Spatio-temporal optical vortices: principles of description and basic properties

These notes appeared as a summary during the initial acquaintance with the structure and properties of spatio-temporal optical vortices (STOV), as these are presented in the current literature. The general approach to the STOV description and characterization is based on the model of scalar paraxial Gaussian wave packet. On this ground, the STOV structures of arbitrary orders are considered as superpositions of spatio-temporal Hermite-Gaussian modes. This representation enables the direct and physically meaningful characterization of the STOV amplitude and phase distributions, their evolution upon free propagation and in optical systems, internal energy flows and the orbital angular momentum. The topologically determined asymmetry and the difference between the "energy center" and "probability center" [Phys. Rev. A 107, L031501 (2023)] is discussed and qualitatively interpreted. Methods for the STOV generation and diagnostics are outlined, and the main properties of non-Gaussian (Bessel-type) STOVs are briefly described. The notes may be useful as initial introduction to the STOV-associated ideas and their extraordinary properties.