**2. Origin, main properties and practical applications of optical vortices**

The singularity of the radiation field phase *S* is identified by the term "optical vortex", which can appear in the complex function exp{*iS*} representing a monochromatic light wave [10]. The amplitude of scalar wave field *A* (and, correspondingly, its intensity *I=*|*A*|2 ) in the point of the vortex location approaches zero. The phase of radiation *S* changes its value by 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., branching of interference fringes), coinciding with zero points of the intensity.

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

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 equal to zero:

$$\oint\_{\Gamma} \nabla\_{\perp} S(\mathfrak{p}, z) d\mathfrak{p} = 2\pi m,\tag{1}$$

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

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".

any type of management of the singular phase could be required.

148 Adaptive Optics Progress

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

This chapter is dedicated to wavefront reconstruction and adaptive phase correction of a

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‐

1

laser mode.

) in the

vortex laser beam, which is generated in the form of the Laguerre-Gaussian *LG*<sup>0</sup>

tive mirror. Conclusions summarize the abovementioned research results.

**2. Origin, main properties and practical applications of optical vortices**

The singularity of the radiation field phase *S* is identified by the term "optical vortex", which can appear in the complex function exp{*iS*} representing a monochromatic light wave [10]. The amplitude of scalar wave field *A* (and, correspondingly, its intensity *I=*|*A*|2

point of the vortex location approaches zero. The phase of radiation *S* changes its value by

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 example, [16]):

$$
\frac{\partial A}{\partial z} - \frac{i}{2k} \frac{\partial^2 A}{\partial r^2} = 0,\tag{2}
$$

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‐ tor. In the process of radiation propagation in the medium vortices appear, travel in space, and disappear (are annihilated).

Laguerre-Gaussian laser beams *LGn <sup>m</sup>* are related to the familiar class of vortex beams and are used most often in experiments with optical vortices [7-9]. They are the eigen-modes of the homogeneous quasi-optical parabolic equation (2) [17], so they do not change their form un‐ der the free space propagation and lens transformations. The correspondent solution of equation (2) in cylindrical coordinates (*r*, *φ*, *z*) has the following form:

$$A(r,\sqrt{\cdot}z) = A\_0 \frac{w\_0}{z\nu} \left(\frac{r}{z\nu}\right)^m \mathcal{O}\_m(\wp) L\_m \left(2\frac{r^2}{z\nu^2}\right) \exp\left(-\frac{r^2}{w^2} + i\frac{zr^2}{z\_0\nu^2} - i(2n+m+1)\text{arctg}\frac{z}{z\_0}\right),\tag{3}$$

The typical transverse size of the beam *w* in (3) is determined by the relation *w*<sup>2</sup> =*w*<sup>0</sup> <sup>2</sup> [1+*z*<sup>2</sup> / *z*0 2 ] where, in its turn, *w*<sup>0</sup> is the transverse beam size in the waist and *z*0 =*kw*<sup>0</sup> <sup>2</sup> /2 is the typical waist length. The radial part of distribution (3) includes the generalized Laguerre polyno‐ mial

$$L\_{\
u}^{\
m}(\mathbf{x}) = \frac{e^{\mathbf{x}}}{\mathbf{x}^{\
m} n!} \frac{d^{\
n}}{d\mathbf{x}^{\
m}} \{\mathbf{x}^{\
m+m} e^{-\mathbf{x}}\}.\tag{4}$$

case at *n*=0, *m*=1 (*LG*<sup>0</sup>

1

**Figure 1.** Phase surface shape of a Laguerre-Gaussian beam carrying an optical vortex.

**Figure 2.** The intensity distribution in the Laguerre-Gaussian laser beam *LGn*

an obliquely incident plane wave at *n*=0, *m*=1, Φ*m*=exp(iφ).

ture of the beam in the experiment.

mode) the intensity distribution has a doughnut-like form that is

Adaptive Optics and Optical Vortices http://dx.doi.org/10.5772/53328 151

*<sup>m</sup>* and the picture of its interference with

shown in Figure 2. Figure 2 also presents a picture of the interference of the given vortex laser beam with the obliquely incident monochromatic plane wave. In the picture, fringe branching is observed in the beam center with the "fork" formation (fringe birth) typical for screw dislocation [7-9]. At the arbitrary *m* number the quantity of fringes that are born corre‐ sponds with this number, the double, triple, and other "forks" are formed. The presence of "forks" in the interference patterns of such kind is the standard evidence of the vortex na‐

In addition to an item responsible for wavefront curvature and transversely-uniform Gouy phase, the angular factor Φ*m*(*φ*) contributes to the phase part of distribution (3). The angular part of formula (3) is represented in the form of a linear combination of harmonic functions

$$\mathcal{O}\_m(\varphi) = c\_1 \cdot \mathcal{Q}\_m(\varphi) + c\_2 \cdot \mathcal{Q}\_{-m}(\varphi) \tag{5}$$

where Ω*m*(*φ*)=exp(*imφ*), *φ*=arctg(*y*/*x*) is the azimuth angle in the transverse plane. The *c*1 and *c*2 constants determine the beam character and the presence of singularity in it.

Let's consider two cases of the angular function Φ*m*(*φ*) distribution from (5) at *m*>0. In the first case, when *c*1=1 and *c*2=0, we have Φ*m*(*φ*)=exp(*imφ*). The form of helicoidal phase surface assigned by the *mφ* function (we do not take into account the wavefront curvature in (3) de‐ termining the beam broadening or narrowing only) in the vortex Laguerre-Gaussian laser beam is presented in Figure 1. This phase does not depend on *r* at the given *φ* and rises line‐ arly with *φ* increasing. In the phase surface the spatial break of *mλ* (or 2π*m* radians) depth is present. Under the complete circular trip around the optical axis on the phase surface it is impossible to get to the starting point. As it has been commented above, such a shape of the phase factor is what causes the singular, vortex behavior of the beam. The positive or nega‐ tive sign of *m* determines right or left curling of the phase helix.

On the optical axis, in the vortex center, the intensity is zero, resulting from the behavior of the radial dependence of (3) and generalized Laguerre polynomial (4). The beam intensity distribution in the transverse plane, as it is seen from (3), is axially-symmetrical (modulus of *A* depends on *r* only) and visually represents the system of concentric rings. In the simplest case at *n*=0, *m*=1 (*LG*<sup>0</sup> 1 mode) the intensity distribution has a doughnut-like form that is shown in Figure 2. Figure 2 also presents a picture of the interference of the given vortex laser beam with the obliquely incident monochromatic plane wave. In the picture, fringe branching is observed in the beam center with the "fork" formation (fringe birth) typical for screw dislocation [7-9]. At the arbitrary *m* number the quantity of fringes that are born corre‐ sponds with this number, the double, triple, and other "forks" are formed. The presence of "forks" in the interference patterns of such kind is the standard evidence of the vortex na‐ ture of the beam in the experiment.

tor. In the process of radiation propagation in the medium vortices appear, travel in space,

used most often in experiments with optical vortices [7-9]. They are the eigen-modes of the homogeneous quasi-optical parabolic equation (2) [17], so they do not change their form un‐ der the free space propagation and lens transformations. The correspondent solution of

*<sup>w</sup>* <sup>2</sup> )exp( <sup>−</sup> *<sup>r</sup>* <sup>2</sup>

The typical transverse size of the beam *w* in (3) is determined by the relation *w*<sup>2</sup> =*w*<sup>0</sup>

waist length. The radial part of distribution (3) includes the generalized Laguerre polyno‐

*d n*

In addition to an item responsible for wavefront curvature and transversely-uniform Gouy phase, the angular factor Φ*m*(*φ*) contributes to the phase part of distribution (3). The angular part of formula (3) is represented in the form of a linear combination of harmonic functions

where Ω*m*(*φ*)=exp(*imφ*), *φ*=arctg(*y*/*x*) is the azimuth angle in the transverse plane. The *c*1 and

Let's consider two cases of the angular function Φ*m*(*φ*) distribution from (5) at *m*>0. In the first case, when *c*1=1 and *c*2=0, we have Φ*m*(*φ*)=exp(*imφ*). The form of helicoidal phase surface assigned by the *mφ* function (we do not take into account the wavefront curvature in (3) de‐ termining the beam broadening or narrowing only) in the vortex Laguerre-Gaussian laser beam is presented in Figure 1. This phase does not depend on *r* at the given *φ* and rises line‐ arly with *φ* increasing. In the phase surface the spatial break of *mλ* (or 2π*m* radians) depth is present. Under the complete circular trip around the optical axis on the phase surface it is impossible to get to the starting point. As it has been commented above, such a shape of the phase factor is what causes the singular, vortex behavior of the beam. The positive or nega‐

On the optical axis, in the vortex center, the intensity is zero, resulting from the behavior of the radial dependence of (3) and generalized Laguerre polynomial (4). The beam intensity distribution in the transverse plane, as it is seen from (3), is axially-symmetrical (modulus of *A* depends on *r* only) and visually represents the system of concentric rings. In the simplest

*c*2 constants determine the beam character and the presence of singularity in it.

tive sign of *m* determines right or left curling of the phase helix.

] where, in its turn, *w*<sup>0</sup> is the transverse beam size in the waist and *z*0 =*kw*<sup>0</sup>

*<sup>w</sup>* <sup>2</sup> <sup>+</sup> *<sup>i</sup>*

*zr* <sup>2</sup>

equation (2) in cylindrical coordinates (*r*, *φ*, *z*) has the following form:

*<sup>m</sup>*(2 *r* 2

*<sup>m</sup>*(*x*)= *<sup>e</sup> <sup>x</sup> x mn* !

*Φm*(*φ*)*L <sup>n</sup>*

*L <sup>n</sup>*

*<sup>m</sup>* are related to the familiar class of vortex beams and are

*<sup>z</sup>*0*<sup>w</sup>* <sup>2</sup> <sup>−</sup>*i*(2*<sup>n</sup>* <sup>+</sup> *<sup>m</sup>* <sup>+</sup> 1)arctg

*<sup>d</sup> <sup>x</sup> <sup>n</sup>* (*<sup>x</sup> <sup>n</sup>*+*me* <sup>−</sup>*<sup>x</sup>*). (4)

*Φm*(*φ*)=*c*<sup>1</sup> ⋅*Ωm*(*φ*) + *c*<sup>2</sup> ⋅*Ω*−*m*(*φ*), (5)

*z z*0

), (3)

<sup>2</sup> /2 is the typical

<sup>2</sup> [1+*z*<sup>2</sup> /

and disappear (are annihilated).

*A*(*r*, *φ*; *z*)= *A*<sup>0</sup>

150 Adaptive Optics Progress

*z*0 2

mial

Laguerre-Gaussian laser beams *LGn*

*w*0 *w* ( *r w* ) *m*

**Figure 1.** Phase surface shape of a Laguerre-Gaussian beam carrying an optical vortex.

**Figure 2.** The intensity distribution in the Laguerre-Gaussian laser beam *LGn <sup>m</sup>* and the picture of its interference with an obliquely incident plane wave at *n*=0, *m*=1, Φ*m*=exp(iφ).

infrequent and correspond to some forbidden atomic and molecular transitions. However, generating the beam carrying the optical vortex, one can readily obtain the light radiation beam with quantum orbital moment. Such beams can be used in investigations of all kinds of polarized light. For example, the photon analogy of spin-orbital interaction of electrons can be studied and in general it is possible to organize the search for new optical interac‐ tions. As the *m*-factor can acquire arbitrary values, any part of the beam (even one photon) can carry an unlimited amount of information coded in the topological charge. Thus the density of information in a channel where coding is realized with the use of orbital moment could be as high as compared with a channel with coding of the spin states of a photon. Be‐ cause only two circular polarization states of the photon are possible, one photon can trans‐ mit only one bit of information. Presently, optical vortices have generated a great deal of interest in optical data processing technologies, namely, the coding/decoding in optical com‐ munication links in free space [20, 21], optical data processing [22], optical interconnects

The next practical application of optical vortices is optical micromanipulations and construc‐ tion of so called optical traps, i.e. areas where the small (a few micrometers) particles can be locked in [25, 26]. Progress in the development of such traps allows the capture of particles of low and large refraction indexes [27]. Presently, this direction of research finds further

It is also possible to use optical vortices to register objects with small luminosity located near a bright companion. Shadowing the bright object by a singular phase screen results in the formation of a window, in which the dim object is seen. The optical vortex filtration of such a kind was proposed in [31]. Using this method the companion located at 0.19 arcsec near

The possibility to use this method to detect planets orbiting bright stars was also illustrated by astronomers [33, 34]. Vortex coronagraphy is now undergoing further development [35,

It was proposed to use optical vortices to improve optical measurements and increase the fidelity of optical testing [39, 40], for investigations in high-resolution fluorescence microsco‐ py [41], optical lithography [42, 43], quantum entanglement [44, 45, 46], Bose–Einstein con‐

Optical vortices show interesting properties in nonlinear optics [48]. For example, in [49, 50, 51] it was predicted that the phase conjugation at SBS of vortex beams is impossible due to the failure of selection of the conjugated mode. For a rather wide class of the vortex laser beams a novel and interesting phenomenon takes place which can be called the phase trans‐ formation at SBS. In essence there is only one Stokes mode, the amplification coefficient of which is maximal and higher than that of the conjugated mode. In other words, the non-con‐ jugated mode is selected of in the Stokes beam. The principal Gaussian mode, which is or‐ thogonal to the laser vortex mode, is an example of such an exceptional Stokes mode. The cause of this phenomenon is in the specific radial and azimuth distribution of the vortex la‐ ser beam. It is interesting that the hypersound vortices are formed in the SBS medium in ac‐

times greater [32].

Adaptive Optics and Optical Vortices http://dx.doi.org/10.5772/53328 153

the object was theoretically differentiated with intensity of radiation 2×105

36]. There are a number of examples of non-astronomical applications [37, 38].

[23], and quantum optics information processing [24].

continuation [28, 29, 30].

densates [47].

**Figure 3.** The intensity distribution in the Laguerre-Gaussian laser beam *LGn <sup>m</sup>* and the picture of its interference with an obliquely incident plane wave at *n*=0, *m*=1, Φ*m*=sin(iφ).

Let's consider one more case when *c*1=-*c*2=1/2*i* in (5), then Φ*m*(*φ*)=sin(*mφ*). Under such condi‐ tions, according to (3) the intensity distribution has no axial symmetry depending on *φ*. In Figure 3 the beam intensity distribution in the transverse plane at Φ*m*=sin*φ* for *n*=0, *m*=1 is shown. The phase portrait of the beam is also given in Figure 3. The half-period shift of in‐ terference fringes in the lower half plane as compared to the upper one demonstrates the phase asymmetry with respect to the *x* axis. The fringe numbers in the lower and upper half planes are the same, i.e. the fringe birth does not take place. It is seen that the edge rather than screw phase dislocation occurs here since the phase distribution is step-like with a break of π (instead of 2π!) radians. The given example of the Laguerre-Gaussian beam dem‐ onstrates that at Φ*m*=sin*φ* the beam is not the vortical in nature. There is no singular point in the beam transverse section but there is a particular line (*y*=0) where the intensity is zero. At *m*>1 in the beam there are *m* such lines passing through the optical axis and dividing the transverse beam section by 2*m* equal sectors. The radiation phase in each sector is uniform and differs in *π* in the neighboring sectors.

Light beams with optical vortices currently attract considerable attention. This attention is encouraged by the extraordinary properties of such beams and by the important manifesta‐ tions of these properties in many applications of science and technology.

It is known for a long time that light with circular polarization possesses an orbital moment. For the single photon its quantity equals ±*ћ*, where *ћ* is the Planck constant. However only relatively recently it was shown [18, 19] that light can have an orbital moment irrespective of its polarization state if its azimuth phase dependence is of the form *S*=*mφ* where *φ* is azi‐ muth coordinate in the transverse cross section of the beam and *m* is the positive or negative integer. The authors [18] supposed that the moment of each photon is defined by the formu‐ la *L=mћ*. As this phase dependence is the characteristic feature of the helical wavefront (the form of which is presented in Figure 1) so the beam carrying the optical vortex and possess‐ ing such phase front has to own the non-zero orbital moment *mћ* per photon. The quantity of *m* is defined by the topological charge of optical vortex.

The concept of orbital moment is not new. It is well known that multipole quantum jumps can results in the emission of radiation with orbital moment. However, such processes are infrequent and correspond to some forbidden atomic and molecular transitions. However, generating the beam carrying the optical vortex, one can readily obtain the light radiation beam with quantum orbital moment. Such beams can be used in investigations of all kinds of polarized light. For example, the photon analogy of spin-orbital interaction of electrons can be studied and in general it is possible to organize the search for new optical interac‐ tions. As the *m*-factor can acquire arbitrary values, any part of the beam (even one photon) can carry an unlimited amount of information coded in the topological charge. Thus the density of information in a channel where coding is realized with the use of orbital moment could be as high as compared with a channel with coding of the spin states of a photon. Be‐ cause only two circular polarization states of the photon are possible, one photon can trans‐ mit only one bit of information. Presently, optical vortices have generated a great deal of interest in optical data processing technologies, namely, the coding/decoding in optical com‐ munication links in free space [20, 21], optical data processing [22], optical interconnects [23], and quantum optics information processing [24].

**Figure 3.** The intensity distribution in the Laguerre-Gaussian laser beam *LGn*

Let's consider one more case when *c*1=-*c*2=1/2*i* in (5), then Φ*m*(*φ*)=sin(*mφ*). Under such condi‐ tions, according to (3) the intensity distribution has no axial symmetry depending on *φ*. In Figure 3 the beam intensity distribution in the transverse plane at Φ*m*=sin*φ* for *n*=0, *m*=1 is shown. The phase portrait of the beam is also given in Figure 3. The half-period shift of in‐ terference fringes in the lower half plane as compared to the upper one demonstrates the phase asymmetry with respect to the *x* axis. The fringe numbers in the lower and upper half planes are the same, i.e. the fringe birth does not take place. It is seen that the edge rather than screw phase dislocation occurs here since the phase distribution is step-like with a break of π (instead of 2π!) radians. The given example of the Laguerre-Gaussian beam dem‐ onstrates that at Φ*m*=sin*φ* the beam is not the vortical in nature. There is no singular point in the beam transverse section but there is a particular line (*y*=0) where the intensity is zero. At *m*>1 in the beam there are *m* such lines passing through the optical axis and dividing the transverse beam section by 2*m* equal sectors. The radiation phase in each sector is uniform

Light beams with optical vortices currently attract considerable attention. This attention is encouraged by the extraordinary properties of such beams and by the important manifesta‐

It is known for a long time that light with circular polarization possesses an orbital moment. For the single photon its quantity equals ±*ћ*, where *ћ* is the Planck constant. However only relatively recently it was shown [18, 19] that light can have an orbital moment irrespective of its polarization state if its azimuth phase dependence is of the form *S*=*mφ* where *φ* is azi‐ muth coordinate in the transverse cross section of the beam and *m* is the positive or negative integer. The authors [18] supposed that the moment of each photon is defined by the formu‐ la *L=mћ*. As this phase dependence is the characteristic feature of the helical wavefront (the form of which is presented in Figure 1) so the beam carrying the optical vortex and possess‐ ing such phase front has to own the non-zero orbital moment *mћ* per photon. The quantity

The concept of orbital moment is not new. It is well known that multipole quantum jumps can results in the emission of radiation with orbital moment. However, such processes are

tions of these properties in many applications of science and technology.

of *m* is defined by the topological charge of optical vortex.

an obliquely incident plane wave at *n*=0, *m*=1, Φ*m*=sin(iφ).

152 Adaptive Optics Progress

and differs in *π* in the neighboring sectors.

*<sup>m</sup>* and the picture of its interference with

The next practical application of optical vortices is optical micromanipulations and construc‐ tion of so called optical traps, i.e. areas where the small (a few micrometers) particles can be locked in [25, 26]. Progress in the development of such traps allows the capture of particles of low and large refraction indexes [27]. Presently, this direction of research finds further continuation [28, 29, 30].

It is also possible to use optical vortices to register objects with small luminosity located near a bright companion. Shadowing the bright object by a singular phase screen results in the formation of a window, in which the dim object is seen. The optical vortex filtration of such a kind was proposed in [31]. Using this method the companion located at 0.19 arcsec near the object was theoretically differentiated with intensity of radiation 2×105 times greater [32]. The possibility to use this method to detect planets orbiting bright stars was also illustrated by astronomers [33, 34]. Vortex coronagraphy is now undergoing further development [35, 36]. There are a number of examples of non-astronomical applications [37, 38].

It was proposed to use optical vortices to improve optical measurements and increase the fidelity of optical testing [39, 40], for investigations in high-resolution fluorescence microsco‐ py [41], optical lithography [42, 43], quantum entanglement [44, 45, 46], Bose–Einstein con‐ densates [47].

Optical vortices show interesting properties in nonlinear optics [48]. For example, in [49, 50, 51] it was predicted that the phase conjugation at SBS of vortex beams is impossible due to the failure of selection of the conjugated mode. For a rather wide class of the vortex laser beams a novel and interesting phenomenon takes place which can be called the phase trans‐ formation at SBS. In essence there is only one Stokes mode, the amplification coefficient of which is maximal and higher than that of the conjugated mode. In other words, the non-con‐ jugated mode is selected of in the Stokes beam. The principal Gaussian mode, which is or‐ thogonal to the laser vortex mode, is an example of such an exceptional Stokes mode. The cause of this phenomenon is in the specific radial and azimuth distribution of the vortex la‐ ser beam. It is interesting that the hypersound vortices are formed in the SBS medium in ac‐ cordance with the law of topologic charge conservation. The predicted effects have been completely confirmed experimentally [52, 53, 54].

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

Tatarsky and von Karman modifications of the Kolmogorov model [61].

the conservation law of topological charge (or orbital angular moment) [7-9].

*ik*

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

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

**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.

<sup>2</sup> (*ε*˜ <sup>−</sup>1)*A*=0, (7)

Adaptive Optics and Optical Vortices http://dx.doi.org/10.5772/53328 155
