Diffraction-free beams in fractional Schrödinger equation

Area of Science:

• Quantum mechanics
• Nonlinear optics
• Mathematical physics

Background:

• The fractional Schrödinger equation (FSE) describes quantum systems with long-range interactions.
• Gaussian beams are fundamental solutions in wave propagation studies.
• Understanding beam dynamics in FSE is crucial for applications in optics and quantum physics.

Purpose of the Study:

• To analytically and numerically investigate the propagation of 1D and 2D Gaussian beams in the FSE without a potential.
• To explore the effects of linear chirp on beam trajectories and diffraction properties.
• To introduce and analyze the Talbot effect for diffractionless beams within the FSE framework.

Main Methods:

• Analytical solutions for Gaussian beam propagation in FSE.
• Numerical simulations to verify analytical findings.
• Mathematical analysis of beam splitting, deflection, and diffraction.

Main Results:

• 1D Gaussian beams split into two nondiffracting beams without chirp.
• 2D Gaussian beams exhibit conical diffraction without chirp.
• Chirped 1D beams deflect along specific trajectories, independent of chirp.
• Chirped 2D beams deflect along diffraction cones, with direction dependent on chirp.
• Both 1D and 2D beams demonstrate diffractionless and uniform propagation.
• The Talbot effect for diffractionless beams in FSE is introduced.

Conclusions:

• Gaussian beams exhibit unique diffractionless and splitting/deflecting behaviors in the FSE.
• The study reveals controllable beam steering via chirp in 2D propagation.
• The findings extend the understanding of wave phenomena in fractional media and introduce a new Talbot effect.