Finite-bandwidth Kramers-Kronig relations for acoustic group velocity and attenuation derivative applied to encapsulated microbubble suspensions

Kramers-Kronig (KK) analyses of experimental data are complicated by the conflict between the inherently bandlimited data and the requirement of KK integrals for a complete infinite spectrum of input information. For data exhibiting localized extrema, KK relations can provide accurate transforms over finite bandwidths due to the local-weighting properties of the KK kernel. Recently, acoustic KK relations have been derived for the determination of the group velocity (cg) and the derivative of the attenuation coefficient (alpha') (components of the derivative of the acoustic complex wave number). These relations are applicable to bandlimited data exhibiting resonant features without extrapolation or unmeasured parameters. In contrast to twice-subtracted finite-bandwidth KK predictions for phase velocity and attenuation coefficient (components of the undifferentiated wave number), these more recently derived relations for cg and alpha' provide stricter tests of causal consistency because the resulting shapes are invariant with respect to subtraction constants. The integrals in these relations can be formulated so that they only require the phase velocity and attenuation coefficient data without differentiation. Using experimental data from suspensions of encapsulated microbubbles, the finite-bandwidth KK predictions for cg and alpha' are found to provide an accurate mapping of the primary wave number quantities onto their derivatives.     

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.     

On the applicability of Kramers-Kronig relations for ultrasonic attenuation obeying a frequency power law

In the recent literature concern has been raised regarding the validity of Kramers-Kronig relations for media with ultrasonic attenuation obeying a frequency power law. It is demonstrated, however, that the Kramers-Kronig dispersion relations for application to these types of media are available. The developed dispersion relations are compared with measurements on several liquids, and agreement is found to better than 1 m/s over the experimentally available bandwidth. A discussion regarding the validity of these dispersion relations, in particular how the dispersion relations relate to the so-called Paley-Wiener conditions, forms the conclusion.     

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.     

Time-dependent seafloor acoustic backscatter (10-100 kHz)

A time-dependent model of the acoustic intensity backscattered by the seafloor is described and compared with data from a calibrated, vertically oriented, echo-sounder operating at 33 and 93 kHz. The model incorporates the characteristics of the echo-sounder and transmitted pulse, and the water column spreading and absorption losses. Scattering from the water-sediment interface is predicted using Helmholtz-Kirchhoff theory, parametrized by the mean grain size, the coherent reflection coefficient, and the strength and exponent of a power-law roughness spectrum. The composite roughness approach of Jackson et al. [J. Acoust. Soc. Am. 79, 1410-1422 (1986)], modified for the finite duration of the transmitted signal, is used to predict backscatter from subbottom inhomogeneities. It depends on the sediment's volume scattering and attenuation coefficients, as well as the interface characteristics governing sound transmission into the sediment. Estimation of model parameters (mean grain size, roughness spectrum strength and exponent, volume scattering coefficient) reveals ambiguous ranges for the two spectral components. Analyses of model outputs and of physical measurements reported in the literature yield practical constraints on roughness spectrum parameter settings appropriate for echo-envelope-based sediment classification procedures.     

Measuring the power-law exponent of an atmospheric turbulence phase power spectrum with a Shack Hartmann wave-front sensor

For non-Kolmogorov turbulence we develop a differential angle-of-arrival fluctuation coefficient, which is the ratio between the transverse and longitudinal differential angle-of-arrival variances, and a slope structure-correlation coefficient, which is the ratio between the transverse and longitudinal differences of the slope correlation function and the slope structure function, to measure the power-law exponent of a phase power spectrum with a Shack-Hartmann wave-front sensor: The differential arrival-of-angle fluctuation coefficient and the slope structure-correlation coefficient are both related to power-law exponent beta and are independent of strength parameter rho(0) of the turbulence. We compare the methods developed and use them to evaluate beta in recently completed horizontal atmospheric experiments for 1000-m laser beam propagation.     

Kramers-Kronig analysis on the real refractive index of porous media in the terahertz spectral range

We present a terahertz time-domain experimental technique for the detection of scattering from porous media. The method for detection of the scattering enables one to make a decision when Fresnel or Kramers-Kronig (K-K) analysis can be applied for a porous medium. In this study the real refractive index of a tablet is calculated using the conventional K-K dispersion relation and also using a singly subtractive K-K relation, which are applied to the extinction coefficient obtained from the Beer-Lambert law. The advantage of the K-K analysis is that one gets estimates both for absolute refractive index and also dispersion of the porous tablet, whereas Fresnel analysis provides only the absolute value of the index.     

多模梯度折射率塑料光纤带宽的分析

结合WKB法对多模梯度折射率塑料光纤(GI-POF)色散和带宽进行了综合分析,研究了在GI-POF中材料色散对带宽影响的程度和规律,给出较佳折射率指数(g)的范围,并说明实测带宽高于理论带宽的原因.     

Statistical analysis of sound transmission results obtained on the New Jersey continental shelf

Experiments have been conducted near the site of AMCOR Borehole 6010 on the New Jersey Shelf to evaluate propagation predictability in sandy shallow-water environments. The influence of a nonlinear frequency dependence of the sediment volume attenuation in the uppermost sediment layer at this location is examined. Previously it was determined that a frequency power-law exponent of 1.5 was required for the best modeling of experimental results over the band 50-1000 Hz. The approach here references the attenuation to an accepted value at 1 kHz and makes extensive comparisons between measurements and calculations, to determine a power-law exponent of 1.85+/-0.15.     

Spectral shape of relaxations in silica glass

Precise low-frequency light scattering experiments on silica glass are presented, covering a broad temperature and frequency range ( 9 GHz相似文献   

The dependence of time-domain speed-of-sound measurements on center frequency, bandwidth, and transit-time marker in human calcaneus in vitro

Time-domain speed-of-sound (SOS) measurements in calcaneus are effective predictors of osteoporotic fracture risk. High attenuation and dispersion in bone, however, produce severe distortion of transmitted pulses that leads to ambiguity of time-domain SOS measurements. An equation to predict the effects of system parameters (center frequency and bandwidth), algorithm parameters (pulse arrival-time marker), and bone properties (attenuation coefficient and thickness) on time-domain SOS estimates is derived for media with attenuation that varies linearly with frequency. The equation is validated using data from a bone-mimicking phantom and from 30 human calcaneus samples in vitro. The data suggest that the effects of dispersion are small compared with the effects of frequency-dependent attenuation. The equation can be used to retroactively compensate data. System-related variations in SOS are shown to decrease as the pulse-arrival-time marker is moved toward the pulse center. Therefore, compared with other time-domain measures of SOS, group velocity exhibits the minimum system dependence.     

Effect of frequency dependent processes on pulsed Doppler sample volume

The change in the frequency spectrum of ultrasound pulses as a result of frequency dependent attenuation and scatter is known to alter the spectrum of Doppler signals from blood flow parallel to the beam axis. It is shown here that the change in pulse shape accompanying this pulse spectrum change as a result of these processes, together with the velocity dispersion accompanying frequency dependent attenuation and linked to it by the Kramers-Kronig relations, will change the pulsed Doppler sample volume shape and position.     

典型大气窗口太赫兹波传输特性和信道分析

在已有大气传输模型的基础上,发展了新的太赫兹波大气传输衰减与色散模型,对宽频太赫兹波在真实大气中传输的衰减和色散特性进行了数值模拟研究.改进太赫兹时域光谱技术,对0.3—2.0 THz频段太赫兹波的大气传输特性进行了透射光谱测量,并得到了一组连续吸收参数.比对发现实验窗口区强度和吸收峰的位置都与计算结果符合得很好.据此选取了三个可行的信道:340,410和667 GHz窗口区,利用线性色散理论和无线通信原理分别从物理上精确地计算了这些信道的群速色散参数和信道容量,并分析了影响最大传输数据率的因素-天线增益.研究结果表明:太赫兹波大气传输1 km时,这三个信道群速色散很小,信号不易被展宽;最大传输速率达十几Gbps,高于单模光纤,但需要更高的天线增益.     

Mode coupling effects in optical fibres: perturbative solution of the time-dependent power flow equation

Some new aspects of the perturbative solution of the time-dependent power flow equation are investigated. The roles of the amplitude in the fibre perturbation, of the intrinsic attenuation coefficient and of the launching condition are stressed. The reaching of the steady state is studied with particular attention in order to give predictions on attenuation and bandwidth of real optical links.     

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.     

Application of a multivariate statistical technique to interpreting data from multichannel equipment for the example of the KLEM spectrometer

A multivariate statistical procedure for solving problems of estimating physical parameters on the basis of data from measurements with multichannel equipment is described. Within the multivariate procedure, an algorithm is constructed for estimating the energy of primary cosmic rays and the exponent in their power-law spectrum. They are investigated by using the KLEM spectrometer (NUCLEON project) as a specific example of measuring equipment. The results of computer experiments simulating the operation of the multivariate procedure for this equipment are given, the proposed approach being compared in these experiments with the one-parameter approach presently used in data processing.     

Optical and structural parameters of the ZnO thin film grown by pulsed filtered cathodic vacuum arc deposition

Transparent conductive ZnO film was deposited on glass substrate by pulsed filtered cathodic vacuum arc deposition (PFCVAD). Optical parameters such as absorption coefficient α, the refractive index n, energy band gap E_g and dielectric constants have been determined using different methods. Kramers-Kronig and dispersion relations were employed to determine the complex refractive index and dielectric constants using reflection data in the ultraviolet-visible-near infrared regions. The spectra of the dielectric coefficient were used to calculate the energy band gap and the value was 3.24 eV. The experimental energy band gap was found to be 3.22 eV for 357 nm thick ZnO thin film. The envelope method was also used to calculate the refractive index and the data were consistent with K-K relation results. The structure of the film was analyzed with an x-ray diffractometer and the film was polycrystalline in nature with preferred (002) orientation.     

Inversion symmetry and exact critical exponents of dissipating waves in the sandpile model

By an inversion symmetry, we show that in the Abelian sandpile model the probability distribution of dissipating waves of topplings that touch the boundary of the system shows a power-law relationship with critical exponent 5/8 and the probability distribution of those dissipating waves that are also last in an avalanche has an exponent of 1. Our extensive numerical simulations not only support these predictions, but also show that inversion symmetry is useful for the analysis of the two-wave probability distributions.     

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.     

Dynamic scaling of the critical ultrasonic attenuation in binary liquids: evidence from broadband spectrometry

Between 200 kHz and 500 MHz, the ultrasonic attenuation spectrum of the ethanol/dodecane mixture of critical composition has been measured at various temperatures near the critical one. The measured broadband spectra are evaluated in light of the Ferrell-Bhattacharjee dynamic scaling theory. The experimental findings largely fit the theoretical spectral function if suitable crossover corrections are made using dynamic light-scattering data. The measured scaling function follows the predictions of the recent Folk-Moser renormalization-group theory even better. Combining our ultrasonic attenuation coefficient data with density data from the literature, the asymptotic critical behavior of the thermal expansion coefficient and of the specific heat has been determined and found consistent with the two-scale factor universality. The sonic attenuation data have also been evaluated to yield the adiabatic coupling constant g of the dynamic scaling theory. The unusually small absolute value |g|=0.1 resulted, in fairly good agreement with |g|=0.13 as following from thermodynamic data.