Topological charge of axial superposition of Gaussian optical vortices

The topological charge of a finite superposition of optical vortices with a Gaussian envelope is considered. It is shown theoretically and numerically that in the initial plane the topological charge of such a superposition is equal to the number of zeros of a complex polynomial of degree n, where n is the maximum topological charge of optical vortices in the superposition located in a unit radius disk including its boundary. When propagating in free space, the topological charge of such a superposition is always equal to n. If the modulus of the coefficient of the superposition term with the topological charge equal to k is greater than the sum of the moduli of all other coefficients of the superposition, then k zeros lie in the unit radius disk and the topological charge of the entire superposition in the initial plane is equal to k (k ≤ n). If all coefficients of the superposition are equal in modulus, then in the initial plane the topological charge is equal to half (n/2), but during propagation the topological charge is again equal to n. In this case, additional zeros of the superposition of optical vortices are formed almost immediately at a distance much smaller than the wavelength from the initial plane and at a distance from the optical axis greater than the radius of the limiting aperture of the initial field.