Exact and approximate analytical time-domain Green's functions for space-fractional wave equations.

The Chen-Holm and Treeby-Cox wave equations are space-fractional partial differential equations that describe power law attenuation of the form α(ω)≈α0|ω|y. Both of these space-fractional wave equations are causal, but the phase velocities differ, which impacts the shapes of the time-domain Green's functions. Exact and approximate closed-form time-domain Green's functions are derived for these space-fractional wave equations, and the resulting expressions contain symmetric and maximally skewed stable probability distribution functions. Numerical results are evaluated with ultrasound parameters for breast and liver at different times as a function of space and at different distances as a function of time, where the reference calculations are computed with the Pantis method. The results show that the exact and approximate time-domain Green's functions contain both outbound and inbound propagating terms and that the inbound component is negligible a short distance from the origin. Exact and approximate analytical time-domain Green's functions are also evaluated for the Chen-Holm wave equation with power law exponent y = 1. These comparisons demonstrate that single term analytical expressions containing stable probability densities provide excellent approximations to the time-domain Green's functions for the Chen-Holm and Treeby-Cox wave equations.

Linear and nonlinear ultrasound simulations using the discontinuous Galerkin method.

A nodal discontinuous Galerkin (DG) code based on the nonlinear wave equation is developed to simulate transient ultrasound propagation. The DG method has high-order accuracy, geometric flexibility, low dispersion error, and excellent scalability, so DG is an ideal choice for solving this problem. A nonlinear acoustic wave equation is written in a first-order flux form and discretized using nodal DG. A dynamic sub-grid scale stabilization method for reducing Gibbs oscillations in acoustic shock waves is then established. Linear and nonlinear numerical results from a two-dimensional axisymmetric DG code are presented and compared to numerical solutions obtained from linear and Khokhlov-Zabolotskaya-Kuznetsov-based simulations in FOCUS. The numerical results indicate that these nodal DG simulations capture nonlinearity, thermoviscous absorption, and diffraction for both flat and focused pistons in homogeneous media.

Approximate analytical time-domain Green's functions for the Caputo fractional wave equation.

The Caputo fractional wave equation [Geophys. J. R. Astron. Soc. 13, 529-539 (1967)] models power-law attenuation and dispersion for both viscoelastic and ultrasound wave propagation. The Caputo model can be derived from an underlying fractional constitutive equation and is causal. In this study, an approximate analytical time-domain Green's function is derived for the Caputo equation in three dimensions (3D) for power law exponents greater than one. The Green's function consists of a shifted and scaled maximally skewed stable distribution multiplied by a spherical spreading factor 1/(4πR). The approximate one dimensional (1D) and two dimensional (2D) Green's functions are also computed in terms of stable distributions. Finally, this Green's function is decomposed into a loss component and a diffraction component, revealing that the Caputo wave equation may be approximated by a coupled lossless wave equation and a fractional diffusion equation.

The ac Stark effect in nitric oxide induced by rapidly swept continuous wave quantum cascade lasers.

A large ac Stark effect has been observed when nitric oxide, at low pressure in a long optical path (100 m) Herriot cell, is subjected to infrared radiation from a rapidly swept, continuous wave infrared quantum cascade laser. As the frequency sweep rate of the laser is increased, an emission signal induced by rapid passage occurs after the laser frequency has passed through the resonance of 1-0 R(11.5)(3/2 /)molecular absorption line. At very high sweep rates a laser field-induced splitting of the absorptive part of the signal is observed, due to the ac Stark effect. This splitting is related to the Autler-Townes mixing of the e, f lambda doublet components of the 1-0 R(11.5)(3/2) transition, which lie under the Doppler broadened envelope.

Observation of infrared free-induction decay and optical nutation signals from nitrous oxide using a current modulated quantum cascade laser.

Free induction decay (FID), optical nutation, and rapid passage induced signals in nitrous oxide, under both optically thin and optically thick conditions, have been observed using a rapid current pulse modulation, or chirp, applied to the slow current ramp of a quantum cascade (QC) laser. The variation in optical depth was achieved by increasing the pressure of nitrous oxide in a long path length multipass absorption cell. This allows the variation of optical depth to be achieved over a range of low gas pressures. Since, even at the highest gas pressure used in the cell, the chirp rate of the QC laser is faster than the collisional reorientation time of the molecules, there is minimal collisional damping, allowing a large macroscopic polarization of the molecular dipoles to develop. This is referred to as rapid passage induced polarization. The resultant FID signals are enhanced due to the constructive interference between the field within the gas generated by the slow ramp of the laser (pump), and that of the fast chirp of the laser (probe) signal generated by pulse modulation of the continuously operating QC laser. The FID signals obtained at large optical depth have not been observed previously in the mid-infrared regions, and unusual oscillatory signals have been observed at the highest gas pressures used.

Sub-Doppler spectra of infrared hyperfine transitions of nitric oxide using a pulse modulated quantum cascade laser: rapid passage, free induction decay, and the ac Stark effect.

Using a low power, rapid (nsec) pulse-modulated quantum cascade (QC) laser, collective coherent effects in the 5 µm spectrum of nitric oxide have been demonstrated by the observation of sub-Doppler hyperfine splitting and also Autler-Townes splitting of Doppler broadened lines. For nitrous oxide, experiments and model calculations have demonstrated that two main effects occur with pulse-modulated (chirped) quantum cascade lasers: free induction decay signals, and signals induced by rapid passage during the laser chirp. In the open shell molecule, NO, in which both Λ-doubling splitting and hyperfine structure occur, laser field-induced coupling between the hyperfine levels of the two Λ-doublet components can induce a large ac Stark effect. This may be observed as sub-Doppler structure, field-induced splittings, or Autler-Townes splitting of a Doppler broadened line. These represent an extension of the types of behaviour observed in the closed shell molecule nitrous oxide, using the same apparatus, when probed with an 8 µm QC laser.

Laser ablation isotope ratio mass spectrometry for enhanced sensitivity and spatial resolution in stable isotope analysis.

Stable isotope analysis permits the tracking of physical, chemical, and biological reactions and source materials at a wide variety of spatial scales. We present a laser ablation isotope ratio mass spectrometry (LA-IRMS) method that enables δ(13)C measurement of solid samples at 50 µm spatial resolution. The method does not require sample pre-treatment to physically separate spatial zones. We use laser ablation of solid samples followed by quantitative combustion of the ablated particulates to convert sample carbon into CO(2). Cryofocusing of the resulting CO(2) coupled with modulation in the carrier flow rate permits coherent peak introduction into an isotope ratio mass spectrometer, with only 65 ng carbon required per measurement. We conclusively demonstrate that the measured CO(2) is produced by combustion of laser-ablated aerosols from the sample surface. We measured δ(13)C for a series of solid compounds using laser ablation and traditional solid sample analysis techniques. Both techniques produced consistent isotopic results but the laser ablation method required over two orders of magnitude less sample. We demonstrated that LA-IRMS sensitivity coupled with its 50 µm spatial resolution could be used to measure δ(13) C values along a length of hair, making multiple sample measurements over distances corresponding to a single day's growth. This method will be highly valuable in cases where the δ(13)C analysis of small samples over prescribed spatial distances is required. Suitable applications include forensic analysis of hair samples, investigations of tightly woven microbial systems, and cases of surface analysis where there is a sharp delineation between different components of a sample.

Capillary absorption spectroscopy for high temporal resolution measurements of stable carbon isotopes in soil and plant-based systems.

Capillary Absorption Spectroscopy (CAS) is a relatively new analytical technique for performing stable isotope analysis. Here, we demonstrate the utility of CAS by recording and quantifying variation in 13C in controlled and biologically relevant applications. We calibrated CAS system response to increased 13CO2, with an observed ∼4‰ increase in measured Δ13C for each 0.03 ppm shift in 13CO2 concentration. We leveraged this calibration to quantify rates of biogeochemical processes using a 13C tracer. For example, we monitored microbial respiration of 13C-glucose within an agricultural soil at 10 s quantification intervals and results demonstrated 8.6% ± 0.4 of added glucose was converted to 13CO2 within 1.5 h of incubation. We expanded the demonstration by adapting a rhizobox to permit continuous monitoring of 13CO2 in a soil (as distinct from plant) headspace to track the timing and quantify respiration rates of fresh plant photosynthate and observed a 3.5 h delay between plant exposure to a13CO2 tracer and the first signs of respiration by soil biota. These experiments highlight CAS is effective in producing high temporal resolution quantification of 13CO2 and demonstrate potential applications.

Passive standoff detection of RDX residues on metal surfaces via infrared hyperspectral imaging.

Hyperspectral images of galvanized steel plates, each containing a stain of cyclotrimethylenetrinitramine (RDX), were recorded using a commercial long-wave infrared imaging spectrometer. Demonstrations of passive RDX chemical detection at areal dosages between 16 and 90 microg/cm(2) were carried out over practical standoff ranges between 14 and 50 m. Anomaly and target detection algorithms were applied to the images to determine the effect of areal dosage and sensing distance on detection performance for target RDX. The anomaly detection algorithms included principal component analysis, maximum autocorrelation factors, and principal autocorrelation factors. Maximum difference factors and principal difference factors are novel multivariate edge detection techniques that were examined for their utility in detection of the RDX stains in the images. A target detection algorithm based on generalized least squares was applied to the images, as well, to see if the algorithm can identify the compound in the stains on the plates using laboratory reflection spectra of RDX, cyclotetramethylenetetranitramine (HMX), and 2,4,6-trinitrotoluene (TNT) as the target spectra. The algorithm could easily distinguish between the nitroaromatic (TNT) compound and the nitramine (RDX, HMX) compounds, and, though the distinction between RDX and HMX was less clear, the mean weighted residuals identified RDX as the stain on the plate. Improvements that can be made in this detection technique are discussed in detail. As expected, it was found that detection was best for short distances and higher areal dosages. However, the target was easily detected at all distances and areal dosages used in this study.

Fractal ladder models and power law wave equations.

The ultrasonic attenuation coefficient in mammalian tissue is approximated by a frequency-dependent power law for frequencies less than 100 MHz. To describe this power law behavior in soft tissue, a hierarchical fractal network model is proposed. The viscoelastic and self-similar properties of tissue are captured by a constitutive equation based on a lumped parameter infinite-ladder topology involving alternating springs and dashpots. In the low-frequency limit, this ladder network yields a stress-strain constitutive equation with a time-fractional derivative. By combining this constitutive equation with linearized conservation principles and an adiabatic equation of state, a fractional partial differential equation that describes power law attenuation is derived. The resulting attenuation coefficient is a power law with exponent ranging between 1 and 2, while the phase velocity is in agreement with the Kramers-Kronig relations. The fractal ladder model is compared to published attenuation coefficient data, thus providing equivalent lumped parameters.

Space-time duality and high-order fractional diffusion.

Superdiffusion, characterized by a spreading rate t^{1/α} of the probability density function p(x,t)=t^{-1/α}p(t^{-1/α}x,1), where t is time, may be modeled by space-fractional diffusion equations with order 1<α<2. Some applications in biophysics (calcium spark diffusion), image processing, and computational fluid dynamics utilize integer-order and fractional-order exponents beyond this range (α>2), known as high-order diffusion or hyperdiffusion. Recently, space-time duality, motivated by Zolotarev's duality law for stable densities, established a link between time-fractional and space-fractional diffusion for 1<α≤2. This paper extends space-time duality to fractional exponents 1<α≤3, and several applications are presented. In particular, it will be shown that space-fractional diffusion equations with order 2<α≤3 model subdiffusion and have a stochastic interpretation. A space-time duality for tempered fractional equations, which models transient anomalous diffusion, is also developed.

Causal impulse response for circular sources in viscous media.

The causal impulse response of the velocity potential for the Stokes wave equation is derived for calculations of transient velocity potential fields generated by circular pistons in viscous media. The causal Green's function is numerically verified using the material impulse response function approach. The causal, lossy impulse response for a baffled circular piston is then calculated within the near field and the far field regions using expressions previously derived for the fast near field method. Transient velocity potential fields in viscous media are computed with the causal, lossy impulse response and compared to results obtained with the lossless impulse response. The numerical error in the computed velocity potential field is quantitatively analyzed for a range of viscous relaxation times and piston radii. Results show that the largest errors are generated in locations near the piston face and for large relaxation times, and errors are relatively small otherwise. Unlike previous frequency-domain methods that require numerical inverse Fourier transforms for the evaluation of the lossy impulse response, the present approach calculates the lossy impulse response directly in the time domain. The results indicate that this causal impulse response is ideal for time-domain calculations that simultaneously account for diffraction and quadratic frequency-dependent attenuation in viscous media.

Analytical time-domain Green's functions for power-law media.

Frequency-dependent loss and dispersion are typically modeled with a power-law attenuation coefficient, where the power-law exponent ranges from 0 to 2. To facilitate analytical solution, a fractional partial differential equation is derived that exactly describes power-law attenuation and the Szabo wave equation ["Time domain wave-equations for lossy media obeying a frequency power-law," J. Acoust. Soc. Am. 96, 491-500 (1994)] is an approximation to this equation. This paper derives analytical time-domain Green's functions in power-law media for exponents in this range. To construct solutions, stable law probability distributions are utilized. For exponents equal to 0, 1/3, 1/2, 2/3, 3/2, and 2, the Green's function is expressed in terms of Dirac delta, exponential, Airy, hypergeometric, and Gaussian functions. For exponents strictly less than 1, the Green's functions are expressed as Fox functions and are causal. For exponents greater than or equal than 1, the Green's functions are expressed as Fox and Wright functions and are noncausal. However, numerical computations demonstrate that for observation points only one wavelength from the radiating source, the Green's function is effectively causal for power-law exponents greater than or equal to 1. The analytical time-domain Green's function is numerically verified against the material impulse response function, and the results demonstrate excellent agreement.

A time-space decomposition method for calculating the nearfield pressure generated by a pulsed circular piston.

A time-space decomposition approach is derived for numerical calculations of the transient nearfield pressure generated by a circular piston. Time-space decomposition analytically separates the temporal and spatial components of a rapidly converging single integral expression, thereby converting transient nearfield pressure calculations into the superposition of a small number of fast-converging spatial integrals that are weighted by time-dependent factors. Results indicate that, for the same peak error value, time-space decomposition is at least one or two orders of magnitude faster than the Rayleigh-Sommerfeld integral, the Schoch integral, the Field II program, and the DREAM program. Time-space decomposition is also faster than methods that directly calculate the impulse response by at least a factor of 3 for a 10% peak error and by a factor of 17 for a 1% peak error. The results show that, for a specified maximum error value, time-space decomposition is significantly faster than the impulse response and other analytical integrals evaluated for computations of transient nearfield pressures generated by circular pistons.

Regenerative feedback resonant circuit to detect transient changes in electromagnetic properties of semi-insulating materials.

A prototype regenerative feedback resonant circuit has been developed for measuring the transient spectral response due to perturbations in properties of various electromagnetic materials. The circuit can accommodate a variety of cavity resonators, shown here in the 8 GHz range, with passive quality factors (Qstat) as high as 7000 depending upon material loading. The positive feedback enhanced dynamic quality factors (Qdyn) of resonator∕material combinations in the regenerative circuit are on the order of 10(7)-10(8). The theory, design, and implementation of the circuit is discussed along with real-time monitored example measurements of effects due to photon-induced charge carriers in high-resistivity silicon wafers and magnetic-field induced perturbations of yttrium-iron garnet.

Development and regulatory strategies for drug and diagnostic co-development.

At the 5th FDA-Drug Industry Association (DIA) Workshop on 'Pharmacogenomics in Drug Development and Regulatory Decision Making', track four focused on the current thinking and issues in the co-development of therapeutic drugs or biologics, and their companion diagnostic products. Identification and validation of genomic and other biomarkers are becoming important components of drug-development strategies, and recent successes show the power of personalized approaches to change the benefit-risk paradigm for new drugs.

An annular superposition integral for axisymmetric radiators.

A fast integral expression for computing the nearfield pressure is derived for axisymmetric radiators. This method replaces the sum of contributions from concentric annuli with an exact double integral that converges much faster than methods that evaluate the Rayleigh-Sommerfeld integral or the generalized King integral. Expressions are derived for plane circular pistons using both continuous wave and pulsed excitations. Several commonly used apodization schemes for the surface velocity distribution are considered, including polynomial functions and a "smooth piston" function. The effect of different apodization functions on the spectral content of the wave field is explored. Quantitative error and time comparisons between the new method, the Rayleigh-Sommerfeld integral, and the generalized King integral are discussed. At all error levels considered, the annular superposition method achieves a speed-up of at least a factor of 4 relative to the point-source method and a factor of 3 relative to the generalized King integral without increasing the computational complexity.

A fast near-field method for calculations of time-harmonic and transient pressures produced by triangular pistons.

Analytical expressions are demonstrated for fast calculations of time-harmonic and transient near-field pressures generated by triangular pistons. These fast expressions remove singularities from the impulse response, thereby reducing the computation time and the peak numerical error with a general formula that describes the near-field pressure produced by any triangular piston geometry. The time-domain expressions are further accelerated by a time-space decomposition approach that analytically separates the spatial and temporal components of the numerically computed transient pressure. Applied to a Hanning-weighted input pulse, time-space decomposition converts each spatio-temporal integral into six spatial integral evaluations at each field point. Time-harmonic and transient calculations are evaluated for an equilateral triangle with sides equal to four wavelengths, and the resulting errors are compared to pressures obtained with exact and approximate implementations of the impulse response method. The results show that the fast near-field method achieves smaller maximum errors and is consistently faster than the impulse response and methods that approximate the impulse response.

An efficient grid sectoring method for calculations of the near-field pressure generated by a circular piston.

An analytical expression is derived for time-harmonic calculations of the near-field pressure produced by a circular piston. The near-field pressure is described by an efficient integral that eliminates redundant calculations and subtracts the singularity, which in turn reduces the computation time and the peak numerical error. The resulting single integral expression is then combined with an approach that divides the computational grid into sectors that are separated by straight lines. The integral is computed with Gauss quadrature in each sector, and the number of Gauss abscissas in each sector is determined by a linear mapping function that prevents large errors from occurring in the axial region. By dividing the near-field region into 10 sectors, the raw computation time is reduced by nearly a factor of 2 for each expression evaluated in this grid. The grid sectoring approach is most effective when the computation time is reduced without increasing the peak error, and this is consistently accomplished with the efficient integral formulation. Of the four single integral expressions evaluated with grid sectoring, the efficient formulation that eliminates redundant calculations and subtracts the singularity demonstrates the smallest computation time for a specified value of the maximum error.

Semiactive infrared remote sensing: a practical prototype and field comparison.

A semiactive method of Fourier-transform infrared (FTIR) remote sensing has been developed and field tested. The method replaces the sender telescope of an active technique with an extended, heated broadband source. The output of the extended source (a commercial griddle) is not collimated and thus facilitates alignment by having the detector optics simply point at the griddle. The present source fills the detector's field of view at 100 m and maintains a temperature approximately 80 K warmer than ambient. In field tests with live CO releases, the method was approximately 30 times less sensitive than active methods, but approximately 30 times more sensitive than passive methods, with far greater sensitivity in the midwave infrared.