Higher-order Coefficients Research Articles

Numerical Simulation of Highly-Nonlinear Dispersion-Shifted Fiber Optical Parametric Gain Spectrum with Fiber Loss and Higher-Order Dispersion Coefficients

The previous study investigated the fiber parameters on the analytical one-pump fiber optical parametric amplifier (FOPA) gain spectrum of a loss-free highly-nonlinear dispersion-shifted fiber (HNL-DSF) and got the optimum results for each parameter. However, the FOPA gain of the combination of all the optimum values of the considered parameters was not reported. Hence, this paper intends to investigate the analytical FOPA gain of the combination of all the optimum values of the considered parameters from the previous study. Later, the analytical gain was compared with the numerical gain from the fourth-order Runge-Kutta (RK4) method and Matlab built-in function, ode45. Next, the effect of fiber loss and higher order dispersion coefficients such as fourth and sixth-order dispersion coefficients were studied. It is found that RK4 gives a smaller error and the gain reduces while the bandwidth remains same in presence of fiber loss. The fourth-order dispersion coefficient broadens the bandwidth a bit while maintaining the gain and there are two narrow-band gains generated to the left and right-side of the broad-band gain. The sixth-order dispersion coefficient just shifts the two narrow-band gains toward or away from the broadband gain depending on the positive or negative signs of the sixth-order dispersion coefficient.

Peripheral Refraction and Aberration Profiles with Multifocal Lenses.

The amount of central or peripheral myopic shift, as induced by different multifocal contact lenses when viewing objects at distance or near, may provide insights on the potential efficacy for slowing eye growth. The present study aims to compare peripheral refraction and higher-order aberration profiles of four multifocal contact lenses with a single vision control lens. Thirty-five myopes (age 21.2 ± 2.1 years) completed the trial, of whom 16 wore Air Optix Aqua and Proclear Multifocal Distance and Near (Group 1, spherical equivalent: -2.90 ± 0.95D), whereas 19 wore Air Optix Aqua, Air Optix Multifocal, and PureVision Multifocal (Group 2, spherical equivalent: -2.95 ± 0.78D). Refraction and aberration profiles with lenses were measured using the BHVI-EyeMapper with (-2.00 to -5.00D in 1.00D steps) and without (+1.00D fogging) accommodation. Data were quantified using M2/4 (2nd and 2nd + 4th order), J0, J45, and higher-order aberration coefficients coma C[3, 1] and spherical aberration C[4, 0]. The center-distance lens exhibited a relative peripheral myopic shift in M2/4 and J0, positive on-axis C[4, 0], negative on-axis C[3, 1] and on-axis M4 was less negative for accommodative demands ≤-3.00D (P < .05). Inversely, the center-near lenses showed a relative peripheral hyperopic shift in M2/4 and J0, negative on-axis C[4, 0], positive on-axis C[3, 1] and on-axis M4 was more negative for demands of -2.00 and -3.00D (P < .05). Independent of lens type, relative peripheral M4 significantly decreased during accommodation. Accounting for C[4, 0], a greater change in relative M profiles and accommodative responses was found for multifocal lenses. Based on the hypothesis that myopic retinal defocus counters eye growth, center-near multifocal lenses exhibited the preferred on-axis features, i.e., producing a central myopic shift at near compared to the control. The center-distance lens exhibited preferred off-axis features, producing relative peripheral myopia, which increased further during accommodation.

Numerical Simulation of Highly-Nonlinear Dispersion-Shifted Fiber Optical Parametric Gain Spectrum with Fiber Loss and Higher-Order Dispersion Coefficients

Numerical Simulation of Highly-Nonlinear Dispersion-Shifted Fiber Optical Parametric Gain Spectrum with Fiber Loss and Higher-Order Dispersion Coefficients

Spectral properties of nonlinearly chirped fiber Bragg gratings for optical communications

Numerical investigation of spectral properties of nonlinearly chirped grating under strain is made. Calculation is performed using Matlab code based on solving the coupled mode equations using transfer matrix method. Our findings show that, by optimizing the linear and higher-order nonlinear chirp coefficients, the reflection bandwidth, side lobes, and group delay could be tunable with the increment of the applied tension. Indeed, the fiber grating chirp is tunable by the applied tension represented by the increase of the reflection spectrum bandwidth and the decrease of the group delay ripples. Besides, the time delay curve typically exhibits two behaviors within the rejection bandwidth: a linear variation for large wavelengths and a quadratic variation for short wavelengths. Therefore, with such kind of grating, it is possible to compensate both linear and nonlinear dispersion in high speed optical networks.

(3+1)D Quasiparticle Anisotropic Hydrodynamics for Ultrarelativistic Heavy-Ion Collisions.

We present the first comparisons of experimental data with phenomenological results from (3+1)D quasiparticle anisotropic hydrodynamics (aHydroQP). We compare particle spectra, average transverse momentum, and elliptic flow. The dynamical equations used for the hydrodynamic stage utilize aHydroQP, which naturally includes both shear and bulk viscous effects. The (3+1)D aHydroQP evolution obtained is self-consistently converted to hadrons using anisotropic Cooper-Frye freeze-out. Hadron production and decays are modeled using a customized version of therminator 2. In this first study, we utilized smooth Glauber-type initial conditions and a single effective freeze-out temperature T_{FO}=130 MeV with all hadronic species in full chemical equilibrium. With this rather simple setup, we find a very good description of many heavy-ion observables.

Posterior corneal shape: Comparison of height data from 3 corneal topographers

To compare the ability of 3 clinical corneal topographers to describe the posterior corneal shape. University Medical Center Groningen, the Netherlands. Prospective observational study. Corneas of healthy participants were measured twice with a dual Scheimpflug instrument (Galilei G2), a scanning-slit system (Orbscan IIz), and a single Scheimpflug instrument (Pentacam HR). Height data describing the posterior corneal shape were fit with Zernike polynomials. Mean values with standard deviations (SD), test-retest variability (coefficient of repeatability [CoR]), and interdevice variability were determined for the defocus Z(2,0), astigmatism Z(2,-2) and Z(2,2), and higher-order terms coma Z(3,-1) and Z(3,1), trefoil Z(3,-3) and Z(3,3), and spherical aberration Z(4,0) coefficients for 5.5mm and 8.0mm diameters. For the 5.5mm diameter, CoRs ranged from 0.3 to 4.3μm with the dual Scheimpflug instrument, 1.6 to 5.2μm with the scanning-slit system, and 0.3 to 2.0μm with the singleScheimpflug instrument. The CoR was similar for the Scheimpflug instruments (P=.43) but poorer for the scanning-slit system (P<.001). The CoRs of the Scheimpflug instruments were smaller than the corresponding population SD for defocus, cardinal astigmatism, coma, and spherical aberration. The scanning-slit system failed to provide 8.0mm diameter data. There was a significant bias (interdevice variability) between the Scheimpflug instruments in the higher-order coefficients at both diameters. Repeatability in assessing the posterior corneal shape was generally good for the Scheimpflug instruments but poor for the scanning-slit system. Interdevice variability between the Scheimpflug instruments compromised the interchangeability of higher-order coefficients. For astigmatism, CoR and 95% limitsof agreement of the Scheimpflug instruments typically corresponded to 0.1 diopter per astigmatism term.

Compressive sensing based spherical harmonics decomposition of a low frequency sound field within a cylindrical cavity.

A low frequency sound field within a cylindrical cavity can be well approximated by a sparse set of Fourier-Bessel series in a spherical coordinate system. The approximation accuracy can be guaranteed as long as the series coefficients are well estimated by use of spherical microphone arrays (SMA). Conventional methods like spherical Fourier transform and Helmholtz equation least square require a large number of sensors, and it is difficult to estimate the high order coefficients in the presence of sensor noise. To cope with these issues, compressive sensing (CS) is utilized and compared with the conventional methods. In this study, the estimate of the high order coefficients from contaminated measurements is examined, and the effect of modal behavior is analyzed. Numerical simulations show that sensors can be significantly saved especially for high order cavity modes due to the low sparsity of coefficients which is also stable as the modal frequency increases, and the overall accuracy is also improved. The optimal SMA configuration for the use of CS is studied. A large SMA with omni-microphone deployed by the Fliege sampling scheme is suggested. The CS results are finally validated through the good simulated reconstructions of cylindrical cavity modes.

Renormalized coupling constants for the three-dimensional scalar λ ϕ 4 field theory and pseudo-ε-expansion

Renormalized coupling constants g_{2k} that enter the critical equation of state and determine nonlinear susceptibilities of the system possess universal values g*_{2k} at the Curie point. They are calculated, along with the ratios R_{2k} = g_{2k}/g_4^{k-1}, for the three-dimensional scalar \lambda\phi^4 field theory within the pseudo-\epsilon-expansion approach. Pseudo-\epsilon-expansions for g*_6, g*_8, R*_6, and R*_8 are derived in the five-loop approximation, numerical estimates are presented for R*_6 and R*_8. The higher-order coefficients of the pseudo-\epsilon-expansions for the sextic coupling are so small that simple Pade approximants turn out to be sufficient to yield very good numerical results. Their use gives R*_6 = 1.650 while the most recent lattice estimate is R*_6 = 1.649(2). For the octic coupling pseudo-\epsilon-expansions are less favorable from the numerical point of view. Nevertheless, Pade-Borel resummation leads in this case to R*_8 = 0.890, the number differing only slightly from the values R*_8 = 0.871, R*_8 = 0.857 extracted from the lattice and field-theoretical calculations.

Phase transitions in BaTiO3 under uniaxial compressive stress: Experiments and phenomenological analysis

The relative permittivity of polycrystalline BaTiO3 was measured from –150 °C to 250 °C at compressive bias stresses up to −500 MPa. Mechanical loading shifted the rhombohedral-orthorhombic, orthorhombic-tetragonal, and tetragonal-cubic phase transition temperatures and produced a pronounced broadening of the dielectric softening in the vicinity of all three transitions. The inter-ferroelectric rhombohedral-orthorhombic and orthorhombic-tetragonal phase transitions were found to be less stress sensitive than the ferroelectric-paraelectric transition occurring between tetragonal and cubic phases at the Curie point. The application of compressive stress resulted in a strong suppression of the relative permittivity, such that at the highest applied stress of −500 MPa, the permittivity in the single phase regions away from the phase transitions was found to display only a weak dependence on temperature between −100 °C and 125 °C. The experimental observations closely followed the predictions of a 2-4-6 Landau polynomial wherein the dielectric stiffness and higher-order dielectric stiffness coefficients are linear functions of uniaxial stress.

All-order dispersion cancellation and energy-time entangled state.

Dispersion cancellation with an energy-time entangled photon pair in Hong-Ou-Mandel (HOM) interference is one phenomenon that reveals the nonclassical nature of the entangled photon pair. This phenomenon has been observed in materials with very weak dispersions. If the higher-order dispersion coefficient is non-negligible, then the experiment must be modified to realize dispersion cancellation. All-order dispersion cancellation using balanced dispersion was suggested by Steinberg. However, the same phenomenon is expected to occur even if a photon pair is not entangled. This behaviour can be explained by path indistinguishability with identical dispersion. To achieve an all-order dispersion experiment that cannot be explained classically, we modified the experiment and performed another all-order dispersion cancellation experiment that cannot be explained by identical dispersion. This is the first demonstration of nonclassical all-order dispersion cancellation.

On the Summation of Divergent, Truncated, and Underspecified Power Series via Asymptotic Approximants

A compact and accurate solution method is provided for problems whose infinite power series solution diverges and/or whose series coefficients are only known up to a finite order. The method only requires that either the power series solution or some truncation of the power series solution be available and that some asymptotic behavior of the solution is known away from the series' expansion point. Here, we formalize the method of asymptotic approximants that has found recent success in its application to thermodynamic virial series where only a few to (at most) a dozen series coefficients are typically known. We demonstrate how asymptotic approximants may be constructed using simple recurrence relations, obtained through the use of a few known rules of series manipulation. The result is an approximant that bridges two asymptotic regions of the unknown exact solution, while maintaining accuracy in-between. A general algorithm is provided to construct such approximants. To demonstrate the versatility of the method, approximants are constructed for three nonlinear problems relevant to mathematical physics: the Sakiadis boundary layer, the Blasius boundary layer, and the Flierl-Petviashvili monopole. The power series solution to each of these problems is underspecified since, in the absence of numerical simulation, one lower-order coefficient is not known; consequently, higher-order coefficients that depend recursively on this coefficient are also unknown. The constructed approximants are capable of predicting this unknown coefficient as well as other important properties inherent to each problem. The approximants lead to new benchmark values for the Sakiadis boundary layer and agree with recent numerical values for properties of the Blasius boundary layer and Flierl-Petviashvili monopole.

Optimizing bandwidth of fiber optical parametric amplifier with different combinations of higher-order dispersion coefficient signs

This paper investigates the influence of different combinations of second-order (β2 ), fourth-order (β4 ) and sixth-order (β6 ) dispersion coefficient signs (negativity/positivity) to the fiber optical parametric amplifier gain performance. The numerical simulation has exploited the fourth-order Runge-Kutta method to solve the coupled amplitude equations which basically represent the parametric process of four-wave mixing. In the normal regime at which β2 is positive, the fiber optical parametric amplifier exhibits a poor gain spectrum and it is well performed once the pump is presumed to be in anomalous regime i.e. β2 is negative. The effect of β4 is significant when the signal wavelength is further from the pump wavelength and a wide amplification bandwidth is produced when the fiber have positive β4 in the anomalous regime. As for β6 , unfortunately in this simulation its influences are hardly can be seen probably because of its small value regardless its sign, but, from phase-mismatch equation it is shown that its impact is prominent when the signal is positioned way far from the pump. All in all, this shown that the signs combination of the higher-order dispersion coefficients are significant in order to optimize the FOPA amplification bandwidth.

Image-based gradient non-linearity characterization to determine higher-order spherical harmonic coefficients for improved spatial position accuracy in magnetic resonance imaging

PurposeSpatial position accuracy in magnetic resonance imaging (MRI) is an important concern for a variety of applications, including radiation therapy planning, surgical planning, and longitudinal studies of morphologic changes to study neurodegenerative diseases. Spatial accuracy is strongly influenced by gradient linearity. This work presents a method for characterizing the gradient non-linearity fields on a per-system basis, and using this information to provide improved and higher-order (9th vs. 5th) spherical harmonic coefficients for better spatial accuracy in MRI. MethodsA large fiducial phantom containing 5229 water-filled spheres in a grid pattern is scanned with the MR system, and the positions all the fiducials are measured and compared to the corresponding ground truth fiducial positions as reported from a computed tomography (CT) scan of the object. Systematic errors from off-resonance (i.e., B0) effects are minimized with the use of increased receiver bandwidth (±125kHz) and two acquisitions with reversed readout gradient polarity. The spherical harmonic coefficients are estimated using an iterative process, and can be subsequently used to correct for gradient non-linearity. Test-retest stability was assessed with five repeated measurements on a single scanner, and cross-scanner variation on four different, identically-configured 3T wide-bore systems. ResultsA decrease in the root-mean-square error (RMSE) over a 50cm diameter spherical volume from 1.80mm to 0.77mm is reported here in the case of replacing the vendor's standard 5th order spherical harmonic coefficients with custom fitted 9th order coefficients, and from 1.5mm to 1mm by extending custom fitted 5th order correction to the 9th order. Minimum RMSE varied between scanners, but was stable with repeated measurements in the same scanner. ConclusionsThe results suggest that the proposed methods may be used on a per-system basis to more accurately calibrate MR gradient non-linearity coefficients when compared to vendor standard corrections.

Vertex radius measurement of an off-axis parabola with a three-ball spherometer

A spherometer is often used to precisely measure the radius of curvature of a spherical surface. It can also measure the vertex radius of a more complex surface such as an off-axis parabola (OAP). This paper provides a reliable algorithm to find the vertex radius of an OAP by solving a few equations based on the test geometry. This algorithm can also be easily expanded to any conic surface with high-order aspheric coefficients. The algorithm was verified by measuring an 8-inch diameter OAP and comparing the results with its known prescription. Results show good agreement. An example of measuring the vertex radius of a 4-m diameter OAP is also presented. In addition to this, a calculation was done to show that the coma and astigmatism are independent of the clocking of the spherometer on the optic.

Solitons, Lax pair and infinitely-many conservation laws for a higher-order nonlinear Schrödinger equation in an optical fiber

Under investigation in this paper is a higher-order nonlinear Schrödinger equation in an optical fiber. Lax pair and infinitely-many conservation laws are derived via the symbolic computation. By virtue of the Darboux transformation, one-, two- and three-soliton solutions are derived. Propagation and interaction of the solitons are illustrated graphically: Velocity of the soliton is linearly related to the higher-order dispersion coefficients, while amplitude of the soliton does not depend on them at all. Width of the one soliton increases with the decrease of the intensity of the complex eigenvalue parameter. Head-on interaction between the two bidirectional solitons as well as overtaking and oscillating interaction between the two unidirectional solitons are presented. For the interactions among the three solitons, we display two head-on and one overtaking interactions along with three overtaking interactions. Graphical analysis shows that each interaction between the two solitons is elastic, and each interaction among the three solitons is pairwise elastic.

A reduced-rank approach for implementing higher-order Volterra filters

The use of Volterra filters in practical applications is often limited by their high computational burden. To cope with this problem, many strategies for implementing Volterra filters with reduced complexity have been proposed in the open literature. Some of these strategies are based on reduced-rank approaches obtained by defining a matrix of filter coefficients and applying the singular value decomposition to such a matrix. Then, discarding the smaller singular values, effective reduced-complexity Volterra implementations can be obtained. The application of this type of approach to higher-order Volterra filters (considering orders greater than 2) is however not straightforward, which is especially due to some difficulties encountered in the definition of higher-order coefficient matrices. In this context, the present paper is devoted to the development of a novel reduced-rank approach for implementing higher-order Volterra filters. Such an approach is based on a new form of Volterra kernel implementation that allows decomposing higher-order kernels into structures composed only of second-order kernels. Then, applying the singular value decomposition to the coefficient matrices of these second-order kernels, effective implementations for higher-order Volterra filters can be obtained. Simulation results are presented aiming to assess the effectiveness of the proposed approach.

Non-equilibrium of charged particles in swarms and plasmas—from binary collisions to plasma effects

In this article we show three quite different examples of low-temperature plasmas, where one can follow the connection of the elementary binary processes (occurring at the nanoscopic scale) to the macroscopic discharge behavior and to its application. The first example is on the nature of the higher-order transport coefficient (second-order diffusion or skewness); how it may be used to improve the modelling of plasmas and also on how it may be used to discern details of the relevant cross sections. A prerequisite for such modeling and use of transport data is that the hydrodynamic approximation is applicable. In the second example, we show the actual development of avalanches in a resistive plate chamber particle detector by conducting kinetic modelling (although it may also be achieved by using swarm data). The current and deposited charge waveforms may be predicted accurately showing temporal resolution, which allows us to optimize detectors by adjusting the gas mixture composition and external fields. Here kinetic modeling is necessary to establish high accuracy and the details of the physics that supports fluid models that allows us to follow the transition to streamers. Finally, we show an example of positron traps filled with gas that, for all practical purposes, are a weakly ionized gas akin to swarms, and may be modelled in that fashion. However, low pressures dictate the need to apply full kinetic modelling and use the energy distribution function to explain the kinetics of the system. In this way, it is possible to confirm a well established phenomenology, but in a manner that allows precise quantitative comparisons and description, and thus open doors to a possible optimization.

Binary Black Hole Mergers in the First Advanced LIGO Observing Run

The first observational run of the Advanced LIGO detectors, from September 12, 2015 to January 19, 2016, saw the first detections of gravitational waves from binary black hole mergers. In this paper we present full results from a search for binary black hole merger signals with total masses up to $100 M_\odot$ and detailed implications from our observations of these systems. Our search, based on general-relativistic models of gravitational wave signals from binary black hole systems, unambiguously identified two signals, GW150914 and GW151226, with a significance of greater than $5\sigma$ over the observing period. It also identified a third possible signal, LVT151012, with substantially lower significance, and with an 87% probability of being of astrophysical origin. We provide detailed estimates of the parameters of the observed systems. Both GW150914 and GW151226 provide an unprecedented opportunity to study the two-body motion of a compact-object binary in the large velocity, highly nonlinear regime. We do not observe any deviations from general relativity, and place improved empirical bounds on several high-order post-Newtonian coefficients. From our observations we infer stellar-mass binary black hole merger rates lying in the range $9-240 \mathrm{Gpc}^{-3} \mathrm{yr}^{-1}$. These observations are beginning to inform astrophysical predictions of binary black hole formation rates, and indicate that future observing runs of the Advanced detector network will yield many more gravitational wave detections.

Anisotropy of a cubic ferromagnet at criticality.

Critical fluctuations change the effective anisotropy of cubic ferromagnet near the Curie point. If the crystal undergoes phase transition into orthorhombic phase and the initial anisotropy is not too strong, reduced anisotropy of nonlinear susceptibility acquires at T_{c} the universal value δ_{4}^{*}=2v^{*}/3(u^{*}+v^{*}) where u^{*} and v^{*} are coordinates of the cubic fixed point on the flow diagram of renormalization group equations. In the paper, the critical value of the reduced anisotropy is estimated within the pseudo-ε expansion approach. The six-loop pseudo-ε expansions for u^{*}, v^{*}, and δ_{4}^{*} are derived for the arbitrary spin dimensionality n. For cubic crystals (n=3) higher-order coefficients of the pseudo-ε expansions obtained turn out to be so small that use of simple Padé approximants yields reliable numerical results. Padé resummation of the pseudo-ε series for u^{*}, v^{*}, and δ_{4}^{*} leads to the estimate δ_{4}^{*}=0.079±0.006, indicating that detection of the anisotropic critical behavior of cubic ferromagnets in physical and computer experiments is certainly possible.

Improvement of quality of voice conversion based on spectral differential filter using STRAIGHT-based Mel-cepstral coefficients

In our previous research, we proposed a technique for converting the speech individuality of speech of an arbitrary input speaker into a specified speaker’s one. This technique used the neural network (NN) for many-to-one mapping, which was trained from the pairs of multiple source speakers and a target speaker. Using the NN, we design a filter based on the difference of the spectra of the source and the target speech to directly convert the waveform of the input speaker into the target speaker’s one. An advantage of the proposed method is that the direct waveform conversion alleviates the quality degradation caused by the F0 extraction error. To improve this proposed method, we use mel-cepstral coefficients extracted by STRAIGHT, the high-quality tool of speech analysis and synthesis. The higher order components of the cepstral coefficients determine the detailed shape of the spectral envelope, but the prediction accuracy of the high-order coefficients might be worse than that of the lower-order ones. Thus, we investigated the effect of the order of cepstral components to find the condition that gave better conversion quality. Additionally, we exploited the dynamic features to reduce the discontinuity of the frame and improve the converted speech quality.