This paper discusses the sharp focusing of homogeneously polarized laser beams in a uniaxial crystal. It is shown analytically and numerically that, when homogeneously polarized radiation is sharply focused along the axis of a crystal, two foci are formed, corresponding to the ordinary and extraordinary rays. The introduction of a vortex phase into the incident beam results in the formation of zero intensity at the center of the focus of the ordinary rays and a longitudinal electric-field component at the center of the focus of the extraordinary rays. The contribution of the longitudinal component is greater, the sharper the focusing. With circular polarization, larger amplification of the longitudinal component is observed, and its intensity becomes comparable with that of the transverse components. This results in the formation of a focus of the extraordinary rays with a flat vertex.

The methods of generation of distributions with vortical phase singularity are considered. Especially such topics were attended as using of the fractional Fourier transformation and transformation of Airy similar distributions.

In the article it is shown that the birefringence effect is well visualized at the expense of special structure of a laser beam with using Bessel laser beams. On the base of diffraction image variances of Bessel beams on birefringence surface determination of various parameters of a crystal with unknown characteristics is possible.

In this article propagation of laser beams in anisotropic crystals at an angle to a crystal axis also is analytically and numerically investigated. For an uniaxial crystal with use of a method of a stationary phase the analytical expressions showing dependence of astigmatic distortion from polarisation of the falling bunch are received. Also it is shown that a laser beam propagating at an angle to a crystal axis undergoes the birefringence effect which can disappear in accordance with of a plane of an inclination of a beam and a plane of linear polarisation. For Gaussian beams the expression connecting parametres of a beam and a crystal at which there is visual division of ordinary and extraordinary beams is received.

Astigmatic transformation of one dimensional distributions, such as Airy functions, is considered. Various types of astigmatic converters, including one cylindrical lens are considered. As a result of such transformations it is possible to form matrices of optical vortices of a various configuration, which, in particular, expands possibilities of an optical micromanipulation. Results of modelling and natural experiments are discussed.