Orbital angular momentum of an astigmatic Hermite-Gaussian beam

An explicit formula for the normalized orbital angular momentum (OAM) of an elliptical Hermite-Gaussian (HG) beam of orders $(0, n)$ focused by a cylindrical lens is obtained. In modulus, this OAM can be both greater and smaller than $n$. If the cylindrical lens focuses not an elliptical, but a conventional HG beam, the latter will also have an OAM that can be both larger and smaller in modulus than that of an elliptical HG beam. For $n = 0$, this beam converts to an astigmatic Gaussian beam, but, as before, it will still have OAM. With the help of two interferograms, a phase of the astigmatic Gaussian beam is reconstructed, which is then used to calculate the normalized OAM. The values of the OAM calculated by the theoretical formula and using a hybrid method combining modeling with experiment differ only by $6 \%$.