Caustics of the vortex beams generated by vortex lenses and vortex axicons.

In this work, the propagation of vortex beams is treated using a catastrophe theory approach. Analytic expressions are deduced to describe caustic surfaces produced by vortex lenses and vortex axicons. The obtained analytics allow us to explain the formation of the shadow region along the optical axis for vortex beams using geometric optics (previously, the zero axial intensity was explained just by diffraction effects).

Reconstruction of an optical surface from a given source-target map.

We propose a new method for the reconstruction of a reflecting (refracting) surface from a given source-target map defining the relationships between the directions of incident and reflected (refracted) rays. In the proposed method, the optical surface is represented as an envelope of a set of paraboloids (reflecting surface) or ellipsoids (refracting surface). This representation allows the problem of design of an optical surface to be reduced to the reconstruction of a function from its total differential.

Calculating the energy spectrum of complex low-dimensional heterostructures in the electric field.

An algorithm for solving the steady-state Schrödinger equation for a complex piecewise-constant potential in the presence of the E-field is developed and implemented. The algorithm is based on the consecutive matching of solutions given by the Airy functions at the band boundaries with the matrix rank increasing by no more than two orders, which enables the characteristic solution to be obtained in the convenient form for search of the roots. The algorithm developed allows valid solutions to be obtained for the electric field magnitudes larger than the ground-state energy level, that is, when the perturbation method is not suitable.