Conservation of the half-integer topological charge on propagation of a superposition of two Bessel-Gaussian beams

We theoretically show that by continuously varying the amplitude of a constituent beam in the superposition of two Bessel-Gaussian (BG) beams with topological charges (TCs) m and n the TC of the entire superposition can be controlled. Between the constituent beams, there is a topological competition. For some parameters of the BG beams, TC of the superposition equals n, while for other parameters, it is equal to m. In the center of such a superposition (on the optical axis), there is an m-fold degenerate intensity null (optical vortex with the TC of m < n), whereas n-m optical vortices with the unitary TC are in the beam periphery. We demonstrate that when the amplitudes of the constituent beams tend to be equal, peripheral vortices "move" to infinity. In this case, superposition of two BG beams has a half-integer TC, and this TC is conserved on propagation in space. The fractional part of the TC is, however, "hidden" in the infinity ("hidden" phase). Conservation of the half-integer TC of an optical vortex on propagation is shown.