Orbital angular momentum of structurally stable laser beams

We analyze different types of structurally stable paraxial laser beams characterized by a finite number of cross-section intensity nulls. In more detail, we study structurally stable beams whose complex amplitude is described by an nth degree complex polynomial with a Gaussian envelope. We derive a relationship for the orbital angular momentum that the beams carry and show that given arbitrary complex coefficients of the polynomial, their topological charge is defined by the polynomial degree, n. By way of illustration, we analyze a rotating Laguerre-Gaussian beam whose topological charge is described by the radial index of a Laguerre polynomial. We show that this beam has non-zero on-axis intensity, with all intensity nulls arranged on a line on the right of the optical axis, rotating anticlockwise by an angle of π/2 upon the free space propagation.