Helical Ince-Gaussian laser beams as superpositions of Hermite-Gaussian beams

In this work, we theoretically and numerically analyze helical Ince-Gaussian (hIG) modes, hIG_p,q(x,y,ε). We derive explicit analytical relationships to describe the ε-dependence (where ε is the ellipticity parameter) of the orbital angular momentum (OAM) of hIG_p,q(x,y,ε) modes at p=2,3,4,5. The derivation procedure relies on expansions of the hIG modes in terms of Hermite-Gaussian modes. It is shown that in the general case, the OAM is an even function of ε and exhibits no monotonic behavior with ε varying from zero to plus and minus infinity.