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.     

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.     

Focal properties of cylindrically polarized axisymmetric Bessel-modulated Gaussian beams by a high NA parabolic mirror

Focusing properties of the cylindrical vector axisymmetric Bessel-modulated Gaussian beam with quadratic radial phase dependence (QBG beam) in high numerical aperture parabolic mirror system is investigated theoretically by vector diffraction theory. Results show that intensity distribution in focal region can be altered considerably by beam parameter μ and polarization angle. The tightly focused cylindrically polarized axial symmetric Bessel-modulated Gaussian beams by a high numerical aperture parabolic mirror have possible applications in particle acceleration, optical trapping and manipulating, single molecule imaging and high resolution imaging microscopy.     

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.     

Cylindrical vector axisymmetric Bessel-modulated Gaussian beam

Focusing properties of the cylindrical vector axisymmetric Bessel-modulated Gaussian beam with quadratic radial dependence (QBG beam) in high numerical aperture system is investigated theoretically by vector diffraction theory. Results show that intensity distribution in focal region can be altered considerably by beam parameter μ and polarization angle. Polarization angle may adjust transverse intensity distribution, for instance from one focal spot to one ring shape. While μ alters axial intensity distribution remarkably, focal splitting may occur with tunable focal shift, and real value μ also may induce local intensity minimum. For certain case, with increasing imaginary value μ, transverse focal spot shrinks accompanied with higher full width half maximum of axial intensity distribution.     

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.     

Tightly focusing of spirally polarized Quadratic Bessel Gaussian beam through a dielectric interface

Based on vector diffraction theory, the tight focusing properties of spirally polarized axisymmetric Bessel-modulated Gaussian beam through a dielectric interface on high numerical aperture (NA) are investigated theoretically. The optical intensity distribution in the focal region of high NA objective lens is investigated in detail by numerical calculations. The results show that the focal shift induced in the focal region is by mismatch of refractive indices across the dielectric interface. It is also found that the optical intensity in focal region of spirally polarized Quadratic Bessel Gaussian (QBG) beam can be altered considerably by changing spiral parameter C that indicates the polarization spiral degree of the incident beam.     

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.     

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.     

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.     

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.     

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.     

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.     

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.     

Tight focusing effect of annular obstructed Bessel-modulated Gaussian beam

Based on vectorial diffraction theory, the effect of annular apodization on tightly focused azimuthally polarized Bessel-modulated Gaussian beam (QBG) are investigated theoretically. The numerical results show that the intensity distribution in focal region of the incident beam can be altered considerably by changing beam parameter (μ) and introducing annular apodization (δ). Beam parameter induces the focal splitting in transverse direction, while annular apodization leads to change in focal pattern along optical axis of the focusing system. More interesting, the focal splitting may be in continuous in certain case of incident beam propagating through aligned optical system which is suitable for application such as optical manipulation and optical trapping.     

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 of flattened Gaussian beams by an annular lens

Focusing properties of flattened Gaussian beams (FGBs) passing through an annular lens is studied based on the Collins formula. It is found that the on-axis irradiance distributions of focused FGBs are unsymmetrical with respect to the geometrical focal plane even for large values of Fresnel number F_w associated with the beam, so that there exist focal shifts in general. Detailed numerical results show the dependence of focal shifts on the beam and system parameters. Focal shifts of FGBs by a lens without central obscuration and focal shifts of Gaussian beams by an annular lens can be treated as special cases of the obscure ratio =0 and beam order N=0, respectively. Furthermore, focal shifts of plane waves by an annular lens can also be treated as a special case of N=0 and F_w→∞.     

Focal shift and focal switch of partially polarized Gaussian Schell-model beams passing through a system with the aperture and spherically aberrated lens separated

Based on the beam coherent-polarization (BCP) matrix approach and propagation law of partially coherent beams, the focal shift and focal switch of partially polarized Gaussian Schell-model (PGSM) beams passing through a system with the aperture and spherically aberrated lens separated is studied in detail. Our main attention is focused on the effect of spherical aberration and partial coherence on the focal shift and focal switch of PGSM beams. It is shown that for polarizer-free case there is no focal switch of PGSM beams, the focal shift of PGSM beams is closely related with spherical aberration coefficient C₄, auto-coherence length σ_a, truncation parameter δ and relative position s/f between the aperture and lens in general, and is independent of the cross-coherence length σ_c. After inserting a polarizer the focal switch can take place. Numerical calculation results are given to illustrate how the spherical aberration and partial coherence affect the focal shift and focal switch of PGSM beams.     

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.