This book examines two-dimensional light fields (sinc-beams, Airy beams, Hankel beams, and Bessel beams with fractional order), in which the reverse energy flow is formed near isolated intensity nulls. That is, near those points where the energy (power) density is zero. A reverse canonical energy flow has also been found in three-dimensional nonparaxial vector light fields with linear, azimuthal, and hybrid polarizations. The reverse energy flow is formed in the near field, in the presence of evanescent waves, as well as in the region of a sharp focus, where there is no evanescent wave. Interestingly, the reverse energy flow takes place in the near field of a Hertzian linear point dipole. This book is devoted to an interesting and largely unexplored effect in optics: the reverse energy flow of light. It examines two-dimensional and three-dimensional nonparaxial vector light fields, in which a reverse canonical energy flow is formed in certain spatial regions. The propagation of coherent monochromatic light is described in several ways: by decomposition into a plane-wave spectrum and by Rayleigh–Sommerfeld vector integrals. Projections of the electric vector in a sharp focus are described using Richards–Wolf theory and Debye integrals. The book presents precise analytical expressions for the longitudinal projections of the canonical energy flow vector, which strictly prove the presence of a reverse flow in the light fields under consideration. The book includes the results that the authors obtained in 2024-2025. The book will be of interest to a wide range of scientists, engineers working in the field of optics, photonics, laser physics, opto-information technologies, and optical instrumentation. It can also be useful for bachelors and masters in the specialties applied mathematics and physics, applied mathematics and informatics, optics and graduate students specializing in these areas.

Victor Kotlyar is Head of the Laboratory at Image Processing Systems Institute of the National Research Center “Kurchatov Institute” and Professor of computer science at Samara National Research University. He received his M.S., Ph.D., and Dr.Sc. degrees in physics and mathematics from Samara State University (1979), Saratov State University (1988), and Moscow Central Design Institute of Unique Instrumentation, the Russian Academy of Sciences (1992). He is Co-Author of 400 scientific papers, 10 books, and 7 inventions. His current interests are diffractive optics and singular optics.

Alexey Kovalev has graduated (2002) from Samara National Research University, majoring in Applied Mathematics. He received his doctor in physics and maths degree in 2012. He is Senior Researcher of Laser Measurements Laboratory at Image Processing Systems Institute of the National Research Center “Kurchatov Institute” and Associate Professor of computer science at Samara National Research University. He is Co-Author of more than 270 scientific papers and 4 book. His research interests are mathematical diffraction theory and optical vortices.

Sergey Stafeev (b. 1985) received master’s degree in applied mathematics and physics in Samara National Research University (2009). He received his Ph.D. in 2012. He is Researcher of Laser Measurements Laboratory at the Image Processing Systems Institute, NRC “Kurchatov Institute.” His scientific interests include diffractive optics, nonuniform polarization, tight focusing.