Generation of an optical vortex immediately behind a spherical lens upon its illumination by circularly polarized light

We analyze various types of structurally stable paraxial laser beams with a finite number of intensity nulls. In more detail, we study structurally stable beams with complex amplitude equal to a complex polynomial of degree n with a Gaussian envelope. We obtain an expression for the orbital angular momentum of such beams and show that for arbitrary complex polynomial coefficients, the topological charge of such beams is equal to the degree of the polynomial, n. As an example of such beams, a rotating Laguerre-Gaussian beam is investigated, whose topological charge is equal to the radial index of the Laguerre polynomial. On the optical axis, such a beam has a nonzero intensity, while all intensity nulls lie on a straight line to the right from the optical axis and rotate counterclockwise by an angle of π/2 upon free-space beam propagation.