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Radially polarized Lorentz-Gauss vortex beam with sine-azimuthal variation wavefront was proposed and its focusing properties were investigated by vector diffraction theory. The effect of phase parameter on focal pattern is very considerable, and some novel intensity pattern comes into being under certain focusing condition, including multiple-peaks, crescent shape, multiple intensity lines, intensity cross. Especially, the symmetry characteristics are systematical on increasing phase parameter. For case of even number of phase parameter, there are two symmetric axes for whole focal pattern, while there is only one symmetric axis for odd number of phase parameter. When topological charge is small, the intensity maximum shifts within small distance range, while intensity maximum may switch for higher topological charge in focal pattern evolution process.

Focusing properties of linearly polarized Lorentz-Gaussian vortex beams was investigated by vector diffraction theory, from which topological charge determination method is proposed. Results show that the focal pattern can be altered considerably by topological charge, and for certain condition, the number of dark spots along one coordinate direction equals to topological charge, and the number of intensity lines crossing focal pattern along the other coordinate direction minus topological charge is one. The intensity lines across focal pattern get weaker for higher topological charge, which leads to the boundaries among dark spots become blurring. Focal shift value of the most out dark peak increases on increasing topological charge, and the dependence curve is monotonous. Topological charge can be obtained according to dependence curve even for fractional topological charge.

Based on the diffractive vector theory, the focusing properties of annular cylindrical vector beams generated by the novel polarized convertor are investigated. By the combination of the vortex phase plate eimφ and the phase variety parameter cosφ + ihsinφ which exists naturally in the CV beams, the intensity pattern at the focus can be tailored. Flattop spot and light intensity with pure transverse and longitudinal component are formed in the focal region. More interestingly, tunable dark focal spot is also achieved. By changing h, the light intensity in the center of spot can be changed at will while the outer ring of spot remains the same. This works are important for microscope imaging and optical lithography, particle manipulation, electric acceleration etc.

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.

Half-space amplitude modulation was introduced into the linearly polarized cosh-Gaussian beams with sine-azimuthal variation wavefront, and corresponding tunable focal patterns were investigated in detail. Simulation results show that focal pattern can be changed remarkably by the amplitude modulation. And under condition of different beam parameters and phase parameter that indicates the sine-azimuthal variation wavefront, the focal pattern evolution also differs considerably. Some novel focal shape may appear, including cross-shape, multiple-peak focal pattern, multiple intensity lines, wheel focal pattern, bell shape, and swallowtail focal spots. Symmetry of the whole focal pattern can be altered remarkably, the whole focal pattern evolves into uniaxial symmetry about perpendicular axes for odd and even phase parameter, respectively.

The pure phase plate was introduced to alter the half-space phase value of the incident linearly polarized hyperolic-cosine-Gaussian (cosh-Gaussian) beams with sine-azimuthal variation wavefront, and focusing properties of the cosh-Gaussian beams by half space phase plate were investigated in detail. Simulation results show that focal pattern can be altered very considerably by the phase plate under condition of different beam parameters, numerical aperture, and phase parameter that indicates the phase change frequency on increasing azimuthal angle. Symmetry of the whole focal pattern can be altered remarkably, polysymmetrical focal patterns evolve into nonsymmetrical focal patterns. And some novel focal shape may appear, including cross-shape, multiple-peak focal pattern, multiple intensity lines, wheel focal pattern, swallowtail focal pattern, and dark hollow focal spots.