Gauss Optics and Gauss Sum on an Optical Phenomena

In the previous article (Found. Phys. Lett. 16:325–341, 2003), we showed that a reciprocity of the Gauss sums is connected with the wave and particle complementary. In this article, we revise the previous investigation by considering a relation between the Gauss optics and the Gauss sum based upon the recent studies of the Weil representation for a finite group.     

Axial intensity distribution and focal shifts of focused partially coherent conical Bessel–Gauss beams

Based on the generalized Huygens–Fresnel diffraction integral and the theory of partially coherent light, the axial intensity distribution and the focal shifts of focused partially coherent conical Bessel–Gauss beams are investigated. The dependence of the maximum on-axis intensity and the associated focal shift on the Fresnel number and the coherence parameter are examined and illustrated numerically. The results related to the J ₀-correlated beams and the fully coherent Bessel–Gauss beams are obtained as particular cases of this study.     

菲涅耳衍射公式误差相位的讨论

根据波动光学的理论,对菲涅耳近场衍射公式误差相位的影响进行了分析。以单缝衍射为例,讨论了误差相位与计算精度的联系。结合菲涅耳线波带片焦面光强分布的计算,对比了菲涅耳衍射公式与基尔霍夫衍射公式的计算结果,说明在不满足误差相位近似条件下使用菲涅耳衍射公式带来的影响。最后对影响波带片焦面光强分布的几个因素进行了讨论。     

利用菲涅耳波射线方程求解晶体光学问题的研究

利用晶体光学基本方程推导出菲涅耳波射线方程，通过波射线方程并结合麦克斯韦方程组求解晶体光学问题。     

Comparison of thermodynamically consistent charge carrier flux discretizations for Fermi–Dirac and Gauss–Fermi statistics

We compare three thermodynamically consistent Scharfetter–Gummel schemes for different distribution functions for the carrier densities, including the Fermi–Dirac integral of order 1/2 and the Gauss–Fermi integral. The most accurate (but unfortunately also most costly) generalized Scharfetter–Gummel scheme requires the solution of an integral equation. Since one cannot solve this integral equation analytically, several modified Scharfetter–Gummel schemes have been proposed, yielding explicit flux approximations to the implicit generalized flux. The two state-of-the-art modified fluxes used in device simulation software are the diffusion-enhanced flux and the inverse activity coefficient averaging flux. We would like to study which of these two modified schemes approximates the implicit flux better. To achieve this, we propose a new method to solve the integral equation numerically based on Gauss quadrature and Newton’s method. This numerical procedure provides a highly accurate reference flux, enabling us to compare the quality of the two modified Scharfetter–Gummel schemes. We extend previous results (Farrell in J Comput Phys 346:497–513, 2017a) showing that the diffusion-enhanced ansatz leads to considerably lower flux errors for the Blakemore approximation to the physically more relevant Fermi–Dirac and Gauss–Fermi statistics.     

On an integral equation arising in the transport of radiation through a slab involving internal reflection

In an earlier contribution to this journal [M.M.R. Williams, Eur. Phys. J. B 47, 291 (2005)], we derived an integral equation for the transmission of radiation through a slab of finite thickness which incorporated internal reflection at the surfaces. Here we generalise the problem to the case when there is a source on each face and the reflection coefficients are different at each face. We also discuss numerical and analytic solutions of the equation discussed in [M.M.R. Williams, Eur. Phys. J. B 47, 291 (2005)] when the reflection is governed by the Fresnel conditions. We obtain numerical and graphical results for the reflection and transmission coefficients, the scalar intensity and current and the emergent angular distributions at each face. The incident source is either a mono-directional beam or a smoothly varying distribution which goes from isotropic to a normal beam. Of particular interest is the philosophy of the numerical solution and whether a direct numerical approach is more effective than one involving a more elegant analytical solution using replication and the Hilbert problem. We also develop the solution of this problem using diffusion theory and compare the results with the exact transport solution. An erratum to this article is available at .     

线宽误差对菲涅耳透镜衍射效率的影响

基于标量衍射理论,首先得出存在线宽误差情况下4阶菲涅耳透镜衍射效率的解析式;并以 4阶、8阶和16阶菲涅耳透镜为例,进行计算机模拟计算并系统分析线宽误差对菲涅耳透镜衍射效率的影响。通过实验测量16阶菲涅耳透镜的线宽误差,分析其对16阶菲涅耳透镜衍射效率的影响。模拟计算结果对多阶菲涅耳透镜的制作提供重要的理论指导。     

Truncated Hermite–Gauss series expansion and its application

The rect function is expanded into a truncated Hermite–Gauss (H–G) series and its optimization is analyzed. As an application example, a propagation expression for a cosh-Gaussian (ChG) beam through a slit is derived and the results are compared with the direct numerical integration of the Fresnel diffraction integral and with the complex Gaussian function expansion.     

椭圆高斯光束产生近似无衍射Bessel-Gauss光

研究了轴棱锥聚焦像散椭圆高斯光束的光场分布特性,根据菲涅耳衍射积分理论导出了椭圆高斯光束经轴棱锥衍射后的光场分布,通过数值积分给出椭圆高斯光束经轴棱锥聚焦后的近轴光场强度分布情况,将其与圆高斯光束产生的近似Bessel-Gauss场进行比较,发现椭圆高斯光束经轴棱锥聚焦后的光束在一定的传播距离内也具有无衍射特性,且轴上光强分布与圆高斯光束产生的Bessel-Gauss光束的轴上光强分布具有相似的形式,而这种无衍射光场的强度在垂直于光轴的平面上不再是柱对称分布。根据近轴球面波产生近似Bessel光束的最大无衍射距离公式计算了椭圆Bessel-Gauss光束在子午面和弧矢面上的最大无衍射距离,整个光束的无衍射距离由入射到轴棱锥上的椭圆光斑短轴方向的尺寸决定。     

Diffraction of entangled photon pairs by ultrasonic waves

In this paper, we have presented and established a new theoretical formulation of photon optics based on photon path and Feynman path integral idea. We have used Feynman path integral approach to discuss Fraunhofer, Fresnel diffraction of single photon and entangled photon pairs by ultrasonic wave and obtained the following results: i) quantum state and probability distribution of single photon and entangled photon pairs by Fraunhofer and Fresnel ultrasonic diffraction, ii) oblique incidence Raman-Nath and Bragg diffraction conditions, iii) total correlation state and its probability distribution. Our calculation results are in agreement with the experiment results. Comparing one-photon and two-photon diffraction effects by ultrasonic waves, we have found that two-photon diffraction by ultrasonic waves is also a sub-wavelength diffraction.     

A Simulation of Near Field Optics by Three Dimensional Volume Integral Equation of Classical Electromagnetic Theory

A numerical simulation code for three dimensional problems of near-field optics has been developed using the volume integral equation with the moment method. The object is assumed to be continuous and macroscopic dielectric and can be treated by macroscopic Maxwell#x0027;s equations. The code can treat the large-scale moment method matrix that is obtained by the discretization of the volume integral equation. The resultant matrix equation is solved by an iteration method called the generalized minimum residual method with reasonable computational cost for simple problems of near field optics. Simulation of a simplified model of a scanning near-field optical microscope has been performed and basic polarization characteristics of the system have been investigated in detail. The code is also applied to the collection-mode of a photon scanning tunneling microscope, where the incident wave is the evanescent wave, and basic relation between near-field and far field i.e., output image, is recognized.     

PO-TDIE混合方法在天线电磁兼容问题中的应用

应用物理光学-时域积分方程(PO-TDIE)混合方法对电大尺寸平台天线电磁兼容问题进行分析.详细推导PO-TDIE混合方法的一般表达式,并给出计算复杂度.数值算例表明,方法提高了传统时间递推方法(MOT)的计算效率,能够有效应用于电磁兼容分析中.     

On focal shift and phase Fresnel lens

A physical interpretation of the focal shift is given based on geometrical optics. The results of an analytical study of the focal shift depending on the Fresnel numbers are presented. The effect of the aperture diameter and the width of the central zone of the phase Fresnel lens on the distribution of the light intensity along the axis upon illumination by a convergent light beam is analyzed.     

A Simple Design Approach of a Fresnel Lens for Creating Uniform Light-Emitting Diode Light Distribution Patterns

Abstract

Devising an efficient method for distributing high-radiance light-emitting diode emissions onto target surfaces is a continuing challenge. Most current design methods are mathematically complex and require intricate optimizations. In this article, a simple and highly accurate geometric optics analysis is described for creating a free-form total internal reflection collimator lens and a Fresnel exit lens, which can be fabricated easily for producing a specific intensity distribution. A powerful application is the ability to create various high-efficiency and uniform illumination patterns from a standard widely used MR16 light bulb by simply attaching a Fresnel exit lens onto the face of the bulb.     

Solving the hypersingular boundary integral equation in three-dimensional acoustics using a regularization relationship

Regularization of the hypersingular integral in the normal derivative of the conventional Helmholtz integral equation through a double surface integral method or regularization relationship has been studied. By introducing the new concept of discretized operator matrix, evaluation of the double surface integrals is reduced to calculate the product of two discretized operator matrices. Such a treatment greatly improves the computational efficiency. As the number of frequencies to be computed increases, the computational cost of solving the composite Helmholtz integral equation is comparable to that of solving the conventional Helmholtz integral equation. In this paper, the detailed formulation of the proposed regularization method is presented. The computational efficiency and accuracy of the regularization method are demonstrated for a general class of acoustic radiation and scattering problems. The radiation of a pulsating sphere, an oscillating sphere, and a rigid sphere insonified by a plane acoustic wave are solved using the new method with curvilinear quadrilateral isoparametric elements. It is found that the numerical results rapidly converge to the corresponding analytical solutions as finer meshes are applied.     

Weyl Ordering of Two-Mode Fresnel Operator and Its Applications

By using the Weyl ordering operator formula and the Weyl transformation rule, we derive Weyl ordering of the two-mode Fresnel operator, and then obtain its matrix element in the coordinate representation, which is the integral kernel of the generalized two-mode Fresnel transformation in classical optics. By means of the optical operator method, we obtain a decomposition of the two-mode Fresnel operator’s Weyl ordering and an operator identity.     

Symmetrical group theory for mathematical complexity reduction of digital holograms

This work presents the use of mathematical group theory through an algorithm to reduce the multiplicative computational complexity in the process of creating digital holograms. An object is considered as a set of point sources using mathematical symmetry properties of both the core in the Fresnel integral and the image, where the image is modeled using group theory. This algorithm has multiplicative complexity equal to zero and an additive complexity (k ? 1) × N for the case of sparse matrices and binary images, where k is the number of pixels other than zero and N is the total points in the image.     

X射线布拉格-菲涅耳光学元件的实验研究

用光学全息方法在涂有光刻胶的多层膜表面上的形成布拉格－菲涅耳元件所需的波带片图形,通过离子束刻蚀方法将波带片图形转写到多层膜上,完成元件的制作,给出布拉格－菲涅耳光学元件测量结果,测量结果表明上述方法适于制作布拉格－菲涅耳元件。     

An integral equation and Monte Carlo study of homo- and hetero- nuclear square-well diatomic fluids

Square-well homo-nuclear and hetero-nuclear diatomic fluids are studied using the Ornstein-Zernike equation and the recently proposed RHNC-VM closure. Monte Carlo canonical simulations have been performed to complete recent literature simulation data. The integral equation thermodynamic and structural results are compared with these and literature simulation data at three elongations over a large range of densities and temperatures. The RHNC-VM theory agrees excellently with the simulation thermodynamic and structural results. Its accuracy revealed slight errors in simulation data in work by Lisal and Nezbeda [1999, Molec. Phys., 96, 335]. The data have been re-simulated.     

A general <Emphasis Type="Italic">Z</Emphasis>-scan theory

A novel Z-scan theory based on the solution of the nonlinear paraxial wave equation, completed by the Huygens–Fresnel principle is introduced. This theory is valid for the general case, i.e. for thick samples and large nonlinearities including both nonlinear refraction and absorption. In both limiting cases of thin sample and weak nonlinearity, predictions of this model are in good agreement with theories not using parabolic approximation for the beam profile. It is shown that the widely used parabolic approximation leads to inadequate results when evaluating Z-scan measurements.