Power Law Absorption Research Articles

Optimizing parametric acoustic arrays for high-fidelity low-frequency signals

Parametric acoustic array (PAA) can leverage the nonlinear behavior of sound travelling through the nonlinear medium (e.g., air, water, human tissues) to generate low-frequency broadband high-directivity sound due to the self-demodulation of finite-amplitude modulated ultrasonic waves. However, while propagating, self-demodulated signals (hereon message) interact one with another, leading to message deformation and increase of the total harmonic distortion (THD). This makes it problematic to use powerful PAA to control complex low-frequency signals. This work explores a method for reducing THD by optimizing the input signal using various digital signal processing algorithms. The sustainability of the required spectral components with distance and the spatial boundaries of the operation of the proposed optimization algorithm are also investigated experimentally with a 520-channel hexagonal PAA and numerically. The simulations were done via k-wave acoustic toolbox considering nonlinearities and power law absorption.

A static memory method for modelling time-fractional power law absorption

The attenuation of ultrasound propagation through tissue is known to follow a frequency power-law in the time domain. This can be modelled as a loss operator in the equation of state within the Euler equations that takes the form of a fraction time derivative operator. This can be treated through a second order accurate transfer between the fractional time derivative and a fractional Laplacian spacial operator in order to avoid storage of the full time history. This is used for example by the k-Wave toolbox. As an alternate finite history methods have been suggested. Building on recent work, here, we re-write the time fractional derivative as a finite sum using a recursion relation. This allows the time fractional derivative to be directly computed with a static memory requirement. For homogeneous media, the advantage of this over the fractional Laplacian methods may not initially be obvious with a base higher memory requirement and computational cost with only a small increase in accuracy. However, upon introducing a heterogeneous medium with regions of different power law attenuation; the increased computational cost of the time fractional method is contained exclusively within the pre-computation, while the increased cost for the fractional Laplacian is applied to each time step.

Temporal and spectral study of the X-ray pulsar 2S 1553–542 during the 2021 outburst

We study the timing and spectral properties of the X-ray pulsar 2S 1553–542 using the NuSTAR and NICER during the outburst in January–February 2021. During the outburst, the spin period of the neutron star was $$P\sim 9.2822$$ s based on NuSTAR data. The pulse profiles are studied using different NICER observations, which implies that the profile is more or less sinusoidal with a single peak and the beaming patterns are dominated mainly by the pencil beam. The NICER spectra of the source are studied for different days of the outburst. They can be well described by a model consisting of a blackbody emission, power law and photoelectric absorption component. The variation of spectral parameters with luminosity is studied over the outburst. The photon index shows anti-correlation with luminosity below the critical luminosity, which implies that the source was accreting in the sub-critical accretion regime during the NICER observations. We also report the anti-correlation between pulsed fraction (PF) and luminosity of the 2S 1553–542 using NICER observations. The evolution of spin-up rate with luminosity is studied during the outburst, which implies that both are strongly correlated. The torque-luminosity model is applied to estimate the magnetic field at different spin-up rates. The magnetic field is estimated to be $${\simeq }2.56 \times 10^{12}$$ G from the torque-luminosity model using the source distance of 20 kpc. The magnetic field is also estimated using the critical luminosity, consistent with our findings.

Broadband pulses in shear wave elastography

Usually pulses used for shear wave elastography and ARFI are considered to be monochromatic; however, there are advantages to using broadband pulses and including the absorption characteristics of the tissues. A new broadband formulation for volume acoustic radiation forces is presented that accounts for the pulse shape and the changes it encounters propagating through tissues with frequency power law absorption properties. This equation reduces to the well know formulation for the continuous wave case derived by W. Nyborg. Ideal tone bursts employed in elastography have the same time average intensity tissue as continuous wave intensities, irrespective of length. Tone bursts in viscous media have turn-on and turn-off transients called “precursors” and “postcursors,” which can increase pulse amplitudes at the ends of the pulses [Szabo, JASA 45(1): 124–130 (1969)]. By accounting for absorption across a broader bandwidth pulse spectrum, these transient effects can be reduced. Their effects and the potential for pulse shaping are discussed.

On nonlinear effects in holographic-modulated ultrasound

Holographic acoustic lenses (HALs), also known as acoustic holograms, are used for generating unprecedented complex focused ultrasound (FU) fields. HALs store the phase profile of the desired wavefront, which is used to reconstruct the acoustic pressure field when illuminated by a single acoustic source. Nonlinear effects occur as the sound intensity increases, leading to distorted and asymmetric waveforms. Here, the k-space pseudospectral method is used to perform homogeneous three-dimensional nonlinear acoustic simulations with power law absorption. An in-depth analysis is performed to study the evolution of holographic-modulated FU fields produced by HALs as the excitation amplitude increases. It is shown that nonlinear waveform distortion significantly affects the reconstruction of the pressure pattern when compared to the linear condition. Diffraction and nonlinear effects result in an asymmetric waveform with distinct positive and negative pressure patterns at the target plane. Peak positive pressure distribution becomes more localized around the areas with the highest nonlinear distortion. The peak signal-to-distortion ratio (PSDR) at the target plane falls while the nonuniformity index (NUI) rises. As a result of harmonic generation, the heat deposition distribution becomes highly localized with a significant increase in the NUI. Nonlinear effects have also been shown to flatten the peak negative pressure distribution while having minimal effect on the PSDR or NUI. However, nonlinear effects are shown to be critical for accurately predicting cavitation zones. Findings will pave the way for HALs implementation in high-intensity applications and prompt the incorporation of nonlinear acoustics into the notion of computer-generated holography.

Deep XMM-Newton Observations of an X-ray Weak Broad Absorption Line Quasar at z = 6.5

Abstract We report X-ray observations of the most distant known gravitationally lensed quasar, J0439+1634 at z = 6.52, which is also a broad absorption line (BAL) quasar, using the XMM-Newton Observatory. With a 130 ks exposure, the quasar is significantly detected as a point source at the optical position with a total of 358 − 19 + 19 net counts using the EPIC instrument. By fitting a power law plus Galactic absorption model to the observed spectra, we obtain a spectral slope of Γ = 1.45 − 0.09 + 0.10 . The derived optical-to-X-ray spectral slope α ox is − 2.07 − 0.01 + 0.01 , suggesting that the X-ray emission of J0439+1634 is weaker by a factor of 18 than the expectation based on its 2500 Å luminosity and the average α ox versus luminosity relationship. This is the first time that an X-ray weak BAL quasar at z > 6 has been observed spectroscopically. Its X-ray weakness is consistent with the properties of BAL quasars at lower redshift. By fitting a model including an intrinsic absorption component, we obtain intrinsic column densities of N H = 2.8 − 0.6 + 0.7 × 10 23 cm − 2 and N H = 4.3 − 1.5 + 1.8 × 10 23 cm − 2 , assuming a fixed Γ of 1.9 and a free Γ, respectively. The intrinsic rest-frame 2–10 keV luminosity is derived as (9.4–15.1) × 1043 erg s−1, after correcting for lensing magnification (μ = 51.3). The absorbed power-law model fitting indicates that J0439+1634 is the highest redshift obscured quasar with a direct measurement of the absorbing column density. The intrinsic high column density absorption can reduce the X-ray luminosity by a factor of 3–7, which also indicates that this quasar could be a candidate intrinsically X-ray weak quasar.

Power law attenuation of shear waves in fractal media

There are two mechanisms for attenuation of mechanical waves: absorption and multiple scattering. We explore the second mechanism for shear waves in the context of MR elastography. The theory for attenuation was first given in the seismic field (O’Doherty & Anstey, Geophys. Prosp. 1971). Later phase was included, and the mean field concept was applied (Banik et al., Geophys. 1985). Then, a more rigorous theory which also deals with random fractals was developed (Solna, SIAM J ApplMath, 2003; Garnier & Solna, Multiscale Model Simul, 2009). We have developed this theory further by allowing fractality over a limited length scale and also shown that it gives results that are indistinguishable from those of power law absorption models like the fractional Kelvin-Voigt model (Holm & Sinkus, JASA, 2010). This makes it a challenge to understand whether attenuation in a real medium is due to one or the other mechanism or both. Measurement of dispersion has been done for polystyrene microspheres embedded in agarose which is nearly lossless up to 1000 Hz. The phase velocity has a power law variation with exponent up to d = 0.24 (Lambert et al., PRL 2015). According to the theory this is consistent with a fractional dimension which is d less than the Euclidean dimension.

Lévy flights with power-law absorption.

We consider a particle performing a stochastic motion on a one-dimensional lattice with jump lengths distributed according to a power law with exponent μ+1. Assuming that the walker moves in the presence of a distribution a(x) of targets (traps) depending on the spatial coordinate x, we study the probability that the walker will eventually find any target (will eventually be trapped). We focus on the case of power-law distributions a(x)∼x(-α) and we find that, as long as μ<α, there is a finite probability that the walker will never be trapped, no matter how long the process is. This result is shown via analytical arguments and numerical simulations which also evidence the emergence of slow searching (trapping) times in finite-size system. The extension of this finding to higher-dimensional structures is also discussed.

Eventual Self-similarity of Solutions for the Diffusion Equation with Nonlinear Absorption and a Point Source

This paper is concerned with the transient dynamics described by the solutions of the reaction-diffusion equations in which the reaction term consists of a combination of a superlinear power-law absorption and a time-independent point source. In one space dimension, solutions of these problems with zero initial data are known to approach the stationary solution in an asymptotically self-similar manner. Here we show that this conclusion remains true in two space dimensions, while in three and higher dimensions the same conclusion holds true for all powers of the nonlinearity not exceeding the Serrin critical exponent. The analysis requires dealing with solutions that contain a persistent singularity and involves a variational proof of existence of ultra-singular solutions, a special class of self-similar solutions in the considered problem.

Modeling power law absorption and dispersion in viscoelastic solids using a split-field and the fractional Laplacian.

The absorption of compressional and shear waves in many viscoelastic solids has been experimentally shown to follow a frequency power law. It is now well established that this type of loss behavior can be modeled using fractional derivatives. However, previous fractional constitutive equations for viscoelastic media are based on temporal fractional derivatives. These operators are non-local in time, which makes them difficult to compute in a memory efficient manner. Here, a fractional Kelvin-Voigt model is derived based on the fractional Laplacian. This is obtained by splitting the particle velocity into compressional and shear components using a dyadic wavenumber tensor. This allows the temporal fractional derivatives in the Kelvin-Voigt model to be replaced with spatial fractional derivatives using a lossless dispersion relation with the appropriate compressional or shear wave speed. The model is discretized using the Fourier collocation spectral method, which allows the fractional operators to be efficiently computed. The field splitting also allows the use of a k-space corrected finite difference scheme for time integration to minimize numerical dispersion. The absorption and dispersion behavior of the fractional Laplacian model is analyzed for both high and low loss materials. The accuracy and utility of the model is then demonstrated through several numerical experiments, including the transmission of focused ultrasound waves through the skull.

Counterpropagation of waves with shock fronts in a nonlinear tissue-like medium

A numerical model for describing the counterpropagation of one-dimensional waves in a nonlinear medium with an arbitrary power law absorption and corresponding dispersion is developed. The model is based on general one-dimensional Navier-Stokes equations with absorption in the form of a time-domain convolution operator in the equation of state. The developed algorithm makes it possible to describe wave interactions in the presence of shock fronts in media like biological tissue. Numerical modeling is conducted by the finite difference method on a staggered grid; absorption and sound speed dispersion are taken into account using the causal memory function. The developed model is used for numerical calculations, which demonstrate the absorption and dispersion effects on nonlinear propagation of differently shaped pulses, as well as their reflection from impedance acoustic boundaries.

Modeling nonlinear ultrasound propagation in heterogeneous media with power law absorption using a k-space pseudospectral method

The simulation of nonlinear ultrasound propagation through tissue realistic media has a wide range of practical applications. However, this is a computationally difficult problem due to the large size of the computational domain compared to the acoustic wavelength. Here, the k-space pseudospectral method is used to reduce the number of grid points required per wavelength for accurate simulations. The model is based on coupled first-order acoustic equations valid for nonlinear wave propagation in heterogeneous media with power law absorption. These are derived from the equations of fluid mechanics and include a pressure-density relation that incorporates the effects of nonlinearity, power law absorption, and medium heterogeneities. The additional terms accounting for convective nonlinearity and power law absorption are expressed as spatial gradients making them efficient to numerically encode. The governing equations are then discretized using a k-space pseudospectral technique in which the spatial gradients are computed using the Fourier-collocation method. This increases the accuracy of the gradient calculation and thus relaxes the requirement for dense computational grids compared to conventional finite difference methods. The accuracy and utility of the developed model is demonstrated via several numerical experiments, including the 3D simulation of the beam pattern from a clinical ultrasound probe.

Efficient large-scale ultrasound simulation using the k-space pseudospectral method

Computational acoustics offers a powerful tool for investigating the interaction between ultrasound waves and the human body. However, in many common ultrasound settings, performing realistic simulations is computationally difficult due to the large size of the tissue volume compared to the size of the acoustic wavelength. This is particularly true in the case of high-intensity focused ultrasound where large diameter transducers are used to tightly focus ultrasound waves, often deep within the tissue. Here, an efficient model for large-scale ultrasound simulation is presented. The model is based on coupled first-order acoustic equations valid for nonlinear wave propagation in heterogeneous media with power law absorption. The equations are discretized using the k-space pseudospectral method and encoded using advanced programming techniques for parallel computer architectures. This allows the efficient simulation of nonlinear ultrasound propagation in three-dimensions over hundreds of wavelengths. Applications to both diagnostic and therapeutic ultrasound are discussed, and the results from several simulation examples are presented.

A k-space Green’s function solution for acoustic initial value problems in homogeneous media with power law absorption

An efficient Green's function solution for acoustic initial value problems in homogeneous media with power law absorption is derived. The solution is based on the homogeneous wave equation for lossless media with two additional terms. These terms are dependent on the fractional Laplacian and separately account for power law absorption and dispersion. Given initial conditions for the pressure and its temporal derivative, the solution allows the pressure field for any time t>0 to be calculated in a single step using the Fourier transform and an exact k-space time propagator. For regularly spaced Cartesian grids, the former can be computed efficiently using the fast Fourier transform. Because no time stepping is required, the solution facilitates the efficient computation of the pressure field in one, two, or three dimensions without stability constraints. Several computational aspects of the solution are discussed, including the effect of using a truncated Fourier series to represent discrete initial conditions, the use of smoothing, and the properties of the encapsulated absorption and dispersion.

Photoacoustic tomography in absorbing acoustic media using time reversal

The reconstruction of photoacoustic images typically neglects the effect of acoustic absorption on the measured time domain signals. Here, a method to compensate for acoustic absorption in photoacoustic tomography is described. The approach is based on time-reversal image reconstruction and an absorbing equation of state which separately accounts for acoustic absorption and dispersion following a frequency power law. Absorption compensation in the inverse problem is achieved by reversing the absorption proportionality coefficient in sign but leaving the equivalent dispersion parameter unchanged. The reconstruction is regularized by filtering the absorption and dispersion terms in the spatial frequency domain using a Tukey window. This maintains the correct frequency dependence of these parameters within the filter pass band. The method is valid in one, two and three dimensions, and for arbitrary power law absorption parameters. The approach is verified through several numerical experiments. The reconstruction of a carbon fibre phantom and the vasculature in the abdomen of a mouse are also presented. When absorption compensation is included, a general improvement in the image magnitude and resolution is seen, particularly for deeper features.

Modeling power law absorption and dispersion for acoustic propagation using the fractional Laplacian

The efficient simulation of wave propagation through lossy media in which the absorption follows a frequency power law has many important applications in biomedical ultrasonics. Previous wave equations which use time-domain fractional operators require the storage of the complete pressure field at previous time steps (such operators are convolution based). This makes them unsuitable for many three-dimensional problems of interest. Here, a wave equation that utilizes two lossy derivative operators based on the fractional Laplacian is derived. These operators account separately for the required power law absorption and dispersion and can be efficiently incorporated into Fourier based pseudospectral and k-space methods without the increase in memory required by their time-domain fractional counterparts. A framework for encoding the developed wave equation using three coupled first-order constitutive equations is discussed, and the model is demonstrated through several one-, two-, and three-dimensional simulations.

XMM-Newton observations of the diffuse X-ray emission in the starburst galaxy NGC 253

Aims: We present a study of the diffuse X-ray emission in the halo and the disc of the starburst galaxy NGC 253. Methods: After removing point-like sources, we analysed XMM-Newton images, hardness ratio maps and spectra from several regions in the halo and the disc. We introduce a method to produce vignetting corrected images from the EPIC pn data, and we developed a procedure that allows a correct background treatment for low surface brightness spectra, using a local background, together with closed filter observations. Results: Most of the emission from the halo is at energies below 1 keV. In the disc, also emission at higher energies is present. The extent of the diffuse emission along the major axis of the disc is 13.6 kpc. The halo resembles a horn structure and reaches out to ~9 kpc perpendicular to the disc. Disc regions that cover star forming regions, like spiral arms, show harder spectra than regions with lower star forming activity. Models for spectral fits of the disc regions need at least three components: two thermal plasmas with solar abundances plus a power law and galactic foreground absorption. Temperatures are between 0.1 and 0.3 keV and between 0.3 and 0.9 keV for the soft and the hard component, respectively. The power law component may indicate an unresolved contribution from X-ray binaries in the disc. The halo emission is not uniform, neither spatially nor spectrally. The southeastern halo is softer than the northwestern halo. To model the spectra in the halo, we needed two thermal plasmas with solar abundances plus galactic foreground absorption. Temperatures are around 0.1 and 0.3 keV. A comparison between X-ray and UV emission shows that both originate from the same regions.

X-ray edge problem of graphene

The x-ray edge problem of graphene with the Dirac fermion spectrum is studied. At half-filling the linear density of states suppresses the singular response of the Fermi liquid, while away from half-filling the singular features of the Fermi liquid reappear. The crossover behavior as a function of the Fermi energy is examined in detail. The exponent of the power-law absorption rate depends both on the intra- and intervalley scattering, and it changes as a function of the Fermi energy, which may be tested experimentally.

Nonlinear and linear wave equations for propagation in media with frequency power law losses

The Burgers, KZK, and Westervelt wave equations used for simulating wave propagation in nonlinear media are based on absorption that has a quadratic dependence on frequency. Unfortunately, most lossy media, such as tissue, follow a more general frequency power law. The authors first research involved measurements of loss and dispersion associated with a modification to Blackstock’s solution to the linear thermoviscous wave equation [J. Acoust. Soc. Am. 41, 1312 (1967)]. A second paper by Blackstock [J. Acoust. Soc. Am. 77, 2050 (1985)] showed the loss term in the Burgers equation for plane waves could be modified for other known instances of loss. The authors’ work eventually led to comprehensive time-domain convolutional operators that accounted for both dispersion and general frequency power law absorption [Szabo, J. Acoust. Soc. Am. 96, 491 (1994)]. Versions of appropriate loss terms were developed to extend the standard three nonlinear wave equations to these more general losses. Extensive experimental data has verified the predicted phase velocity dispersion for different power exponents for the linear case. Other groups are now working on methods suitable for solving wave equations numerically for these types of loss directly in the time domain for both linear and nonlinear media.

Exploring the spectral properties of faint hard X-ray sources with $\vec XMM$-$\vec Newton$

We present a spectroscopic study of 41 hard X-ray sources detected serendipitously with high significance (>5σ in the 2–10 keV band) in seven EPIC performance/verification phase observations. The large collecting area of EPIC allows us to explore the spectral properties of these faint hard X-ray sources with 2 < F2-10 < 80 10-14 erg cm-2 s-1 even though the length of the exposures are modest (~20 ks). Optical identifications are available for 21 sources of our sample. Using a simple power law plus Galactic absorption model we find an average value of the photon index Γ ~ 1.6–1.7, broadly consistent with recent measurements made at similar fluxes with ASCA and with Chandra stacked spectral analyses. We find that 31 out of 41 sources are well fitted by this simple model and only eight sources require absorption in excess of the Galactic value. Interestingly enough, one third of these absorbed sources are broad line objects, though with moderate column densities. Two sources in the sample are X-ray bright optically quiet galaxies and show flat X-ray spectra. Comparing our observational results with those expected from standard synthesis models of the cosmic X-ray background (CXB) we find a fraction of unabsorbed to absorbed sources larger than predicted by theoretical models at our completeness limit of F2-10 ~ 5 10-14 erg cm-2 s-1. The results presented here illustrate well how wide-angle surveys performed with EPIC on board XMM-Newton allow population studies of interesting and unusual sources to be made as well as enabling constraints to be placed on some input parameters for synthesis models of the CXB.