**Author details**

Thus, the phase surface of the distorted *LG*<sup>0</sup>

front sensor and a flexible deformable mirror.

of a high quality optical vortex.

**7. Conclusions**

178 Adaptive Optics Progress

1 mode is corrected in the closed-loop adaptive

<sup>1</sup> mode with a well-

optical system, including the bimorph piezoceramic mirror and the Hartmann-Shack wave‐ front sensor with the singular reconstruction technique. Experiments demonstrate the abili‐ ty of the bimorph mirror to correct the optical vortex in a practical sense, namely, to focus the doughnut-like beam into a beam with a bright axial spot that considerably increases the Strehl ratio and optical system resolution. Since the phase break is not reproduced exactly on the flexible corrector surface, the off-axis vortices can appear in far field at the beam periphery.

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‐

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

determined 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

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

possible to obtain a singular beam very close to a Laguerre-Gaussian *LG*<sup>0</sup>


2 International Science and Technology Center, Russia
