Diffraction-free beams in fractional Schrödinger equation

Fractional effects, such as fractional quantum Hall effect1, fractional Talbot effect2, fractional Josephson effect3 and other effects coming from the fractional Schrodinger equation (FSE)4, introduce seminal phenomena in and open new areas of physics. FSE, developed by Laskin5,6,7, is a generalization of the Schrodinger equation (SE) that contains fractional Laplacian instead of the usual one, which describes the behavior of particles with fractional spin8. This generalization produces nonlocal features in the wave function. In the last decade, research on FSE was very intensive9,10,11,12,13,14,15,16,17. Even though the equivalence of SE and paraxial wave equation is known for long, not until 2015 was the concept of FSE introduced in optics18, by Longhi. In the paper, he realized a scheme to explore FSE in optics, by using aspherical optical cavities. He has found eigenmodes of a massless harmonic oscillator, the dual Airy functions13,18. Airy wave function, the eigenmode of the standard SE in free space, does not diffract during propagation. This feature was also firstly discovered in quantum mechanics19 and then brought into optics20. In light of peculiar properties of an Airy beam, which include self-acceleration, self-healing and the absence of diffraction, a lot of attention has been directed to accelerating diffractionless beams in the last decade. For more information, the reader is directed to review article21. On the other hand, earlier literature on FSE mostly focused on mathematical aspects of the eigenvalue problem of different potentials, such as the massless harmonic oscillator. Although the harmonic potential and other potentials22,23,24,25 are of high interest, here we are not interested in the eigenvalue problem of potentials. Rather, we focus on the dynamics of beams in FSE without any potential. Even the simplest problem of what happens in FSE without a potential is still interesting to be explored. Are there nondiffracting solutions? Do such solutions accelerate during propagation? Are the solutions self-healing? These questions are addressed in this paper. In this article we investigate the dynamics of waves in both one-dimensional (1D) and two-dimensional (2D) FSE without a potential. To avoid complexity of the mathematical problem of fractional derivatives, we take Gaussian beams as an example and make the analysis in the inverse space. Methods introduced here apply to other beams as well. An approximate but accurate analytical solution to the problem is obtained, which agrees well with the corresponding numerical simulation. An alternative method, based on the factorization of wave equation, is also introduced. Even though the overall analysis appears deceptively simple, the results obtained point to profound changes in the propagation of beams in FSE, as compared to the regular SE. We discover that a Gaussian beam without chirp splits into two diffraction-free Gaussian beams in the 1D case and undergoes conical diffraction in the 2D case. If the input Gaussian beam is chirped, it also propagates diffraction-free and exhibits uniform motion. This uniform propagation is not much affected by the chirp. Along the way, we also introduce the fractional Talbot effect of diffractionless beams in FSE.