New expressions for field distribution in the focal region

In an earlier paper dealing with the flat-topped light beams [Y. Li, Opt. Lett. 27 (2002) 1007], it is shown that the flat-topped beams can be expressed as 1 − [1 − exp(−ξ²)]^M, where ξ is a dimensionless parameter and M is a non-negative number. The binomial expansion of this express contains only lowest-order Gaussian modes; this situation makes it possible to develop a new formulation of diffraction of converging spherical wave at an aperture in a plane opaque screen if the Gaussian mode expansion is employed to describe the boundary values of the screen.     

非傍轴矢量高斯光束单缝衍射的严格理论

提出非傍轴矢量光束衍射严格的电磁场理论,并以非傍轴矢量高斯光束的单缝衍射为例加以说明.与基于矢量瑞利-索末菲衍射积分推出的解析结果做了数值计算比较,表明了二者的一致性.对理论的进一步推广做了简要讨论. 关键词： 严格的电磁场衍射理论 矢量瑞利-索末菲衍射积分 非傍轴矢量高斯光束的单缝衍射     

Construction of the two-electron contribution to the Fock matrix by numerical integration

A novel method to numerically calculate the Fock matrix is presented. The Coulomb operator is re-expressed as an integral identity, which is discretized. The discretization of the auxiliary t dimension separates the x, y, and z dependencies transforming the two-electron Coulomb integrals of Gaussian-type orbitals (GTO) to a linear sum of products of two-dimensional integrals. The s-type integrals are calculated analytically and integrals of the higher angular-momentum functions are obtained using recursion formulae. The contributions to the two-body Coulomb integrals obtained for each discrete t value can be evaluated independently. The two-body Fock matrix elements can be integrated numerically, using common sets of quadrature points and weights. The aim is to calculate Fock matrices of enough accuracy for electronic structure calculations. Preliminary calculations indicate that it is possible to achieve an overall accuracy of at least 10^?12 E _h using the numerical approach.     

A method for determining foil thicknesses in TEM by using convergent beam electron diffraction under weak beam conditions

In the present work, convergent beam electron diffraction was studied in zirconium (a material of intermediate atomic number) at 300 keV, under weak beam diffraction conditions. For a particular thickness, the details in an observed low order disc were matched to those calculated using the multibeam dynamical theory. This presents the possibility of determining foil thickness over a wide range, with an estimated experimental accuracy of ≈7% or less. In contrast to other convergent beam techniques, the present method, which uses weak beam conditions, can employ commonly-occurring low order reflections to extract thicknesses.

A simple equation based on the two beam approximation, is derived to determine foil thickness (to within ≈ 10%) without resorting to detailed image matching. This equation can be used for a rough estimate of foil thickness while carrying out TEM observations.     

A peculiarity of diffraction measurements using a Gaussian laser beam

Numerical modelling of the Gaussian beam diffraction by a wire near the Fraunhofer limit is presented. Two cases are considered: the case of the wire being on the axis of the beam and the case of the wire being off-axis. For the case where the wire is on the axis, the displacements of the diffracted minima are calculated. For the off-axis case, the changes in the positions of the maxima and minima and the power density are calculated. The conditions are determined at which the measurements made using a Gaussian beam can be calculated using the formula for plane waves. This work arises in connection with research being performed into the Doppler spectrum of big particles.     

Effects of different correlations of partially coherent electromagnetic beam on three-dimensional spectral intensity distribution in the focal region

The effects of position-dependent correlations between the x- and y-components of the partially coherent incident electromagnetic beam on the spectral intensity distribution in the focal plane and along the z-axis of an optical imaging system with a high-numerical aperture are studied. The electromagnetic Gaussian Schell-model (GSM) beam is employed as an example to illustrate the analysis.     

Diffraction of spherical waves at an annular aperture in the use of the boundary diffraction wave theory: A comparison of different diffraction integral approaches

The integral expression for divergent spherical waves diffracted at an annular aperture is derived based on the theory of the boundary diffraction wave. The expressions for divergent spherical waves diffracted at a circular aperture and a disk, and the axial field are treated as the special cases of our general one. Numerical calculation results for axial and transversal intensity distributions are given to compare our results with the Kirchhoff diffraction integral, first and second Rayleigh diffraction integrals. As expected, our results are in agreement with those in the use of the Kirchhoff diffraction integral, but the computer time is reduced greatly by using the boundary diffraction wave theory. The four diffraction formulae are shown to be consistent for axial and transversal intensity distributions, if the source and observation points are positioned equally from the aperture, or the observation point is located enough far from the aperture. Otherwise, the mean value of the first and second Rayleigh diffraction integrals is equal to the result of the boundary diffraction wave theory.     

圆孔衍射光强分布的数值计算

利用数值积分的方法,计算了由点光源经圆孔衍射形成的光强分布,分别得到夫琅禾费及菲涅耳衍射图样,宽定量分析了满足夫琅禾费衍射的条件。     

一种气象观测数据求导的新方法

气象观测数据不可避免地带有观测误差,而对带有误差的离散数据求导是一个不适定的反问题.为了解决这一长期困扰气象工作者的问题,利用吉洪诺夫正则化思想,提出了矩形区域内观测数据的二维一阶偏导数重构算法.通过系列模拟观测数据试验检验了该算法的性能,表明该算法有效且计算精度高.另外,应用该算法对气象观测资料进行客观分析,结果表明：该算法作为一种新的客观分析方法是可行的,并且还能提高小尺度天气系统的识别能力. 关键词： 数值微分 吉洪诺夫正则化 客观分析     

New results in the use of a “reverse engineering” method in cosmologies with scalar field

The method of “reverse engineering” for designing potentials in cosmologies with “quintessence” scalar field is systematically used for several types of cosmologies (through the time behavior of the scale factor). The general recipe is introduced and then applied when matter other than the scalar field is present, and for tachyonic scalar fields. The possibility of using this method for prescribing initial data in numerical simulations in cosmology is investigated.      

Coherence vortex evolution of partially coherent vortex beams in the focal region

Taking Gaussian Schell-model vortex beams as an example of partially coherent vortex beams, we study the evolution and subwavelength structures of coherence vortices in the focal region. The dependence of coherence vortices on the truncation parameter δ, the normalized coherence length ε and the topological charges m is illustrated numerically. It is found that during the evolution process the innermost m coherence vortices and the saddle points near the coherence vortices always remain in the focal plane.     

Development of a protocol for small beam bi-dimensional dose distribution measurements with radiochromic films

Gafchromic™ films have become popular due to their ease of use and their near water equivalence. This last property is crucial for stereotactic small beam dosimetry as demonstrated in recent papers. An accurate bi-dimensional dose measurement with Gafchromic™ films is very challenging mainly because of the non-uniformity response of flatbed scanners (used for films digitalization) and their own non-uniformity. The first proposal of this work is to develop bi-dimensional protocol for small beams and evaluate the associated uncertainty. The second proposal is to validate this protocol for the bi-dimensional measurements of treatment plans performed with the CyberKnife^® system.First, the uniformity of an Epson V700 flatbed scanner and a batch of EBT3 Gafchromic™ films has been investigated. A “four films” dosimeter was designed to reduce the errors (statistic and systematic) due to their non-uniformity. Then, the “four films” dosimeter protocol in both a homogeneous (RW3 material) and heterogeneous (RW3, lung-like and bone-like materials) phantoms has been used to measure the bi-dimensional dose distributions of three simple CyberKnife^® treatment plans. Two tumor locations (middle of the lung and near lung/bone interface) were considered for the heterogeneous phantom. These plans were achieved with the 10 mm fixed collimator and modeled with the PENELOPE Monte Carlo code in order to calculate accurate dose distributions. Finally, the “four films” bi-dimensional dose distributions were compared to the PENELOPE Monte Carlo simulations.Regarding the uncertainty associated to the bi-dimensional dose measurement protocol, the relative standard deviation σ_D on the dose was 1.2% in the range from 0.5 to 4.0 Gy. Regarding the protocol validation on CyberKnife^® treatment plans, a very good agreement was found with all measurement points passing the {3% - 3 mm} Gamma Index criteria.     

Intensity distribution in the focal region of a “doughnot” mode

We study the intensity distribution of a TEM₀₁₊₁₀ mode propagating through a single diffraction limited lens. The theoretical intensity distributions along the optical path remain similar until the truncation effects, due to the lens diameter, are negligible. The focal distribution is however very sensitive to truncation, and this is relevant for industrial applications of high power lasers. An experimental system to measure the intensity distribution at the focal plane is described and its results are compared with the theory.     

聚焦高斯涡旋光束焦区电场和磁场的偏振奇点

以高斯涡旋光束为例,研究了聚焦场中电场和磁场的偏振奇点变化规律.结果表明,高斯涡旋光束经透镜聚焦后,在焦区存在二维和三维电场和磁场的偏振奇点,其位置一般不重合.适当改变与焦面的距离以及光阑截断参数等出现圆偏振奇点的移动、产生和湮没.不同二维和三维电场和磁场圆偏振奇点湮没所对应的各参数临界值不同.在二维电场中,几何焦面上会有V点的产生.     

非平衡磁控溅射系统磁场的半解析法

应用电磁场分析中的一种新方法——半解析法,对一种典型的非平衡磁控溅射系统进行了磁场分析. 结果表明,应用半解析法计算,求解变量少、方法简单,而且计算精度高,该方法相对于等效源法有严密的理论依据,标量位函数的表达式为级数解析式,有利于场强的计算,更有利于优化设计. 关键词： 非平衡磁控溅射 半解析法 标量磁位 磁场强度     

A detailed study through the focal region of near-threshold single-shot femtosecond laser ablation nano-holes in borosilicate glass

A detailed study of the morphology of nano-craters drilled in borosilicate glass by single shot femtosecond laser ablation near the ablation threshold has been performed by scanning electron microscopy, atomic force microscopy and scanning electron microscopy imaging after focused ion beam sectioning. The influence of the numerical aperture (NA = 0.4 and 0.8), the pulse energy (16 nJ < E_p < 600 nJ) and the position of the specimen surface into the focal region were systematically investigated, leading to nanometric or micrometric scales in every spatial dimension. The nanocrater’s size is not restricted by the diffraction limit but determined by the laser pulse stability and the material properties. If the beam is focused inside the glass, two craters are drilled, shaping very distinct morphologies. Their dimensions have been studied in details and different relationships have been proposed for the evolutions of the depths and of the various diameters of these craters as functions of the pulse energy, the numerical aperture and the position of specimen surface in the beam-material interaction region. It is suggested that the long, thin conical profile with very high aspect ratio of the secondary craters is due to a spontaneous reshaping of the beam which transforms the incoming Gaussian pulse into a Gaussian-Bessel pulse. As proposed in the developed model the geometry of the second craters seems to be connected with the one of the main craters.     

非平衡磁控溅射系统磁场的半解析法

应用电磁场分析中的一种新方法——半解析法，对一种典型的非平衡磁控溅射系统进行了磁场分析. 结果表明，应用半解析法计算,求解变量少、方法简单，而且计算精度高，该方法相对于等效源法有严密的理论依据，标量位函数的表达式为级数解析式，有利于场强的计算，更有利于优化设计.     

Phase anomaly and phase singularities of the field in the focal region of high-numerical aperture systems

The phase behavior of the three Cartesian components of the electric field in the focal region of a high-numerical aperture focusing system is studied. The Gouy phase anomaly and the occurence of phase singularities are examined in detail. It is found that the three field components exhibit different behaviors.     

The rigorous electromagnetic theory of the diffraction of vector beams by a circular aperture

The rigorous electromagnetic theory of the diffraction of vector beams by an aperture is proposed and numerically evaluated by the diffraction of vector Gaussian beams by a circular aperture. The results are compared with those, by using the vector Rayleigh–Sommerfeld diffraction integrals and angular-spectrum expression theory, showing a good consistency. The numerical calculation shows that the result calculated by the rigorous theory is much more precise than those calculated by the integral method in diffraction near field, and a highly consistency reaches by the three methods in diffraction far field. A further extension of the theory is discussed.