Focusing of a vortex carrying beam with Gaussian background by a lens in the presence of spherical aberration and defocusing

Diffraction pattern formed by a lens for a vortex containing truncated beam with Gaussian background, and in the presence of spherical aberration and defocusing has been studied by Fresnel–Kirchhoff diffraction integral. For the study, two different values of topological charge are selected. Compensation of aberration in the presence of appropriate value of defocusing is investigated. Presence of spherical aberration results in an increase in the size of the dark core of the diffraction pattern. Results are also presented for the encircled energy. Some results are also presented for the influence of truncation parameter of the beam, on the point spread function at different observation planes.     

Intensity distribution properties of Gaussian vortex beam propagation in atmospheric turbulence

By using wave optics numerical simulation, the intensity-hole effect, beam spreading and wandering properties of Gaussian vortex beam propagation in atmospheric turbulence are investigated quantitatively. It is found that an intensity hole in the center of the beam pattern appears gradually as a Gaussian vortex beam propagates. The size of the intensity hole increases with the increase of the topological charge of the vortex phase. However, the intensity hole could to some extent be filled with optical energy by atmospheric turbulence, especially in strong turbulence. The radius of the intensity hole first decreases and then increases with the growth of turbulence strength. The effective radius of vortex beam with larger topological charge is greater than with a smaller topological charge. But the topological charge has no evident influence on beam wandering.     

涡旋光束在湍流大气中的传输特性

根据广义的惠更斯-菲涅耳原理, 研究了涡旋光束在湍流大气中的传输特性。研究结果表明, 涡旋光束在湍流大气中传输时, 截面光强会从空心分布转化为高斯分布。光束所带的拓扑电荷数以及大气湍流均会影响光强分布的变化。研究结果还表明, 涡旋光束能够抑制大气湍流对光束扩展的影响, 这一现象得到了实验上的证实。通过杨氏双缝干涉的方法, 还研究了涡旋光束经过湍流大气传输后的拓扑电荷数。研究发现, 涡旋光束经过湍流大气后, 拓扑电荷数将发生波动。     

高斯涡旋光束的光束传输因子和峭度参数

基于强度二阶矩定义, 导出了高斯涡旋光束光束传输因子即M² 因子的解析表达式, 高斯涡旋光束的M² 因子唯一取决于拓扑电荷数n. 数值计算表明, 高斯涡旋光束的M² 因子随着拓扑电荷数n的增大而增大. 基于强度高阶矩, 还导出了高斯涡旋光束经傍轴ABCD光学系统传输时峭度参数的解析表达式, 高斯涡旋光束的峭度参数取决于拓扑电荷数n、参数δ、矩阵元A和矩阵元D. 在自由空间传输时, 高斯涡旋光束的峭度参数仅取决于拓扑电荷数n和参数δ. 自由空间传输时, 高斯涡旋光束峭度参数的变化规律为: 峭度参数随参数δ的增大先减小而后趋向于一最小值, 随拓扑电荷数n的增大而减小. 这一研究有助于高斯涡旋光束的实际应用.     

Propagation of Gaussian background vortex beams diffracted at a half-plane screen

Taking the Gaussian background vortex beam with topological charge +2 as a typical example, a closed-form expression for vortex Gaussian beams passing through a half-plane screen is derived and used to study the propagation dynamics of on-axis and off-axis vortex diffracted beams, and to compare with the case of the free-space propagation. It is shown that there may exist many phase singularities or no phase singularity of vortex diffracted Gaussian beams in the diffraction field. Number and position of phase singularities are dependent on the vortex position at the source plane and propagation distance. The creation, motion and annihilation of phase singularities in the diffraction field may appear by varying the vortex position and propagation distance.     

拉盖尔高斯涡旋光束的传输

针对拉盖尔高斯涡旋光束,推导了其传输后目标平面上光电场的解析表达式,理论研究表明,传输一段距离后, 对于拉盖尔高斯光束的光斑大小的描述,高斯光斑尺寸已经不再适用.如果采用光强最亮处的半径来表示目标平面上的光斑大小则比较方便. 除了传输中的衍射导致光束展宽以外, 横截面上光束的相位分布也发生了独特的变化. 等相位线由原来的射线转化为弧线,拓扑电荷数为正时,弧线朝顺时针方向弯曲,拓扑电荷数为负时,弧线朝逆时针方向弯曲. 关键词： 涡旋光束 传输 光斑尺寸 相位分布     

Detecting the topological charge of vortex beams using an annular triangle aperture

The Fraunhofer diffraction pattern of a vortex beam by an annular triangle aperture is analyzed theoretically and experimentally. It is found that the pattern of the far-field diffraction intensity distribution exhibits a triangular lattice array, which becomes much clearer with the increase of the ratio of the inner to the outer side of the annular triangle aperture. The number of spot points of any external side of the triangular lattice array minus one is just equal to the topological charge value of the measured optical vortex. For the vortex beam with negative topological charge, the triangular diffraction pattern after the annular triangle aperture will be rotated by 180° in relation to the case of the positive topological charge. Based on the above properties, we propose a simple and feasible method to determine the magnitude and sign of the topological charge of an optical vortex beam.     

Radially polarized hollow Gaussian beam with on-axis spiral optical vortex

The focusing properties of radially polarized hollow Gaussian beam (HGB) with on-axis spiral optical vortex are investigated theoretically by vector diffraction theory. The phase wavefront of HGB is the function of radial coordinate. Calculation results show that the focusing properties can be altered considerably by beam order of HGB, topological charge of the on-axis optical vortex, and phase parameter that characterizes the radial phase wavefront distribution. Higher topological charge induces focal evolution from one focal spot to annular focal pattern in transverse direction, while phase parameter can lead to focal shift along optical axis remarkably. In addition, focal shift direction can also be adjusted by changing varying direction of phase parameter.     

Propagation dynamics of optical vortices in Laguerre–Gaussian beams

We have calculated the propagation dynamics of an initial off-axis vortex with topological charge 1 in Laguerre–Gaussian background beams , which are examples of background beams with non-generic dislocation surfaces, on which the real and imaginary parts of the light field are zero. When initially a vortex with broad core (e.g., r-vortex) is embedded in the background beam, the dislocation surfaces are destroyed during propagation and two vortices with opposite charge are created per dislocation surface in planes perpendicular to the propagation direction. For a vortex with narrow core (e.g., point vortex) diffraction is important and leads to the birth of more than two vortices per dislocation surface. These results are also valid for other background beams with dislocation surfaces, e.g., Hermite–Gaussian and Ince–Gaussian beams. We investigated experimentally the spatial evolution of the intensity distribution of an initial off-axis vortex with narrow core and topological charge 1 in background beams. The experimental results are in good agreement with the calculated intensity distributions.     

Structure of a tightly focused vortex beam in the presence of primary coma

Structure of a tightly focused circularly polarized vortex beam in the presence of primary coma has been studied by using Debye-Wolf diffraction integral. The role of topological charge and handedness of the polarization on the focused structure has been discussed. Results have been presented for the total intensity and squares of the polarization components for left-, and right circularly polarized vortex beams. Impact of coma, on the dark core of an azimuthally polarized non-vortex beam is also investigated and compared with the dark core of a circularly polarized vortex beam. The presence of comatic aberration in the focusing system results in a positional shift of the high intensity lobes, and reduction of the intensity on one side of the center. Effect of coma on the focused structure has also been discussed in the context of STED microscopy.     

涡旋光束的产生与干涉

分别从理论上和实验上研究了涡旋光束的产生和干涉现象.理论上分析了分数阶和整数阶涡旋光束同球面波以及平面波的干涉情况,并从实验上得出了其干涉图形.实验结果和理论模拟基本上一致.研究表明,随着涡旋光束拓扑荷数的变化,干涉图形也会产生变化.这一现象可用于测定分数阶涡旋光束的拓扑荷数.     

椭圆涡旋光束在海洋湍流中的传输特性

本文采用分步相位屏方法来仿真椭圆涡旋光束在海洋中的实际传输情况,并对椭圆涡旋光束在海洋湍流中的传输光强和闪烁因子进行了仿真。研究发现,椭圆涡旋光束在海洋传输过程中,光斑会发生明显的旋转,同时光斑会产生暗核且暗核个数与光束的拓扑荷数相等。一个拓扑荷数为m的相位奇点会分裂成m个拓扑荷数为1的相位奇点,并且海洋湍流越强,光斑受到的干扰越严重。研究还发现,在较弱的海洋湍流中,随着传输距离的增加,椭圆涡旋光束的闪烁因子会低于高斯光束和涡旋光束的闪烁因子,而且在远距离处拓扑荷数越大闪烁因子降低越明显,同时也发现,传播一段距离后涡旋光束的闪烁因子会低于高斯光束的闪烁因子。在较强湍流中,椭圆涡旋光束的闪烁因子会交叠在一起。对于不同强度的海洋湍流,随着均方温度耗散率的增大,椭圆涡旋光束的轴上点闪烁因子也增大。在同一传输距离处,束腰宽度越小的椭圆涡旋光束闪烁因子越小。     

离轴拉盖尔-高斯涡旋光束传输中的光斑演变

在产生涡旋光束过程中, 固体激光器所输出的光束中心 很难与螺旋相位板的中心完全对准, 实际出射的光束为离轴涡旋光束. 在衍射理论的基础上, 对离轴涡旋光束的传输进行了研究, 推导了离轴涡旋光束传输一段距离后电场和光强的解析表达式.研究表明, 与理想的涡旋光束不同, 离轴涡旋光束具有非对称性的光强分布, 在传输过程中光斑除了展宽外, 涡旋暗核还会发生移动. 拓扑电荷数的大小只影响到光束的展宽, 拓扑电荷数为正时, 暗核沿着逆时针切线方向移动; 拓扑电荷数为负时, 暗核沿着顺时针的切线方向移动, 该结果对长距离探测涡旋光束的对准问题起到指导作用.     

Measuring the topological charge of optical vortices with an axicon

We analyzed the spatial spectrum of the diffraction intensity pattern of an ideal Bessel beam and found an implicit rule that the number of the bright rings in the spatial spectrum is equal to the topological charge of the Bessel beam. The radius of the bright and dark rings has some relation with the topological charge and can be determined accurately. It provides us with a new way for measuring the topological charge of an optical vortex through its diffraction intensity pattern after an axicon. The results of simulation coincide with the theory.     

Orbital angular momentum density and spiral spectra of Lorentz–Gauss vortex beams passing through a single slit

Based on the Hermite–Gaussian expansion of the Lorentz distribution and the complex Gaussian expansion of the aperture function, an analytical expression of the Lorentz–Gauss vortex beam with one topological charge passing through a single slit is derived. By using the obtained analytical expressions, the properties of the Lorentz–Gauss vortex beam passing through a single slit are numerically demonstrated. According to the intensity distribution or the phase distribution of the Lorentz–Gauss vortex beam, one can judge whether the topological charge is positive or negative. The effects of the topological charge and three beam parameters on the orbital angular momentum density as well as the spiral spectra are systematically investigated respectively. The optimal choice for measuring the topological charge of the diffracted Lorentz–Gauss vortex beam is to make the single slit width wider than the waist of the Gaussian part.     

利用全息法在偶氮聚合物薄膜中记录涡旋光场

利用全息技术在偶氮聚合物薄膜中记录了拓扑荷数q=–1,1,2,4的涡旋光场,并将记录的原始叉形光栅与计算全息光栅进行对比,对不同拓扑荷数涡旋光的记录速率和偶氮材料的可重复擦写性能进行了测试;记录完成后,将复现涡旋光与高斯光束干涉,并与原始涡旋光和原始叉形光栅对比,分析了记录质量.实验结果表明:高阶涡旋光场的全息叉形光栅会在记录过程中发生劈裂,轻微劈裂的涡旋光束仍维持一个稳定的环状结构;全息记录过程中不同拓扑荷数的涡旋光束记录速率较为统一,偶氮材料可经过上百次的擦写而不出现疲劳;再现涡旋光与原始涡旋光在光强分布结构上保持高度一致,再现涡旋光的干涉条纹与原始涡旋全息光栅保持高度一致,涡旋光及其携带的拓扑荷信息可被有效记录和读取.     

Effects of low order atmosphere-turbulence aberrations on the vortex carrying Gaussian beam

The normalized intensity distributions at the focal plane of a vortex carrying Gaussian beam propagating in weak/middle turbulent atmosphere channel with z-tilt aberration, defocus aberration, astigmatism aberration or total turbulent aberration are discussed by numerical calculation. Our results show that the effect of z-tilt aberration on the intensity distribution of optical vortex beam is main effect of total turbulent aberration. In weak turbulent region, the effects of defocus and astigmatism aberration on the intensity distribution of a vortex carrying Gaussian beam can be ignored. In middle turbulent region, the effect of z-tilt aberration is still the most significant, but the effects of defocus and astigmatism aberrations, specially the effect of astigmatism on the center dark core of the intensity distribution at focal plane, can no longer be disregarded. Our results also show that for three low order aberrations, the beam with values of the topological charge, the beam have larger beam-radius and undergo smaller effects of z-tilt turbulent aberration on the doughnut distribution. For defocus aberration, the radius of the center dull of beam intensity increases with the values of topological charges increasing. And for astigmatism aberration, the beam with odd number topological charges, the center dull speck of beam intensity becomes to the bright speck. But for even number topological charges, the center dull speck of beam intensity maintains the dull one.     

Angular momentum density of a Gaussian vortex beam

The Gaussian vortex beam is assumed to be linearly polarized.The analytical expression of the electric field of a linearly polarized Gaussian vortex beam propagating in free space is derived by using the vectorial Rayleigh-Sommerfeld integral formulae.The propagating magnetic field of the linearly polarized Gaussian vortex beam is presented by taking the curl of the electric field.By employing the electromagnetic field of the linearly polarized Gaussian vortex beam beyond the paraxial approximation,the analytical expression of the angular momentum density of the linearly polarized Gaussian vortex beam is derived.The three components of the angular momentum density of a linearly polarized Gaussian vortex beam are demonstrated in the reference plane.The effects of the linearly polarized angle and the topological charge on the three components of the angular momentum density are investigated.To acquire the more longitudinal angular momentum density requires such an optimal choice that the linearly polarized angle is set to be zero and the topological charge increases.This research is useful to the optical trapping,the optical guiding,and the optical manipulation.     

The effect of tilt aberration and astigmatism of turbulent atmosphere on the intensity distribution of a vortex carrying Gaussian beam

The effect of tilt and astigmatism aberration of the turbulent atmosphere on the intensity distribution of a focused vortex carrying Gaussian beam was investigated based on the extended Fresnel-Kirchhoff diffraction integral and the quadratic approximation of phase structure function. Our results have shown that the intensity distribution on the focal plane in the effect of tilt aberration changes with the turbulent strength, the propagation distance and the topological charge of the initial beam. The propagation distance is larger, the focal spot size will be larger, and the central dip will be less deep. It is also noticed that the intensity distribution of a beam with single topological charge is affected more by tilt aberration in comparison to the beam with double topological charge. The effect of astigmatism on the intensity distribution is quite less than that of tilt aberration. For focused beam propagation in atmosphere, the effect of the tilt aberration on the intensity distribution of optical vortex beam is the main effect of total turbulent aberration.     

Generation of a hollow Gaussian beam and its anomalous behavior

I attempted to produce a laser beam with no light in the central line of the beam, by Fresnel diffraction of the Gaussian beam by a spiral zone plate (SZP). The study imparts an arbitrary-order phase singularity on the light field. The experimental results show that the optical vortex radius depends on the singularity’s integer order n (also termed topological charge, or order of the dislocation). Also, the radius of maximum intensity is shown to depend on the singularity number. Anomalous behavior of the spectra at phase singularity produced by various zone plates near the focus of a converging Gaussian beam is also studied using a simple experimental technique. It is found that the spectrum of the beam on the SZP (spiral zone plate) consisting of a single spectral profile shows almost vanishing intensity at all frequencies with a tendency of splitting into two peaks at the on-axis point and shows redshift and blueshift around the phase singularity. A comparative study of the anomalous behavior of the spectra at phase singularity produced by various SZPs, with varying phase singularity order n, near the focus of a converging Gaussian laser beam is studied.