It is common knowledge that light or light rays in vacuum or uniform media travel along straight lines with velocity $c$ or $c/n$, where $c=299,792,458 m/s$ (exactly to the meter!) and $n$ is the index of refraction of the medium [1]. It is a less common knowledge that light or light beams of different forms can follow curved paths, can accelerate or decelerate, and even come to a standstill in certain media [2]. While it is expected that such a propagation of light more easily happens in strange non-uniform media, it is nonetheless true that it can happen in free space as well, and that the trajectories of such light beams can be arbitrary. This field of physical optics, known as the field of non-diffracting self-accelerating beams, started in the 21st century [3] and is flourishing by now. Although in some aspects quite complex, in essence it is a linear physical theory that allows treatment in the inverse or Fourier space. In this presentation we will address its beginnings, essential features, most prominent examples – and how to complicate matters by including nonlinearity. The examples include Airy, Bessel, Weber, Mathieu, and other beams, as well as examples based on the fractional Schrödinger equation [4, 5].
References
[1] M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge UP, 1999); B.E.A. Saleh M.C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley & Sons, 2019).
[2] H.E. Hernandez-Figueroa, E. Recami, and M.Z. Zambroni-Rached, eds., Non-diffracting Waves (Wiley-VCH, 2015).
[3] G. A. Siviloglou and D. N. Christodoulides, Accelerating Finite Energy Airy Beams, Opt. Lett. 32, 979 (2007); M.A. Bandres, I. Kaminer, M. Segev and D.N. Christodoulides, Accelerated Optical Beams, OPTICS & PHOTONICS NEWS, JUNE 2013.
[4] S. Longhi, Fractional Schrödinger Equation in Optics, Opt. Lett. 40, 1117 (2015); N. Laskin, Fractional Quantum Mechanics (World Scientific, 2018).
[5] Y.Q. Zhang, X. Liu, M.R. Belić, W.P. Zhong, Y.P. Zhang, and M. Xiao, Propagation Dynamics of a Light Beam in a Fractional Schrödinger Equation, PRL 115, 180403 (2015);
$\phantom{[5]} $ Y.Q. Zhang, H. Zhong, M.R. Belić, and Y.P. Zhang, Guided Self-Accelerating Airy Beams—A Mini-Review. Appl. Sci. 7, 341 (2017).