Sergey G. Volotovskiy

An analytical approach is developed for calculating a phase diffraction axicon with angularly variable spatial frequency that generates a prescribed light curve without iteratively solving an inverse design problem: the axicon frequency function is obtained directly from the polar equation of the target curve. When the axicon is combined with a lens, the intended curve is formed in the focal plane (Fraunhofer region), whereas in the Fresnel zone the intensity pattern is closely related to the curve’s evolute. We systematize this correspondence for rings, ellipses, spirals and polygonal curves, and derive analytical parameter bounds associated with the appearance of inflection points that qualitatively reshape both the evolute and the diffracted field (including hybrid curve–evolute patterns and “rose”-type contours). The qualitative outcome depends on the curve class/type. For a convex curve without inflections the expected curve is formed in in the far field (or in the focal plane), whereas the Fresnel intensity envelope follows the evolute. If the curve has peculiarities, especially inflection points with unbounded evolute branches, hybrid curve–evolute structure appears, and the focal pattern may itself resemble the evolute. For ellipses and polygonal families we analytically identify parameter intervals that mark this transition. The approach is verified experimentally for polygonal axicons, with attention to fabrication tolerances and to matching camera planes with the simulated propagation distances. Measured intensity distribution patterns agree with the corresponding simulations for both convex contours and curves that contain inflection points. The results are of fundamental interest and of practical value for optical trapping and laser surface structuring, where a compact phase element should deliver a controlled bright contour at a chosen working distance.

In this paper, we investigate the possibility of recognizing a superposition of wave aberrations from a focal pattern based on a matrix of partial sums. Due to the peculiarities of the focal pattern, some types of the considered superpositions are recognized ambiguously from the intensity pattern in the focal plane by standard error-reduction algorithms. It is numerically shown that when recognizing superpositions of Zernike functions from the intensity pattern in the focal plane, the use of step-by-step optimization in combination with the Levenberg–Marquardt algorithm yields good results only with an initial approximation close to the solution. In some cases, the root mean square reaches 0.3, which is unacceptable for precise detection in optical systems that require prompt correction of aberrations in real time. Therefore, to overcome this drawback, an algorithm was developed that considers partial sums, which made it possible to increase the convergence range and achieve unambiguous recognition results for aberrations (root mean square does not exceed 10⁻⁸) described by superpositions of Zernike functions up to n = 5.

We propose a method for generating three-dimensional light fields with given intensity and phase distributions using purely phase transmission functions. The method is based on a generalization of the well-known approach to the design of diffractive optical elements that focus an incident laser beam into an array of light spots in space. To calculate purely phase transmission functions, we use amplitude encoding, which made it possible to implement the designed elements using a single spatial light modulator. The generation of light beams in the form of rings, spirals, Lissajous figures, and multi-petal “rose” distributions uniformly elongated along the optical axis in the required segment is demonstrated. It is also possible to control the three-dimensional structure of the intensity and phase of the shaped light fields along the propagation axis. The experimentally generated intensity distributions are in good agreement with the numerically obtained results and show high potential for the application of the proposed method in laser manipulation with nano- and microparticles, as well as in laser material processing.

We propose to use a refractive conical axicon for generating azimutally polarized beams. We investigate polarization states of optical rays passing through an interface between optical media, and also polarization transformation with a refractive axicon. We develop a software for raytracing which correctly processes polarization states of the rays and visualizes ellipses of polarization. The polarization state is described in the Jones notation and based on the energy conservation law. We derive and implement formulas for calculating the Jones vector in different bases, as well as trans-ferring the Jones vector from one basis to another. Algorithms for displaying polarization ellipses on one plane for beams that are not plane-parallel have been developed. Ray paths in a three-dimensional axicon are calculated and shown with due regard for polarization.

The use of a multichannel wavefront sensor matched with phase Zernike functions to determine the type and magnitude of aberration in the analyzed wavefront is investigated. The approach is based on stepwise compensation of wavefront aberrations based on a dynamically tunable spatial light modulator. As criteria for successful detection, not only the magnitude of the correlation peak, but also the maximum intensity, compactness, and orientation of the distribution in each diffraction order are considered. On the basis of numerical simulation, the efficiency of the proposed approach is shown for detecting both weak and strong (up to a wavelength) wavefront aberrations.

In this paper, we consider the comparative formation of perfect optical vortices in the non-paraxial mode using various optical elements: non-paraxial and parabolic toroidal vortex lenses, as well as a vortex axicon in combination with a parabolic lens. The theoretical analysis of the action of these optical elements, as well as the calculation of caustic surfaces, is carried out using a hybrid geometrical-optical and wave approach. Numerical analysis performed on the basis of the expansion in conical waves qualitatively confirms the results obtained and makes it possible to reveal more details associated with diffraction effects. Equations of 3D-caustic surfaces are obtained and the conditions of the ring radius dependence on the order of the vortex phase singularity are analyzed. In the non-paraxial mode, when small light rings (several tens of wavelengths) are formed, a linear dependence of the ring radius on the vortex order is shown. The revealed features should be taken into account when using the considered optical elements forming the POV in various applications.

We propose binary diffractive optical elements, combining several axicons of different types (axis-symmetrical and spiral), for the generation of a 3D intensity distribution in the form of multiple vector optical ‘bottle’ beams, which can be tailored by a change in the polarization state of the illumination radiation. The spatial dynamics of the obtained intensity distribution with different polarization states (circular and cylindrical of various orders) were investigated in paraxial mode numerically and experimentally. The designed binary axicons were manufactured using the e-beam lithography technique. The proposed combinations of optical elements can be used for the generation of vector optical traps in the field of laser trapping and manipulation, as well as for performing the spatial transformation of the polarization state of laser radiation, which is crucial in the field of laser-matter interaction for the generation of special morphologies of laser-induced periodic surface structures.

We performed a detailed comparative study of the parametric high degree (cubic, fourth, and fifth) power phase apodization on compensation defocusing and chromatic aberration in the imaging system. The research results showed that increasing the power degree of the apodization function provided better independence (invariance) of the point spread function (PSF) from defocusing while reducing the depth of field (DOF). This reduction could be compensated by increasing the parameter α; however, this led to an increase in the size of the light spot. A nonlinear relationship between the increase in the DOF and spot size was shown (due to a small increase in the size of the light spot, the DOF can be significantly increased). Thus, the search for the best solution was based on a compromise of restrictions on the circle of confusion (CoC) and DOF. The modeling of color image formation under defocusing conditions for the considered apodization functions was performed. The subsequent deconvolution of the resulting color image was demonstrated.