Focal shift in radially polarized hollow Gaussian beam

Focal shift in radially polarized hollow Gaussian beam (HGB) with radial wavefront distribution is investigated theoretically. The wavefront phase distribution is cosine function of radial coordinate. Simulation results show that the intensity distribution in focal region of the radially polarized HGB can be adjusted considerably by the beam order of HGB n and cosine parameter C that indicates the phase change degree. On increasing C, focus can shift along optical axis and focal pattern changes remarkably. Focus may move in different direction under different condition. Focal shift distance fluctuates on increasing C, and fluctuation amplitude also increases simultaneously. In addition, threshold value of C for focal shift from one side to the other side of the paraxial focal plane differs for different n.     

Focal shift of radially polarized axisymmetric Bessel modulated Gaussian beam with radial variance phase plate

Focal shift of the radially polarized axisymmetric QBG beam with radial variance phase plate is investigated theoretically by vector diffraction theory. Axisymmetric Bessel modulated Gaussian beam with quadratic radial dependence (QBG beam) has attracted much attention recently. Calculation results show that focus shifts considerably by changing the phase parameter C that indicates the radial phase variance speed. Under condition of beam parameter μ of radially polarized axisymmetric QBG beam, there is one focal spot that shifts far away from optical aperture on increasing C. When increases the value of beam parameter, there may occur two focal peaks that also shift remarkably on increasing C.     

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.     

Focal shift in radially polarized beam with high NA lens axicon by using radial cosine phase wavefront

Focal shift in radially polarized beam with radial cosine phase wavefront are investigated theoretically by vector diffraction theory. The wavefront phase distribution is cosine function of radial coordinate. Simulation results show that when the radially polarized beam with radial cosine wavefront phase is focused, the focal pattern differs considerably with frequency parameter in the cosine function term. On increasing C, focus can shift along optical axis and focal pattern changes remarkably. Focus may move in different direction under different condition. Focal shift distance fluctuates on increasing C, and fluctuation amplitude also increases simultaneously. In addition, focal shift direction can also be adjusted by changing frequency parameter in cosine function.     

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.     

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.     

Focusing of hyperbolic-cosine-Gaussian beam with a non-spiral vortex

The focusing properties of the hyperbolic-cosine-Gaussian beam, which contains an optical vortex induced by a non-spiral phase plate, are investigated numerically. The phase plate alters phase distribution linearly in a half-section part of the hyperbolic-cosine-Gaussian beam, which results in one non-spiral optical vortex. Results show that the phase variation rate of the non-spiral phase plate influences focal intensity distribution considerably, and some novel focal patterns, such as line focal spot, “H”-shape focal spot, and intensity peaks array, may occur. Focal shift, focal split, and focal switch appear in focal evolution with increasing phase variation rate. The focal evolution differs for different parameters in cosh parts of the hyperbolic-cosine-Gaussian beam. For big parameters in cosh parts, the value of local intensity peaks is comparable to that of maximum intensity peaks.     

Focusing of radially polarized beam with radial cosine phase wavefront

Radially polarized beam has gained much interest recently due to its properties and applications. In this article, the focusing properties of radially polarized beam with radial cosine phase wavefront are investigated theoretically. Results show that when the radially polarized beam with radial cosine wavefront phase is focused, the focal pattern differs considerably with frequency parameter in the cosine function term. In the high numerical aperture focusing system, focal shift occurs, and novel focal patterns evolve considerably, for instance, from only one peak to two or multiple overlapping peaks. In addition, peak intensity ratio of radially polarized component to longitudinal polarized component in the focal region fluctuates smoothly for low-frequency parameter, then drops sharply, and comes back remarkably with increasing frequency parameter. Simultaneously focal shift increases slowly, and then decreases, suddenly, focal shift sign changes that results from focal switch phenomenon, and then fluctuates.     

Generation of sub wavelength focal spot with large depth of focus generated by radially polarized beam

We investigate the focusing properties of a radially polarized Bessel Gaussian beam by a high numerical aperture (NA) lens based on vector diffraction theory. We observe that our proposed system generates a sub wavelength focal spot of 0.42λ having large uniform focal depth of 6.45λ. The authors expect such a long depth of focus have great potential for use in optical, biological, high-resolution and atmospheric sciences.     

Focal shift of apodized truncated hyperbolic-cosine-Gaussian beam

Focal shift of hyperbolic-cosine-Gaussian beams induced by pure phase apodizer was investigated theoretically. The pure phase apodizer consists of three concentric zones: center circle zone, inner annular zone and outer annular zone. And the phase variance of the inner annular zone is adjustable. Results show that intensity peak moves far from optical aperture and then shrinks sharply for certain radii of zones with increasing phase variance of the inner annular zone. Simultaneously, one new intensity peak occurs near optical aperture, moves far from the optical aperture, and then becomes intensity maximum peak, and repeats the evolution process of the former intensity peak. Tunable focal shift occurs with focal switch. Decreasing the phase variance can change the move direction of the intensity peaks. In addition, the maximum distance between the two intensity peaks can be altered by beam parameters of cosh parts, and the distance value increases and then decreases with increasing beam parameters of cosh parts for certain radii of zones. Tunable focal shift is also discussed to construct optical tweezers.     

Focusing properties of the hyperbolic-cosine-Gaussian beam induced by phase plate

Focusing properties of hyperbolic-cosine-Gaussian beam induced by phase plate were investigated theoretically in this Letter. The phase plate consists of three concentric zones: center circle zone, inner and outer annular zones with 0, π, 0 phase variations, respectively. Numerical simulations show that the evolution of the focal shape is altered considerably by changing the radii of zones and the decentered parameters in cosh parts of the light beam. Increase inner radius of the outer annular zone, focal spot broadens along optical axis, splits into two peaks, and then the two peaks combine back into one peak. In this process, the focal depth increases and then decreases three times, so three focal depth peaks occur. And the distance between the two peaks for bigger inner radius of outer annular zone is very small. With decreasing decentered parameters in cosh part of the beam, position of the three focal depth peaks shift toward to small radius, and the peak values also fall simultaneously.     

Focal shifting of transversely polarized beam using cosine phase plate

Focal shift of transversely polarized beams induced by cosine phase masks are investigated theoretically by vector diffraction theory. Results show that when the transversely polarized beam with radial cosine wavefront phase is focused, the focal pattern differs considerably with frequency parameter in the cosine function term. Increasing the value of frequency parameter in the cosine part of the phase mask, focal shift may occur, simultaneously, the focal shift direction may change. Moreover, by altering frequency parameter or phase variation parameter of the phase mask will change the energy distributions of maximum intensity peak and other small intensity peaks. And novel focal patterns also evolve considerably, such as from only one peak to five of multiple peaks. The tunable focal shift can be used to construct controllable optical tweezers.     

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.     

Focal shift of three-portion concentric piecewise cylindrical vector beam

Focal shift of a concentric piecewise cylindrical vector beam is investigated by means of vector diffractive theory. A section of the beam consists of three concentric portions. The center circle portion and outer annular portion are radial polarized, and the inner annular portion is generalized polarized. The phase of the inner annular portion is tunable. For certain geometric parameters of the beam, the focal shift occurs and can be adjusted by phase shift and polarized direction of the inner annular portion. For some cases, a three-peak focal pattern may occur, and peaks shift in the same direction, and one local minimum also shifts. For some other cases, a two-peak focal pattern may occur with shifting peaks. The tunable quantitative focal shift is investigated in detail.     

Phase filter design for sharper focus of radially polarized beam

Based on vector diffraction theory, a phase-only filter with cosine function is proposed and a sharper focal spot of radially polarized beam is obtained on the focal plane. The function parameters of the filter are optimized and a series of optimized filters for different Strehl ratio S are given. The optimization results show that there is a significant improvement on obtaining sharper focal spot with this new type of filter. Compared with other results, this type of filters has superior supperresolution effect and higher energy utilization ratio. The numerical calculation results about the phase filter may facilitate new approaches to get superresolution.     

大数值孔径光子筛偏振特性研究

根据角谱理论建立不同偏振照明条件下的光子筛矢量衍射模型。在此基础上,对入射光分别为线偏振光、径向偏振光、切向偏振光三种特殊偏振状态下的光子筛聚焦光强分布进行了模拟分析。研究结果表明,对于大数值孔径光子筛,入射光的偏振特性将对光子筛聚焦光强分布产生巨大影响。线偏振光将使聚焦光斑沿偏振方向拉伸,切向偏振光产生的聚焦光斑具有"中空"结构,而径向偏振光所产生的聚焦光斑呈较为规则的圆形,且其焦深优于线偏照明情况。在激光直写及高分辨成像等光子筛典型应用中采用径向偏振照明将进一步提高系统分辨力。     

Focusing properties of radially polarized Bessel-like beam with radial cosine phase wavefront by a high numerical aperture objective

By the use of the vector diffraction theory, the focusing properties of radially polarized Bessel-like beam with radial cosine wave front phase through a high numerical-aperture (NA) lens are investigated theoretically and numerically in this work. The wave front phase distribution is a cosine function of radial coordinate. Calculation results show that focus shift is considerably influenced by changing frequency parameter C and topological charge. An increase on the focus shift C, the total intensity pattern changes remarkably and it focuses along the optical propagation axis. Thus, the focal shift direction can be adjusted by the change of the frequency parameter in cosine function. In this paper, the generation of the focal spot allows an increase in focal length in the axial direction of the incident beam propagation. Under higher numerical aperture (NA = 0.95), the effect of the frequency parameter and topological charge on the focal pattern gets stronger.     

Effect of binary phase plate on incident radially polarized hollow Gaussian beam

Effect of a pure binary phase filter on the three-dimensional light intensity distributions in focal region is theoretically investigated in this paper. The results show that the proposed binary phase filter may induce flat top profile with large depth of focus and reduced focal spot by properly adjusting the geometrical parameters of the binary phase filter of incident radially polarized hollow Gaussian beam (HGB) beam. On the other hand, the HGB beam benefits the most from the use of an annulus. Such kind of system is potentially useful for lithography, imaging, optical data storage, optical trapping, optical excitation of molecules, or coupling to optical fibers.     

Focal patterns of higher order hyperbolic-cosine-Gaussian beam with one optical vortex

Intensity distribution in focal region plays an important role in many optical systems. In this paper, the focal patterns of higher order hyperbolic-cosine-Gaussian (HOCG) beam in focal plane were investigated. The HOCG beam contains one spiral optical vortex on its optical axis. Results show that the focal pattern can be altered considerably by beam order of the incident HOCG beam, and some novel focal patterns may occur, including foursquare focal pattern, cross-shaped dark focal focus, foursquare intensity peaks chain, and multiple intensity peaks array. Focal pattern evolution principle on increasing beam order also differs remarkably under condition of different topological charge of the optical vortex and displacement parameter associated with the cosh parts. And like topological charge and displacement parameter, the beam order and numerical aperture may affect focal pattern.     

Spirally polarized axisymmetric Bessel-modulated Gaussian beam

Focusing properties of the spirally polarized axisymmetric Bessel-modulated Gaussian beam with quadratic radial dependence (QBG beam) are investigated theoretically by vector diffraction theory. Calculation results show that intensity distribution in focal region can be altered considerably by beam parameter μ and spiral parameter C that indicates polarization spiral degree. For certain real value μ, focal spot can evolve from one focal spot to one focal ring, spherical crust focal shape, then two focal rings on increasing spiral parameter C. It was found that under condition of different μ, evolution principle of focal pattern differs very remarkably on increasing C. And some novel focal shapes may appear, including rhombic shape, quadrangular shape, two-spherical crust focus shape, two-peak shape, one dark hollow focus, two dark hollow focuses pattern, triangle dark hollow focus.