Angular Spectrum Decomposition Research Articles

A model for passive underwater noise suppression by bubble curtains surrounding point or line sources in shallow water

Underwater noise generated by pile driving, rotating machinery, or towers supporting offshore wind turbines may disturb marine life and inhibit detection of coastal activities via passive sonar and seismic sensors. Noise abatement techniques have therefore been proposed to limit the propagation of such noise into the far field, and many of these employ a curtain of freely-rising bubbles or tethered encapsulated bubbles to surround the towers [Lee et al., J. Acoust. Soc. Am. 131, 3507(A) (2012)]. An analytic model, based on a Green's function approach, is presented for the passive noise suppression provided by a discrete number of bubbles surrounding submerged point sources or pulsating cylindrical towers above horizontally-stratified layers of sediment. The sediment layers are modeled as viscoelastic media and the Green's function is derived via angular spectrum decomposition [Hay et al., J. Acoust. Soc. Am. 129, 2477(A), (2011)]. Simulations in which the bubbles are assumed to react independently to the incident field will be compared to those in which bubble-bubble interaction is taken into account. The effects of bubble size distributions and void fractions on noise suppression will be investigated for different source configurations. [This work was supported by the Department of Energy under Grant DE-EE0005380.]

Green’s functions for a volume source in an elastic half-space

Green's functions are derived for elastic waves generated by a volume source in a homogeneous isotropic half-space. The context is sources at shallow burial depths, for which surface (Rayleigh) and bulk waves, both longitudinal and transverse, can be generated with comparable magnitudes. Two approaches are followed. First, the Green's function is expanded with respect to eigenmodes that correspond to Rayleigh waves. While bulk waves are thus ignored, this approximation is valid on the surface far from the source, where the Rayleigh wave modes dominate. The second approach employs an angular spectrum that accounts for the bulk waves and yields a solution that may be separated into two terms. One is associated with bulk waves, the other with Rayleigh waves. The latter is proved to be identical to the Green's function obtained following the first approach. The Green's function obtained via angular spectrum decomposition is analyzed numerically in the time domain for different burial depths and distances to the receiver, and for parameters relevant to seismo-acoustic detection of land mines and other buried objects.

A model for noise radiated by submerged piles and towers in littoral environments

Pile driving in shallow water during the construction of bridges and other structures can produce transient broadband noise of sufficient intensity to kill fish and disturb marine mammals. Sustained tonal noise radiated by towers supporting offshore wind turbines contains energy in frequency bands that may inhibit detection of coastal activities via passive sonar and seismic sensors. Understanding the generation and propagation of underwater noise due to pile driving and wind farms is important for determining the best strategies for mitigating the environmental impact of these noise sources. An analytic model, based on a Green's function approach, is presented for the sound radiated in the water column by a submerged cylindrical structure embedded in horizontally stratified layers of sediment. The sediment layers are modeled as viscoelastic media and the Green's function is derived via angular spectrum decomposition. Noise radiation due to both vibration of the structure and impulses delivered to the sediment is considered. Contributions to the pressure field in the water column due to radiation directly into the water, radiation from the sediment into the water, and Scholte waves propagating along the sediment-water interface will be discussed. [Work supported by the ARL:UT IR&D program.]

Green’s function approach to modeling spherical bubble pulsation between viscoelastic layers.

Experiments have shown that bubbles oscillating near surfaces of elastic media experience a translational force whose direction, toward or away from the surface, depends on elastic moduli and standoff distance. [Brujan et al., J. Fluid. Mech. 433, 251–314 (2001).] Bubble translation has also been observed in ex vivo blood vessels following acoustic excitation. [Chen et al., J. Acoust. Soc. Am. 127, 1977–(2010).] A model is developed for a stationary spherical bubble pulsating between two parallel viscoelastic layers. A Green’s function for particle displacement is derived via angular spectrum decomposition and is used to calculate the translational force on a bubble due to pressure reflections from the layers. The predicted dependence of the direction of the translational force on elastic properties and standoff distance is in qualitative agreement with the reported measurements. The reflected pressure is also incorporated in a Rayleigh–Plesset equation for the nonlinear pulsation of the bubble. Evaluation of the Green’s function on the surfaces of the elastic layers yields their displacements due to the bubble pulsations. Predicted surface displacements exhibit characteristics in agreement with wall displacements observed in blood vessels by Chen et al. [J. Acoust. Soc. Am. 125, 2680–(2009).] [Work supported by NIH DK070618.]

Equations for Electromagnetic Radiation Transfer in Dielectric Random Media with Effects of Near Fields and Opposite Wave Streams' Interference

A radiative transfer theory for electromagnetic wave multiple scattering in dielectric random media with effects of near fields is presented. The theory is based on Sommerfeld-Weyl angular spectrum decomposition of wave amplitudes and technique of Dyson and Bethe-Salpeter equations for the ensemble averaged angular spectrum amplitudes and the coherence matrix of angular specrum amplitudes, respectively. We derive the four integral equations’ system, for two autocoherences of waves propagating forward and backward with respect to embedding parameter into the medium slab and two cross-coherences of evanescent waves decaying in opposite directions. In so doing the equations for autocoherences describe basically the usual radiative transfer with wave intensity multiple scattering by random inhomogeneities (scatterers) and free path between them. In the contrary the equations for cross-coherences demonstrate a new effect of elementary evanescent waves’ virtually exciting inside an inhomogeneity in pairs of opposite decaying derections and contribution of these evanescent pairs to wave energy flux.

Tomographic imaging of an ultrasonic field in a plane by use of a linear array: theory and experiment

Quantitative ultrasonic characterization of in-homogeneous and anisotropic materials is often difficult due to undesired phenomena such as beam steering and phase aberration of the insonifying field. We introduce a method based on tomographic reconstruction techniques for the visualization of an ultrasonic field using a linear array rotated in a plane. Tomographic reconstruction of the ultrasonic field is made possible through the phase-sensitive nature of the tall, narrow piezoelectric elements of a linear array that act as parallel line integrators of the pressure field. We validate the proposed imaging method through numerical simulations of propagated ultrasonic fields based upon the angular spectrum decomposition technique. We then demonstrate the technique with experimental measurements of two textile composites and a reference water path. We reconstruct images of the real and imaginary parts of a transmitted 2 MHz ultrasonic field that are then combined to reconstruct images of the power and unwrapped phase. We also construct images of the attenuation and phase shift for several regions of the composites. Our results demonstrate that tomographic imaging of an ultrasonic field in a plane using a rotated linear array can potentially improve ultrasonic characterization of complex materials.

Analytic design and visualization of multiple surface plasmon resonance excitation using angular spectrum decomposition for a Gaussian input beam

We propose an exact design, analysis, and visualization method for multiple surface plasmon resonance (MSPR) mode excitation phenomena for a structure composed of an optimized-thickness polymethyl-methacrylate layer and a gold thin-film layer. The proposed simulation method is based on a recursive transfer matrix method (R-TMM) and Gaussian angular spectrum decomposition. Our method illustrates, under the Kretchmann-Raether attenuated total reflection (ATR) geometry, the response for an angle-modulated Gaussian incident beam. To verify the simulation results we also performed experiments to excite MSPR modes under the ATR geometry. Our fast and exact R-TMM with the Gaussian angular spectrum method can be widely applied to the design and analysis of metal- and dielectric-composed thin film structures.

Optical scattering spectroscopy by using tightly focused supercontinuum

We present analysis of forward optical scattering spectra of a single spherical object illuminated with tightly focused white light supercontinuum based on angular spectrum decomposition and Mie's scattering theory. The experimental results agree with the theoretical calculations.

Simple analysis of the energy density distribution of the diffracted ultraviolet beam from a fiber Bragg grating phase mask.

The energy density distribution of the diffracted ultraviolet field from a phase mask used for fiber Bragg grating fabrication is efficiently analyzed at short-to-intermediate distances by application of a thin phase screen model and the angular spectrum decomposition method. A direct comparison with the results obtained from a rigorous coupled-wave model shows good agreement.

Estimation of the influence of finite dielectric substrates on the far-field pattern of an array of metallic scatterers in the infrared

The far-field scattered in the infrared by an arrangement of metallic structures deposited on a dielectric wafer is estimated in this paper. The scattering is modelled by using operators that describe the far field obtained under the regime applicable for the Babinet's principle in its vectorial approach and the Stratton–Chu approximation. The far-field scattered by an arrangement of thin gold layers over a dielectric wafer under infrared illumination is computed. The model assumes a normally incident vectorial Gaussian beam focused over the arrangement plane. An angular spectrum decomposition of the field is done. Then, every plane wave is scattered by the whole structure: arrangement + substrate layer. The reflexions in the substrate layer and the arrangement action have been taken into account in an operator formalism. Numerical results estimating the influence of substrate thickness on the pattern are obtained.

Image formation of confocal microscopes using Lorentz's reciprocity theorem

A general expression for the image formation of confocal scanning microscopes is derived within the framework of vector optics. The derivation is based on a reciprocity principle in the form of a closed surface integral over electromagnetic fields, the so-called Lorentz reciprocity theorem. The approach comprises the reflection- and the transmission-type setup. It is assumed that single-mode optical fibers serve as pinholes. Different types of polarization of both, illumination and detection and the non-polarized case are investigated. It shows that, with the help of Lorentz's reciprocity theorem, the detection characteristics are given by emission characteristics and vice versa, although the assumptions concerning the optical setup are very general. The result is formulated with the help of angular spectrum decomposition. As the spectra may contain evanescent waves, the approach may be applied also to scanning near-field microscopes working in the illumination mode.

Application of the angular spectrum approach to compute the far-field pressure

The angular spectrum decomposition of an acoustic source can be used to compute the radiation pressure in three dimensional radiation problems. Then, the aim of this paper is to demonstrate that the discrete far-field pressure can be computed from the two-dimensional (2D) discrete Fourier transform (DFT) of the normal velocity. In fact, using the asymptotic expression of the Rayleigh integral, it will be shown that the analytical far-field pressure is given by the 2D Fourier transform of the normal velocity. Using the discrete form of this expression, guidelines for selection of the sampling interval and the size of baffle in which the source is mounted will be given. The case of a rectangular source mounted in a rigid baffle and radiating into water will be considered.

Modeling of received ultrasound signals from finite planar targets

Presents a computational solution to the ultrasonic target-dependent spectral distortion for a finite planar target (FT-SD). The FT-SD predicts the spectrum of the output signal from the receiving transducer generated in response to the insonification of a finite planar specular target, as a function of incident angle. The transducer is assumed to be operating in pulse-echo mode. The development is made via scalar diffraction theory, in terms of the angular spectrum decomposition of the acoustic source field. Numerical simulations of the FT-SD are presented for several reflector sizes and orientations, insonified by a planar circular piston transducer. Experimental verification of results for a finite square planar reflector is given for a range of reflector orientations.

Comments on "Efficient angular spectrum decomposition of acoustic sources-parts I and II"[with reply

Problems with the angular spectrum analysis of a recent pair of papers (see ibid., vol. 40, p. 238-49, 1993 and vol. 40, p. 250-57, 1993)are described. An alternative numerically correct approach is described and is used to efficiently and accurately re-compute a result considered in the original papers. The original authors give their reply to the comments.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Efficient angular spectrum decomposition of acoustic sources. I. Theory

The angular spectrum decomposition is evaluated in terms of plane wave angular range, angular resolution, and spatial aliasing error using two-dimensional FFT (2-D FFT). The algorithm makes possible the source plane decomposition of normal velocity and pressure fields radiated by transducers of arbitrary shape, with significantly faster results achievable for planar sources. Although the angular spectrum is equally applicable to fields far from the transducer, the efficient calculation is derived specifically for fields in or very close to the source plane. An antialiasing algorithm is proposed that allows the source to be discretized with fewer sample points for a given accuracy than required with simple discretization techniques. Guidelines for the selection of sampling interval, discretization size, etc. are developed on an application-specific basis and indicate the best ratio of numerical accuracy to computational cost.

Efficient angular spectrum decomposition of acoustic sources. II. Results

For pt.I see ibid., vol.40, no.3, p.238-49 (1993). Results of angular spectrum computations are presented for three applications: general decomposition, pressure field calculation, and modeling of pulse-echo measurements. Wherever possible, analytical (exact) results are used as reference for the decomposition results. This permits the accuracy of the angular spectrum decomposition to be evaluated for specific choices of M and N where M is the number of samples across the maximum characteristic length, d, of the source region and N is the total number of samples along each side of the source decomposition plane, and for both sampling techniques. Based on the results for the three applications, guidelines for choosing M and N are presented in a graphical format.

Evaluation of angle-dependent spectral distortion for infinite, planar elastic media via angular spectrum decomposition

The angle-dependent spectral distortion for an infinite planar interface (IP-ASD) is examined for the case of fluid–fluid and fluid–solid (elastic) interfaces. The results are based on computation of the angular spectrum (plane wave) decomposition of the acoustic pressure field as radiated by an equivalent image source transducer, for each frequency of interest. The exact incident-angle-dependent reflection coefficient of the interface may thus be used in the computation, versus approximate methods previously required [D. P. Orofino and P. C. Pedersen, J. Acoust. Soc. Am. 92, 2883–2899 (1992)]. Comparisons of the previous approximate IP-ASD results to the current exact results are made. IP-ASD results for elastic parallel plates are also presented for a variety of plate orientations and geometries. For the special case of normal incidence, a comparison of the IP-ASD result to a previously published result is made. The IP-ASD is additionally examined for a reduced subset of plane wave components that approximate the radiated pressure field. The error in the IP-ASD versus the number of plane waves in the subset is discussed.

Shock wave production in medical ultrasound—A new computational approach

At high intensities, the presence of nonlinear propagation effects can influence the performance of medical ultrasound devices. In general, models that can fully account for the combined effects of tissue nonlinearities, tissue attenuation, and beam diffraction effects are quite complex, require extensive computational time, and provide little physical insight as to the interaction of these processes. A novel, computationally efficient model has been developed that accounts for the effects of nonlinearities, attenuation, and diffraction. The model propagates a circularly symmetric beampattern forward in small axial increments. Over the increment, the fundamental and harmonics are diffracted and attenuated linearly using angular spectrum decomposition techniques. Then, the complex amplitudes of the fundamental and harmonics are altered according to theory of plane-wave propagation in a medium with nonlinear behavior. This hybrid, incremental algorithm is shown to reproduce theoretical Gaussian and focused piston beampatterns described by others, and also matches experimental data from focused ultrasound beampatterns measured in the present study. The new hybrid algorithm is computationally efficient and provides some insight into the relative importance of nonlinearities, attenuation, and focusing in medical imaging and lithotripsy conditions. [Work supported by NIH.]

The use of frequency weighting in multifrequency holography

Backpropagating an acoustical field that is measured on a planar surface using angular spectrum decomposition [J. Goodman, Introduction to Fourier Optics (McGraw‐Hill, New York, 1968)] can be used to reconstruct a source region or a scattering region. The reconstruction algorithm is inherently monochromatic and hence the range resolution is typically worse than the lateral resolution. In diffraction tomography, the same backpropagation algorithm is used and the summation over views is used to improve the overall resolutions [A. Devaney, Ultrason. Imag. 4, 336–350 (1982)]. In the case of B‐scan imaging, the reverse is true. The use of short pulse provides a good range resolution while the lateral resolution is limited by the beamwidth. It is therefore advantageous to combine the wideband approach with the backpropagation. The use of multiple frequencies in backpropagating the acoustical field has shown to improve the range roolution [e.g., T. J. Teo and J. M. Reid, Acoust. Imag. 14, 143–153 (1985)]. The two techniques can therefore be combined by suitably weighting the frequency distribution with the amplitude and phase spectra of a pulse. We will also show the effect of pulse shape on the image reconstructed and range resolution in particular.

Evanescence in Liquid Crystals

Abstract The existence of a special type of electromagnetic oscillations in liquid crystals is brought out. The Saupe-Maier form of the Lorentz-Lorenz equation for nematics is used to show that there are four electromagnetic modes for each incident frequency as opposed to the commonly accepted two modes. For a nematic occupying the half space z > the results of an angular spectrum decomposition show that the phase velocity of the inhomogeneous evanescent waves are smaller than that of the associated homogeneous wave. Implications of the existence of evanescent waves in the smectic and cholesteric phases are discussed.