Vector Helmholtz-Gauss and vector Laplace-Gauss beams

We demonstrate the existence of vector Helmholtz-Gauss (vHzG) and vector Laplace-Gauss beams that constitute two general families of localized vector beam solutions of the Maxwell equations in the paraxial approximation. The electromagnetic components are determined starting from the scalar solutions of the two-dimensional Helmholtz and Laplace equations, respectively. Special cases of the vHzG beams are TE and TM Gaussian vector beams, nondiffracting vector Bessel beams, polarized Bessel-Gauss beams, modes in cylindrical waveguides and cavities, and scalar Helmholtz-Gauss beams. The general expression of the vHzG beams can be used straightforwardly to obtain vector Mathieu-Gauss and vector parabolic-Gauss beams, which to our knowledge have not yet been reported.     

超短脉冲复宗量厄米高斯光束

从超短高斯脉冲光束出发,根据复宗量厄米函数的性质,用理论解析推导的方法,给出了一组新的超短脉冲光束的解析解,称为超短复宗量厄米高斯脉冲光束。此脉冲光束解的每个频率分量都是复宗量厄米高斯光束,时间脉冲的形状是任意的,具有相同的衍射距离参量,并且可以描述短于一个光学周期的超短脉冲。对这种超短脉冲光束及其在自由空间中的传输过程进行了较为细致的研究,分析了超短复宗量厄米高斯脉冲光束的轴上光强、光强的横向分布、等衍射性质、脉冲极性反转、脉冲延迟等基本性质。讨论了引入缓变包络近似后出现的时空奇异性。     

Nonparaxial TM and TE beams in free space

Expressions for the fields of TM and TE laser beams in free space that are rigorous solutions to Maxwell's equations are given in a closed form. The electric and the magnetic fields are both expressed in terms of nonparaxial elegant Laguerre-Gaussian beams that are exact solutions of the Helmholtz equation. These solutions involve well-known functions, such as spherical Bessel and associated Legendre functions. Radially and azimuthally polarized beams of arbitrary order are considered, and the lowest-order radially polarized beam (TM(01) beam) is investigated in detail.     

超短脉冲复宗量拉盖尔-高斯光束

从超短高斯脉冲光束出发。根据拉盖尔多项式的性质,用理论解析推导的方法。给出了一组新的超短脉冲光束的解析解,称为超短复宗量拉盖尔-高斯脉冲光束。此脉冲光束解的每个频率分量都是复宗量拉盖尔-高斯光束。时间脉冲的形状是任意的,具有相同的衍射距离参量,并且可以描述短于一个光学周期的超短脉冲。对这种超短脉冲光束及其在自由空间中的传输过程进行了较为细致的研究,分析了超短复宗量拉盖尔一高斯脉冲光束的轴上光强、光强的横向分布、衍射性质、脉冲极性反转、脉冲延迟等。讨论了引入缓变包络近似后出现的时空奇异性。     

Localized light waves: Paraxial and exact solutions of the wave equation (a review)

Simple explicit localized solutions are systematized over the whole space of a linear wave equation, which models the propagation of optical radiation in a linear approximation. Much attention has been paid to exact solutions (which date back to the Bateman findings) that describe wave beams (including Bessel-Gauss beams) and wave packets with a Gaussian localization with respect to the spatial variables and time. Their asymptotics with respect to free parameters and at large distances are presented. A similarity between these exact solutions and harmonic in time fields obtained in the paraxial approximation based on the Leontovich-Fock parabolic equation has been studied. Higher-order modes are considered systematically using the separation of variables method. The application of the Bateman solutions of the wave equation to the construction of solutions to equations with dispersion and nonlinearity and their use in wavelet analysis, as well as the summation of Gaussian beams, are discussed. In addition, solutions localized at infinity known as the Moses-Prosser “acoustic bullets”, as well as their harmonic in time counterparts, “X waves”, waves from complex sources, etc., have been considered. Everywhere possible, the most elementary mathematical formalism is used.     

Transformation of singular light beams at nondegenerated four-wave mixing in dye solutions

An analysis of a frequency transformation of singular light beams by the dynamic hologram at a frequency nondegenerated four-wave mixing in dye solutions has been carried out. The transformation of optical vortices of the first and second order from the infrared spectrum to the visible one has been demonstrated experimentally with simultaneous inversion of the topological charge.     

Second-harmonic generation in sound beams reflected from, and transmitted through, immersed elastic solids

Second-harmonic generation in sound beams reflected from and transmitted through thick isotropic elastic solids is investigated experimentally. Measurements of diffraction patterns are compared with a theoretical model based on integral solutions for harmonic generation in sound beams. The solutions are connected by classical linear theory for reflection and transmission at fluid-solid interfaces. Nonspecular phenomena associated with rapid phase variations near critical angles are accurately described. The principal restriction is that the solid is sufficiently thick that internal reflections may be ignored.     

NATURAL VIBRATIONS OF LAMINATED COMPOSITE BEAMS BY USING MIXED FINITE ELEMENT MODELLING

A six-node, plane-stress mixed finite element model has been developed by using Hamilton's energy principle for the natural vibrations of laminated composite beams. Continuity of the transverse stress and displacement fields has been enforced through the thickness of the laminated beam in the formulation for proper modelling. The transverse stress components have been invoked as the nodal degrees of freedom by applying elasticity relations. Natural frequencies of laminated composite beams obtained through the present formulation have been shown to be in good agreement with the data available in the literature. Various mode shapes have also been presented as benchmark solutions.     

Real-time diffraction gratings in DDI-methanol solutions

Nondegenerate four-wave mixing has been performed in DDI-methanol solutions using ruby laser as writing beams. The reading beam was either a pulsed one at 538.8 nm or a continuous one at 632.8 nm. Contributions from both saturation and thermal gratings were separated temporally and evaluated.     

Elegant Inceben Gaussian beams in a quadratic-index medium

Elegant Ince—Gaussian beams, which are the exact solutions of the paraxial wave equation in a quadratic-index medium, are derived in elliptical coordinates. These kinds of beams are the alternative form of standard Ince—Gaussian beams and they display better symmetry between the Ince-polynomials and the Gaussian function in mathematics. The transverse intensity distribution and the phase of the elegant Ince—Gaussian beams are discussed.     

Nonparaxial elegant Laguerre-Gaussian beams

Closed-form nonparaxial expressions for optical beams are useful to calculate the fields produced by tightly focused laser beams. Such expressions for elegant Laguerre-Gaussian (eLG) beams that are exact solutions of the Helmholtz equation are introduced. These solutions are expressed as linear combinations of a finite number of analytic functions that involve spherical Bessel functions and associated Legendre functions of complex arguments. In the paraxial limit, the expressions proposed have the property to reduce to the well-known eLG beams.     

基于焦散线方法的自加速光束设计

以艾里光束为代表的自加速光束是一类在自由空间中具有弯曲传播特性的新型特殊光束.这类光束因其具有无衍射、自加速和自修复等奇异特性引起了人们的广泛关注,有望应用于光学微粒操控、激光微加工、全光路由和超分辨成像等诸多领域.由于艾里光束只能沿着抛物线的轨迹传播,限制了其在实际应用中的灵活性,因而设计出能够沿着不同轨迹传播的自加速光束是这一研究领域的关键问题,而基于焦散线方法的自加速光束设计是解决该问题的有效途径之一.这一方法是将设计的传播轨迹与光学焦散线联系起来,通过分析形成该焦散线所需的光线簇构造出对应的初始场分布.基于该原理并经过不断发展,不同类型的自加速光束相继得以实现,并且借助维格纳函数还可以同时实现实空间和傅里叶空间的自加速光束设计,为自加速光束的应用提供了更多的可能性.本文对基于焦散线方法的自加速光束设计原理和进展进行全面介绍.     

A Formula of Propagating Beams Driven by Few-Cycle Gaussian Pulse in Dispersive Media

A formula is developed to describe the propagation of beams driven by few-cycle Gaussian pulse in a media with group velocity dispersion （GVD）. With the method, the spatiotemporal evolution of the pulsed beam can be straightforwardly quantified as long as the monochromatic beam solutions in free space, which have been widely investigated in previous works, are known. The method makes it possible to analytically deal with the few-cycle pulsed beams with transverse profiles other than the Gaussian one, which is, to our knowledge, the one mainly investigated previously, in GVD media.     

Third-Order Nonlinear Integrated Device Based on an Effective Graded-Index Waveguide for All-Optical Multistability

We present a theoretical study of all-optical multistability in an integrated device composed by a step-index nonlinear Kerr waveguide with a variable width film, providing an effective graded-index profile. The self- and cross-action of nonlinear Gaussian counterpropagating beams in this structure and a feedback controlling mechanism based on transverse modal coupling generate a multistable behavior. The results have been obtained and explained on the basis of the exact solutions of the variational problem for the counterpropagating Gaussian beams. Finally, in order to make clear the possibilities of this device, novel applications of the all-optical multistable integrated device are proposed for smart sensing.     

Transformation of nonlinear sawtooth waves in a medium with refractive inhomogeneities

Theexact solutions of intensive sawtooth waves and beams in a medium with a sound speed inhomogeneity have been found. It has been shown that the focusing on the medium inhomogeneity is strongly pronounced for a single pulse, resulting in the formation of narrow high-amplitude areas. It has been determined that the refractive infomogeneity can lead to a shift of the amplitude maximum of the initial focused beam and the transformation of the converging beam into a dispersing one.     

Nonparaxial vortex vector solitons in a nonlinear cubic medium

The propagation of narrow three-dimensional wave beams in a nonlinear cubic medium has been analyzed taking into account all vector components of the wave field, including radial, angular, and longitudinal ones. Soliton solutions to the Maxwell equations in the form of vortex solitons are found, and their properties are investigated.     

Third-Order Nonlinear Integrated Device Based on an Effective Graded-Index Waveguide for All-Optical Multistability

We present a theoretical study of all-optical multistability in an integrated device composed by a step-index nonlinear Kerr waveguide with a variable width film, providing an effective graded-index profile. The self- and cross-action of nonlinear Gaussian counterpropagating beams in this structure and a feedback controlling mechanism based on transverse modal coupling generate a multistable behavior. The results have been obtained and explained on the basis of the exact solutions of the variational problem for the counterpropagating Gaussian beams. Finally, in order to make clear the possibilities of this device, novel applications of the all-optical multistable integrated device are proposed for smart sensing.     

Exact nonparaxial beams of the scalar Helmholtz equation

It is shown that three-dimensional nonparaxial beams are described by the oblate spheroidal exact solutions of the Helmholtz equation. For what is believed to be the first time, their beam behavior is investigated and their corresponding parameters are defined. Using the fact that the beam width of the family of paraxial Gaussian beams is described by a hyperbola, we formally establish the connection between the physical parameters of nonparaxial spheroidal beam solutions and those of paraxial beams. These results are also helpful for investigating exact vector nonparaxial beams.     

Volkov Solution for Two Laser Beams and ITER

We find the solutions of the Dirac equation for two plane waves (laser beams) and we determine the modified Compton formula for the scattering of two photons on an electron. The practical meaning of the two laser beams is, that two laser beams impinging on a target which is constituted from material in the form a foam, can replace 100–200 laser beams impinging on a normal target and it means that the nuclear fusion with two laser beams is realistic in combination with the thermonuclear reactor such as ITER.     

Self-action of light beams in nonlinear media: soliton solutions

The behaviour of light beams in nonlinear media (self-focussing of optical beams as well as self-modulation and transmission of optical pulses) is considered in the case when the nonlinear polarization contains susceptibilities of third- and fifth-order. The soliton solutions are obtained for the nonlinear equations deduced.