Topological Charge of Co-Axial Superposition of Gaussian Optical Vortices

In this work, we analyze the topological charge (TC) of finite superposition of optical vortices (OVs) with a Gaussian envelope. In the source plane, the superposition under study is theoretically and numerically shown to have the TC equal to the number of zeros of a complex polynomial of degree n, where n is the largest TC of the constituent OVs found inside and on a unit-radius circle. Meanwhile upon free space propagation, the TC of the superposition always equals n. We reveal that if, in absolute values, the coefficient of a superposition term with TC = k is larger than the sum of all the rest superposition coefficients, then k zeros occur inside the unit-radius circle, with the total TC of the superposition being equal to k (k ≤ n) in the source plane. If all the coefficients are equal to each other in the absolute value, then, in the source plane, TC takes a value of n/2, before returning to the value of n upon propagation. In this case, extra zeros of the superposition of OVs occur almost at once, at a subwavelength distance from the source plane, with the distance from the optical axis being larger than the radius of an aperture limiting the source field.