Three different types of astigmatic Hermite-Gaussian beams with orbital angular momentum

Abstract

In this work, we study three different types of astigmatic (anisotropic) Hermite-Gaussian (aHG) beams whose complex amplitude in the Fresnel diffraction zone is described by the complex argument of a Hermite polynomial of degree (n, 0). The first-type beam is a circularly symmetric optical vortex with topological charge n that has passed through a cylindrical lens. The outgoing optical vortex ‘splits’ into n first-order optical vortices, carrying an orbital angular momentum (OAM) per photon of n. The second-type beam is a Hermite-Gaussian (HG) beam of order (n, 0) generated by passing an elliptic HG beam through a cylindrical lens, which acts by imprinting an OAM into the original HG beam. The OAM of such a beam is a sum of vortex and astigmatic components and can reach large values (tens and hundreds of thousands per photon). We derive an exact formula to describe the OAM of these aHG beams. Under certain conditions, the zero intensity lines of the aHG beam ‘merge’ into an n-fold degenerate intensity null on the optical axis, with the OAM of the beam becoming equal to n. The third type is an elliptical optical vortex with topological charge n that has passed through a cylindrical lens. With a special choice of the ellipticity degree (1:3), the beam retains its structure upon propagation, with the on-axis degenerate intensity null not ‘splitting’ into n optical vortices. The beam is shown to carry a fractional OAM not equal to n. Using intensity distributions of the aHG beam in the foci of two cylindrical lenses, we measure a normalized OAM of the elliptic aHG beam, with the deviation from a theoretical estimate being as low as 7% (experimental OAM-13.62 and theoretical OAM-14.76).