涂层目标1.06μm偏振双向反射分布函数测量方法研究

为了获取复杂涂层目标的光学散射特性,介绍了一套全自动偏振双向反射分布函数测量系统,可以测量半球空间内几乎所有角度下的偏振双向反射分布函数。采用双波片旋转法实现了对光的偏振调制,利用离散傅里叶变换反演出了涂层目标的Muller矩阵。给出了某涂层样品的测量结果与模型的计算比较。结果表明,这种方法是一种测量涂层目标偏振双向反射分布函数的有效方法。     

光束分数傅里叶变换的Wigner分布函数分析方法

利用Wigner分布函数(WDF)方法，对光束的分数傅里叶变换特性进行了研究.以厄米 高斯(H G)光束为例，导出了H G光束在分数傅里叶变换面上光强分布的解析公式和H G光束在分数傅里叶变换面上束宽的解析计算公式.通过数值计算研究了H G光束光强随分数傅里叶变换阶数变化的规律.研究表明：选取适当的分数傅里叶变换阶数p，在x，y方向可以得到相等束宽的对称光强分布. 关键词： Wigner分布函数 厄米 高斯(H G)光束 分数傅里叶变换     

含介质的单透镜和双透镜分数傅里叶变换光学系统

根据菲涅耳衍射公式,导出了含介质的单透镜与双透镜系统输出面的光场复振幅分布,并与分数傅里叶变换定义式比较,得到了实现分数傅里叶变换的系统结构参量.运用矩阵光学方法,导出上述系统的ABCD矩阵,并与分数傅里叶变换矩阵比较,研究发现两种方法所得结果一致.     

光学分数傅里叶变换及其应用

介绍了分数傅里叶变换的定义及其性质,给出了分数傅里叶变换的光学实验结构,并对分数傅里叶变换在光学信息处理领域中的应用进行了综述。     

规范光学分数傅里叶变换下的光学传递函数

基于规范光学分数傅里叶变换,从光学传递函数概念出发,根据菲涅耳衍射、夫琅和费衍射定义,应用线性系统理论,分别给出了菲涅耳与夫琅和费衍射系统在规范分数傅里叶变换下的光学传递函数数学表达式。证明了其具有分数傅里叶变换的级联性,指出常规傅里叶变换下的光学传递函数为分数传递函数的特例。所得结果对光学分数傅里叶变换在信息处理、像质评价等方面的应用具有实用价值。     

分数傅里叶变换面上余弦-高斯光束的变换特性

 利用Wigner分布函数的方法，研究了余弦-高斯光束的分数傅里叶变换特性。导出了余弦-高斯光束在分数傅里叶变换面上光强分布和束宽的解析计算公式，并对此进行了数值模拟计算。研究表明：分数傅里叶变换阶数对余弦-高斯光束的光强分布有明显影响，余弦-高斯光束的轴上光强随分数傅里叶变换阶数呈周期性变化，束宽随分数傅里叶变换阶数也呈周期性变化，周期为2；对给定调制参数的余弦-高斯光束，通过适当选取分数傅里叶变化阶数可以获得平顶的光强分布。     

Wigner分布函数及非整数傅里叶变换

本文研究了Wigner分布函数的特性,利用Wigner分布函数的旋转,将整数域傅里叶变换推广到了非整数域,描述了自由空间光场的演变过程.     

分数傅里叶变换滤波优化算法及滤波器设计

分数傅里叶变换（FrFT, fractional Fourier transform）是经典傅里叶变换的一种表现形式，可理解为在时 频平面中坐标轴系以原点为轴逆时针旋转一定的角度。通过数学推导，对能否利用分数傅里叶变换进行信号滤波，滤波的优化算法如何，以及滤波器有哪些设计结构等问题进行深入的研究，指出分数傅里叶变换适用于非平稳信号滤波。采用Matlab进行了数值仿真实验。实验结果表明：在信号滤波方面，由于傅里叶变换在处理某些数据时有局限性，因此分数傅里叶变换与傅里叶变换相比具有显著的优势。最后给出FrFT滤波器的设计思想。     

用Wigner分布函数方法研究平顶多高斯光束的分数傅里叶变换

本文用Wigner分布函数方法分析了平顶多高斯光束的分数傅里叶变换。导出了分数傅里叶变换面上光的强度、束宽、远场发散角、M2因子和K参数的解析表达式。通过数值模拟研究了分数傅里叶变换阶数对平顶多高斯光束传输性质的影响。     

针对磁共振射频脉冲的时频域分析方法比较研究

射频脉冲可实现样本自旋体系的精确操控,进而产生预期的核磁共振(NMR)信号,在NMR信号产生过程中扮演重要角色.该文分别采用短时傅里叶变换(STFT)、连续小波变换(CWT)和维格纳-威利分布(WVD)几种时频域分析方法对射频脉冲(优化形状脉冲)进行特性分析和比较.结果表明,三种方法各自具有优缺点,结合各自优势对射频脉冲进行各种方法分析,可以更好地理解复杂脉冲的幅度、相位特性在时频域的分布情况.该文的研究方法将为直观理解复杂射频脉冲对自旋体系的作用机制提供参考.     

函数的H-导数和分形动力学

 给出了函数的Hausdorff测度与H-导数。从而，为开展自给的分形动力学提供了解析的数学工具。同时，给出了分形动力学的演化方程组并做了初步讨论与分析。     

A generalized Weyl—Wigner quantization scheme unifying P-Q and Q-P ordering and Weyl ordering of operators

By extending the usual Wigner operator to the s-parameterized one as 1/4π2 integral (dyduexp [iu(q-Q)+iy(p-P)+is/2yu]) from n=- ∞ to ∞ with s beng a,real parameter,we propose a generalized Weyl quantization scheme which accompanies a new generalized s-parameterized ordering rule.This rule recovers P-Q ordering,Q-P ordering,and Weyl ordering of operators in s = 1,1,0 respectively.Hence it differs from the Cahill-Glaubers’ ordering rule which unifies normal ordering,antinormal ordering,and Weyl ordering.We also show that in this scheme the s-parameter plays the role of correlation between two quadratures Q and P.The formula that can rearrange a given operator into its new s-parameterized ordering is presented.     

Global existence and blow up of solutions for Cauchy problem of generalized Boussinesq equation

We study the Cauchy problem of generalized Boussinesq equation u_tt−u_xx+(u_xx+f(u))_xx=0, where f(u)=±|u|^p or ±|u|^p−1u, p>1. By introducing a family of potential wells we obtain invariant sets, vacuum isolating and threshold result of global existence and nonexistence of solution.     

Dynamical Localization of Quantum Walks in Random Environments

We study shock statistics in the scalar conservation law ∂ _t u+∂ _x f(u)=0, x∈ℝ, t>0, with a convex flux f and spatially random initial data. We show that the Markov property (in x) is preserved for a large class of random initial data (Markov processes with downward jumps and derivatives of Lévy processes with downward jumps). The kinetics of shock clustering is then described completely by an evolution equation for the generator of the Markov process u(x,t), x∈ℝ. We present four distinct derivations for this evolution equation, and show that it takes the form of a Lax pair. The Lax equation admits a spectral parameter as in Manakov (Funct. Anal. Appl. 10:328–329, 1976), and has remarkable exact solutions for Burgers equation (f(u)=u ²/2). This suggests the kinetic equations of shock clustering are completely integrable.     

Inverse problem for the Grad-Shafranov equation with affine right-hand side

For a given domain ω ⋐ ℝ² with boundary γ = ∂ω, we study the cardinality of the set $ \mathfrak{A}_\eta \left( \Phi \right) $ \mathfrak{A}_\eta \left( \Phi \right) of pairs of numbers (a, b) for which there is a function u = u _(a,b): ω → ℝ such that ∇² u(x) = au(x) + b ⩾ 0 for x ∈ ω, u| _γ = 0, and ||∇u(s)| − Φ(s) ⩽ η for s ∈ γ. Here η ⩾ 0 stands for a very small number, Φ(s) = |∇(s)| / ∫ _γ |∇v| d γ, and v is the solution of the problem ∇² v = a ₀ v + 1 ⩾ 0 on ω with v| _γ = 0, where a ₀ is a given number. The fundamental difference between the case η = 0 and the physically meaningful case η > 0 is proved. Namely, for η = 0, the set $ \mathfrak{A}_\eta \left( \Phi \right) $ \mathfrak{A}_\eta \left( \Phi \right) contains only one element (a, b) for a broad class of domains ω, and a = a ₀. On the contrary, for an arbitrarily small η > 0, there is a sequence of pairs (a _j , b _j ) ∈ $ \mathfrak{A}_\eta \left( \Phi \right) $ \mathfrak{A}_\eta \left( \Phi \right) and the corresponding functions u _j such that ‖f _u ^j+1‖ − ‖f _u ^j ‖ > 1, where ‖f _u ^j = max _x∈ω |f _u ^j (x)| and f _u ^j (x) = a _j u _j (x) + b _j . Here the mappings f _u ^j : ω → ℝ necessarily tend as j → ∞ to the δ-function concentrated on γ.     

一类广义强阻尼Sine-Gordon方程的整体解

同时考虑黏性效应及外阻尼作用研究了一类广义强阻尼Sine-Gordon方程-利用Galerkin方法,首先证明了该方程在初值u(x,0)∈H¹₀(Ω),u_t(x,0)∈L²(Ω)的条件下初边值问题存在整体弱解u(x,t),并证明了整体弱解关于初始条件具有 关键词： Sine-Gordon型方程 强阻尼 Galerkin方法 整体解     

Essentially different distributions of current in the inverse problem for the Grad-Shafranov equation

The paper answers a question debated by physicists for many years. It is proved that, for almost equal gradients of the magnetic flux u at its zero-level curve ∂ω, which is the piecewise smooth boundary of a simply-connected domain ω ⋐ ℝ², the inverse problem for the Grad-Shafranov equation of plasma equilibrium in a tokamak (in the cylindrical approximation) admits essentially different profiles of distributions f _u : ω ∋ (x, y) ↦ f(u(x, y)) = u _xx (x, y) + u _yy (x, y) ⩾ 0 in the class of third-order polynomials f(u) = Σ _m=0³ a _m u ^m .     

The Squeezing Effect of Three-Mode Operator as an Extension from Two-Mode Squeezing Operator

We theoretically study the squeezing effect in a 3-wave mixing process, generated by the operator S₃ o exp[m(a₁a₂-a₁^fa₂^f)+n(a₁a₃-a₁^fa₃^f)]S_{3}\equiv \exp[\mu(a_{1}a_{2}-a_{1}^{\dagger}a_{2}^{\dagger})+\nu(a_{1}a_{3}-a_{1}^{\dagger}a_{3}^{\dagger})]. The corresponding 3-mode squeezed vacuum state in Fock space and its uncertainty relation are presented. It turns out that S ₃ may exhibit enhanced squeezing. By virtue of integration within an ordered product (IWOP) of operators, we also give the S ₃’s normally ordered expansion. Finally, we calculate the Wigner function of 3-mode squeezed vacuum state by using the Weyl ordering invariance under similar transformations.     

Solutions of Doubly and Higher Order Iterated Equations

Michael Fisher once studied the solution of the equation f(f(x))=x ²–2. We offer solutions to the general equation f(f(x))=h(x) in the form f(x)=g(ag ^–1(x)) where a is in general a complex number. This leads to solving duplication formulas for g(x). For the case h(x)=x ²–2, the solution is readily found, while the h(x)=x ²+2 case is challenging. The solution to these types of equations can be related to differential equations.     

Weak and Partial Symmetries of Nonlinear PDE in Two Independent Variables

Abstract

Nonclassical infinitesimal weak symmetries introduced by Olver and Rosenau and partial symmetries introduced by the author are analyzed. For a family of nonlinear heat equations of the form u _t = (k(u) u _x)_x + q(u), pairs of functions (k(u), q(u)) are pointed out such that the corresponding equations admit nontrivial two-dimensional modules of partial symmetries. These modules yield explicit solutions that look like u(t, x) = F (θ(t) x + φ(t)) or u(t, x) = G(f(x) + g(t)).