Hermite–Haussian laser beams with orbital angular momentum

We consider vortex Hermite–Gaussian modes (VHG-modes) with their complex amplitude being proportional to an $n$-th order Hermite polynomial dependant on a real parameter $a$. When $|a| < 1$, there are n isolated intensity nulls on the horizontal axis in the beam’s cross-section. These nulls generate optical vortices with a topological charge of $+1$ $(a <0)$ or $-1$ $(a > 0)$. If $|a| > 1$, the VHB-mode has analogous isolated nulls on the vertical axis. When $|a| = 1$, all $n$ isolated nulls appear on the optical axis in the center of the beam and generate an $n$-th order optical vortex. In this case, the VHG-mode coincides with a Laguerre–Gaussian mode of order $(0, n)$. For $a = 0$, the VHG-mode coincides with a Hermite–Gaussian mode of order $(0, n)$. We calculate the orbital angular momentum of the VHB-modes, which depends on a parameter a and varies from $0$ (at $a = 0$ and $a \to\infty$) to $n$ (at $a = 1$).