On a time-domain representation of the Kramers-Kronig dispersion relations

Acoustic Radiation Forces at the Crossroads of Ultrasound Exposimetry, HIFU, and Elastography

The applications of the acoustic radiation force (ARF) continue to multiply and extend from elastography into high-intensity focused ultrasound (HIFU), diagnostic imaging, lithotripsy, sonochemistry, levitation, and microsonics yet fundamental principles remain shrouded in mystery. A well-known and popular equation often used for calculating ARF in elastography is in conflict with the equation commonly employed for calculating ARF for determining acoustic power in radiation force balances (RFBs). Controversies have sparked debate for over a century concerning the physical mechanisms underlying ARFs. For over four decades, the science of ultrasound exposimetry has steadily improved and has provided clues in terms of accumulated data about the characteristics of transmitted ultrasound fields. Concurrently, the availability and capability of predicting these fields have improved significantly. The author draws on these sources to re-examine the physical principles behind ARFs. Conflicts are shown to stem from idealized configurations and simplistic assumptions. By more fully accounting for the pulse shape and spectrum, the effect of frequency power law attenuation, diffraction, and nonlinearity, more accurate equations are developed for ARFs for practical applications which are more consistent with exposimetry observations. Simulations compare well to corrected 1.5 MHz RFB data. While some questions await resolution, the approach presented here settles several longstanding conflicts and provides a new broadband framework for future ARF work.

Analytical Methods for Causality Evaluation of Photonic Materials

We comprehensively review several general methods and analytical tools used for causality evaluation of photonic materials. Our objective is to call to mind and then formulate, on a mathematically rigorous basis, a set of theorems which can answer the question whether a considered material model is causal or not. For this purpose, a set of various distributional theorems presented in literature is collected as the distributional version of the Titchmarsh theorem, allowing for evaluation of causality in complicated electromagnetic systems. Furthermore, we correct the existing material models with the use of distribution theory in order to obtain their causal formulations. In addition to the well-known Kramers–Krönig (K–K) relations, we overview four further methods which can be used to assess causality of given dispersion relations, when calculations of integrals involved in the K–K relations are challenging or even impossible. Depending on the given problem, optimal approaches allowing us to prove either the causality or lack thereof are pointed out. These methodologies should be useful for scientists and engineers analyzing causality problems in electrodynamics and optics, particularly with regard to photonic materials, when the involved mathematical distributions have to be invoked.

Random propagation times for ultrasonics through polyethyilene

Low power ultrasonics are used for testing high density polyethylene pipe material. Attenuation and velocity give valuable information on the material in situ and without damages. In this paper we revisit recent data in the frequency band (4,10) megahertz. We prove that propagation is equivalent to random delays following stable probability laws. Moreover, the emergence of a companion noise non-detectable by devices is compliant with the law of conservation of energy.

Study of Electronic Structure, Thermal Conductivity, Elastic and Optical Properties of α, β, γ-Graphyne

In recent years, graphyne was found to be the only 2D carbon material that has both sp and sp² hybridization. It has received significant attention because of its great potential in the field of optoelectronics, which arises due to its small band gap. In this study, the structural stability, electronic structure, elasticity, thermal conductivity and optical properties of α, β, γ-graphynes were investigated using density functional theory (DFT) systematically. γ-graphyne has the largest negative cohesive energy and thus the most stable structure, while the β-graphyne comes 2nd. Both β and γ-graphynes have sp-sp, sp-sp² and sp²-sp² hybridization bonds, of which γ-graphyne has shorter bond lengths and thus larger Young's modulus. Due to the difference in acetylenic bond in the structure cell, the effect of strain on the electronic structure varies between graphynes: α-graphyne has no band gap and is insensitive to strain; β-graphyne's band gap has a sharp up-turn at 10% strain, while γ-graphyne's band gap goes up linearly with the strain. All the three graphynes exhibit large free carrier concentration and these free carriers have small effective mass, and both free carrier absorption and intrinsic absorption are found in the light absorption. Based on the effect of strain, optical properties of three structures are also analyzed. It is found that the strain has significant impacts on their optical properties. In summary, band gap, thermal conductivity, elasticity and optical properties of graphyne could all be tailored with adjustment on the amount of acetylenic bonds in the structure cell.

Extending the Utility of the Parabolic Approximation in Medical Ultrasound Using Wide-Angle Diffraction Modeling

Wide-angle parabolic models are commonly used in geophysics and underwater acoustics but have seen little application in medical ultrasound. Here, a wide-angle model for continuous-wave high-intensity ultrasound beams is derived, which approximates the diffraction process more accurately than the commonly used Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation without increasing implementation complexity or computing time. A method for preventing the high spatial frequencies often present in source boundary conditions from corrupting the solution is presented. Simulations of shallowly focused axisymmetric beams using both the wide-angle and standard parabolic models are compared to assess the accuracy with which they model diffraction effects. The wide-angle model proposed here offers improved focusing accuracy and less error throughout the computational domain than the standard parabolic model, offering a facile method for extending the utility of existing KZK codes.

Inverse problems in cancellous bone: estimation of the ultrasonic properties of fast and slow waves using Bayesian probability theory

Quantitative ultrasonic characterization of cancellous bone can be complicated by artifacts introduced by analyzing acquired data consisting of two propagating waves (a fast wave and a slow wave) as if only one wave were present. Recovering the ultrasonic properties of overlapping fast and slow waves could therefore lead to enhancement of bone quality assessment. The current study uses Bayesian probability theory to estimate phase velocity and normalized broadband ultrasonic attenuation (nBUA) parameters in a model of fast and slow wave propagation. Calculations are carried out using Markov chain Monte Carlo with simulated annealing to approximate the marginal posterior probability densities for parameters in the model. The technique is applied to simulated data, to data acquired on two phantoms capable of generating two waves in acquired signals, and to data acquired on a human femur condyle specimen. The models are in good agreement with both the simulated and experimental data, and the values of the estimated ultrasonic parameters fall within expected ranges.

Is the Kramers-Kronig relationship between ultrasonic attenuation and dispersion maintained in the presence of apparent losses due to phase cancellation?

Phase cancellation effects can compromise the integrity of ultrasonic measurements performed with phase sensitive receiving apertures. A lack of spatial coherence of the ultrasonic field incident on a phase sensitive receiving array can produce inaccuracies of the measured attenuation coefficient and phase velocity. The causal (Kramers-Kronig) link between these two quantities in the presence of phase distortion is investigated using two plastic polymer materials, Plexiglas and Lexan, that exhibit attenuation coefficients that increase linearly with frequency, in a fashion analogous to that of soft tissue. Flat and parallel plates were machined to have a step of a thickness corresponding to an integer number of half wavelengths within the bandwidth investigated, 3 to 7 MHz. Insonification of the stepped portion of each plate produces phase cancellation artifacts at the receiving aperture and, therefore, in the measured frequency dependent attenuation coefficient. Dispersion predictions using two different forms of the Kramers-Kronig relations were performed for the flat and the stepped regions of each plastic plate. Despite significant phase distortion and a detection system sensitive to these aberrations, the Kramers-Kronig link between the apparent attenuation coefficient and apparent phase velocity dispersion remains intact.

Kramers-Kronig analysis of attenuation and dispersion in trabecular bone

A restricted-bandwidth form of the Kramers-Kronig dispersion relations is applied to in vitro measurements of ultrasonic attenuation and dispersion properties of trabecular bone specimens from bovine tibia. The Kramers-Kronig analysis utilizes only experimentally measured properties and avoids extrapolation of ultrasonic properties beyond the known bandwidth. Compensation for the portions of the Kramers-Kronig integrals over the unknown bandwidth is partially achieved by the method of subtractions, where a subtraction frequency acts as an adjustable parameter. Good agreement is found between experimentally measured and Kramers-Kronig reconstructed dispersions. The restricted-bandwidth approach improves upon other forms of the Kramers-Kronig relations and may provide further insight into how ultrasound interacts with trabecular bone.

Causality-imposed (Kramers-Kronig) relationships between attenuation and dispersion

Causality imposes restrictions on both the time-domain and frequency-domain responses of a system. The Kramers-Kronig (K-K) relations relate the real and imaginary parts of the frequency-domain response. In ultrasonics, K-K relations often are used to link attenuation and dispersion. We review both integral and differential forms of the frequency-domain K-K relations that are relevant to theoretical models and laboratory measurements. We consider two methods for implementing integral K-K relations for the case of finite-bandwidth data, namely, extrapolation of data and restriction of integration limits. For the latter approach, we discuss the accuracy of K-K predictions for specific classes of system behavior and how the truncation of the integrals affects this accuracy. We demonstrate the accurate prediction of attenuation and dispersion using several forms of the K-K relations relevant to experimental measurements of media with attenuation coefficients obeying a frequency power law and media consisting of resonant scatterers. We also review the time-causal relations that describe the time-domain consequences of causality in the wave equation. These relations can be thought of as time-domain analogs of the (frequency-domain) K-K relations. Causality-imposed relations, such as the K-K and time-causal relations, provide useful tools for the analysis of measurements and models of acoustic systems.

Transient propagation in media with classical or power-law loss

This paper addresses the problem of small-signal transient wave propagation in media whose absorption coefficient obeys power-law frequency dependence, i.e., alpha infinity omega n. Our approach makes use of previously derived relations between the absorption and dispersion based on the Kramers-Kronig relations. This, combined with a recently obtained solution to a causal convolution wave equation enable expressions to be obtained for one-dimensional transient propagation when n is in the range 0 < n < 3. For n = 2, corresponding to no dispersion, straightforward analytical expressions are obtained for a delta-function and a sinusoidal step function sources and these are shown to correspond to relations previously derived. For other values of n, the effects of dispersion are accounted for by using Fourier transforms. Examples are used to illustrate the results for normal and anomalous dispersive media and to examine the question as to the conditions under which the effects of dispersion should be accounted for, especially for wideband ultrasound pulses of the type used in B-mode tissue imaging. It is shown that the product of the attenuation and total propagation path can be used as a criterion for judging whether dispersion needs to be accounted for.

Frequency-domain wave equation and its time-domain solutions in attenuating media

Our purpose in this paper is to describe the wave propagation in media whose attenuation obeys a frequency power law. To achieve this, a frequency-domain wave equation was developed using previously derived causal dispersion relations. An inverse space and time Fourier transform of the solution to this algebraic equation results in a time-domain solution. It is shown that this solution satisfies the convolutional time-domain wave equation proposed by Szabo [J. Acoust. Soc. Am. 96, 491-500 (1994)]. The form of the convolutional loss operator contained in this wave equation is obtained. Solutions representing the propagation of both plane sinusoidal and transient waves propagating in media with specific power law attenuation coefficients are investigated as special cases of our solution. Using our solution, comparisons are made for transient one-dimensional propagation in a medium whose attenuation is proportional to frequency with recently obtained numerical solutions of Szabo's equation. These show good agreement.

Including dispersion and attenuation directly in the time domain for wave propagation in isotropic media

When sound propagates in a lossy fluid, causality dictates that in most cases the presence of attenuation is accompanied by dispersion. The ability to incorporate attenuation and its causal companion, dispersion, directly in the time domain has received little attention. Szabo [J. Acoust. Soc. Am. 96, 491-500 (1994)] showed that attenuation and dispersion in a linear medium can be accounted for in the linear wave equation by the inclusion of a causal convolutional propagation operator that includes both phenomena. Szabo's work was restricted to media with a power-law attenuation. Waters et al. [J. Acoust. Soc. Am. 108, 2114-2119 (2000)] showed that Szabo's approach could be used in a broader class of media. Direct application of Szabo's formalism is still lacking. To evaluate the concept of the causal convolutional propagation operator as introduced by Szabo, the operator is applied to pulse propagation in an isotropic lossy medium directly in the time domain. The generalized linear wave equation containing the operator is solved via a finite-difference-time-domain scheme. Two functional forms for the attenuation often encountered in acoustics are examined. It is shown that the presence of the operator correctly incorporates both, attenuation and dispersion.

Differential forms of the Kramers-Krönig dispersion relations

Differential forms of the Kramers-Krönig dispersion relations provide an alternative to the integral Kramers-Krönig dispersion relations for comparison with finite-bandwidth experimental data. The differential forms of the Kramers-Krönig relations are developed in the context of tempered distributions. Results are illustrated for media with attenuation obeying an arbitrary frequency power law (alpha(omega) = alpha0 + alpha1(absolute value of omega)y). Dispersion predictions using the differential dispersion relations are compared to the measured dispersion for a series of specimens (two polymers, an egg yolk, and two liquids) exhibiting attenuation obeying a frequency power law (1.00 < or = y < or = 1.99), with very good agreement found. For this form of ultrasonic attenuation, the differential Kramers-Krönig dispersion prediction is found to be identical to the (integral) Kramers-Krönig dispersion prediction.