Two-dimensional asymmetric Laguerre-Gaussian diffraction-free beams

We find asymmetric diffraction-free Laguerre-Gaussian solutions of the two-dimensional paraxial wave equation in free space. They are constructed by the Laguerre polynomials and trigonometric functions, and described by the radial and angular mode numbers, and modulation depth. Different optical field distributions of the asymmetric diffraction-free Laguerre-Gaussian beams are presented and analyzed. By selecting appropriate angular mode numbers, we identify novel types of asymmetric diffraction-free beams, obtained by the linear superposition of solutions, such as asymmetric vortex and asymmetric necklace-ring beams. Intensity distribution of the asymmetric diffraction-free beams at four times Rayleigh range with different mode parameters is presented analytically and simulated numerically. This research provides theoretical basis for possible experimental verification and utilization of asymmetric diffraction-free beams in free space or uniform transparent dielectrics.