Airy传递矩阵法与偏压下多势垒结构的准束缚能级

使用Airy函数和传递矩阵方法精确计算了一维定态薛定谔方程，并推广到多势垒结构，求解出有/无偏压作用的2,3势垒结构的准束缚能级，进一步研究了有/无偏压作用的2,3势垒结构的准束缚能级与有效质量和外加电压的关系，并对结论的正确性进行了验证． 另外，文中 还指出了有些文章中关于Airy传递矩阵法与计算偏压下多势垒结构的准束缚能级的错误陈述 . 关键词： 准束缚能级 有/无偏压作用的多势垒结构 Airy函数 透射系数     

Discrete Polynuclear Growth and Determinantal Processes

We consider a discrete polynuclear growth (PNG) process and prove a functional limit theorem for its convergence to the Airy process. This generalizes previous results by Prähofer and Spohn. The result enables us to express the F ₁ GOE Tracy- Widom distribution in terms of the Airy process. We also show some results, and give a conjecture, about the transversal fluctuations in a point to line last passage percolation problem. Furthermore we discuss a rather general class of measures given by products of determinants and show that these measures have determinantal correlation functions.     

Gaussian modulated Ai- and Bi-based solutions of the 2D PWE: a comparison

We present a comparison of the evolution features, in terms of the intensity moments up to the second order, of what are here referred to as Ai-Gauss and Bi-Gauss wave functions, which originate from source functions consisting of Gaussian-like modulated Airy patterns (of the first and second kind). Both have already been considered in the literature, the former being in particular analysed in detail. A paraxial-optics oriented view of the cos-like Airy–Hardy integrals, which stand out as a generalization of the well-known Airy integral, is also developed.     

Higher-Order Airy Scaling in Deformed Dyck Paths

We introduce a deformed version of Dyck paths (DDP), where additional to the steps allowed for Dyck paths, ‘jumps’ orthogonal to the preferred direction of the path are permitted. We consider the generating function of DDP, weighted with respect to their half-length, area and number of jumps. This represents the first example of an exactly solvable two-dimensional lattice vesicle model showing a higher-order multicritical point. Applying the generalized method of steepest descents, we see that the associated two-variable scaling function is given by the logarithmic derivative of a generalized (higher-order) Airy integral.     

The Pearcey Process

The extended Airy kernel describes the space-time correlation functions for the Airy process, which is the limiting process for a polynuclear growth model. The Airy functions themselves are given by integrals in which the exponents have a cubic singularity, arising from the coalescence of two saddle points in an asymptotic analysis. Pearcey functions are given by integrals in which the exponents have a quartic singularity, arising from the coalescence of three saddle points. A corresponding Pearcey kernel appears in a random matrix model and a Brownian motion model for a fixed time. This paper derives an extended Pearcey kernel by scaling the Brownian motion model at several times, and a system of partial differential equations whose solution determines associated distribution functions. We expect there to be a limiting nonstationary process consisting of infinitely many paths, which we call the Pearcey process, whose space-time correlation functions are expressible in terms of this extended kernel.     

Rainbow refractrometry: On the validity domain of Airy’s and Nussenzveig’s theories

To measure the temperature of individual droplets or the average temperature in a section of a spray, the analysis of the light scattered around the rainbow angle provides an attractive approach. Up to now, the analysis of recorded rainbow signals has been carried out in the framework of the full Lorenz-Mie theory or of the Airy theory. In this paper, we consider four approaches (Lorenz-Mie, Debye, Airy and Nussenzveig approaches) to compute the light scattered around the rainbow angle, and we compare them in terms of accuracy and time-consumption. It is shown that the Complex Angular Momentum (CAM) theory proposed by Nussenzveig, modified by using empirical coefficients, allows one to accurately compute the light scattered around the rainbow angle in a large angular domain for particles with diameters as small as 10 μm.     

The Lyapunov Exponent of Products of Random 2×2 Matrices Close to the Identity

We study products of arbitrary random real 2×2 matrices that are close to the identity matrix. Using the Iwasawa decomposition of SL(2,?), we identify a continuum regime where the mean values and the covariances of the three Iwasawa parameters are simultaneously small. In this regime, the Lyapunov exponent of the product is shown to assume a scaling form. In the general case, the corresponding scaling function is expressed in terms of Gauss’ hypergeometric function. A number of particular cases are also considered, where the scaling function of the Lyapunov exponent involves other special functions (Airy, Bessel, Whittaker, elliptic). The general solution thus obtained allows us, among other things, to recover in a unified framework many results known previously from exactly solvable models of one-dimensional disordered systems.     

Enhancement of Autofocusing Performance for Circular Arrayed‐Airy Vortex Beams

Numerically and experimentally circular arrayed‐Airy vortex beams (CAAVBs) are demonstrated, of which propagation dynamics and autofocusing performance are investigated. In contrast to the classical circular Airy beams, the focal intensity of CAAVBs can be abruptly enhanced several orders of magnitude by introducing vortex arrays (VAs) and increasing Airy arrays number, and the autofocusing performance markedly increases in the focal plane due to introducing VAs. The position of the focal point can be readily controlled by varying the circular arrayed Airy radius is also demonstrated. Experimental results agree with the numerical calculations. These advantages could be beneficial for potential applications in optical trapping, the generation of high intensity laser, medical treatment, and nonlinear effect.     

Specular and vortical Airy beams

A new kind of truncated Airy beams is investigated and discussed. These beams are a superposition of shifted and truncated Airy functions and its specular counterparts, where zeroes or extremal points of the Airy function are chosen as a truncation point. The specular Airy beams are smooth at the truncation point and produce a diffraction pattern similar to Hermite-Gaussian modes. Under propagation in Fresnel zone, specular Airy beams demonstrate a symmetrical acceleration in opposite sides and the beam divergence is proportional to the traveled distance squared. The astigmatic mode converter transforms a two-dimensional specular Airy beam into a quasi-annular field with a nonzero orbital angular momentum. Vortical Airy beams based on truncated Airy functions are also discussed. These beams are similar to Laguerre-Gaussian modes, while their annular structure is changed during propagation.     

A relation between multipath group velocity, mode number, and ray cycle distance

Weston's ray invariant or "characteristic time" in a range-dependent environment is exactly equivalent to the Wentzel-Kramers-Brillouin phase integral for ducted normal modes. By considering a ray element it is shown that the ray invariant can also be written in terms of ray cycle distance and cycle time. This leads to a useful formula for group velocity in terms of cycle distance and mode number. Drawing a distinction between the ray and wave interpretation, the Airy phase (i.e., the existence of a group velocity minimum) can be included in this approach. Favorable comparisons are made with group velocities derived from a normal mode model. The relationship is valid for variable sound speed and variable bathymetry, and this is demonstrated numerically. The formula is applicable to active sonar, multipath pulse shape, target signatures, reverberation, tomography, and underwater communications.     

Propagation dynamics of finite-energy Airy beams in nonlocal nonlinear media

We investigate periodic inversion and phase transition of normal and displaced finite-energy Airy beams propagating in nonlocal nonlinear media with the split-step Fourier method. Numerical simulation results show that parameters such as the degree of nonlocality and amplitude have profound effects on the intensity distribution of the period of an Airy beam. Nonlocal nonlinear media will reduce into a harmonic potential if the nonlocality is strong enough, which results in the beam fluctuating in an approximately cosine mode. The beam profile changes from an Airy profile to a Gaussian one at a critical point, and during propagation the process repeats to form an unusual oscillation. We also briefly discus the two-dimensional case, being equivalent to a product of two one-dimensional cases.     

Riemann–Hilbert approach to multi-time processes: The Airy and the Pearcey cases

We prove that matrix Fredholm determinants related to multi-time processes can be expressed in terms of determinants of integrable kernels à la Its–Izergin–Korepin–Slavnov (IIKS) and hence related to suitable Riemann–Hilbert problems, thus extending the known results for the single-time case. We focus on the Airy and Pearcey processes. As an example of applications we re-deduce a third order PDE, found by Adler and van Moerbeke, for the two-time Airy process.     

A systematic approach to the calculation of tricritical temperature with application to the nematic-smecticA phase transition

Using the Landau expansion equations have been found that allow for systematic calculations of tricritical temperature for a given molecular model. The theory has been applied to nematic-smecticA (NA) phase transition of liquid crystals. It has been shown exactly that the NA tricritical temperature depends only on the couplings between the two lowest order translational order parameters and the orientational degrees of freedom. Numerical results of Mayer and Lubensky for the NA tricritical point have been derived exactly. Also a stability condition of the obtained solution has been discussed.Alexander von Humboldt Foundation Fellow, 1985–1986     

Electron spin resonance of photolytically reduced pyrazine

A mapping between the exactly soluble forced oscillator and the general vibrationally inelastic scattering problem is shown to yield a new uniform approximation based on generalized Laguerre polynomials. Computations are reported for collinear He-H₂ collisions in which H₂ is represented by harmonic and Morse oscillators. The results show that the Laguerre approximation avoids the known failings of the existing Airy and Bessel uniform approximations.     

A statistical study of the analysis of congested spectra by total spectrum fitting

Heavily overlapped, or congested spectra often display much structure but few individual “lines.” Methods have been devised for analyzing such spectra through nonlinear least-squares fitting of the intensity as a function of wavelength or wavenumber. Such total spectrum fitting (TSF) methods are examined statistically for a simple diatomic model and compared with the standard “measure-assign-fit” (MAF) approach in use since the dawn of spectroscopy. Monte Carlo computations on typically 1000 synthetic spectra at a time verify that the predictions of the variance-covariance matrix are reliable under many circumstances. However in regions where the P and R branches double up, the predicted standard errors in the key spectroscopic constants rise sharply and greatly exceed estimates from the Monte Carlo ensemble statistics. In the same regions, the MAF method actually gives better precision. However, for imperfectly overlapped R and P branches, the MAF standard errors are typically three times larger than the TSF values; moreover, the MAF statistical errors are dwarfed by bias. The TSF approach, while clearly superior in these tests, has a practical drawback: it, too, can give significant bias if the spectra are analyzed with an incorrect model, as illustrated here through calculations employing the wrong function to describe the spectral lineshape.     

Airy Functions Over Local Fields

Airy integrals are very classical but in recent years they have been generalized to higher dimensions and these generalizations have proved to be very useful in studying the topology of the moduli spaces of curves. We study a natural generalization of these integrals when the ground field is a non-archimedean local field such as the field of p-adic numbers. We prove that the p-adic Airy integrals are locally constant functions of moderate growth and present evidence that the Airy integrals associated with compact p-adic Lie groups also have these properties.     

Three-Airy autofocusing beams

We numerically and experimentally demonstrate that a three-Airy autofocusing beam can be generated by superposing three deformed two-dimensional(2 D) Airy beams with a triangle symmetry. When the initial angle between two wings of the deformed 2 D Airy beams increases, such a three-Airy autofocusing beam exhibits that the focusing length decreases and the intensity contrast at the focal point changes. Moreover, after introducing an optical vortex phase, this three-Airy autofocusing beam displays a transverse rotation in propagation. The rotation angle is determined by the topological charge of the vortex and the initial wing angle. Our results may have some potential applications in optical manipulation.     

High-resolution phase imaging of phase singularities in the focal region of a lens

Subwavelength-resolution phase images of phase dislocations at the focal region of a 20x , 0.4-N.A. lens have been obtained by use of an optical fiber interferometer with a tapered probe in one arm. A phase-stepping algorithm is used to determine a quantitative value of the phase at each point in the scan, clearly showing the presence of edge dislocations between the Airy rings of the diffraction pattern near the lens focus, as well as four isolated screw-type singularties caused by astigmatism in the lens.     

叶轮机械三元跨声速势流多重网格分解因式格式数值解

基于作者本人发展的用于分析求解叶轮机械三元跨声速势流的AF2格式,并结合Jameson发展的用于求解二元跨声速势流的MAD格式,本文提出了一种用于快速求解叶轮机械三元跨声速势流的新的全隐式多重网格分解因式格式(MAF格式)并用其对西德DFVLR单级轴流式压气机转子三元跨声速流场进行了求解计算。     

在反常色散区艾里脉冲与光孤子相互作用规律的研究

本文采用分步傅里叶法,研究了在反常色散区孤子和艾里脉冲相互作用的规律,并且对相互作用后的孤子和艾里脉冲各自的强度、时域和时移进行了MATLAB仿真.通过仿真发现光孤子和艾里脉冲在光纤中相互重叠时,交叉相位调制(XPM)就会建立并且这种调制会影响孤子和艾里脉冲的性质.在相互作用过程中,孤子的形状保持不变,但是受到艾里脉冲自加速特性的影响孤子会发生偏移.艾里脉冲受XPM的影响会转化为孤子,传播方向也会发生偏移.可见,XPM使得艾里脉冲和孤子各自的性质都相互影响着对方.艾里脉冲和孤子的时域也会受到XPM的影响,使得原本不相同的脉冲形状都转变为含有一个主峰和一个次峰的相似结构,并且主峰和次峰的位置和脉冲宽度也大致相同,这也是艾里脉冲能够转换为孤子的一个依据.另外本文还模拟了不同输入强度r下的孤子和艾里脉冲的变化情况,模拟发现不管是艾里脉冲还是孤子时移都随着输入强度r的增大而增大,并且它们的变化趋势都是一样的,同时模拟还发现在相同的的r值下,时移也会随着a值的增大而增大.