Orbital angular momentum of an astigmatic Gaussian laser beam

We show that an elliptic Gaussian beam focused by a cylindrical lens can be represented as a linear combination of a countable number of only even angular harmonics with both positive and negative topological charge. For the orbital angular momentum of an astigmatic Gaussian beam, an exact expression is obtained in the form of a converging series of the Legendre functions of the second kind. It is shown that at some conditions only terms with the positive or negative topological charge are retained in this series.