Книга: Topological charge of optical vortices

The book is devoted to the consideration of unusual laser beams - vortex or singular beams. They are called vortex because their energy flow rotates in a spiral around the optical axis when the beam propagates along this axis. And they are called singular because in the cross section of such beams there are isolated points with zero intensity, in which the phase is not defined. These points are singular (unusual points). Vortex laser beams are described by two integral (averaged) characteristics: orbital angular momentum (OAM) and topological charge (TC). The OAM of the beam shows with what moment of force the light will act on a microparticle placed at the focus of the vortex laser beam. And the TC of the beam shows how many complete jumps by 2π the phase will acquire in the cross section of the beam when it is traversed along a closed contour that encompasses the entire beam. If the OAM of the beam is preserved during the propagation of a scalar paraxial laser beam, then the TС is also preserved if we take into account the phase singularities located at the periphery in the beam cross section (at infinity). The OAM normalized to the beam power can be either an integer or a fractional number. And the TC of the beam (except for the initial plane and at infinity) is always integer. In the book, the TC of the axial superposition of Laguerre-Gauss (LG) vortex beams is calculated. It turned out to be equal to the maximal TC of the beams in the superposition. Therefore, if the amplitude of the light field is expanded according to the basis of the LG beams in an infinite series, then the TC of such a light field is infinite. The book contains examples of vortex beams with infinite TC. Note that the OAM of the beam is determined by the amplitude and phase, and TC is determined only by the phase of the beam. Therefore, there are light beams in which the TC is zero, and the OAM is nonzero (for example, elliptical Gaussian beams). And vice versa, there are beams in which the OAM is equal to zero, and the TC is nonzero. The