Focusing properties of Gaussian beam with mixed screw and conical phase fronts

Vector diffraction theory is employed to investigate the focusing properties of the Gaussian beams with mixed screw and conical phase fronts. Numerical simulations show that the Gaussian beams with screw-conical phase fronts are different from both the ordinary Laguerre-Gaussian beams and the higher-order Bessel beams. Rather than forming the ring-shaped intensity distributions characteristic of optical vortices, focusing the Gaussian beams with screw-conical phase fronts produce non-symmetric spiral intensity distributions at the focal plane. The intensity distribution forms a counter-clockwise non-symmetric screw path around the focus. The rotation of intensity distributions was observed in the focal plane. The gradient force patterns of these beams focused with high NA are also investigated. The results show that the gradient force pattern shape depends principally on parameter topological charge n of the phase distribution. The gradient force pattern expands with increase in the parameter m of the phase distribution. Therefore, one can change the topological charge n or the parameter m of the phase mask to construct the tunable optical trap to meet different requirements. Its potential application might include rotational positioning of particles and accumulation of smaller non-symmetric particles towards the focus.     

会聚高斯光束焦点附近的三维光分布状态

根据惠更斯 菲涅耳原理 ,推导出了会聚高斯光束经圆形孔径衍射后焦点附近的三维光分布状态级数表达式 ,并与经典的均匀光波理论进行了比较。讨论了会聚高斯光束焦移与光学系统及高斯光束参数的关系。给出了会聚高斯光束焦点附近的等照度线图。结果表明 ,对于大菲涅耳数系统来说 ,其光强分布具有对称性     

Propagation and focal shift of J0-correlated Schell-model beams

Based on the propagation law of partially coherent beams, the propagation equations of J₀-correlated Schell-model (JSM) beams through a paraxial optical ABCD system are derived and expressed in closed-form. The free-space propagation and focusing of JSM beams are regarded two special cases of our result. The focusing properties, in particular, the focal shift of JSM beams are studied, and illustrative numerical results are given. It is found that the focal shift of JSM beams, which is determined by an algebraic equation of 3rd order, increases with decreasing Fresnel number N and increasing parameter σ.     

Phase singularity annihilation of focused partially coherent beams

Expressions for focused Gaussian Schell-model (GSM) beams are derived and used to study the annihilation and subwavelength structures of phase singularities in the focal region, and to compare with the case of fully coherent Gaussian beams. It is found that the truncation parameter δ and normalized coherence length ε both affect the presence and spatial distribution of phase singularities in the focal plane. Additionally, during the creation and annihilation process the saddle point near the phase singularity does not disappear in the focal plane for GSM beams.     

The intensity distribution of hollow Gaussian beams focused by a lens with spherical aberration

We developed an expression that describes the hollow Gaussian beams (HGBs) passing through a spherically aberrated lens by using the Collins formula. The radial intensity distribution in both spherical aberration SA free lens, lens that exhibits relatively large in both positive spherical aberration PSA, and negative spherical aberration NSA is calculated. Numerical calculations are made and the results show that the PSA and NSA have a strong influence on the intensity distribution especially at the focus. The study showed remarkable results for which there is no hollow Gaussian beam at a large NSA along the optical axis at the focus. In addition, we found that the DSS, and w_r of focused hollow Gaussian beams in the focal region depend not only on the beam radius, and beam order; but also on the spherical aberration.     

Generation and tight-focusing properties of cylindrical vector circular Airy beams

Tight-focusing properties of cylindrical vector circular Airy beams [i.e., azimuthally polarized (AP) circular Airy beam and radially polarized (RP) circular Airy beam] passing through a high numerical aperture thin lens are investigated in detail. It is found that a super long subwavelength dark channel with full width at half maximum about 0.49λ and depth of focus (DOF) about 52λ can be achieved near the focal region for the case of tight focusing of an AP circular Airy beam, and a super long needle with DOF about 27.5λ of strong longitudinally polarized field can be obtained near the focal region for the case of tight focusing of a RP circular Airy beam. Furthermore, we report experimental generation of an AP circular Airy. Our results will be useful for atom guiding and trapping, particle acceleration and fluorescent imaging.     

Polarization and intensity properties of converging J0-correlated Schell-model beams

Based on the cross-spectral density matrix, closed form result for propagation equation of electromagnetic J₀-correlated Schell-model beams (EJSMBs) through a paraxial optical system is obtained and the focusing properties are studied. Both numerical calculation and physical interpretation are obtained. It is found that a tunable dark hollow area, which has potential applications in optical trapping, can be obtained by altering the coherence parameter and the focal length. It is also shown that even though the original field is unpolarized, the beams can become fully polarized in the focal region with its width being tunable by changing the coherence parameter. The relevance of this work to applications such as coherent detection in optical communication is also discussed.     

4pi聚焦系统中振幅和相位调制的径向偏振涡旋光束聚焦特性的研究

基于Richards-Wolf矢量衍射积分公式, 研究了径向偏振涡旋光束在振幅和相位调制下的4pi聚焦特性.振幅调制是通过振幅滤波实现, 即改变入射光束起始积分值达到调节,相位调制是通过添加相位延迟角δ 的液晶相位延迟器来改变入射光束的偏振态.模拟结果显示,随着振幅的减小, 4pi聚焦系统焦点附近的光轴上呈现出多光球结构; 而相位调制对焦点附近的光强分布产生拉伸作用, 即调节入射光束的拓扑核m和相位延迟器的延迟角δ,可以得到特殊的光强分布. 随着相位δ增大, m=0产生的多光球结构慢慢向光链结构转变,最终变成暗通道;而m=1产生的光链结构慢慢变成光球结构; m=2产生的暗通道变成光球和光链叠加的结构, 这种特殊聚焦光束在光学微操纵领域具有潜在的应用价值. 关键词： 物理光学 偏振 光链 4pi聚焦系统     

Coherence vortex evolution of partially coherent vortex beams in the focal region

Taking Gaussian Schell-model vortex beams as an example of partially coherent vortex beams, we study the evolution and subwavelength structures of coherence vortices in the focal region. The dependence of coherence vortices on the truncation parameter δ, the normalized coherence length ε and the topological charges m is illustrated numerically. It is found that during the evolution process the innermost m coherence vortices and the saddle points near the coherence vortices always remain in the focal plane.     

三环非均匀混合偏振同轴矢量光束的聚焦特性

基于Richards-Wolf矢量衍射积分公式,数值分析了同轴三环非均匀混合偏振矢量光束经过高数值孔径透镜的聚焦特性。该矢量光束由同轴三环局域线偏振矢量光束通过一个相位延迟角为δ的液晶相位延迟器产生,光束偏振变为包含线偏振、圆偏振和椭圆偏振的混合态。同轴三环局域线偏振矢量光束的偏振分布是由径向向内偏振的外环光束、径向向外偏振的内环光束和线偏振方向与径向方向夹角为φ2的中环光束构成。数值模拟结果显示该混合偏振矢量光束的聚焦强度分布与参数φ2和相位延迟角δ密切相关,当选取适当的φ2和δ时,在焦平面附近产生沿光轴方向的三维多点光俘获结构——暗光链,这在光学微操纵领域具有潜在的应用价值。     

Focal shift and focal switch of flat-topped Mathieu-Gaussian beams passing through an apertured lens system

By introducing a hard aperture function into a finite sum of complex Gaussian functions, an approximate analytical expression predominating the distribution of axial intensity for the flat-topped Mathieu-Gauss (FTMG) beams passing through a system with the aperture and lens separated has been derived. The focal shift and the focal switch effect of FTMG beams passing through the system is studied in detail. Numerical calculations have shown that the position of real focal plane is not coincident with the geometrical focus but is somewhat shifted toward the lens. The focal shift and focal switch of FTMG beams take place when the relative separation s/f = 1 by a suitable choice of beam parameter and truncation parameter, for example, the beam parameter is smaller than its corresponding critical value or the truncation parameter is between its two corresponding critical values.     

Tight focusing effect of annular obstructed incident beam in the focal region of high NA lens

Focusing properties of the azimuthally polarized axisymmetric Bessel-modulated Gaussian beam with Quadratic Bessel Gaussian (QBG beam) and annular aperture are investigated theoretically by vector diffraction theory. Simulation results show that the intensity distribution in focal region of the azimuthally polarized axisymmetric QBG beam can be shifted along optical axis considerably by changing parameter (C). On introducing annular aperture (δ), focal pattern at the focus extends along optical axis. In this paper, we have shown the generation of focal hole and focal shifting in the axial direction of incident beam propagating through aligned optical system which is suitable for application such as optical manipulation and optical trapping.     

Concentric piecewise azimuthally polarized beam by a pure phase plate

Focusing properties of the azimuthally polarized beam induced by a pure phase plate are investigated theoretically. The pure phase plate consists of two concentric portions, one center circle portion and one outer annular portion, through which the azimuthally polarized beam passed evolves into concentric piecewise azimuthally polarized beam. When the phase shift of the center portion is π, one ring focus may evolve into novel focal patterns with increasing radius of the center circle portion, such as cylindrical crust focus, two-ring focus, and three-ring focus. And if the geometrical parameters are unchanged, focal patterns also changes considerably with tunable phase of the center portion. Ring focus shifts along the optical axis on the increasing phase. Some optical gradient force distributions and dependence of focal shift on phase shift are also illustrated. This kind of concentric piecewise azimuthally polarized beam can be used in optical manipulation technology.     

Radially polarized hyperbolic-cosine-Gaussian beam

Vector beams have attracted much interest recently. In this paper, focusing properties of the radially polarized hyperbolic-cosine-Gaussian beam are investigated. Simulation results show that the focal depth increases with increasing cosine parameter in the cosine term of the beam, while focal spot decreases simultaneously, namely, superresolution occurs. Focal depth increase velocity is quicker for larger cosine parameter, while the transverse focal spot shrinks more quickly for smaller cosine parameter. In addition, for two-portion concentric piecewise radially polarized hyperbolic-cosine-Gaussian beam with π phase shift in center circle portion, focal pattern evolves considerably with increasing cosine parameter, and the evolution principle differs for different radius of the center circle portion. Focal splitting and novel focal spots may appear. This kind of vector beam can be used in optical storage, optical manipulation, and lithography.     

Second-harmonic wave induced by vortex beams with radial and azimuthal polarizations

The author investigated second-harmonic (SH) wave patterns generated by focused vortex beams with radial and azimuthal polarizations. Expressions for electric field components near the focus through a dielectric interface were obtained based on vector diffraction theory. Furthermore, the (110) and (001) planes of a crystal classified in the symmetry group 4?3m were considered as second order nonlinear optical medium to analyze nonlinear polarization and the intensity distributions of the SH wave in the far-field were simulated for topological charge m = 1 and 2. Numerical simulations indicated that the intensity and emission pattern of the SH wave depended principally on the electric field component and distribution near the focus. In particular, for the (001) plane, the intensity of the SH wave was strongly correlated with the existence of a longitudinal electric field.     

Effect of annular apodization and pupil beam parameter in the focal region of high NA lens

Focusing properties of the radially polarized axisymmetric Bessel-modulated Gaussian beam with quadratic radial dependence (QBG beam) and annular aperture are investigated theoretically by vector diffraction theory. Simulation results show that the intensity distribution in the focal region of the radially polarized axisymmetric QBG beam can be adjusted considerably by small beam parameter (μ) and annular aperture (δ). When μ increases, the focal spot may change to focal hole and changes focal pattern remarkably. On introducing annular aperture, focus can split or extends along the optical axis for different μ. In this paper, we have shown the generation of the focal spot, dark focal spot, focal split and increase in focal depth in the axial direction of the incident beam propagating through the aligned optical system.     

Effect of complex phase plate on tight focusing of azimuthally polarized double ring shaped beam

Based on the vector diffraction theory, the effect of complex phase plate on the intensity distribution of TEM₁₁ mode azimuthally polarized Laguerre–Gaussian beam in the focal region of high NA lens is investigated theoretically. It is observed that a properly designed complex phase plate can generate a subwavelength focal hole having FWHM of 0.332λ with an extended focal depth of 54.4λ. We also observe that by using properly designed complex phase plate generates novel focal patterns including splitting of focal holes and multi focus are obtained. The author expects such investigation is worthwhile for optical manipulation and material processing technologies.     

聚焦部分相干涡旋光束的传输和相干特性

根据部分相干光的传输理论,研究了部分相干高斯 贝塞尔涡旋光束通过光阑透镜聚焦后的传输和空间相关性质.数值计算结果表明,涡旋暗核的大小和焦平面上的光谱相干度都取决于入射光的拓扑电荷n、截断参量δ、相对相干长度σg和参量α.当选择适当的参量,在几何焦点附近会出现局域空心光束.研究还发现在焦面上光谱相干度会产生一个或多个的相位奇点（相干涡旋）,而且拓扑电荷和相对相干长度会对相干涡旋的位置和个数产生影响.在相干极限下,相干涡旋可逐渐演变为光学涡旋.     

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.     

Focal depth and focal splitting of truncated hyperbolic-cosine-Gaussian beam induced by phase plate

Focal depth and focal splitting of hyperbolic-cosine-Gaussian beams induced by a phase plate were investigated. The pure phase plate consists of three concentric zones: a center circle zone, an inner annular zone and an outer annular zone. The phase variance of the inner annular zone is adjustable. Simulation results show that the focal depth can be adjusted by changing the radii of zones. With the increase of the inner radius of the outer annular zone, the focal spot broadens along the optical axis and splits into two peaks. Then the two peaks combine back into one peak. There are two critical values for the inner radius of the outer annular zone, at which focal spot changes sharply. The tunable range of the focal depth varies considerably. The phase variance of the inner annular zone affects focal depth also; when the phase variance is π, the effect attains maximum. The parameters of cosh parts of the beam affect both focal splitting and focal depth evidently; focal splitting disappears with increasing parameters of cosh parts, and focal depth increases with increasing the parameters of cosh parts in both the low and the high numerical-aperture optical systems.