Vectorial Hermite--Laguerre--Gaussian beams beyond the paraxial approximation

Starting from the vectorial Rayleigh--Sommerfeld integrals, the free-space propagation expressions for vectorial Hermite--Laguerre--Gaussian (HLG) beams beyond the paraxial approximation are derived. The far-field expressions and the scalar paraxial results are given as special cases of our general expressions. The intensity distributions of vectorial nonparaxial HLG beams are studied and illustrated with numerical examples.     

Vectorial partially coherent flat-topped beams and their nonparaxial propagation

The concept of vectorial partially coherent flat-topped beams is proposed, and their nonparaxial propagation in free space is studied in detail. Based on the vector Rayleigh diffraction integral formulae, the analytical propagation expressions for the cross-spectral density matrix of the polarized vectorial nonparaxial partially coherent flat-topped beams are derived and applied to study the nonparaxial propagation properties of vectorial partially coherent flat-topped beams. The effect of propagation parameters on the intensity and the coherence property of the nonparaxial vectorial partially coherent flat-topped beam is illustrated and analyzed comparatively with the corresponding paraxial results.     

矢量非傍轴双曲余弦-高斯光束

 引入了矢量非傍轴双曲余弦-高斯(ChG)光束概念。使用矢量瑞利-索末菲衍射积分公式推导出了矢量非傍轴ChG光束在自由空间传输的解析公式。矢量非傍轴ChG光束轴上和远场的解析式以及傍轴结果作为一般传输公式的特例给出。研究表明,对矢量非傍轴ChG光束,其非傍轴性主要由f参数决定,但偏心参数会影响其横向光强剖面形状和非傍轴行为。     

Propagation properties of high order axially symmetric polarized beams

The paraxial and non-paraxial propagation properties of high order axially symmetric polarized beams are theoretically studied. The mathematical expressions of the propagation fields for such beams are derived based on vectorial Rayleigh–Sommerfeld formulas. Simulation results are presented for different dimensionless perturbation parameters, which shows the beam propagation intensity distribution and evolution behavior are affected largely by the polarization property of such beams, especially in the non-paraxial regime.     

矢量非傍轴离轴高斯光束的传输

基于矢量瑞利-索末菲衍射积分公式，得出了波动方程的一个解，它代表矢量非傍轴离轴高斯光束，其在自由空间的传输方程表示为解析的结果.矢量非傍轴离轴高斯光束的轴上和远场公式，矢量非傍轴高斯光束的传输方程，以及傍轴的结果都可作为一般表达式的特例而得出.研究表明，f参数对光束的非傍轴特性有重要影响，而离心参数也影响非傍轴行为.与共轴情况不同的是，对离轴情况，在y方向存在场的纵向分量. 关键词： 激光光学 矢量非傍轴离轴高斯光束 瑞利-索末菲衍射积分 f参数 离轴参数     

Propagation properties of vectorial elliptical Gaussian beams beyond the paraxial approximation

Starting from the vectorial Rayleigh diffraction integral formula and without using the far-field approximation, a solution of the wave equation beyond the paraxial approximation is found, which represents vectorial non-paraxial elliptical Gaussian beams in free space. The far-field expressions for non-paraxial Gaussian beams and elliptical Gaussian beams can be regarded as special cases treated in this paper. Some basic propagation properties of vectorial non-paraxial elliptical Gaussian beams, including the irradiance distribution, phase term, beam widths and divergence angles are studied. Numerical results are given and illustrated.     

非傍轴矢量光束二阶矩的发散特性

基于非傍轴矢量光束横截面上光强的精确表示和矢量二阶矩理论,应用严格的角谱表示的矢量衍射理论,对任意非傍轴矢量衍射光束的传输进行了系统的研究.给出了光束的束宽、横向发散角的积分表达式,论证了非傍轴矢量光束二阶矩的发散特性.对非傍轴矢量光束的束宽、发散角的定义及一些相关的问题进行了分析和讨论. 关键词： 矢量衍射理论 光束传输 矢量二阶矩 发散特性     

Partially coherent non-paraxial vectorial Hermite-Gaussian beams and their far-field properities

The far-field theory of partially coherent vectorial non-paraxial beams is expanded and their analytical propagation expressions of the Wigner distribution function matrices and the cross-spectral-density matrices in free space are derived using the far-field approximation. Some interesting cases, in particular the vectorial non-paraxial partially coherent Hermite-Gaussian (HG) beams, are discussed and treated as special cases of our general expressions. It is shown that the f_σ and f parameters play an important role in determining vector and non-paraxiality of partially coherent HG beams. When two parameters are small enough, scalar and paraxial vectorial approximation is allowed; otherwise, non-paraxial vectorial approximation is applied. But the decided parameters additionally affect their far-field divergence angles.     

非傍轴矢量异常空心光束的传输特性

 将Cai提出的异常空心光束的理论模型推广到非傍轴范畴,推导出非傍轴矢量异常空心光束传输的解析表达式,用以研究它在自由空间中的传输特性。研究表明,异常空心光束在传输过程中光束形状会发生变化。与高斯光束不同,非傍轴异常空心光束傍轴近似成立条件依赖于传输距离,这与异常空心光束光强分布随传输距离的变化有关。非傍轴异常空心光束远场的光束质量可用桶中功率来描述,桶中功率随f参数(波长与束腰宽度的比值)的增大而减小。     

Propagation of nonparaxial vector hollow Gaussian beams through a circular aperture

Based on the vectorial Rayleith-Sommerfeld formulae, the nonparaxial propagation properties of the vector hollow Gaussian beams (HGBs) through a circular aperture are studied in detail. We describe the derivation of the integral expressions of the propagation of nonparaxial vector HGBs through a circular aperture. The derived expression is independent the approximation of paraxial and far field, which are valid for either far and near field and for the systems in which aperture radius is comparable to or even smaller than wavelength. And it is also strict integral formula for the light field on the axis. Numerical calculation results indicate that there is no difference between derived formulae and the Collins formulae in the situation of paraxial approximation. Using the formula deduced, we calculate the propagation properties of HGBs. The calculated results indicate that the propagation field of vector hollow Gaussian beams is asymmetric in near field, while the propagation field is symmetric in far field. These research results could well shed light on the further understanding of the vectorial property of HGBs through a circular aperture, and would play a guiding role in the practical application of HGBs.     

用傅里叶分析法研究艾里光束远场传播特性

为了从理论上深入分析新型无衍射光束艾里光束在有限能量条件下的远场传播特性,首先,从决定光波在自由空间传播的一维旁轴波动方程入手,采用傅里叶分析法,结合艾里函数的特殊性质,并利用经过指数衰减的有限能量初始条件,完整给出了有限能量条件下用于精确描述一维艾里光束在自由空间传播特性的波动方程解析解.然后,利用所得到解析解分别对一维和二维艾里光束在自由空间的传播特性进行了研究,重点分析了不同参量条件对艾里光束进行无衍射传播和横向自加速的影响.研究表明:当任意横向尺度为100μm,衰减系数为0.03、0.05、0.07、0.1、0.2时,二维艾里光束无衍射传播距离分别为1 014、624、455、338、193mm;当横向尺度保持不变时,衰减系数越小,艾里光束保持无衍射传播的距离越大;当衰减系数保持不变时,横向尺度越小,艾里光束横向自加速越大.所采用的研究方法也可用于研究艾里光束在介质中的传播特性.     

Accelerating parabolic beams

We demonstrate the existence of accelerating parabolic beams that constitute, together with the Airy beams, the only orthogonal and complete families of solutions of the two-dimensional paraxial wave equation that exhibit the unusual ability to remain diffraction-free and freely accelerate during propagation. Since the accelerating parabolic beams, like the Airy beams, carry infinite energy, we present exact finite-energy accelerating parabolic beams that still retain their unusual features over several diffraction lengths.     

Nonparaxial propagation of vectorial Gaussian beams diffracted at a circular aperture

Based on the vectorial Rayleigh diffraction integral, the nonparaxial propagation of vectorial Gaussian beams diffracted at a circular aperture is studied. The far-field and paraxial cases are treated as special cases of our general result. It is shown that for the apertured case the f parameter still plays an important role in determining the nonparaxiality of vectorial diffracted Gaussian beams, but both the f parameter and truncation affect the beam evolution behavior.     

Phase-flipped Hermite-Gaussian beams and their propagation beyond the paraxial approximation

An extension of phase-flipped Gaussian (PFG) beams to the multi-mode nonparaxial case is made. The recurrence propagation expressions for phase-flipped Hermite-Gaussian (PFHG) beams beyond the paraxial approximation are derived and used to study nonparaxial propagation properties of PFHG beams in free space and through a knife edge and an aperture, and to compare nonparaxial results with paraxial ones. The propagation of paraxial PFHG beams and PFG beams and nonparaxial PFG beams is treated as special cases of nonparaxial PFHG beams. Numerical examples are given to illustrate the theoretical results.     

Propagation of vectorial apertured cosh-Gaussian beams beyond the paraxial approximation

By using the vectorial Rayleigh-Sommerfeld diffraction integrals, the propagation equation of vectorial nonparaxial cosh-Gaussian (ChG) beams in the presence of an aperture is derived and expressed in a closed form. The on-axis, far-field and paraxial cases are studied as special cases. Analytical results are illustrated with numerical examples.     

Application of the Wigner distribution function to complex-argument Hermite– and Laguerre–Gaussian beams beyond the paraxial approximation

The Wigner distribution function (WDF) is applied to study the propagation of complex-argument Hermite–Gaussian (HG) and Laguerre–Gaussian (LG) beams beyond the paraxial approximation. The analytical expressions for their intensity distributions in free-space propagation are derived, which are expressed in terms of Hermite polynomials for nonparaxial complex-argument HG beams and in terms of the sum of finite Hermite polynomials for nonparaxial complex-argument LG beams. A detailed comparison of the WDF approach, series expansion method and paraxial expressions is made, which shows that in the paraxial regime the WDF approach and series expansion method deliver consistent results with that of paraxial expressions. Beyond the paraxial approximation, the WDF approach offers convergent results, whereas the series expansion method has a limited applicable range, within which it gives consistent results with that of WDF approach but beyond which it gives unrealistic and divergent results.     

Nonparaxial propagation of phase-flipped Gaussian beams

This paper derives the closed-form expressions for nonparaxial phase flipped Gaussian （PFG） beams propagating in free space, through a knife edge and an aperture, which enable us to study nonparaxial propagation properties of PFG beams and to compare nonparaxial results with paraxial ones. It is found that the f parameter, offsetting distance of the knife edge and truncation parameter affect the nonparaxial beam propagation properties. Only under certain conditions the paraxial approximation is applicable. The results are illustrated by numerical examples.     

The fractional Fourier transform of Airy beams using Lohmann and quadratic optical systems

Based on the generalized Huygens–Fresnel integral, the analytical expressions for Airy beams through three types of fractional Fourier transform (FRFT) optical systems (Lohmann I, Lohmann II and quadratic graded-index systems) are derived. Using the two analytical expressions, the propagation properties of Airy beams for the three types of FRFT optical systems are discussed in detail with numerical examples. Results indicate that the intensity of Airy beams periodically changes and remains form-invariant with the changing of fractional order p for the three types of FRFT optical systems.     

Propagation properties of three-dimensional flattened Gaussian beams in uniaxially anisotropic crystals

Starting from the paraxial vectorial theory of beams propagating in uniaxially anisotropic media, the analytical propagation equations of three-dimensional flattened Gaussian beams (FGBs) in uniaxial crystals are derived and illustrated numerically. It is shown that, due to the anisotropy of crystals, the ordinary and extraordinary beams originated by incident FGBs propagate with different Rayleigh lengths, thus the linear polarization state and axial symmetry of incident FGBs do not remain during propagation in crystals.     

非傍轴矢量高斯光束的光强表示

 基于瑞利-索末菲衍射积分，未使用任何近似，对非傍轴矢量光束的两种光强表示式，即传统光强公式和时间平均坡印廷矢量的z分量进行了研究。对非傍轴矢量高斯光束详细数值计算结果的比较表明，两种表示式之间的差异，即两者的相对误差与束腰宽度及传输距离和波长的比值有关。对非傍轴矢量高斯光束，若传输距离与波长的比值为10，束腰宽度与波长的比值大于等于0.8，则最大相对误差不到1.5%。因此，传统光强公式是可用的。