Focus evolution of linearly polarized Lorentz beam with sine-azimuthal variation wavefront induced by one on-axis optical vortex

Focus evolution of linearly polarized Lorentz beam with sine-azimuthal variation wavefront induced by one on-axis optical vortex was investigated theoretically in this article. Calculation results show that the focal pattern can be altered considerably by the charge number of on-axis optical vortex under condition of certain beam parameters and phase parameter that indicates the sine phase change frequency on increasing azimuthal angle. And the focal evolution principle differs remarkably for different beam parameters and the phase parameter. In focus evolution process, some novel focal patterns appear, including annular focal pattern, two-peak focal pattern, intensity lines, hexagon containing two peaks, swallowtail shape, multipede shape, and complex focal pattern. Introduction of optical vortex adds one controllable parameter to alter focal pattern, which may extend application of Lorentz beam in some focusing systems.     

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.     

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.     

Focusing properties of spirally polarized hollow Gaussian beam

Focusing properties of spirally polarized hollow Gaussian beam (HGB) are investigated theoretically by vector diffraction theory in this article. Results show that the optical intensity in focal region of spirally polarized HGB can be altered considerably by the beam order, numerical aperture of the focusing system, and spiral parameter that indicates the polarization spiral degree of the spirally polarized HGB. Spiral parameter can induce focal pattern change in axial direction remarkably, while beam order and numerical aperture affect radial foal pattern more obviously. The tunable principle of the focal pattern by spiral parameter differs very considerably under condition of different numerical aperture and beam order. Many novel focal patterns may occur in focal pattern evolution. It was also found that focal shift and focal depth can be altered significantly by spiral parameter and beam order.     

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.     

Focal shift and focusing properties generation by radial cosine phase masks

Focal shift and focusing properties of Gaussian beams induced by radial cosine phase masks are investigated. Results show that focal shift and the energy distribution among intensity peaks are controlled by two different parameters of the radial cosine phase mask. Increasing the value of frequency parameter in the cosine part of the phase mask, focal shift and focal switch 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 six of multiple peaks. The tunable focal shift can be used to construct controllable optical tweezers. In practice, the tunable phase mask can be implemented through liquid crystal spatial light modulator, which can conveniently alter the wavefront phase distribution of the incident laser beam in the control of computer.     

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.     

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.     

Separation of spin angular momentum in space-variant linearly polarized beam

We show that the spin angular momentum (SAM) flux in a space-variant linearly polarized beam can be separated in the focal plane. Such a beam carries only orbital angular momentum (OAM) and develops a net SAM flux upon focusing. The radial splitting of the SAM flux density is mediated by the phase vortex (or OAM) and can be controlled by the topological charge of the phase vortex. Optical trapping experiments verify the separation of the SAM flux density. The proposed approach enriches the manipulation of the angular momentum of light fields and inspires more designs of focus engineering, which would benefit optical micromanipulation of microscopic particles.     

高次方涡旋光束经大数值孔径透镜的聚焦特性

为了获得多种类型的波长量级聚焦光斑,研究了一种新型涡旋光束,高次方涡旋光束经过大数值孔径透镜的聚焦。基于矢量德拜积分公式,理论上研究了线偏振的高次方涡旋光束经过大数值孔径透镜的聚焦特性。研究了涡旋光束的拓扑荷数和幂次方数对聚焦平面光强和电场x分量的相位分布的影响。研究结果表明,通过控制涡旋光束的拓扑荷数和幂次方数可以产生不同类型的聚焦光强分布,例如尺寸约为2个波长大小的实心和空心型聚焦光斑。此外,与普通的涡旋光束聚焦不同,高次方涡旋光束聚焦后的奇点并不在焦点处。这些特殊的聚焦光斑有望在微粒的操控等领域中得到应用。     

The Effect of the Spin Angular Momentum on the Tight-Focusing Vortex Hollow Gaussian Beams

The propagation properties of a hollow Gaussian beam (HGB) carrying on-axis and off-axis vortices through a high numerical aperture lens are investigated. The intensity of the focused beam in the focal plane can be controlled by choosing the different topological charges, the beam order, and the semi-aperture angle. As intrinsic properties, vortex beams possess both spin and orbital angular momenta. The spin angular momenta (SAM) density can be treated as a vector in 3D since it exists in arbitrary orientation during the beam propagation. The vectors of SAM density orientation of the focused beam in 3D rotate around the central axis whose locations mainly rely on the vortices. The magnitude of the SAM density near the focus plane abruptly varies by altering the focal length of the lens. Under tightly focusing condition, two new pairs of vortices generate alternately on x and y axes in the vectorial electric fields, while the topological charges increase by one.     

Polarization vortex spatial optical solitons in Bessel optical lattices

We investigate the formation of polarization vortex spatial optical solitons in optical lattice induced by a non-diffracting Bessel beam. The properties of these solitons in zeroth-order and first-order Bessel lattices with focusing and defocusing Kerr nonlinearity are discussed. It is found that these solitons have some analogies with phase vortex solitons carrying single positive or negative topological charge in these lattices. Besides, these polarization vortex solitons have complicated dynamical characteristic and can be stabilized in some parameter region.     

Tight focusing properties of radially polarized Lorentz–Gaussian beam

Focusing properties of radially polarized Lorentz–Gauss beam with one on-axis optical vortex was investigated by vector diffraction theory. Results show that intensity distribution in the focal region can be altered considerably by charge number of the optical vortex and the beam parameters. Many novel focal patterns may occur, Such as Peak-centered, donut focal shapes which is potentially useful in optical tweezers, material processing and laser printing.     

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.     

Focal shift of cylindrical vector axisymmetric Bessel-modulated Gaussian beam with radial variance phase wavefront

Axisymmetric Bessel-modulated Gaussian beam with quadratic radial dependence (QBG beam) has attracted much attention recently. In this paper, the focal shift of the cylindrical vector QBG beam with radial variance phase wavefront is investigated theoretically by vector diffraction theory. Results show that focus shifts considerably by changing the phase parameter C that indicates the radial phase variance speed. Under condition of small beam parameter μ of cylindrical vector QBG beam, there is one focal peak that shifts far away from optical aperture on increasing C. When μ increases, there may occur two focal peaks that also shift remarkably on increasing C. And it was found that the dependence of focal shift distance on increasing phase parameter is linear. Phase parameter adjusts the focal shift distance, while, polarization angle does not affect focal shift obviously.     

Focus evolution of Gaussian beam induced by phase plate

In this article, vector diffraction theory is employed to investigate the focusing properties of the Gaussian beam through a phase plate. The phase plate may alter the wavefront phase of an incoming beam by topological charge. Both the circular phase distribution and the annular phase distribution plates are investigated. Numerical simulations show that the focal intensity distribution depends on topological charge. With changing topological charge, focal intensity distribution may evolve into ring shape, and some novel focal spots may occur. Focal intensity distribution evolving process with integer topological charge differs considerably from that with fraction topological charge. When the concentric annular phase plate is placed in the laser path, the focal intensity distribution depends on both the inner radius and topological charge. For small inner radius of the phase plate, doughnut-shape focal pattern occurs. With increasing inner radius, the diameter of the doughnut focal pattern decreases, and doughnut shape disappears slowly in some cases.     

Effect of spherical aberration on tightly focused cylindrically polarized vortex beams

In this paper attention is given to the effects of primary spherical aberration on the cylindrical polarized vortex beam based on the vector diffraction theory. It is observed that by properly choosing the polarization angle and topological charge one can obtain many novel focal patterns suitable for optical tweezers, laser printing and material process. However, it is observed that the focusing objective with spherical aberration generates structural modification and positional shift of the generated focal structure.     

Three-Airy autofocusing beams

We numerically and experimentally demonstrate that a three-Airy autofocusing beam can be generated by superposing three deformed two-dimensional(2 D) Airy beams with a triangle symmetry. When the initial angle between two wings of the deformed 2 D Airy beams increases, such a three-Airy autofocusing beam exhibits that the focusing length decreases and the intensity contrast at the focal point changes. Moreover, after introducing an optical vortex phase, this three-Airy autofocusing beam displays a transverse rotation in propagation. The rotation angle is determined by the topological charge of the vortex and the initial wing angle. Our results may have some potential applications in optical manipulation.     

Tight Focusing Properties of Radially Polarized Gaussian Beams with Pair of Vortices

The tight focusing properties of a radially polarized Gaussian beam with a nested pair of vortices having a radial wave front distribution are investigated theoretically by the vector diffraction theory.The results show that the optical intensity in the focal region can be altered considerably by changing the location of the vortices nested in a radially polarized Gaussian beam.It is noted that focal evolution from one annular focal pattern to a highly confined focal spot in the transverse direction is observed corresponding to the change in the location of the optical vortices in the input plane.It is also observed that the generated focal hole or spot lead to a focal shift along the optical axis remarkably under proper radial phase modulation.Hence the proposed system may be applied to construct tunable optical traps for both high and low refractive index particles.     

Focusing properties of cylindrical vector vortex beams with high numerical aperture objective

Focused by a high numerical-aperture objective in free space, the cylindrical vector beam phase-encoded by vortex phase plate with higher topological charge was capable to generate the doughnut-shaped spot in the vicinity of the focal region. The width of the dark focal spot was manipulated by the phase plate with different topological charge. The relationship between the properties of the focal spot and the vortex phase plate was explicitly analyzed for the input beam with different cylindrical vector polarization. Furthermore, the experimental verification was undertaken at the incidence beam λ = 635 nm with the radial and azimuthal polarization. The experimental results are in excellent agreement with the theoretical calculation.