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

In this work, we theoretically and numerically analyze helical Ince-Gaussian (hIG) modes, <i>h</i><i>I</i><i>G</i>_{<i>p</i>,<i>q</i>}(<i>x</i>,<i>y</i>,<i>ε</i>). We derive explicit analytical relationships to describe the <i>ε</i>-dependence (where <i>ε</i> is the ellipticity parameter) of the orbital angular momentum (OAM) of <i>h</i><i>I</i><i>G</i>_{<i>p</i>,<i>q</i>}(<i>x</i>,<i>y</i>,<i>ε</i>) modes at <i>p</i>=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 <i>ε</i> and exhibits no monotonic behavior with <i>ε</i> varying from zero to plus and minus infinity.