Singular optics is a branch of modern physical optics that involves a wide class of effects associated with the phase singularities in wave fields and with the topology of wave fronts. Optical singularities (optical vortices) exhibit some fundamental features absent in the "usual" fields with smooth wave fronts. Namely, optical vortices possess orbital angular momentum, topological charge for helical wave front of beams with well-defined direction of propagation. As a result, an interesting spatial evolution can be generated such as optical vortices "nucleation" and "annihilation" by pairs with participation of phase saddles, often called "optical chemistry." To study the structure of the circular edge dislocation, an isolated dark (zero-amplitude) ring is created within a light beam, with an analytical description of the field. A screw wave-front dislocation has a feature that the spatial structure of the wave front has the form of a helicoid around the dislocation axis. The chapter also describes reflection, refraction, interference and diffraction of OVs. Both frequency up- and down-conversion processes possess essential peculiarities for light beams with OVs. The chapter discusses the topology of wave fronts and vortex trajectories. Gouy phase shift in singular optics is also described in the chapter.

Chapter 4 - Singular optics

Chapter 5 - Optical vortices and vortex solitons

1 What is a liquid crystal 2 1.1 Nematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Cholesterics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Smectics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 Columnar phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.5 Lyotropic liquid crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Introduction to Liquid Crystals

Introduction to liquid crystals

Optical vortices are phase singularities nested in electromagnetic waves that constitute a fascinating source of phenomena in the physics of light and display deep similarities to their close relatives, quantized vortices in superfluids and Bose-Einstein condensates. We present a brief overview of the major advances in the study of optical vortices in different types of nonlinear media, with emphasis on the properties of {\em vortex solitons}. Self-focusing nonlinearity leads, in general, to the azimuthal instability of a vortex-carrying beam, but it can also support novel types of stable or meta-stable self-trapped beams carrying nonzero angular momentum, such as ring-like solitons, necklace beams, and soliton clusters. We describe vortex solitons created by multi-component beams, by parametrically coupled beams in quadratic nonlinear media, and in partially incoherent light, as well as discrete vortex solitons in periodic photonic lattices.

Optical Vortices and Vortex Solitons

The physical background of singular optics dealing with phase singularities in light waves is reported as a new chapter in modern optics. A review of the current status of nonlinear singular optics is presented.

M V Vasnetsov

Papers

Nonlinear singular optics