**1. Introduction**

The achievement of minimal angular divergence of a laser beam is one of the most impor‐ tant problems in laser physics since many laser applications demand extreme concentration of radiation. Under the beam formation in the laser oscillator or amplifier with optically in‐ homogeneous gain medium and optical elements, the divergence usually exceeds the dif‐ fraction limit, and the phase surface of the laser beam differs from the plane surface. However, even if one succeeds in realizing the close-to-plane radiation wavefront at the la‐ ser output, the laser radiation experiences increasing phase disturbances under the propaga‐ tion of the beam in an environment with optical inhomogeneities (atmosphere). These disturbances appear with the wavefront receiving smooth, regular distortions, the trans‐ verse intensity distribution becomes inhomogeneous, and the beam broadens out.

The correction of the laser radiation phase, which is a smooth continuous spatial function, can be performed using a conventional adaptive optical system including a wavefront sen‐ sor and a wavefront corrector. The wavefront sensor performs the measurement (in other words, reconstruction) of the radiation phase surface; then, on the basis of these data, the wavefront corrector (for example, a reflecting mirror with deformable surface) transforms the phase front in the proper way. If all components of the adaptive optical system are in‐ volved in the common circuit with the feedback, then the adaptive system is known as a closed-loop system. The adaptive correction of the wavefront with smooth distortions has a somewhat long history and considerable advances [1, 2, 3, 4, 5, 6].

When a laser beam passes a sufficiently long distance in a turbulent atmosphere, the socalled regime of strong scintillations (intensity fluctuations) is realized. Under such condi‐ tions the optical field becomes speckled, lines appear in the space along the beam axis where the intensity vanishes and the surrounding zones of the wavefront attain a helicoidal (screw) shape. If the intensity in an acnode of the transverse plane is zero, then the phase in

this point is not defined. In view of its screw form, the phase surface in the vicinity of such point has a break, the height of which is divisible by the wavelength. Since the phase is de‐ fined accurate to the addend that is aliquot to 2*π*, it is formally continuous but under a com‐ plete circling on the phase surface around the singular point one cannot reach the starting place. The integration of the phase gradient over some closed contour encircling such singu‐ lar point results in a circulation not equal to zero, in contrast to the null circulation at the usual smoothed-inhomogeneous regular phase distribution. The indicated properties repre‐ sent evidence of strong distortions of the wavefront – screw dislocations or optical vortices. The vortical character of the beam is detected with ease in the experiment after the analysis of the picture of its interference with the obliquely incident plane wave: the interference fringes arise or vanish in the centers of screw dislocations forming peculiar "forks".

2π*m* with the encircling the singularity point clockwise or counter-clockwise, where *m* is the positive or negative integer number known as the vortex "topological charge". In the centre of the vortex (intensity zero point) the phase remains indefinite. Such an optical singularity is the result of the interference of partial components of the wave field with a phase shift, which is initial or acquired during propagation in an inhomogeneous medium. In 3D space the points with zero intensity form zero lines. On these lines the potentiality of phase field is violated; and the regions of "defective" (singular) phase can be considered as vortex strings like the regions with concentrated vorticity, which are considered in the hydrodynamics of ideal liquid. We are interested in a case where the lines of zero intensity have a predomi‐ nantly longitudinal direction, i.e. form a longitudinal optical vortex. The equiphase surface in the vicinity of such a line has the appearance of a screw-like (helicoidal) structure, thread‐ ed on this line. In the interference pattern of the vortex wave under consideration with any regular wave, the vortices are revealed through the appearance of so-called "forks" (i.e.,

Adaptive Optics and Optical Vortices http://dx.doi.org/10.5772/53328 149

branching of interference fringes), coinciding with zero points of the intensity.

"topological defects", "phase singularities", "phase cuts", and "branch cuts".

G

∂ *A* <sup>∂</sup> *<sup>z</sup>* <sup>−</sup> *<sup>i</sup>* 2*k*

where *d***ρ** is the element of the contour Г.

equal to zero:

example, [16]):

Investigations of waves with screw wavefront and methods of their generation were report‐ ed as early as by Bryngdahl [11]. The theory of waves carrying phase singularities was de‐ veloped in detail by Nye and Berry [12, 10], prompting a series of publications dealing with the problem (see [13, 14, 7, 8] and the lists of references therein). The term "optical vortex" was introduced in [15]. Along with the term "optical vortex", the phenomenon is also refer‐ red to as "wavefront screw dislocation". The latter appeared because of similarities between distorted wavefront and the crystal lattice with defects. The following terms are also used:

Thus the indication of the existence of an optical vortex in an optical field is the presence of an isolated point {**r**, *z*} in a plane, perpendicular to the light propagation axis, in which inten‐ sity *I* (**r**, *z*) is approaching zero, phase *S* (**r**, *z*) is indefinite, and integration of the phase gradi‐ ent ∇⊥*S* field over some closed contour Г encircling this point results in a circulation not

^*S zd m* (,) 2 ,

The propagation of slowly-varying complex amplitude of the scalar wave field *A* in the free space is described by the well-known quasi-optical equation of the parabolic type (see, for

∂<sup>2</sup> *A*

where *k*=*2π*/*λ* is wave number, *λ* is the radiation wavelength, *z* is the longitudinal coordi‐ nate corresponding with the beam propagation axis, **r**=**e**x*x*+**e**y*y* is the transverse radius-vec‐

p

Ñ = ò **ρ ρ** Ñ (1)

<sup>∂</sup>*<sup>r</sup>* <sup>2</sup> =0, (2)

Scintillations in the atmosphere especially decrease the efficiency of light energy transporta‐ tion and distort the information carried by a laser beam in issues of astronomy and optical communications. Scintillation effects present special difficulty for adaptive optics, and their correction is one of key trends in the development of state-of-the-art adaptive optical systems.

However it should be noted that the possibility to control the optical vortices (including the means of adaptive optics) presents interest not only for atmospheric optics but for a new op‐ tical field, namely, singular optics [7, 8, 9]. The fact is that optical vortices have very promis‐ ing applications in optical data processing, micro-manipulation, coronagraphy, etc. where any type of management of the singular phase could be required.

This chapter is dedicated to wavefront reconstruction and adaptive phase correction of a vortex laser beam, which is generated in the form of the Laguerre-Gaussian *LG*<sup>0</sup> 1 laser mode. The content of the chapter is as follows. In Section 1 we specify the origin and main proper‐ ties of optical vortices as well as some their practical applications. Section 2 is dedicated to a short description of the origination of optical vortices in a turbulent atmosphere and corre‐ spondent problems of the adaptive optics. In Section 3 some means are given concerning the generation of optical vortices under laboratory conditions, aimed at the formation of a "ref‐ erence" optical vortex with the maximally predetermined phase surface, and the experimen‐ tal results of such formation are illustrated. Section 4 is concerned with vortex beam phase surface registration, based on measurements of phase local tilts using a Hartmann-Shack wavefront sensor and a novel reconstruction technique. In Section 5 experimental results of correction of a vortex beam are demonstrated in the conventional closed-loop adaptive opti‐ cal system including a Hartmann-Shack wavefront sensor and a bimorph deformable adap‐ tive mirror. Conclusions summarize the abovementioned research results.
