Helical Ince-Gaussian modes as superpositions of Hermite-Gaussian modes

We theoretically and numerically investigate helical Ince-Gaussian modes, hIGp,q(x, y, ε). Explicit analytical expressions are derived that describe dependence of the orbital angular momentum of the helical Ince-Gaussian modes at p=2, 3, 4, 5 on the ellipticity parameter ε. For this purpose, the earlier obtained expansions of Ince-Gaussian modes in terms of Hermite-Gaussian modes are used. We demonstrate that in general the orbital angular momentum is an even function of ε, which changes non-monotonically when ε varies from zero to infinity. At zero ε, the orbital angular momentum is equal to the index q of the Ince-Gaussian mode, whereas at ε=∞, the orbital angular momentum is [q(p–q+1)]1/2. Topological charge of the helical Ince-Gaussian mode depends on ε and is equal to the index q at ε=0 and to the index p at ε=∞.