Focal shift of cylindrical vector axisymmetric Bessel-modulated Gaussian beam with radial variance phase wavefront

Axisymmetric Bessel-modulated Gaussian beam with quadratic radial dependence (QBG beam) has attracted much attention recently. In this paper, the focal shift of the cylindrical vector QBG beam with radial variance phase wavefront is investigated theoretically by vector diffraction theory. Results show that focus shifts considerably by changing the phase parameter C that indicates the radial phase variance speed. Under condition of small beam parameter μ of cylindrical vector QBG beam, there is one focal peak that shifts far away from optical aperture on increasing C. When μ increases, there may occur two focal peaks that also shift remarkably on increasing C. And it was found that the dependence of focal shift distance on increasing phase parameter is linear. Phase parameter adjusts the focal shift distance, while, polarization angle does not affect focal shift obviously.     

Tight focusing of J0-correlated Gaussian Schell-model beam through high numerical aperture

Based on the vectorial Debye diffraction theory, the tight focusing of a linearly polarized J₀-correlated Gaussian Schell-model (JGSM) beam through high numerical aperture (NA) is investigated. The components of intensity distributions as well as the 3D degree of polarization of light at the focal plane are depicted by numerical integrations, respectively. It is shown that intensity distributions as well as the degree of polarization of focused field not only strongly depend on the global correlation length of the JGSM beam but also relate to the focusing parameter of NA. It is also indicated that the weight of the longitudinal intensity component would enhance in the focal plane, as long as either the correlation length of the JGSM beam or the focusing parameter of NA increases.     

Focusing properties of spirally polarized hollow Gaussian beam

Focusing properties of spirally polarized hollow Gaussian beam (HGB) are investigated theoretically by vector diffraction theory in this article. Results show that the optical intensity in focal region of spirally polarized HGB can be altered considerably by the beam order, numerical aperture of the focusing system, and spiral parameter that indicates the polarization spiral degree of the spirally polarized HGB. Spiral parameter can induce focal pattern change in axial direction remarkably, while beam order and numerical aperture affect radial foal pattern more obviously. The tunable principle of the focal pattern by spiral parameter differs very considerably under condition of different numerical aperture and beam order. Many novel focal patterns may occur in focal pattern evolution. It was also found that focal shift and focal depth can be altered significantly by spiral parameter and beam order.     

ON PROPERTIES OF q-DIFFERENCE EQUATIONS

In this article,we consider some type of q-difference equations,which have meromorphic solutions with Borel exceptional zeros and poles.We also give a precise result in the finite order case and some f...     

On exponential sums over primes and application in Waring-Goldbach problem

In this paper, we prove the following estimate on exponential sums over primes: Let κ≥1,βκ=1/2 log κ/log2, x≥2 and α=a/q λsubject to (a, q) = 1, 1≤a≤q, and λ∈R. Then As an application, we prove that with at most O(N2/8 ε) exceptions, all positive integers up to N satisfying some necessary congruence conditions are the sum of three squares of primes. This result is as strong as what has previously been established under the generalized Riemann hypothesis.     

系数具有相同级的复线性微-差分方程亚纯解的增长性

The main purpose of this paper is to study the growth of meromorphic solutions of complex linear differential-difference equations L(z, f) =n∑i=0m∑j=0Aij(z)f(j)(z + ci) = 0 or F(z)with entire or meromorphic coefficients, and ci, i = 0,..., n being distinct complex numbers,where there is only one dominant coefficient.     

两共焦抛物导体柱板间的电势和电场

利用导体曲面函数,将求解拉普拉斯方程转变为积分运算,从而得到了在无电荷区域普遍成立的电势表达式,再通过边界条件导出共焦抛物导体柱板间的电场分布,并给出了电场线方程.     

Ultrasonically Initiated Grafting of Maleic Anhydride onto Poly(propylene)

This paper presents the studies on the grafting of MAH onto PP performed in solution by means of ultrasonic initiation. The effect of sonic intensity and monomer concentration on the amount of grafted MAH was investigated. It was found that grafting reaction could be initiated by ultrasound at ambient temperatures in the absence of an initiator. The quantity of product grafted was found to be 1.35%. It has been shown that optimum conditions for grafting are reached at an ultrasonic intensity of 300 W and a MAH concentration of 30%. PP‐g‐MAH is characterized by FTIR and XRD. DSC shows that the crystallinity of PP‐g‐MAH is lower than that of pure PP.