Fractional Fourier transform of astigmatic sine-Gaussian beams and generation of dark hollow light beams with elliptic geometry

In this work, we develop a novel method of creating dark hollow beam with vortex by converting a sine-Gaussian beam (SeGB) with edge-dislocation and astigmatism through using fractional Fourier transform (FrFT) optical system. On the basis of the definition of the FrFT, an analytical transformation formula is derived for an astigmatic SeGB passing through such a transform system. By use of the derived formulae, the changes of the intensity distribution and the corresponding phase properties associated with the transforming astigmatic SeGBs are analytically discussed in detail. It is found that for an input SeGB without astigmatism, there is still a dark line or an edge dislocation associated with the intensity distribution of the FrFT beam along the initial dislocation line, similar to that of the input SeGB. However, when the input SeGB astigmatically passes through an FrFT optical system, the dark line of the intensity distribution of the input SeGB can be converted into a solitary zero point, or in other words, a dark hollow beam with a single-charge vortex can be produced by SeGB with an edge dislocation. The results reveal that the astigmatism plays a critical role in transforming a SeGB into a dark hollow one through the FrFT optical system. Furthermore, some numerical calculation results based on the derived formula are presented and discussed graphically. It is shown that for appropriate beam parameters and carefully adjusting the transform angle of FrFT, dark hollow beams with single-charge vortex and elongated elliptic geometry can be realized with astigmatic SeGBs. The influences of the beam parameters and the transform angle of FrFT optical system on the generation of perfect dark hollow beams are also investigated. The results demonstrate that the linear eccentricity of the dark hollow beam, which is roughly defined as the ratio of semi-minor axis to semi-major one of the intensity pattern, mainly depends on the Fresnel number. And the optimal linear eccentricity may be relatively large under carefully selecting the beam and optical system parameters. Moreover, optimal parameter values corresponding to perfect dark hollow beam configurations which can be experimentally accessed are presented. As is well known, there are two types of pure phase defects or dislocations in the optical fields: one is screw dislocation or vortex and the other is edge-dislocation. Due to their important applications, the propagation dynamics of optical vortices or edge dislocations are extensively studied both theoretically and experimentally. The vortex-edge dislocation interaction is investigated in detail. However, there are fewer reports on the direct conversion between a single edge dislocation and a vortex. Therefore, the results obtained in this paper represent a significant step forward in understanding the transformation dynamics between beams with pure edge dislocation and vortex, and also opens possibilities for their potential applications, e.g., in generating dark hollow beams with elliptic geometry using FrFT systems.