The analytical solution for the propagation of the laser beam with optical vortex through the system of lenses is presented. The optical vortex is introduced into the laser beam (described as Gaussian beam) by spiral phase plate. The solution is general as it holds for the optical vortex of any integer topological charge, the off-axis position of the spiral phase plate and any number of lenses. Some intriguing conclusions are discussed. The higher order vortices are unstable and split under small phase or amplitude disturbance. Nevertheless, we have shown that off-axis higher order vortices are stable during the propagation through the set of lenses described in paraxial approximation, which is untypical behavior. The vortex trajectory registered at image plane due to spiral phase plate shift behaves like a rigid body. We have introduced a new factor which in our beam plays the same role as Gouy phase in pure Gaussian beam.

In this work we present the principles of recovering the phase shift introduced by the phase sample in an optical vortex scanning microscope. In this microscope a Gaussian beam with an embedded vortex illuminates the sample and then is imaged onto the observation plane of the microscope. When the sample interacts with the vortex beam, its internal structure is modified. In the case of a uniform phase change inside the focused vortex spot, the modification manifests itself as a rotation of the phase image. We show that using a vortex microscope setup we can obtain a high accuracy of the measured phase shift. The second advantage of our system is its very precise tracking of the vortex point.

A multi-vortex system nested in a Gaussian beam forms complex paths when it propagates in a free space. We consider a configuration of vortex quadrupole and present a systematic way to control the creation and annihilation of vortex pairs using only one parameter. It is shown that three classes of quadrupole vortex beams can be identified by the number of vortex pairs created or annihilated. The beam symmetry, asymptotic behavior, and stability are analyzed. The main theoretical results are verified experimentally.

We study the system of phase singularities (‘singular skeleton’) formed when a quasi-plane wave (QPW) input beam diffracts at a double-phase-ramp (DPR) converter. The external (OV positions and 3D trajectories) and internal (ellipticity and orientation angle of the equal-intensity ellipses in the OV-core area) singular-skeleton features are investigated both theoretically and in experiments. The results are presented in comparison with the singular skeleton formed by the DPR when the incident beam is Gaussian. In contrast to the limited number of OVs in the synthesized OV chain, divergent 3D OV trajectories and variable OV morphologies depending on their off-axis distances, which is typical for the Gaussian input beam, it is shown that the QPW-generated diffracted beam carries a rectilinear chain of equidistant optical vortices (OV) with identical morphology parameters. Such singular-skeleton configurations can be useful for the applications to metrology and micromanipulation, in particular, for the multi-particle optical trapping and guiding.

In this paper, we characterize the helical beam structure through an analysis of the spiral character of the phase distribution inside a light beam. In particular, we show that a line connected with the 2π phase jump in the Laguerre–Gauss beam can be described by a Fermat’s spiral. We propose a numerical fitting method to determine the parameters of a spiral equation for the phase distribution of the helical beam. Next, we extend the procedure to a vortex beam created by the spiral phase plate and apply it to experimental phase maps, which allows us to recover the phase shift introduced into the object beam in the optical vortex scanning microscope.

We study positions and morphology of optical vortices (OVs) within the field obtained after transmitting a circular single-charged OV-beam through the transparent screen with a rectilinear π -phase step. Experimental results are obtained with the help of a programmable spatial light modulator which is used for the Kummer beam formation and for introduction of the π -step phase difference at a desirable position within the incident beam cross section. The theory based on the Kirchhoff–Fresnel approximation shows a good agreement with the experimental data; peculiar features of the Kummer beam diffraction are elucidated in the course of confrontation against the results involving the Laguerre–Gaussian beam model with the same transverse size and spherical wavefront component. The diffracted field contains a system of interacting OVs (singular skeleton) which demonstrate a regular evolution (migration) within the diffracted beam cross section while the π -phase step translates across the incident beam; depending on the step position, new OV pairs may emerge and/or annihilate in the topological reactions. The morphology parameters of the separate diffracted-field OVs (orientations and form-factors of the near-core equal-intensity ellipses) also depend on the stage of the OV evolution and indicate conditions favorable for the efficient trapping and guiding of microparticles. The results may be useful for the diagnostics of OVs, experimental measurements of phase objects and in micromanipulation techniques.