**3. Optical vortices in turbulent atmosphere and the problem of adaptive correction**

In early investigations [12] it was shown that the presence of optical vortices is a distinctive property of the so called speckled fields, which form when the laser beam propagates in the scattering media. Experimental evidence of the existence of screw dislocations in the laser beam, passed through a random phase plate, were obtained in [55, 56, 57] where topological limitations were also noted of adaptive control of the laser beams propagating in inhomoge‐ neous media.

Turbulent atmosphere can be represented as the consequence of random phase screens. Un‐ der propagation in the turbulent atmosphere the regular optical field acquires rising aberra‐ tions. These aberrations manifest themselves in the broadening and random wandering of laser beams; the intensity distribution becomes non-regular and the wavefront deviates from initially set surface. These deformations of the wavefront can be corrected using adap‐ tive optics. To this end, effective sensors and correctors of wavefront were designed [1-6]. The problem becomes more complicated when the laser beam passes a relatively long dis‐ tance in a weak turbulent medium or if the turbulence becomes too strong. In this case opti‐ cal vortices develop in the beam; the shape of the wavefront changes qualitatively and singularities appear.

The influence of the scintillation effects are determined (see, for example, [2, 4]) by the close‐ ness to unity of the Rytov variance

$$\left\| \sigma\_{\chi}^{2} \approx 0.56k \stackrel{7/6}{\int\_{0}^{L} \mathbf{C}\_{n}^{2}(\mathbf{z}) \mathbf{z}^{5/6} \text{( $\mathbf{z} / L$ )}^{5/6} d\mathbf{z} \right\| \tag{6}$$

where *Cn* 2 (*z*) describes the dependence of structure constant of the refractive index fluctua‐ tions over the propagation path and *L* is the path length. The regime of strong scintillations is not realized when *σχ* 2 <<1.

Figure 4 demonstrates the results of numerical simulation of propagation of a Gaussian laser beam (*λ*=1 µ) in turbulent atmosphere in a model case of invariable structure constant *Cn* 2 =10-14 cm-2/3 for the distance of 1 km when the regime of strong scintillations is realized ac‐ cording to (6). The steady-state equation for the slowly-varying complex field amplitude *A* differs from the equation (2) by the presence of the inhomogeneous term:

$$\frac{\partial A}{\partial z} - \frac{i}{2k} \frac{\partial^2 A}{\partial r^2} + \frac{ik}{2} (\tilde{\varepsilon} - 1) A = 0,\tag{7}$$

where *ε*˜ is the fluctuating dielectric permittivity of the turbulent atmosphere. In numerical simulations we use the finite-difference algorithm of numerical solving of parabolic equa‐ tion (7) described in [58]. It is characterized by an accuracy, which considerably exceeds the accuracy of the widespread spectral methods [59]. The amplitude error of an elementary harmonic solution of the homogeneous equation is equal to zero, whereas the phase error is significantly reduced and proportional to the transverse integration step to the power of six. To take into account the inhomogeneous term of the equation, the splitting by physical proc‐ esses is employed. The effect of randomly inhomogeneous distribution of dielectric permit‐ tivity is allowed for using a model of random phase screens, which is commonly used in calculations of radiation propagation in optically inhomogeneous stochastic media [60]. The spatial spectrum of dielectric permittivity fluctuations is described taking into account the Tatarsky and von Karman modifications of the Kolmogorov model [61].

cordance with the law of topologic charge conservation. The predicted effects have been

**3. Optical vortices in turbulent atmosphere and the problem of adaptive**

In early investigations [12] it was shown that the presence of optical vortices is a distinctive property of the so called speckled fields, which form when the laser beam propagates in the scattering media. Experimental evidence of the existence of screw dislocations in the laser beam, passed through a random phase plate, were obtained in [55, 56, 57] where topological limitations were also noted of adaptive control of the laser beams propagating in inhomoge‐

Turbulent atmosphere can be represented as the consequence of random phase screens. Un‐ der propagation in the turbulent atmosphere the regular optical field acquires rising aberra‐ tions. These aberrations manifest themselves in the broadening and random wandering of laser beams; the intensity distribution becomes non-regular and the wavefront deviates from initially set surface. These deformations of the wavefront can be corrected using adap‐ tive optics. To this end, effective sensors and correctors of wavefront were designed [1-6]. The problem becomes more complicated when the laser beam passes a relatively long dis‐ tance in a weak turbulent medium or if the turbulence becomes too strong. In this case opti‐ cal vortices develop in the beam; the shape of the wavefront changes qualitatively and

The influence of the scintillation effects are determined (see, for example, [2, 4]) by the close‐

tions over the propagation path and *L* is the path length. The regime of strong scintillations

Figure 4 demonstrates the results of numerical simulation of propagation of a Gaussian laser beam (*λ*=1 µ) in turbulent atmosphere in a model case of invariable structure constant *Cn*

=10-14 cm-2/3 for the distance of 1 km when the regime of strong scintillations is realized ac‐ cording to (6). The steady-state equation for the slowly-varying complex field amplitude *A*

(*z* / *L* ) 5/6

(*z*) describes the dependence of structure constant of the refractive index fluctua‐

*dz*, (6)

2

completely confirmed experimentally [52, 53, 54].

**correction**

154 Adaptive Optics Progress

neous media.

singularities appear.

where *Cn*

2

is not realized when *σχ*

ness to unity of the Rytov variance

*σχ*

2 <<1.

<sup>2</sup> ≈0.56*k* 7/6

*∫* 0

differs from the equation (2) by the presence of the inhomogeneous term:

*L Cn* 2 (*z*)*z* 5/6 The fragment of speckled distribution of optical field intensity after the propagation is shown in Figure 4. Dark spots are seen where the intensity vanishes. As it has been noted before, the presence of optical vortices in the beam is easily detected, based on the picture of its interference with an obliquely incident plane wave. The correspondent picture is shown in Figure 4 as well. In the centers of screw dislocations the fringe branching is observed, i.e. the birth or disappearance of the fringes takes place with formation of typical "forks" in the interferogram (compare with Figure 2). There are also zones of edge dislocations (compare with Figure 3). The number, allocation and helicity of the vortices in the beam are random in nature but the vortices are born as well as annihilated in pairs. If the initial beam is regular (vortex-free), then the total topological charge of the vortices in the beam will be equal to zero in each transverse section of the beam along the propagation path in accordance with the conservation law of topological charge (or orbital angular moment) [7-9].

**Figure 4.** Optical vortices in the laser beam after atmospheric propagation: the speckled intensity distribution and the picture of interference of the beam with the obliquely incident plane wave including "forks" denoted by light circles.

One of the first papers dealing with the appearance of optical vortices in laser beams propa‐ gating in randomly inhomogeneous medium was published by Fried and Vaughn in 1992 [62]. They pointed out that the presence of dislocations makes registration of the wavefront more difficult and they considered methods for solving the problem. In 1995 the authors of Ref. [63] encountered this problem in experimental investigations of laser beam propagation in the atmosphere. It was shown that the existence of light vortices is an obstacle for atmos‐ pheric adaptive optical systems. After that it was theoretically shown that screw dislocations give rise to errors in the procedure of wavefront registration by the Shack-Hartmann sensor [64, 65]. Due to zero amplitude of the signal in singular points, the information carried by the beam becomes less reliable and the compensation for turbulent aberrations is less effec‐ tive [66]. Along with [63], the experimental investigation [67] can be taken here as an exam‐ ple where the results of adaptive correction are presented for distortions of beams propagating in the atmosphere.

The problem of a wavefront corrector (adaptive mirror) suitable for controlling a singular phase surface is also topical. In the adaptive optical systems [77, 78] the wavefront correctors were based on the micro-electromechanical system (MEMS) spatial light modulators with the large number of actuators. The results of [77, 78] shown that continuous MEMS mirrors with high dynamic response bandwidth, combined with the interferometric wavefront sen‐ sor, can ensure a noticeable correction of scintillation. However, the MEMS mirrors are char‐ acterized by low laser damage resistance that can considerably limit applications. The bimorph or pusher-type piezoceramics-based flexible mirrors with the modal response func‐ tions of control elements have a much higher laser damage threshold [3-5]. Recently [88] a complicated cascaded imaging adaptive optical system with a number of bimorph piezocer‐ amic mirrors was used to mitigate turbulence effect basing, in particular, on conventional Hartmann-Shack wavefront sensor data. Conventional adaptive compensation was obtained in [88] which proved to be very poor at deep turbulence. The scintillation and vortices may

Adaptive Optics and Optical Vortices http://dx.doi.org/10.5772/53328 157

In the investigations, the results of which are described in this chapter, the development of an algorithm of the Hartmann-Shack reconstruction of vortex wavefront of the laser beam plays a substantial role. The creation of efficient algorithms for the wavefront sensor of vor‐ tex beams implies the experiments under modeling conditions when the optical vortices are artificially generated by special laboratory means. Moreover, as long as the matter concerns the creation of a new algorithm of wavefront reconstruction, it is possible to estimate its ac‐ curacy only under operation with the beam, the singular phase structure of which is known in detail beforehand. The formation of optical beams with the given configuration of phase singularities and their transformations is one of main trends in the novel advanced optical

Thus, the first stage of the research sees the generation of a vortex laser beam with the given topological charge. In our case the role of this beam is played by the single optical vortex, namely, the Laguerre-Gaussian mode. Further, at the second stage, with the help of the Hartmann-Shack wavefront sensor, the task of registration of the vortex beam phase surface is solved using the new algorithm of singular wavefront reconstruction. Finally, at the third stage, the correction of the singular wavefront is undertaken in a closed-loop adaptive opti‐ cal system, including the Hartmann-Shack wavefront sensor and the wavefront corrector in

As it has been indicated above, to examine the accuracy of the wavefront reconstruction al‐ gorithm and its efficiency in the experiment itself a "reference" vortex beam has to be formed with a predetermined phase surface. This is important as, otherwise, it would be im‐ possible to make sure that the algorithm recovers the true phase surface under conditions when robust alternative methods of its reconstruction are missing or unavailable. The La‐

*<sup>m</sup>* can play the role of such "reference" optical vortices.

be one of the causes of this.

branch – singular optics [7-9].

the form of a piezoelectric-based bimorph mirror.

**4. Generation of optical vortex**

guerre-Gaussian vortex modes *LGn*

Since one of the key elements of an adaptive optical system is the wavefront sensor of laser radiation, there is a pressing need to create sensors that are capable of ensuring the required spatial resolution and maximal accuracy of the measurements. In this connection there is necessity need to develop algorithms for measurement of wavefront with screw disloca‐ tions, which are sufficiently precise, efficient and economical given the computing resour‐ ces, and resistant to measurement noises. The traditional methods of wave front measurements [1-6] in the event of the above-mentioned conditions are in fact of no help. The wavefront sensors have been not able to restore the phase under the conditions of strong scintillations [68]. The experimental determination of the location of phase discontinuities itself already generates serious difficulties [69]. In spite of the fact that the construction features of algo‐ rithms of wavefront recovery in the presence of screw dislocations were set forth in a num‐ ber of theoretical papers [68, 69, 70, 71, 72, 73, 74, 75], there were not many published experimental works in this direction. Thus, phase distribution has been investigated in different diffrac‐ tion orders for a laser beam passed through a specially synthesized hologram, designed for generating higher-order Laguerre-Gaussian modes [76]. An interferometer with high spatial resolution was used to measure transverse phase distribution and localization of phase singu‐ larities. The interferometric wavefront sensor was applied also in a high-speed adaptive optical system to compensate phase distortions under conditions of strong scintillations of the coher‐ ent radiation in the turbulent atmosphere [77] as well as when modelling the turbulent path under laboratory conditions [78]. In [77, 78] the local phase was measured, without reconstruct‐ ing the global wavefront that is much less sensitive to the presence of phase residues. The interferometric methods of phase determination are rather complicated and require that several interferograms are obtained at various phase shifts between a plane reference wave and a signal wave. It is noteworthy, however, that in the adaptive optical systems [1-6] the Hartmann-Shack wavefront sensor [79, 80] has a wider application compared with the interferometric sensors including the lateral shearing interferometers [81, 82], the curvature sensor [83, 84, 85], and the pyramidal sensor [86, 87]. The cause of this is just in a simpler and more reliable arrangement and construction of the Hartmann-Shack sensor. However, there have been practically no publications of the results of experimental investigations connected with appli‐ cations of this sensor for measurements of singular phase distributions.

The problem of a wavefront corrector (adaptive mirror) suitable for controlling a singular phase surface is also topical. In the adaptive optical systems [77, 78] the wavefront correctors were based on the micro-electromechanical system (MEMS) spatial light modulators with the large number of actuators. The results of [77, 78] shown that continuous MEMS mirrors with high dynamic response bandwidth, combined with the interferometric wavefront sen‐ sor, can ensure a noticeable correction of scintillation. However, the MEMS mirrors are char‐ acterized by low laser damage resistance that can considerably limit applications. The bimorph or pusher-type piezoceramics-based flexible mirrors with the modal response func‐ tions of control elements have a much higher laser damage threshold [3-5]. Recently [88] a complicated cascaded imaging adaptive optical system with a number of bimorph piezocer‐ amic mirrors was used to mitigate turbulence effect basing, in particular, on conventional Hartmann-Shack wavefront sensor data. Conventional adaptive compensation was obtained in [88] which proved to be very poor at deep turbulence. The scintillation and vortices may be one of the causes of this.

In the investigations, the results of which are described in this chapter, the development of an algorithm of the Hartmann-Shack reconstruction of vortex wavefront of the laser beam plays a substantial role. The creation of efficient algorithms for the wavefront sensor of vor‐ tex beams implies the experiments under modeling conditions when the optical vortices are artificially generated by special laboratory means. Moreover, as long as the matter concerns the creation of a new algorithm of wavefront reconstruction, it is possible to estimate its ac‐ curacy only under operation with the beam, the singular phase structure of which is known in detail beforehand. The formation of optical beams with the given configuration of phase singularities and their transformations is one of main trends in the novel advanced optical branch – singular optics [7-9].

Thus, the first stage of the research sees the generation of a vortex laser beam with the given topological charge. In our case the role of this beam is played by the single optical vortex, namely, the Laguerre-Gaussian mode. Further, at the second stage, with the help of the Hartmann-Shack wavefront sensor, the task of registration of the vortex beam phase surface is solved using the new algorithm of singular wavefront reconstruction. Finally, at the third stage, the correction of the singular wavefront is undertaken in a closed-loop adaptive opti‐ cal system, including the Hartmann-Shack wavefront sensor and the wavefront corrector in the form of a piezoelectric-based bimorph mirror.
