**7. Conclusions**

This chapter is dedicated to research of the possibility to control the phase front of a laser beam carrying an optical vortex by means of linear adaptive optics, namely, in the classic closed-loop adaptive system including a Hartmann-Shack wavefront sensor and a deforma‐ ble mirror. On the one hand, the optical vortices appear randomly under beam propagation in the turbulent atmosphere, and the correction of singular phase front presents a considera‐ ble problem for tasks in atmospheric optics, astronomy, and optical communication. On the other hand, the controllable optical vortices have very attractive potential applications in op‐ tical data processing and many other scientific and practical fields where the regulation of singular phase is needed. This chapter discusses the main properties and applications of op‐ tical vortices, the problem of adaptive correction of singular phase in turbulent atmosphere, the issues of generating the "reference" laser vortex beam, its wavefront sensing and phase correction in the widespread adaptive optical system including a Hartmann-Shack wave‐ front sensor and a flexible deformable mirror.

The vortex beam is generated with help of a spiral phase plate made of fused quartz by ki‐ noform technology. Provided that the optical quality of the spiral phase plate is good, such a means of vortex formation seems to be more preferable as compared with other considered methods of vortex generation with a well-determined phase surface. As a result, it becomes possible to obtain a singular beam very close to a Laguerre-Gaussian *LG*<sup>0</sup> <sup>1</sup> mode with a welldetermined singular phase structure that is necessary for checking the accuracy of subse‐ quent wavefront reconstruction. The developed spiral phase plates are characterized by high laser damage resistance, the good surface profile accuracy and they facilitate formation of a high quality optical vortex.

The vortex phase surface measurement is carried out by a Hartmann-Shack wavefront sen‐ sor which is simpler in design and construction, more reliable and more widespread in vari‐ ous fields of adaptive optics when compared with other types of sensors. The commonly accepted Hartmann-Shack wavefront reconstruction is performed on the basis of the leastmean-square approach. This approach works well in the case of continuous phase distribu‐ tions but is completely unsuitable for singular phase distributions. Therefore a new reconstruction technique has been developed for the reconstruction of singular phase sur‐ face, starting from the measured phase gradients. The measured shifts of focal spots in the hartmannogram are in good agreement with the calculation results. Using new software in the Hartmann-Shack sensor, the reconstruction of the "reference" vortex phase surface has been carried out to a high degree of accuracy.

The vortex laser beam (distorted *LG*<sup>0</sup> 1 mode) is corrected in the closed-loop adaptive system including a Hartmann-Shack wavefront sensor with singular reconstruction technique and a flexible bimorph piezoceramic mirror with 5х5 actuators allocated in the check geometry. The mirror has high laser damage resistance meaning it can operate with powerful laser beams. The purpose of the correction is to eliminate the singularity of the beam to the high‐ est degree possible. Experiments have demonstrated the ability of the bimorph mirror to correct the optical vortex in a practical sense. As a result of phase correction, the doughnutlike beam is focused into a beam with a bright axial spot that considerably increases the Strehl ratio and is important for practical applications. However, since the wavefront break cannot be reproduced exactly by a mirror with a flexible surface, the residual off-axis vorti‐ ces can appear in far field at the beam periphery.

The investigations described above consolidate the actual birth of the experimental field of novel scientific branch – singular adaptive optics.
