Fractional two-parameter parabolic diffraction-free beams

Abstract

We consider two-parameter real-valued parabolic diffraction-free beams. An analytical expression for the complex amplitude of the beam is obtained using the superposition of the Bessel functions. Symmetry properties of beams are analyzed and numerically demonstrated for different cases. The variety of the formed pictures increases significantly in comparison with earlier regarded parabolic beams. Particularly interesting properties exhibit beams with half-integer indices, they have symmetry properties different from the symmetry of beams with integer indices. For an arbitrary fractional index, the beams have a more complex distribution and do not have any symmetry. This fact facilitates the solution of the inverse problem when it is necessary to form similar structures, since just small changes in two parameters of the analytical expression are needed. This is much simpler than the implementation of iterative procedures, which, as a rule, provide essentially non-smooth solutions that are complex in optical implementation. Experimental generation of the fractional two-parameter parabolic beams is implemented for the first time by a spatial light modulator using phase coding of the calculated complex amplitudes. The different types of experimentally generated parabolic beams are in a good agreement with the modeling results.