Rectangular hard-edged aperture diffracted Laguerre–Gaussian beams

Based on the relations between Laguerre–Gaussian (LG) and Hermite–Gaussian (HG) modes and by introduced the complex Gaussian expansion method for two dimensional rectangular aperture, the approximate analytical propagation expressions of the rotational symmetrical LG beams along with their even and odd modes through a paraxial ABCD optical system with rectangular hard-edged aperture are derived. As special cases of the results, the corresponding closed-forms of the circular aperture diffracted LG beams and non-truncated LG beams are also given. Numerical examples are given to prove the validity of this approximate analytical method and illustrate the propagation properties of the rectangular hard-edged aperture diffracted LG beams.     

Studies of propagation characteristics of elliptical Gaussian beams through circular apertured optical systems

An approximate analytical propagation formula of the elliptical Gaussian beams through a paraxial ABCD optical system with a circular aperture is obtained on the basis of the expansion of a hard-edged aperture into a finite sum of complex Gaussian functions. And the parameter characteristics of the truncated beams including the beam propagation factor and the kurtosis parameter are also studied in detail by using the second-order-moments method.     

Propagation of elliptical cosh-Gaussian beams and off-axial elliptical cosh-Gaussian beams in apertured optical systems

Based on the fact that a hard-edged elliptical aperture can be expanded approximately as a finite sum of complex Gaussian functions in tensor form, an analytical propagation expression for an elliptical cosh-Gaussian beam (EChGB) truncated by an elliptical aperture and passing through an axially nonsymmetrical optical system is obtained by use of vector integration. Furthermore, the analytical propagation expression for a truncated off-axial EChGB is derived as an application example. Some numerical simulations are given to illustrate the propagation properties of EChGBs and off-axial EChGBs passing through a free space with an elliptical aperture. PACS 41.85.Ew; 42.60.Jf; 42.25.Bs     

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.     

Cross-spectral density matrix of a random electromagnetic beam propagating through an apertured axially nonsymmetrical optical system

Using the fact that a hard-edged rectangular aperture function can be approximated by a two-dimensional multi-Gaussian series, analytic formulas for the elements of the cross-spectral density matrix of a random electromagnetic beam truncated by a rectangular aperture and passing through an axially nonsymmetrical optical system is derived. The analysis is illustrated by numerical examples relating to changes in the spectral degree of polarization of electromagnetic Gaussian Schell-model beams propagating through free space, focal system and dual-focus system. In particular, the effect of the size of the hard aperture on the spectral degree of polarization of the beam along its optical axis is discussed.     

Propagation of the Hermite–Gaussian beams through misaligned optical system with a circular aperture

By means of expanding a hard-edged aperture into a finite sum of complex Gaussian functions, the approximate analytical formula of one kind of higher-order Gaussian beams called the Hermite–Gaussian beams (HGBs) passing through circular apertured and misaligned optical system is obtained in this paper. The result provides more convenience for studying its propagation than the usual way by using diffraction integral directly. Some numerical simulations are also given for illustrating the propagation properties of the HGBs through the circular apertured optical systems.     

部分相干Airy光束在湍流大气中传输时的偏振特性

研究了部分相干Airy光束在湍流大气中传输时的偏振特性,偏振保持度作为衡量偏振传输效果的一个重要参数.结果表明:部分相干Airy光束在湍流大气中传输足够远时,其偏振度会变回到初始值;而在自由空间中传输,光束的偏振度会保持在某一个特定值;在湍流大气中,当光束传输距离不是很远时,光束对称轴上的偏振度分布为Airy函数,但是当传输足够远时,该偏振度分布逐渐趋向于类高斯状;光束的束腰半径越大,相干长度越长,越有利于光束传输后偏振的保持;存在一个指数截断因子,使得光束的偏振保持度很差.这些结论对于Airy光束在通信领域中的应用具有重要的意义.     

M~2-factor of high-power laser beams through a multi-apertured ABCD optical system

Based on the generalized truncated second-order moments, an approximate analytical formula of the beam propagation factor M~2 of high-power laser beams passing through the optical system with multiple hard-edged apertures is deduced. Numerical examples of the beams passing through an aperture-spatial filter are enclosed, and the influences of amplitude modulations(AMs) and phase fluctuations(PFs) on the beam propagation quality of high-power laser beams passing through the multi-apertured ABCD optical system are considered and discussed. It is shown that PFs are able to degrade the beam propagation quality of laser beams more than AMs when the high-power laser beams passing through the aperture-spatial filter, furthermore, one or two aperture-lens optical systems configured appropriate aperture parameters are both able to upgrade the beam propagation quality of high-power laser beams. The M~2 factor of Gaussian beam passing through the multi-aperture optical system is a special case in this paper.     

光束通过硬边光阑光学系统变换的改进算法

 使用傅里叶级数与有限个复高斯函数之和的乘积作为硬边光阑的窗口函数的近似式，提出了含硬边光阑近轴ABCD光学系统对光束变换的改进算法。以高斯光束的变换为例，推导出了一个表示为初等函数之和的近似解析传输公式。给出了数值计算结果，并与直接积分Collins公式和使用Wen方法的结果作了比较。研究表明，改进算法能兼顾计算精度和机时，并能用于远场和近场。而使用Wen方法时，在近场（小于0.12倍菲涅尔距离）有明显误差。     

Matrix formulation of axis-symmetric laser beams through a paraxial ABCD system containing hard-edged apertures: A comparative study

A matrix formulation is presented, which enables us to study the propagation of axis-symmetric beams through a paraxial optical ABCD system containing hard-edged aperture. Numerical calculation results of super-Gaussian beams passing through a multi-aperture-lens system are given to illustrate the advantage of the method. A comparison of the matrix formulation, complex Gaussian expansion and direct numerical integration of the Collins formula is made, where the propagation of apertured Laguerre-Gaussian beams is chosen as an illustrative example. It is shown that the matrix formulation provides satisfactory results in both Fraunhofer and Fresnel regions, and reduces the computational time greatly in comparison with the direct integration. However, this method is suited only to axis-symmetric optical beams and systems. By using the complex Gaussian expansion discrepancies exist in the near zone closer to the aperture, but usually its computational efficiency is higher than the matrix formulation.     

多艾里光束合成自聚焦光束的实验实现

多光束合成和单光束聚焦一直是提高激光束功率和功率密度的两个重要方法. 结合艾里光束在自由空间中沿弯曲路径传播的特性, 从数值模拟和实验两个方面, 研究了利用多个一维艾里光束合成自由空间自聚焦光束的方法, 并对所得到的模拟和实验结果进行了对比. 采用分步束传播法, 分别模拟了由四个和八个一维艾里光束合成的自聚焦光束在自由空间中的传播过程, 给出了自聚焦光束在传播过程中横向和纵向的光强分布和变化趋势. 采用计算全息和空间光调制器技术实验, 实现了多个一维艾里光束合成的自聚焦光束. 实验中分别测量了四个和八个一维艾里光束合成的自聚焦光束的横向光强分布. 实验结果和理论结果符合得较好. 另外, 为了进一步增大自聚焦光束的功率, 可以增加参与合成的一维艾里光束的数量. 同时, 自聚焦光束的焦距可以通过调整各个一维艾里光束的相对位置进行调节.     

Propagation of truncated modified Laguerre–Gaussian beams

By expanding the circ function into a finite sum of complex Gaussian functions and applying the Collins formula, the propagation of hard-edge diffracted modified Laguerre–Gaussian beams (MLGBs) through a paraxial ABCD system is studied, and the approximate closed-form propagation expression of hard-edge diffracted MLGBs is obtained. The transverse intensity distribution of the MLGB carrying finite power can be characterized by a single bright and symmetric ring during propagation when the aperture radius is very large. Starting from the definition of the generalized truncated second-order moments, the beam quality factor of MLGBs through a hard-edged circular aperture is investigated in a cylindrical coordinate system, which turns out to be dependent on the truncated radius and the beam orders.     

部分相干平顶光束经光阑衍射的偏振特性

根据光束的相干-偏振矩阵和传输理论,对部分相干平顶光束经正多边形光阑衍射的偏振特性进行了系统的研究.给出了部分相干平顶光束偏振度传输公式,并将高斯-谢尔模型光束以及部分相干平顶光束在自由空间传输的偏振度作为特例统一于一般表达式中.研究表明：部分相干平顶光束经光阑衍射的偏振特性与光阑截断参数、光束的空间相干性、衍射角、传输距离、平顶光束的阶数有关. 关键词： 部分相干平顶光束 偏振特性 相干-偏振矩阵 正多边形光阑 光阑衍射     

部分相干平顶光束经光阑衍射的偏振特性

根据光束的相干-偏振矩阵和传输理论，对部分相干平顶光束经正多边形光阑衍射的偏振特性进行了系统的研究.给出了部分相干平顶光束偏振度传输公式，并将高斯-谢尔模型光束以及部分相干平顶光束在自由空间传输的偏振度作为特例统一于一般表达式中.研究表明：部分相干平顶光束经光阑衍射的偏振特性与光阑截断参数、光束的空间相干性、衍射角、传输距离、平顶光束的阶数有关.     

Approximate analytical representation of Wigner distribution function for Gaussian beams passing through ABCD optical systems with hard aperture

Based on the treatment that a rectangular function can be expanded as an approximate sum of complex Gaussian functions with finite numbers, the analytical expression of the Wigner distribution function for a Gaussian beam passing through a paraxial ABCD optical system with hard-edged aperture is obtained. By numerical simulation, it is shown that the effect of the aperture on the Wigner distribution function is prominent. By comparing the analytical results with the numerical integral results, it is shown that this method of expanding hard-edged aperture into Gaussian functions with finite numbers is proper and ascendant. This method could be extended to study the Wigner distribution functions of other light beams passing through a paraxial ABCD optical system with hard-edged aperture.     

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.     

Comparison of two approximate methods for hard-edged diffracted flat-topped light beams

The integral resulted in an infinite series of Bessel functions and expanding a hard aperture into a complex-Gaussians shape are proposed as two methods for studying the propagation properties of the hard-edged diffraction flat-topped light beam. Using the two methods, the corresponding analytical propagation equations of flat-topped light beams through a circular apertured ABCD optical system are obtained. Some numerical calculations and comparative analyses by using the two methods and the diffraction integral formulae are made. It is shown that the first method of an infinite series of Bessel functions is superior to the second of expanding a hard aperture function into a complex-Gaussians shape at the aspect of calculation accuracy, but the second method is superior to the first method at the aspect of the improvement in the calculation efficiency.     

Self‐Imaging Effect Based on Circular Airy Beams

The self‐imaging effect for a superposition of the fundamental circular Airy beams with successively changing radial displacements is investigated. In free space, this self‐imaging along a parabolic trajectory can persist before the focal point. In the linear index potentials with a gradient, the propagation trajectory of circular Airy–Talbot effect can follow some predefined trajectories determined by engineering the index gradient, and the self‐imaging extent can expand provided that each truncated constituent circular Airy beam maintains its form and continues to accelerate before dispersing strongly.     

高斯光束通过有硬边光阑ABCD光学系统的近似解析传输公式

 对高斯光束通过有硬边光阑近轴ABCD光学系统的传输进行了研究,采用将矩形函数展开为复高斯函数的技巧,推导出了近似的解析传输公式。以高斯光束在有光阑情况下自由空间的传输为例进行了数值计算,对所用方法能节约大量机时的优点和计算误差作了讨论,并进一步推广用于平面波和平顶高斯光束,证实了这一方法具有广泛的应用意义。     

Focal shift of focused truncated random electromagnetic beams

The focal shifts of focused truncated random electromagnetic beams are investigated. Based on the complex Gaussian expansion method for a hard-edged aperture function, the analytical propagation formula of cross-spectral density matrix for a random electromagnetic beam focused by an optical system with a thin lens and a circular aperture is derived. The Fresnel numbers related with the beam and system parameters are defined and used to examine focal shifts. The dependence of the focal shifts on the different Fresnel numbers and polarization distribution are discussed in detail with numerical examples.