! can be described by a complex-valued wave 25 function, i'(r) −i!t (r; t) = u(r) e e ; (1) 26 that is characterized by real-valued amplitude and phase profiles, u(r) and '(r), respectively. 27 Eq. (1) can be generalized for vector fields by incorporating separate amplitude and phase 2 Scanning Holographic Camera 28 profiles for each of the Cartesian coordinates. The field propagates according to the wave 29 equation, r2 = k2 ; (2) 30 where the wave number k is the magnitude of the local wave vector, k(r) = r'(r): (3) 31 A hologram is produced by illuminating an object with an incident wave, 0(r; t), whose 32 amplitude and phase profiles are u0(r) and '0(r), respectively. The object scatters some of 33 that wave to produce s(r; t), which propagates to the imaging plane. In-line holography 34 uses the remainder of the incident field as a reference wave that interferes with the scattered 35 field to produce a superposition (r; t) = 0(r; t) + s(r; t) (4) 36 whose properties are recorded. The wave equation then can be used to numerically re- 37 construct the three-dimensional field from its value in the plane. In this way, numerical 38 back-propagation can provide information about the object's position relative to the record- 39 ing plane as well as its size, shape and properties. The nature of the recording determines 40 how much information can be recovered. 41 A. Optical holography: Intensity holograms Optical cameras record the intensity of the field in the plane, and so discard all of the in- formation about the wave's direction of propagation that is encoded in the phase. Interfering 3 Scanning Holographic Camera the scattered wave with a reference field yields an intensity distribution, I(r) = j (r; t)j2 (5a) 2 i'0(r) i's(r) = u0(r) e + us(r) e ; (5b) 42 that blends information about both the amplitude and the phase of the scattered wave into 43 a single scalar field. 44 The properties of the scattered field can be interpreted most easily if the incident field 45 can be modeled as a unit-amplitude plane wave, ikz −i!t 0(r; t) ≈ e e ; (6a) 46 in which case, 2 i's(r) I(r) ≈ 1 + us(r) e : (6b) 47 If, furthermore, the scattering process may be modeled with a transfer function, s(r) = T (r − rs) 0(rs); (6c) 48 then I(r) can be used to estimate parameters of T (r), including the position and properties 49 of the scatterer. 50 This model has proved useful for interpreting in-line holograms of micrometer-scale col- 13 51 loidal particles . Fitting to Eq. (6) can locate a colloidal sphere in three dimensions with 14,15 52 nanometer precision . The same fit yields estimates for the sphere's diameter and re- 15 16 53 fractive index to within a part per thousand . Generalizations of this method work 17{19 54 comparably well for tracking clusters of particles . 4 Scanning Holographic Camera 55 B. Rayleigh-Sommerfeld back propagation 56 The success of fitting methods is based on a priori knowledge of the nature of the scat- 57 terer, which is encoded in the transfer function, T (r). In instances where such knowledge is 58 not available, optical holograms also can be used as a basis for reconstructing the scattered 59 field, s(r; t), in three dimensions. This reconstruction can serve as a proxy for the structure 60 of the sample. 12,20 61 One particularly effective reconstruction method is based on the Rayleigh-Sommerfeld 21 62 diffraction integral . The field, (x; y; 0; t) in the imaging plane, z = 0, propagates to point 21 63 r in plane z as (r; t) = (x; y; 0; t) ⊗ hz(x; y); (7a) 64 where 1 d eikr h (x; y) = : (7b) z 2π dz r 65 is the Rayleigh-Sommerfeld propagator. The convolution in Eq. (7) is most easily computed 66 with the Fourier convolution theorem using the Fourier transform of the propagator, p −iz k2−q2 Hz(q) = e : (8) 67 Equation (7) can be used to numerically propagate a measured wave back to its source, 68 thereby reconstructing the three-dimensional field responsible for the recorded pattern. 69 C. Acoustic holography: Complex holograms 70 Early implementations of acoustic holography resembled optical holography in recording 22 71 the intensity of the sound field . Sound waves of modest intensity, however, are fully 5 Scanning Holographic Camera FIG. 1. (color online) Schematic representation of the scanning acoustic camera. A polargraph composed of two stepping motors and a timing belt translates a microphone across the field of view in a serpentine pattern. A harmonic sound field is created by an audio speaker driven by a signal generator. The signal detected by the microphone is analyzed by a lock-in amplifier to obtain the amplitude u(r) and phase '(r) of the sound's pressure field at each position r = (x; y). 72 characterized by a scalar pressure field, p(r; t), whose amplitude and phase can be measured 73 directly. In that case, Rayleigh-Sommerfeld back-propagation can be used to reconstruct 74 the complex sound field with an accuracy that is limited by instrumental noise and by the 75 size of the recording area. For appropriate systems, the transfer-function model can be used 76 to obtain information about scatterers in the field. 77 Whereas optical holography benefits from highly developed camera technology, implemen- 78 tations of quantitative acoustic holography must confront a lack of suitable area detectors 79 for sound waves. Commercial acoustic transducer arrays typically include no more than a 6 Scanning Holographic Camera 10 80 few dozen elements. Conventional area scanners yield excellent results over small areas 81 but become prohibitively expensive for large-area scans. We therefore introduce flexible and 82 cost-effective techniques for recording complex-valued acoustic holograms. 83 II. SCANNING ACOUSTIC CAMERA 84 Figure1 depicts our implementation of a scanning acoustic camera for holographic imag- 85 ing. A signal generator (Stanford Research Systems DS345) drives an audio speaker at a 86 desired frequency !. The resulting sound wave propagates to a microphone whose output is 87 analyzed by a dual-phase lock-in amplifier (Stanford Research Systems SR830) referenced 88 to the signal generator. Our reference implementation operates at 8 kHz, which corresponds 89 to a wavelength of 42:5 mm. 90 The lock-in amplifier records the amplitude and phase of the pressure field at the mi- 91 crophone's position. We translate the microphone across the x-y plane using a flexible 92 low-cost two-dimensional scanner known as a polargraph that initially was developed for 23 93 art installations . Correlating the output of the lock-in amplifier with the position of the 94 polargraph yields a map of the complex pressure field in the plane. 95 A. Polargraph for flexible wide-area scanning The polargraph consists of two stepper motors (Nema 17, 200 steps/rev) with toothed pulleys (16 tooth, GT2, 7:5 mm diameter) that control the movement of a flexible GT2 timing belt, as indicated in Fig.1. The acoustic camera's microphone (KY0030 high-sensitivity sound module) is mounted on a laser-cut support that hangs under gravity from the middle 7 Scanning Holographic Camera of the timing belt. The motors' rotations determine the lengths, s1(t) and s2(t) of the two chords of timing belt at time t, and therefore the position of the microphone, r(t).