Propagation characteristics of annular laser beams passing through the reflection Bragg grating with deformation

When high-power annular laser beams produced by the unstable resonator pass through the volume Bragg grating (VBG), absorption of light in the VBG will induce a temperature increment, resulting in changes in surface distortion. Considering that the surface distortion of the grating induces index and period differences, the scalar wave equations for the annular laser beams propagating in the VBG have been solved numerically and iteratively using finite-difference and sparse matrix methods. The variation in intensity distributions, the total power reflection coefficient, and the power in the bucket (PIB) for the annular laser beams passing through the reflection VBG with deformation have been analyzed quantitatively. It can be shown that the surface distortion of the VBG and the beam orders of the annular beams affect evidently the intensity distributions, the power reflection coefficient, and the PIB of the output beam. The peak intensity decreases as the deformation of the VBG increases. The total power reflection efficiency decreases significantly with the increase in deformations of the VBG. The PIB of the output beam decreases as the obscuration ratio β and the deformation of the VBG increase. For the given obscuration ratio β, the influence of deformation of reflection VBG on the PIB of the annular beams is more sensitive with increase in distortion of the VBG and decrease in beam order.     

Improvement of the dynamic range of a fiber specklegram sensor based on volume speckle recording in photorefractive materials

In this work, the improvement of the dynamic range of a micro displacement sensor based on fiber specklegrams holographically stored in a photorefractive BSO (Bi₁₂SiO₂₀) crystal is reported. In our experimental setup, a plastic optical fiber (POF) was used to generate a subjective speckle pattern that was recorded in the crystal using a two-wave mixing arrangement. The speckle size was controlled by modifying the diameter of a pupil aperture adjacent to a lens producing the image of the speckle. Fringe patterns were obtained at the output of the system by producing micro displacements of the fiber output end. An increase in the visibility of the fringe patterns was appreciated when the pupil aperture diameter decreased even without controlling the average modulation of the intensity of the light pattern, i.e. when the speckle length increased and the average light modulation simultaneously decreased. This behavior allowed recovering the autocorrelation functions of fringe patterns associated with displacements that initially led to decorrelation, and therefore, significantly to improve the dynamic range of the metrological system. To the best of our knowledge this is the first report about the influence of speckle size on the dynamic range of fiber specklegrams sensors recorded on photorefractive materials.     

Remarks on the Balance Relations for the Two-Dimensional Navier–Stokes Equation with Random Forcing

We use the balance relations for the stationary in time solutions of the randomly forced 2D Navier-Stokes equations, found in [10], to study these solutions further. We show that the vorticity ξ(t,x) of a stationary solution has a finite exponential moment, and that for any the expectation of the integral of over the level-set , up to a constant factor equals the expectation of the integral of over the same set.     

On the Continuity of Pathwise Solutions to Langevin Equations in Infinite Dimensions

We fix a rich probability space (,F,P). Let (H,) be a separable Hilbert space and let be the canonical cylindrical Gaussian measure on H. Given any abstract Wiener space (H,B,) over H, and for every Hilbert–Schmidt operator T: HBH which is (|{}|,)-continuous, where |{}| stands for the (Gross-measurable) norm on B, we construct an Ornstein–Uhlenbeck process : (,F,P)×[0,1](B,|{}|) as a pathwise solution of the following infinite-dimensional Langevin equation d _t =db _t +T( _t )dt with the initial data ₀=0, where b is a B-valued Brownian motion based on the abstract Wiener space (H,B,). The richness of the probability space (,F,P) then implies the following consequences: the probability space is independent of the abstract Wiener space (H,B,) (in the sense that (,F,P) does not depend on the choice of the Gross-measurable norm |{}|) and the space C _B consisting of all continuous B-valued functions on [0,1] is identical with the set of all paths of . Finally, we present a way to obtain pathwise continuous solutions :d _t =

Hamel‐isomorphic images of the unit ball

In this article we consider linear isomorphisms over the field of rational numbers between the linear spaces ?² and ?. We prove that if f is such an isomorphism, then the image by f of the unit disk is a strictly nonmeasurable subset of the real line, which has different properties than classical non‐measurable subsets of reals. We shall also consider the question whether all images of bounded measurable subsets of the plane via a such mapping are non‐measurable (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Limit relations for three simple hyperbolic systems of conservation laws

This paper is concerned with the limit relations from the Euler equations of one‐dimensional compressible fluid flow and the magnetohydrodynamics equations to the simplified transport equations, where the δ‐shock waves occur in their Riemann solutions of the latter two equations. The objective is to prove that the Riemann solutions of the perturbed equations coming from the one‐dimensional simplified Euler equations and the magnetohydrodynamics equations converge to the corresponding Riemann solutions of the simplified transport equations as the perturbation parameterx ε tends to zero. Furthermore, the result can also be generalized to more general situations. Copyright © 2009 John Wiley & Sons, Ltd.     

Young measure solutions of some nonlinear mixed‐type equations

This contribution deals with measure‐valued solutions to two types of nonlinear partial differential equations for which, in general, the results on the existence of classical or weak solutions fail. These are the potential equation for transonic flow and the associated unsteady problem (forward–backward diffusion equation). The solutions are constructed by an iteration scheme (Katchanov method) and additional time discretization (Rothe method) in the second case. The existence is proved in the sense of spatial gradient Young measures. Copyright © 2010 John Wiley & Sons, Ltd.     

Lebesgue积分几何意义的推广

证明了非负有界函数的Lebesgue上积分等于函数下方图形的Lebesgue外测度,其Lebesgue下积分等于函数下方图形的Lebesgue内测度,从而将积分的几何意义从可测情形推广到一般情形.