Topological charge of two parallel Laguerre-Gaussian beams

We analytically, numerically, and experimentally determine a topological charge (TC) of the sum of two axisymmetric off-axis Laguerre-Gaussian (LG) beams with the indices (0, m ) and (0, n ). In particular, we find that at m = n , the combined beam has TC = n , which suggests that the sum of two identical off-axis LG beams has the TC of an individual constituent LG beam. At m < n , the TC of the sum is found to take one of the following four values: TC 1 = ( m + n )/2, TC 2 = TC 1 + 1, TC 3 = TC 1 + 1/2, and TC 4 = TC 1 – 1/2. We also establish rules for selecting one of the four feasible values of TC. For the sum of two on-axis LG beams, TC of the superposition equals the larger constituent TC, i.e. TC = n . Meanwhile following any infinitesimally small off-axis shift, TC of the sum either remains equal to the pre-shift TC or decreases by an even number. This can be explained by an even number of optical vortices (OV) with TC = –1 instantly ‘arriving’ from infinity that compensate for the same number of OV with TC = +1 born in the superposition. We also show that when two LG beams with different parity are swapped in the superposition, the topological charge of the superposition changes by 1. Interestingly, when superposing two off-axis LG beams tilted to the optical axis so that their superposition produces a structurally stable beam, an infinite number of screw dislocations with TC = +1 are arranged along a certain line, with the total TC of the superposition equal to infinity.