Second-order Amplitude Research Articles

Effect of gas velocity on the weakly nonlinear instability of a planar viscous sheet

A weakly nonlinear spatial instability of a two-dimensional planar viscous sheet for sinuous disturbances in a co-flowing inviscid gas stream is investigated theoretically, with an emphasis on the effect of the surrounding gas velocity. The solutions of the second-order interface disturbances are derived and the wave deformation has been computed. The results indicate that the second-order surface disturbance of the fundamental sinuous mode is varicose, which causes the thinning and the subsequent breakup of the liquid sheet. The nonlinear behaviors of the planar sheet are quite sensitive to variations in gas-to-liquid velocity ratio. The deviation of the velocity ratio from the value of unity leads to a larger growth rate, a larger second-order initial amplitude, and a shorter breakup length, and therefore enhances the instability. The growth rates predicted by the present nonlinear analysis according to the shortest breakup length are generally smaller than the linear predictions and can better conform to the experimental measures of Barreras et al. [“Linear instability analysis of the viscous longitudinal perturbation on an air-blasted liquid sheets,” Atomization Sprays 11, 139 (2001)]. Furthermore, the wave deformations of the most unstable disturbances are presented. The nonlinear instability of the planar sheet for a fixed velocity difference is performed. An equal increase of the gas and liquid velocity reduces the spatial growth rate and increases the breakup length, but generally has no influences on the second-order initial amplitude and the wavelength of the disturbance.

Magnetorotational Decay Instability in Keplerian Disks

The saturation of the magnetorotational instability (MRI) in thin Keplerian disks through three-wave resonant interactions is introduced and discussed. That mechanism is a natural generalization of the fundamental decay instability discovered five decades ago for infinite, homogeneous, and immovable plasmas. The decay instability relies on the energy transfer from the MRI to stable slow Alfvén-Coriolis as well as magnetosonic waves. A second-order forced Duffing amplitude equation for the initially unstable MRI as well as two first-order equations for the other two waves are derived. The solutions of those equations exhibit bounded bursty nonlinear oscillations for the MRI as well as unbounded growth for the linearly stable slow Alfvén-Coriolis and magnetosonic perturbations, thus giving rise to the magnetorotational decay instability.

Carrier-envelope-phase-induced asymmetries in double ionization of helium by an intense few-cycle XUV pulse

The carrier-envelope-phase (CEP) dependence of electron angular distributions in double ionization of He by an arbitrarily polarized, few-cycle, intense XUV pulse is formulated using perturbation theory (PT) in the pulse amplitude. Owing to the broad pulse bandwidth, interference of first- and second-order PT amplitudes produces asymmetric angular distributions sensitive to the CEP. The PT parametrization is shown to be valid by comparing with results of solutions of the full-dimensional, two-electron time-dependent Schr\odinger equation for the case of linear polarization.

Weakly nonlinear instability of planar viscous sheets

AbstractA second-order instability analysis has been performed for sinuous disturbances on two-dimensional planar viscous sheets moving in a stationary gas medium using a perturbation technique. The solutions of second-order interface disturbances have been derived for both temporal instability and spatial instability. It has been found that the second-order interface deformation of the fundamental sinuous wave is varicose or dilational, causing disintegration and resulting in ligaments which are interspaced by half a wavelength. The interface deformation has been presented; the breakup time for temporal instability and breakup length for spatial instability have been calculated. An increase in Weber number and gas-to-liquid density ratio extensively increases both the temporal or spatial growth rate and the second-order initial disturbance amplitude, resulting in a shorter breakup time or length, and a more distorted surface deformation. Under normal conditions, viscosity has a stabilizing effect on the first-order temporal or spatial growth rate, but it plays a dual role in the second-order disturbance amplitude. The overall effect of viscosity is minor and complicated. In the typical condition, in which the Weber number is 400 and the gas-to-liquid density ratio is 0.001, viscosity has a weak stabilizing effect when the Reynolds number is larger than 150 or smaller than 10; when the Reynolds number is between 150 and 10, viscosity has a weak destabilizing effect.

Lateralization based on interaural differences in the second-order amplitude modulator

Second-order amplitude modulation is a relatively slow variation of the modulation depth of a first-order amplitude modulation with higher frequency. In contrast to first-order modulation, which appears as a physical component in the stimulus spectrum after half-wave rectification, second-order modulation is not necessarily demodulated by the auditory periphery. For binaural processing of second-order amplitude modulated stimuli it is unknown whether interaural time differences (ITDs) in the second-order modulation result in a lateralized percept. Thus, second-order modulation can serve as a tool to investigate whether demodulation of interaurally delayed components is a prerequisite for lateralization. In most of the psychoacoustic experiments presented here, a 25 Hz sinusoidally amplitude-modulated (SAM) 160 Hz tone was either transposed to 4 kHz by half-wave rectifying this SAM waveform before multiplication with a 4 kHz tone (TSAM), or by adding an offset before multiplication (SAMAM). The experiments revealed an inability to lateralize the SAMAM based on ITDs in the 25 Hz component, whereas subjects could lateralize the TSAM. Given that only the TSAM results in a demodulated 25 Hz component after peripheral auditory processing, this result supports the hypothesis that demodulation is a prerequisite for lateralization, which has consequences for temporal modulation processing in models of binaural interaction.

The Faddeev approach applied to antihydrogen formation

The differential cross sections for antihydrogen formation, in ground and excited states, from energetic antiproton–positronium collisions are computed analytically, using the Faddeev formalism in a second-order approximation. At the considered range of the impact energy, the second-order terms corresponding to the classical Thomas double-scattering processes are dominant. It is also shown that there exists a dynamical interference between two of the second-order partial amplitudes. Both amplitudes correspond to classical double-collision mechanisms in which the laboratory critical angle is 0.54 mrad. The interference between these amplitudes is approximately destructive for the capture into s-states with even parity.

Raman scattering for weakened bonds in the intermediate state: enhancement of low-frequency vibrations

A multimode theory of the Raman scattering in resonance with an electronic transition causing a strong weakening of atomic bonds is proposed. Simple analytical relations between the Fourier transforms of the first- and second-order Raman amplitudes and the absorption are derived. It is predicted that the Raman scattering on low-frequency vibrations will be strongly enhanced. Besides the second-order Raman scattering is also enhanced as compared to the first-order scattering. The Raman excitation profiles show a structure caused by the Airy oscillations. The shapes of the profiles of all vibrations contributing to the weakening bond are the same as a consequence of the strong mode mixing under the virtual vibronic transition.

Topographical alterations of inner retinal activity during systemic hyperoxia–hypercapnia in normal subjects and patients with type 1 diabetes

The purpose of this study was to investigate whether an increase in the circulating oxygen supply can alter inner retinal function, assessed by recordings of multifocal oscillatory potentials. We studied 9 subjects with type 1 diabetes (8 without overt retinopathy, one with 2 microaneurysms) and 10 similar-aged normal subjects. The central 60 degrees of the retina was stimulated by an array of 61 hexagonal elements, and mfOP recordings were obtained while breathing room air or carbogen. First-order kernel analysis of the recordings shows 2 potentials (first-order OP1, OP2), whereas second-order kernel analysis produces 3 potentials (second-order OP1, OP2, OP3). Two methods were used to analyze the results: First, we performed a ring analysis for each subject and measured the amplitudes and latencies of the five potentials. We demonstrate that during carbogen inhalation, the control subjects, but not the patients with diabetes, showed significantly increased second-order OP3 amplitudes, for a retinal ring from around 1.8-13 degrees eccentricity. Secondly, a topographical analysis was performed on the amplitude of the second-order OP3 in all 61 traces (from the average recordings of each subject group), which revealed significant alterations not visible in a ring analysis. A similar topographical analysis of the amplitude of the first-order OP2 revealed a small increase in its amplitude during carbogen inhalation for both subject groups. This study demonstrates that some aspects of inner retinal function are modified by the inhalation of carbogen. The reduced effect of carbogen inhalation on the recordings from the patients with diabetes may be due to compromised vascular perfusion in these subjects.

Second-order amplitudes in loop quantum gravity

We explore some second-order amplitudes in loop quantum gravity. In particular, we compute some second-order contributions to diagonal components of the graviton propagator in the large distance limit, using the old version of the Barrett–Crane vertex amplitude. We illustrate the geometry associated with these terms. We find some peculiar phenomena in the large distance behavior of these amplitudes, related to the geometry of the generalized triangulations dual to the Feynman graphs of the corresponding group field theory. In particular, we point out a possible further difficulty with the old Barrett–Crane vertex: it appears to lead to flatness instead of Ricci flatness, at least in some situations. The observation raises the question whether this difficulty remains with the new version of the vertex.

Perturbation theory analysis of attosecond photoionization

Ionization of an atom by a few-cycle attosecond xuv pulse is analyzed using perturbation theory (PT), keeping terms in the transition amplitude up to second order in the pulse electric field. Within the PT approach, we present an ab initio parametrization of the ionized electron angular distribution (AD) using rotational invariance and symmetry arguments. This parametrization gives analytically the dependence of the AD on the carrier envelope phase (CEP), the polarization of the pulse, and on the ionized electron momentum direction, $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathbf{p}}$. For the general case of an elliptically polarized pulse, we show that interference of the first- and second-order transition amplitudes causes a CEP-dependent asymmetry (with respect to $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathbf{p}}\ensuremath{\rightarrow}\ensuremath{-}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathbf{p}}$) and both elliptic and circular dichroism effects. All of these effects are maximal in the polarization plane and depend not only on the CEP but also on the phase of dynamical atomic parameters that enter our parametrization of the AD. Within the single active electron model of an atom, for an initial $s$ or $p$ state we define all dynamical parameters in terms of radial matrix elements (analytic expressions for which are given for the Coulomb and zero-range potentials). For ionization of the H atom by linearly polarized pulses, our PT results are in excellent agreement with results of numerical solutions of the time-dependent Schr\"odinger equation of Peng et al. [New J. Phys. 10, 025030 (2008)]. Also, our numerical results show that the asymmetries and dichroism effects at low electron energies have a different physical origin from those at high electron energies. Moreover, our results for Gaussian and cosine-squared pulse shapes are in good qualitative agreement. Finally, we show that our analytic formulas may prove useful for determining few-cycle extreme ultraviolet (xuv) pulse characteristics, such as the CEP and the polarization.

Prolonged Multifocal Electroretinographic Implicit Times in the Ocular Ischemic Syndrome

Purpose. To examine retinal function in chronic ocular ischemia using multifocal electroretinography (mfERG). Methods. Thirteen patients with unilateral ocular ischemic syndrome (OIS) underwent assessment of ophthalmic systolic blood pressure by ocular pneumoplethysmography, carotid artery patency by ultrasonography, intraocular pressure (IOP) by applanation tonometry, retinal perfusion by fluorescein angiography, and retinal function by mfERG. Results. Ophthalmic systolic blood pressure was 67.0 +/- 11.6 mm Hg in eyes with OIS and 106.1 +/- 18.0 mm Hg in fellow eyes, whereas IOP was 13.8 +/- 3.2 and 14.4 +/- 1.7 mm Hg, respectively. Summed mfERG implicit times (N1, P1, N2) were prolonged in eyes with OIS, by 7.6%, 6.2%, and 7.5%, respectively, compared with fellow eyes (P < or = 0.0048). The retardation of retinal function was significant outside the macula, whereas the assessment of responses from the central retina was limited by high variance. Second-order kernel (first slice) summed implicit times (N1, P1, N2) were also prolonged in OIS, by 6.6%, 7.3%, and 6.8%, respectively (P < or = 0.0058). Of the amplitudes, only the second-order N2 amplitude was significantly abnormal, being reduced by 23.2% in OIS (P = 0.011). Conclusions. The function of the outer and middle layers of the retina was found to be suppressed in chronic ocular hypoperfusion. The moderate delay in retinal function does not appear to explain the prominent photopic symptom of diffuse glare in bright light, and the delay could be evidence of a functional adaptation that serves to maintain and optimize signaling under conditions of compromised perfusion. (ClinicalTrials.gov number, NCT00403195.).

Complex Fano asymmetry parameters for helium L = 0, 1, 2 autoionizing levels excited by electron impact

Theoretical Fano q-parameters for He autoionizing states excited by electron impact are presented, as extracted from first- and second-order hybrid distorted-wave Born + R-matrix calculations. It is found that the q-parameters for (2s2)1S, (2p2)1D, and (2s2p)1P from the first-order calculations are essentially real quantities, while those from the second-order calculations are complex but with a real part similar to the first-order values. These findings are interpreted in terms of the relative phases of the first- and second-order amplitudes.

Changes in first- and second-order multifocal electroretinography in idiopathic macular hole and their correlations with macular hole diameter and visual acuity

To evaluate the first- and second-order multifocal electroretinography (mfERG) responses in patients with idiopathic macular hole, and their correlations with macular hole diameter measured by optical coherence tomography (OCT) and visual acuity. Twenty-four eyes of 24 patients with idiopathic macular hole underwent mfERG and OCT examinations. The response amplitudes and implicit times of the first- and second-order mfERG were analyzed and compared with 20 age-similar normal control subjects. Correlation analyses between visual acuity, apical and basal diameters of the macular hole, and the first- and second-order mfERG amplitudes and implicit times were performed. The first-order mfERG N1 and P1 amplitudes in the central two concentric rings were reduced in macular hole eyes compared with controls (p < 0.006). For the second-order mfERG, only the N1 mfERG amplitude was significantly reduced at ring 6 in macular hole eyes compared with controls (p = 0.030). Correlation analysis showed that apical diameter of macular hole was significantly correlated with the first-order N1 amplitude of rings 2 to 5 (p < 0.024), the first-order P1 amplitude of rings 2 to 6 (p < 0.05), as well as the second-order P1 mfERG amplitudes of rings 3 to 6 and N1 amplitudes of rings 3 to 5 (p < 0.05). LogMAR visual acuity showed significant correlation with apical diameter of the macular hole (p = 0.002), and also with the first-order P1 amplitude of ring 2 (p = 0.024). In eyes with idiopathic macular hole, reductions in first-order mfERG responses are limited to the central macula, while the second-order mfERG response abnormalities involved more of the peripheral macular region. OCT measurement of apical and not the basal diameter of macular hole correlated with the severity of retinal dysfunction assessed by both mfERG and visual acuity.

Positronium formation as a three-body reaction. II. The second-order nuclear amplitudes

We derive an exact analytic form for the second-order nuclear amplitudes, under the Faddeev three-body approach, which is applicable to the nonrelativistic high energy impact interaction where positronium is formed in the collision of a positron with an atom.

Elastic electron-atom scattering in amplitude-phase representation with application to electron diffraction and spectroscopy

The amplitude-phase representation (APR) [W. E. Milne, Phys. Rev. 35, 863 (1930)] is applied to the relativistic radial Schr\"odinger equation corresponding to the Dirac equations with central potential. The initial conditions, hitherto unspecified, for the nonlinear second-order amplitude equation at the finite radius of a muffin-tin (MT) sphere are established by a semiconvergent method. This opens the possibility of using APR for the calculation of electron scattering phase shifts with a finite MT radius as well as with a large atomic radius, whereby the first wave node is phase of reference equal to a multiple of $\ensuremath{\pi}$, adding nothing to the phase shifts. Furthermore, APR is used for benchmarking wave functions obtained by ordinary differential equation integration of the Dirac equations up to high energy and high orbital quantum number. The present APR procedure is discussed with reference to earlier numerical methods. To complete the physical picture, the paper ends with a discussion on exchange-correlation dependent scattering potential in MT spheres of optimized radii, a crystal potential model whose dependence on radii and energy dependent inner potential has recently been corroborated by low energy electron diffraction in oxides with many atoms per unit cell.

Association between Multifocal ERG Implicit Time Delays and Adaptation in Patients with Diabetes

Local first-order multifocal electroretinogram (mfERG) implicit time (K1-IT) delays have proved to be important and predictive indicators of retinal function in diabetes. To better understand the nature of these delays, the authors examined the spatial association between K1-IT and second-order amplitudes (K2-SNR; a measure of adaptation) in diabetic and control subjects. The authors studied K1-IT, K1 amplitude, and K2-SNRs of responses from 35 retinal zones. These were recorded from 20 diabetic patients without retinopathy, 20 patients with mild to moderate nonproliferative diabetic retinopathy (NPDR), and 30 healthy control subjects. The K1-IT and K2-SNR measurements were then adjusted according to normative and subject median values to reduce or remove the effects of retinal location, intersubject differences, and abnormally small K1 amplitudes. There was no significant association between K1-IT and K2-SNR in the control group (P > 0.05) and only a marginal association in the NoRet group (P = 0.05). In contrast, longer K1-ITs were significantly associated with reduced K2-SNRs in NPDR subjects (P < 0.01). In the NPDR eyes, zones without retinopathic lesions showed a significant association between K1-IT and K2-SNR (P < 0.01). The results suggest that an association between longer K1-IT and reduced K2-SNR (abnormal adaptation) develops after the appearance of NPDR, but this association does not depend on the presence of colocalized retinopathic lesions.

First and second-order kernel multifocal electroretinography abnormalities in acute central serous chorioretinopathy

To evaluate the first and second-order kernel multifocal electroretinogram (mfERG) response abnormalities in patients with acute central serous chorioretinopathy (CSC). This was a cross-sectional observational study in which 45 eyes of 45 patients with acute CSC underwent mfERG recordings. Peak amplitudes and implicit times of the first and second-order kernel responses were analyzed and compared with 20 age-matched normal controls. Correlation analyses were performed between the patients' visual acuity and the first and second-order amplitudes and implicit times. The first-order N1 and P1 mfERG amplitudes in the central three concentric rings were reduced in eyes with acute CSC compared with controls (P < 0.05). The first-order P1 implicit times of the central four rings were also delayed (P < 0.05). For the second-order mfERG response, there were significant reductions in the second-order P1 and N1 amplitudes in rings 3-5 compared with controls (P < 0.05). No significant difference between the second-order P1 and N1 implicit times was found compared with controls (P > 0.05). Correlation analyses showed significant correlations between visual acuity and the first-order N1 response amplitudes of rings 1 and 2, and for the first-order N1 and P1 implicit times of rings 1-4 (P < 0.05). Both first and second-order mfERG response abnormalities occur in eyes with acute CSC. These results suggest that in acute CSC, while outer retinal dysfunction is mostly localized to the central macula, there might be more widespread impairment in adaptive mechanisms of the inner retina or outer plexiform layer dysfunction in the more peripheral macula.

Investigation of the closure and simplified Green's function approximations in second-order distorted-wave calculations for the (e, 2e) processes

Triple-differential cross sections for electron impact ionization of helium at 102 eV are presented for both coplanar and perpendicular plane asymmetric geometries within the framework of second-order distorted-wave theory. The closure approximation and the simplified Green's function approximation, which are currently used in the evaluation of the second-order amplitudes, are investigated and the accuracy of the approximations is examined. It is found that second-order effects are significant for this incident energy and the simplified Green's function approximation is generally more valid than the closure approximation.

Positronium formation as a three-body reaction. I. The second-order positron-electron interaction amplitude

We derive the exact analytic form for the second-order positron-electron interaction term in the Faddeev three-body approach which is applicable in the nonrelativistic high energy region. Although there is no nonintegrable singularity in the six-dimensional integral form of this amplitude, here the basic difficulty arises from the presence of complex nonintegral exponents in the components included in the integrand. Consequently, three brunch cuts must be handled simultaneously. However, by using an integral representation of the gamma function, these brunch cuts are removed from the integrand. Expanding the radial parts of the initial and final wave functions further reduces the second-order positron-electron interaction term to a one-variable integral in terms of Bessel functions of the third kind. The different final closed expressions are ultimately derived in terms of the generalized hypergeometric functions for different regions of the scattering angle.

The Effect of Spatial-Frequency Filtering on the Visual Processing of Global Structure

In three experiments we measured reaction times (RTs) and error rates in identifying the global structure of spatially filtered stimuli whose spatial-frequency content was selected by means of three types of 2-D isotropic filters (Butterworth of order 2, Butterworth of order 10, and a filters with total or partial Gaussian spectral profile). In each experiment, low-pass (LP), bandpass (BP), and high-pass (HP) filtered stimuli, with nine centre or cut-off spatial frequencies, were used. Irrespective of the type of filter, the experimental results showed that: (a) RTs to stimuli with low spatial frequencies were shorter than those to stimuli with medium or high spatial frequencies, (b) RTs to LP filtered stimuli were nearly constant, but they increased in a nonmonotonic way with the filter centre spatial frequency in BP filtered stimuli and with the filter cut-off frequency in HP filtered stimuli, and (c) the identification of the global pattern occurred with all visible stimuli used, including BP and HP images without low spatial frequencies. To remove the possible influence of the energy, a fourth experiment was conducted with Gaussian filtered stimuli of equal contrast power (c(rms) = 0.065). Similar results to those described above were found for stimuli with spatial-frequency content higher than 2 cycles deg(-1). A model of isotropic first-order visual channels collecting the stimulus spectral energy in all orientations explains the RT data. A subsequent second-order nonlinear amplitude demodulation process, applied to the output of the most energetic first-order channel, could explain the perception of global structure of each spatially filtered stimulus, including images lacking low spatial frequencies.