**4. Generation of optical vortex**

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

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.

propagating in the atmosphere.

156 Adaptive Optics Progress

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‐ guerre-Gaussian vortex modes *LGn <sup>m</sup>* can play the role of such "reference" optical vortices.

To create a beam with phase singularities artificially from an initial plane or Gaussian wave, a number of experimental techniques have been elaborated. There are many papers concern‐ ing the various aspects of generation of beams with phase singularities (see, for example, [89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104]). Among other possibilities, we can also refer to several methods for phase singularity creation in the optical beams based on nonlinear effects [105, 106, 107, 108]. The generation of optical vortices is also possible in the waveguides [109, 110, 111]. The adaptive mirrors themselves can be used for the forma‐ tion of optical vortices [112, 113]. In this chapter, though, we dwell only on a number of ways to generate the vortex beams, which allow one to form close-to-"reference" vortices with well-determined singular phase structure that is necessary for the accuracy analysis of the new algorithm of Hartmann-Shack wavefront reconstruction.

**Figure 5.** Configuration of a laser cavity intended for generation of spiral beams [116].

shown [117] that modes of such a resonator are singular beams.

1

mode for conversion into a *LG*<sup>0</sup>

and *HG*10 modes.

**Figure 6.** Intensity distribution of vortex beam generated in the laser cavity in near (left) and far (right) zone [116].

A ring cavity with the Dove's prism can also be used to generate vortex beams. It was

The next way to generate the optical vortex uses a phase (or a mode) converter. Usually it transforms a Hermit-Gaussian mode, generated in the laser, into a corresponding Laguerre-Gaussian mode. This method was first proposed in [118]. In the experiment the authors used a cylindrical lens, the axis of which was placed at an angle of 45º with respect to the *HG*<sup>01</sup>

sition of *HG*<sup>01</sup> mode parallel to the lens axis and a mode perpendicular to the axis. The mode perpendicular to the lens axis passes through a focus, advancing the relative Gouy phase be‐ tween the two modes of *π*/2 as required to form the doughnut mode from uncharged *HG*<sup>01</sup>

In Ref. [91] an expression was derived for an integral transformation of Hermit-Gaussian modes into Laguerre-Gaussian modes in the astigmatic optical system, and it was shown theoretically that passing the beam through the cylindrical lens can perform the conversion. The theory of a *π*/2 mode converter and a *π* mode converter, produced by two cylindrical

mode. The incident mode appears to the lens as a superpo‐

Adaptive Optics and Optical Vortices http://dx.doi.org/10.5772/53328 159

One method for generation of the screw dislocations is by forming the vortex beam immedi‐ ately inside a laser cavity. The authors of [114] were the first to report that the generation of wavefront vortices is possible using a cw laser source. It was shown in [115] that insertion of a non-axisymmetric transparency into the cavity results in generation of a vortex beam. It was reported in [116] that a pure spiral mode can be obtained by introducing a spiral phase element (SPE) into the laser cavity, which selects the chosen mode. The geometry of the cavi‐ ty intended for generation of such laser beams from [116] is shown in Figure 5. Here a rear mirror is replaced by a reflecting spiral phase element, which adds the phase change +2*imφ* after reflecting. As a result of reflection, the phase of a spiral mode -*imφ* changes to +*imφ*. The cylindrical lens inside the cavity is focused on the output coupler. This lens inverts the helicity of the mode back to the field described by exp{-*imφ*} and ensures the generation of the required spiral beam. The beam at the output passes another cylindrical lens and its dis‐ tribution becomes the same as inside the cavity. A pinhole in the cavity ensures the genera‐ tion of a spiral mode of minimal order, i.e., *TEM*<sup>01</sup> mode. It should be stressed that the spiral phase element determines the parameters of the spiral beam within the cavity so that by its variation the parameters of the output beam can be controlled.

This method was tested with a linearly polarized CO2-laser. The reflecting spiral phase ele‐ ment was made of silicon by multilevel etching. It had 32 levels with the entire height of break *λ* that corresponds to *m*=1. Precision of etching was about 3% and deviation of the surface from the prescribed form was less than 20 nm. The reflecting coefficient of the element was great‐ er than 98%; the diameter and length of the laser tube were 11 mm and 65 cm, respectively. The lens inside the cavity with focal length 12.5 cm was focused in the output concave mirror with a radius of 3 m. An identical lens was placed outside the cavity to collimate the beam. In Figure 6 we demonstrate the stable spiral beam obtained in the experiment [116]. The vorti‐ cal nature of the beam is proved not by demonstration of the "fork" in the interferogram but by the doughnut-like intensity distribution in the near and the far zone.

**Figure 5.** Configuration of a laser cavity intended for generation of spiral beams [116].

To create a beam with phase singularities artificially from an initial plane or Gaussian wave, a number of experimental techniques have been elaborated. There are many papers concern‐ ing the various aspects of generation of beams with phase singularities (see, for example, [89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104]). Among other possibilities, we can also refer to several methods for phase singularity creation in the optical beams based on nonlinear effects [105, 106, 107, 108]. The generation of optical vortices is also possible in the waveguides [109, 110, 111]. The adaptive mirrors themselves can be used for the forma‐ tion of optical vortices [112, 113]. In this chapter, though, we dwell only on a number of ways to generate the vortex beams, which allow one to form close-to-"reference" vortices with well-determined singular phase structure that is necessary for the accuracy analysis of

One method for generation of the screw dislocations is by forming the vortex beam immedi‐ ately inside a laser cavity. The authors of [114] were the first to report that the generation of wavefront vortices is possible using a cw laser source. It was shown in [115] that insertion of a non-axisymmetric transparency into the cavity results in generation of a vortex beam. It was reported in [116] that a pure spiral mode can be obtained by introducing a spiral phase element (SPE) into the laser cavity, which selects the chosen mode. The geometry of the cavi‐ ty intended for generation of such laser beams from [116] is shown in Figure 5. Here a rear mirror is replaced by a reflecting spiral phase element, which adds the phase change +2*imφ* after reflecting. As a result of reflection, the phase of a spiral mode -*imφ* changes to +*imφ*. The cylindrical lens inside the cavity is focused on the output coupler. This lens inverts the helicity of the mode back to the field described by exp{-*imφ*} and ensures the generation of the required spiral beam. The beam at the output passes another cylindrical lens and its dis‐ tribution becomes the same as inside the cavity. A pinhole in the cavity ensures the genera‐ tion of a spiral mode of minimal order, i.e., *TEM*<sup>01</sup> mode. It should be stressed that the spiral phase element determines the parameters of the spiral beam within the cavity so that by its

This method was tested with a linearly polarized CO2-laser. The reflecting spiral phase ele‐ ment was made of silicon by multilevel etching. It had 32 levels with the entire height of break *λ* that corresponds to *m*=1. Precision of etching was about 3% and deviation of the surface from the prescribed form was less than 20 nm. The reflecting coefficient of the element was great‐ er than 98%; the diameter and length of the laser tube were 11 mm and 65 cm, respectively. The lens inside the cavity with focal length 12.5 cm was focused in the output concave mirror with a radius of 3 m. An identical lens was placed outside the cavity to collimate the beam. In Figure 6 we demonstrate the stable spiral beam obtained in the experiment [116]. The vorti‐ cal nature of the beam is proved not by demonstration of the "fork" in the interferogram but

the new algorithm of Hartmann-Shack wavefront reconstruction.

158 Adaptive Optics Progress

variation the parameters of the output beam can be controlled.

by the doughnut-like intensity distribution in the near and the far zone.

**Figure 6.** Intensity distribution of vortex beam generated in the laser cavity in near (left) and far (right) zone [116].

A ring cavity with the Dove's prism can also be used to generate vortex beams. It was shown [117] that modes of such a resonator are singular beams.

The next way to generate the optical vortex uses a phase (or a mode) converter. Usually it transforms a Hermit-Gaussian mode, generated in the laser, into a corresponding Laguerre-Gaussian mode. This method was first proposed in [118]. In the experiment the authors used a cylindrical lens, the axis of which was placed at an angle of 45º with respect to the *HG*<sup>01</sup> mode for conversion into a *LG*<sup>0</sup> 1 mode. The incident mode appears to the lens as a superpo‐ sition of *HG*<sup>01</sup> mode parallel to the lens axis and a mode perpendicular to the axis. The mode perpendicular to the lens axis passes through a focus, advancing the relative Gouy phase be‐ tween the two modes of *π*/2 as required to form the doughnut mode from uncharged *HG*<sup>01</sup> and *HG*10 modes.

In Ref. [91] an expression was derived for an integral transformation of Hermit-Gaussian modes into Laguerre-Gaussian modes in the astigmatic optical system, and it was shown theoretically that passing the beam through the cylindrical lens can perform the conversion. The theory of a *π*/2 mode converter and a *π* mode converter, produced by two cylindrical lenses, was described in more details in [92]. Padgett et al. describe in a tutorial paper [119] how a range of Laguerre-Gaussian modes can be produced using two cylindrical lenses starting from the corresponding Hermit-Gaussian modes, and present the clear examples, showing the intensity and phase distributions obtained. The initial higher order Hermit-Gaussian modes can be produced in the laser with intracavity cross-wires. The authors of [96] used a similar technique with a Nd:YAG laser operating at the 100 mW level.

The peculiarity of the interference of two Laguerre–Gaussian modes, having the opposite helicity of the phase, manifests itself in the branching of a fringe in the middle of the beam and formation of a characteristic "fork" with an additional fringe appearing in the centre, as compared with the case of a vortex mode interfering with a plane reference wave (see Figure 2). Such branching of fringes indicates the vortex nature of the investigated beam, while the absence of branching is a manifestation of the regular character of the beam phase surface.

**Figure 7.** The optical scheme for registration of the phase portrait of a laser beam [54]: 1 – dividing parallel-sided

The invention of a branched hologram [89, 93] uncovered a relatively easy way to produce beams with optical vortices from an ordinary wave by using its diffraction on the amplitude diffraction grating. The idea of singular beam formation is based on the holographic princi‐ ple: a readout beam restores the wave, which has participated in the hologram recording. Instead of writing a hologram with two actual optical waves, it is sufficient to calculate the interference pattern numerically and, for example, print the picture in black-and-white or grey scale. The amplitude grating after transverse scaling can, when illuminated by a regu‐

**Figure 8.** The experimental distribution of intensity and phase portrait of the laser mode *LG*<sup>0</sup>

lar wave, reproduce singular beams in diffraction orders.

plate, 2 – mirrors, 3 – lens, 4 – CCD camera.

phase converter [54].

1

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

in far field

161


1 obtained at usage of a

Figure 8 displays the experimental distributions of intensity of the laser mode *LG*<sup>0</sup>

and its picture of interference with an obliquely incident wave in the form of *LG*<sup>0</sup>

Even in the absence of the required initial *HG*10 mode in the laser emission, it is easy to produce artificially a similar configuration by introducing a glass plate in a half of the *TEM*00 beam and achieving the necessary *π* phase shift, and then to apply mode conversion. An example of such doughnut beam creation was reported in [120]. The efficiency of conversion was about 50%. It is possible to use the cylindrical lens mode converter [121] but with production of the initial higher order Hermit-Gaussian mode, by exciting it in an actively stabilized ring cavity, match‐ ing in the Gaussian beam from their titanium sapphire laser. The efficiency was up to 40% for the *LG*<sup>0</sup> 1 mode, and higher order doughnuts could also be obtained easily. The method pro‐ posed in [122] is based on the formation of a pseudo *HGNM* mode, propagating the Gaussian beam of a number of edges of thin glass plates and forming the edge dislocations with the following its astigmatic conversion into a Laguerre-Gaussian mode.

It was reported in [123] that in the event of ideal conversion, the efficiency of Hermit-Gaus‐ sian mode transformation into Laguerre-Gaussian mode is about 99.9%. The spherical aber‐ ration does not reduce the efficiency factor. Typically cylindrical lenses are not perfect and their defects give rise to several Laguerre-Gaussian modes. The superposition of compo‐ nents can be unstable and this means a dependence of intensity on the longitudinal coordi‐ nate. If special means are not employed the precision of lens fabrication is about 5%, in this case the efficiency of beam transformation into Laguerre-Gaussian mode is 95%. Imperfec‐ tions of 10% result in drop of efficiency down to 80%.

In [124, 54] the formation of the Laguerre-Gaussian *LG*<sup>0</sup> 1 or *LG*<sup>1</sup> 1 modes was performed at the output of a pulsed laser-generator of Hermit-Gaussian *HG*01 or *HG*21 modes with the help of a tunable astigmatic *π*/2-converter based on the so-called optical quadrupole [125]. It con‐ sists of two similar mechano-optical modules, each of which incorporates the positive and negative cylindrical lenses with the same focal length and a positive spherical lens. The me‐ chanical configuration of each module can synchronously turn the incorporated cylindrical lenses in the opposite directions with respect to the optical axis, which ensures its rearrange‐ ment. In the initial position the optical forces of cylindrical lenses completely compensate each other and their axes coincide with the main axes of the intensity distribution of the la‐ ser. The distance between the modules is fixed so that the spherical lenses in different mod‐ ules are located at a focal distance from each other and form the optical Fourier transformer.

To study the phase structure of radiation, in [54] use was made of a special interferometer scheme, where the reference beam was produced from a part of the original Laguerre–Gaus‐ sian *LG*<sup>0</sup> 1 or *LG*<sup>1</sup> 1 mode (see Figure 7). As a result, each of the modes interfered with a similar one, but with a topological charge of the opposite sign (the opposite helicity), i.e., with *LG*<sup>0</sup> -1 or *LG*<sup>1</sup> -1 mode. The interference fringe density depended on the thickness of the plane-paral‐ lel plate 1 and could be additionally varied by inclining the mirrors 2 (see figure 7).

The peculiarity of the interference of two Laguerre–Gaussian modes, having the opposite helicity of the phase, manifests itself in the branching of a fringe in the middle of the beam and formation of a characteristic "fork" with an additional fringe appearing in the centre, as compared with the case of a vortex mode interfering with a plane reference wave (see Figure 2). Such branching of fringes indicates the vortex nature of the investigated beam, while the absence of branching is a manifestation of the regular character of the beam phase surface.

lenses, was described in more details in [92]. Padgett et al. describe in a tutorial paper [119] how a range of Laguerre-Gaussian modes can be produced using two cylindrical lenses starting from the corresponding Hermit-Gaussian modes, and present the clear examples, showing the intensity and phase distributions obtained. The initial higher order Hermit-Gaussian modes can be produced in the laser with intracavity cross-wires. The authors of

Even in the absence of the required initial *HG*10 mode in the laser emission, it is easy to produce artificially a similar configuration by introducing a glass plate in a half of the *TEM*00 beam and achieving the necessary *π* phase shift, and then to apply mode conversion. An example of such doughnut beam creation was reported in [120]. The efficiency of conversion was about 50%. It is possible to use the cylindrical lens mode converter [121] but with production of the initial higher order Hermit-Gaussian mode, by exciting it in an actively stabilized ring cavity, match‐ ing in the Gaussian beam from their titanium sapphire laser. The efficiency was up to 40% for

mode, and higher order doughnuts could also be obtained easily. The method pro‐

1 or *LG*<sup>1</sup> 1

mode (see Figure 7). As a result, each of the modes interfered with a similar

modes was performed at the


posed in [122] is based on the formation of a pseudo *HGNM* mode, propagating the Gaussian beam of a number of edges of thin glass plates and forming the edge dislocations with the

It was reported in [123] that in the event of ideal conversion, the efficiency of Hermit-Gaus‐ sian mode transformation into Laguerre-Gaussian mode is about 99.9%. The spherical aber‐ ration does not reduce the efficiency factor. Typically cylindrical lenses are not perfect and their defects give rise to several Laguerre-Gaussian modes. The superposition of compo‐ nents can be unstable and this means a dependence of intensity on the longitudinal coordi‐ nate. If special means are not employed the precision of lens fabrication is about 5%, in this case the efficiency of beam transformation into Laguerre-Gaussian mode is 95%. Imperfec‐

output of a pulsed laser-generator of Hermit-Gaussian *HG*01 or *HG*21 modes with the help of a tunable astigmatic *π*/2-converter based on the so-called optical quadrupole [125]. It con‐ sists of two similar mechano-optical modules, each of which incorporates the positive and negative cylindrical lenses with the same focal length and a positive spherical lens. The me‐ chanical configuration of each module can synchronously turn the incorporated cylindrical lenses in the opposite directions with respect to the optical axis, which ensures its rearrange‐ ment. In the initial position the optical forces of cylindrical lenses completely compensate each other and their axes coincide with the main axes of the intensity distribution of the la‐ ser. The distance between the modules is fixed so that the spherical lenses in different mod‐ ules are located at a focal distance from each other and form the optical Fourier transformer. To study the phase structure of radiation, in [54] use was made of a special interferometer scheme, where the reference beam was produced from a part of the original Laguerre–Gaus‐

one, but with a topological charge of the opposite sign (the opposite helicity), i.e., with *LG*<sup>0</sup>

lel plate 1 and could be additionally varied by inclining the mirrors 2 (see figure 7).


following its astigmatic conversion into a Laguerre-Gaussian mode.

tions of 10% result in drop of efficiency down to 80%. In [124, 54] the formation of the Laguerre-Gaussian *LG*<sup>0</sup>

[96] used a similar technique with a Nd:YAG laser operating at the 100 mW level.

the *LG*<sup>0</sup> 1

160 Adaptive Optics Progress

sian *LG*<sup>0</sup> 1 or *LG*<sup>1</sup> 1

or *LG*<sup>1</sup>

Figure 8 displays the experimental distributions of intensity of the laser mode *LG*<sup>0</sup> 1 in far field and its picture of interference with an obliquely incident wave in the form of *LG*<sup>0</sup> -1 mode.

**Figure 7.** The optical scheme for registration of the phase portrait of a laser beam [54]: 1 – dividing parallel-sided plate, 2 – mirrors, 3 – lens, 4 – CCD camera.

**Figure 8.** The experimental distribution of intensity and phase portrait of the laser mode *LG*<sup>0</sup> 1 obtained at usage of a phase converter [54].

The invention of a branched hologram [89, 93] uncovered a relatively easy way to produce beams with optical vortices from an ordinary wave by using its diffraction on the amplitude diffraction grating. The idea of singular beam formation is based on the holographic princi‐ ple: a readout beam restores the wave, which has participated in the hologram recording. Instead of writing a hologram with two actual optical waves, it is sufficient to calculate the interference pattern numerically and, for example, print the picture in black-and-white or grey scale. The amplitude grating after transverse scaling can, when illuminated by a regu‐ lar wave, reproduce singular beams in diffraction orders.

Using the description of the singular wave amplitude (2), one can easily calculate the pattern of interference of such wave with a coherent plane wave tilted by the angle *γ* with respect to the *z* axis. The calculated interference pattern depends on the angle *γ* between the interfer‐ ing waves and corresponds to two well-known holographic schemes: on-axis [126] when *γ*=0 and off-axis [127] holograms. The spiral hologram (or spiral zone plate) realized under the on-axis scheme suffers from all the disadvantages inherent to the on-axis holograms, namely, the lack of spatial separation of the reconstructed beams from the directly transmit‐ ted readout beam. Therefore the on-axis spiral holograms have not found wide application unlike the off-axis computer-generated holograms [128].

Under interference between the plane wave and the optical vortex with unity topologic charge the transmittance of amplitude diffraction grating varies according to

$$T = \left[\frac{1}{2} \left[1 - \cos\left(\left(\frac{2\pi x}{\Lambda}\right) - \text{arctg}\left(\frac{y}{\Lambda}\right)\right)\right]\right]^2. \tag{8}$$

**Figure 10.** The set-up scheme for formation of the optical vortex:1 – He-Ne laser; 2 – collimator; 3 – optical plane

Adaptive Optics and Optical Vortices http://dx.doi.org/10.5772/53328 163

The experimental set-up scheme for formation of the optical vortex with the help of comput‐ er-synthesized amplitude grating is shown in Figure 10. The experimental set-up consists of a system for forming the collimated laser beam (*λ*=0.633 µ), a Mach-Zehnder interferometer and a registration system of the far field beam intensity and interference pattern. The system for forming the collimated beam includes the He-Ne laser 1 and a collimator 2 consisting of two lenses forming the Gaussian beam with a plane wave front. The Mach-Zehnder interfer‐ ometer consists of two plane plates 3 and two mirrors 4. The computer-synthesized ampli‐ tude diffraction grating 5 is inserted into one of the arms of the interferometer. The lens 6 focuses the radiation onto the screen of CCD-camera 7. When blocking or admitting the ref‐ erence beam from the second arm of the interferometer, the CCD camera registers the far field intensity of the vortex beam or its interference pattern with the reference beam, respec‐ tively. Varying the angle between both these beams allows one to vary the interference

After passing the beam through the optical scheme, the central peak (0-th diffraction order) is formed in the far-field zone. It concerns the non-scattered component of the beam that has passed through the grating. Less intensive two doughnut-shaped lateral peaks are formed symmetrically from the central peak. The lateral peaks represent the optical vortices and have a topological charge equal to the value and opposite to the sign. In the 1st order of dif‐ fraction there is only 16.7 % of energy penetrating the grating in an ideal scenario. In the ex‐ periment this part of energy is equal to about 10% owing to the imperfect structure of the

For registration of the optical vortex it is necessary to cut off the unnecessary diffraction or‐ ders. The pictures of doughnut-like intensity distribution of the optical vortex (the lateral peak) in far field and its interference pattern are shown in Figure 11. The rigorous proof of that the obtained lateral peaks bear the optical vortices is the availability of typical "fork" in the interference pattern. The formed vortex in Figure 11, as the vortices in Figures 6 and 8, is

mode that is seen from their comparison with Figure 2.

1

plate, 4 – reflecting plane mirror; 5 – amplitude grating forming the optical vortex; 6 – lens; 7 – CCD camera.

fringe density.

grating and its incomplete transmittance.

rather different from the ideal *LG*<sup>0</sup>

where Λ=*λ*/*γ* is the grating period. When the basic Gaussian mode passes through the gra‐ ting and is focused by the lens, the *LG*<sup>0</sup> 1 and *LG*<sup>0</sup> -1 modes occur in the far field in the 1st and – 1st orders of diffraction, respectively. The period of the grating should be equal to 100-200 microns to separate the orders of diffraction properly in the actual experiment.

The two simplest ways to fabricate the amplitude diffraction gratings in the form of comput‐ er-synthesized holograms are as follows. The first involves the printing of an image onto a transparency utilized in laserjet printers. The second approach consists in photographing an inverted image, printed on a sheet of white paper, onto photo-film. Fragments of images of the gratings with the profile (8) obtained upon usage of the laser transparency with the reso‐ lution of 1200 ppi as well as the photo-film are shown in Figure 9 [129, 130, 131]. The usage of the photo-film is more preferable since it gives higher quality of the vortex to be formed and greater power conversion coefficient into the required diffraction order.

**Figure 9.** Magnified fragment of the amplitude grating in the experiment with laser transparency (left) and photofilm (right).

Using the description of the singular wave amplitude (2), one can easily calculate the pattern of interference of such wave with a coherent plane wave tilted by the angle *γ* with respect to the *z* axis. The calculated interference pattern depends on the angle *γ* between the interfer‐ ing waves and corresponds to two well-known holographic schemes: on-axis [126] when *γ*=0 and off-axis [127] holograms. The spiral hologram (or spiral zone plate) realized under the on-axis scheme suffers from all the disadvantages inherent to the on-axis holograms, namely, the lack of spatial separation of the reconstructed beams from the directly transmit‐ ted readout beam. Therefore the on-axis spiral holograms have not found wide application

Under interference between the plane wave and the optical vortex with unity topologic

*<sup>Λ</sup>* ) <sup>−</sup>arctg(

where Λ=*λ*/*γ* is the grating period. When the basic Gaussian mode passes through the gra‐

1st orders of diffraction, respectively. The period of the grating should be equal to 100-200

The two simplest ways to fabricate the amplitude diffraction gratings in the form of comput‐ er-synthesized holograms are as follows. The first involves the printing of an image onto a transparency utilized in laserjet printers. The second approach consists in photographing an inverted image, printed on a sheet of white paper, onto photo-film. Fragments of images of the gratings with the profile (8) obtained upon usage of the laser transparency with the reso‐ lution of 1200 ppi as well as the photo-film are shown in Figure 9 [129, 130, 131]. The usage of the photo-film is more preferable since it gives higher quality of the vortex to be formed

**Figure 9.** Magnified fragment of the amplitude grating in the experiment with laser transparency (left) and photo-

*y <sup>x</sup>* )) <sup>2</sup>

, (8)


charge the transmittance of amplitude diffraction grating varies according to

<sup>2</sup> <sup>1</sup>−cos(( <sup>2</sup>*π<sup>x</sup>*

1 and *LG*<sup>0</sup>

microns to separate the orders of diffraction properly in the actual experiment.

and greater power conversion coefficient into the required diffraction order.

unlike the off-axis computer-generated holograms [128].

*<sup>T</sup>* <sup>=</sup> <sup>1</sup>

ting and is focused by the lens, the *LG*<sup>0</sup>

162 Adaptive Optics Progress

film (right).

**Figure 10.** The set-up scheme for formation of the optical vortex:1 – He-Ne laser; 2 – collimator; 3 – optical plane plate, 4 – reflecting plane mirror; 5 – amplitude grating forming the optical vortex; 6 – lens; 7 – CCD camera.

The experimental set-up scheme for formation of the optical vortex with the help of comput‐ er-synthesized amplitude grating is shown in Figure 10. The experimental set-up consists of a system for forming the collimated laser beam (*λ*=0.633 µ), a Mach-Zehnder interferometer and a registration system of the far field beam intensity and interference pattern. The system for forming the collimated beam includes the He-Ne laser 1 and a collimator 2 consisting of two lenses forming the Gaussian beam with a plane wave front. The Mach-Zehnder interfer‐ ometer consists of two plane plates 3 and two mirrors 4. The computer-synthesized ampli‐ tude diffraction grating 5 is inserted into one of the arms of the interferometer. The lens 6 focuses the radiation onto the screen of CCD-camera 7. When blocking or admitting the ref‐ erence beam from the second arm of the interferometer, the CCD camera registers the far field intensity of the vortex beam or its interference pattern with the reference beam, respec‐ tively. Varying the angle between both these beams allows one to vary the interference fringe density.

After passing the beam through the optical scheme, the central peak (0-th diffraction order) is formed in the far-field zone. It concerns the non-scattered component of the beam that has passed through the grating. Less intensive two doughnut-shaped lateral peaks are formed symmetrically from the central peak. The lateral peaks represent the optical vortices and have a topological charge equal to the value and opposite to the sign. In the 1st order of dif‐ fraction there is only 16.7 % of energy penetrating the grating in an ideal scenario. In the ex‐ periment this part of energy is equal to about 10% owing to the imperfect structure of the grating and its incomplete transmittance.

For registration of the optical vortex it is necessary to cut off the unnecessary diffraction or‐ ders. The pictures of doughnut-like intensity distribution of the optical vortex (the lateral peak) in far field and its interference pattern are shown in Figure 11. The rigorous proof of that the obtained lateral peaks bear the optical vortices is the availability of typical "fork" in the interference pattern. The formed vortex in Figure 11, as the vortices in Figures 6 and 8, is rather different from the ideal *LG*<sup>0</sup> 1 mode that is seen from their comparison with Figure 2.

not greater than the wavelength of light. This aim is realized by jumps of the optical path length not less than the even number of half wavelengths. These jumps form the lines divid‐ ing the kinoform into several zones. In boundaries of each zone the optical path length can be constant (there are two levels of binary phase elements), they can change discretely (*n*level phase elements), or they can change more continuously (in an ideal phase element *n* approaches infinity). With an increase of *n* the phase efficiency of the element increases as well as its ability to control light properly. The application of kinoforms facilitates a reduc‐ tion in the number of optical elements in the system by combining optical properties of sev‐ eral elements into one kinoform. Thus, these optical methods offer broad potential for anyone who wants to obtain beams with desired properties and to generate beams with op‐

The fabrication of spiral (or helicoidal) phase plates techniques has progressed in recent years [139, 140, 141]. We will describe the generation of a doughnut Laguerre-Gaussian *LG*<sup>0</sup>

mode with the help of a spiral phase plate [129, 130, 142, 143, 144] manufactured with etch‐ ing of the fused quartz substrate using kinoform technology. Quartz displays a high damage threshold at *λ*=0.3-1.3 *μ*, high uniformity of chemical composition and refractive index *n* that

The fabrication of a kinoform spiral phase plate of fused quartz is performed as follows [142]. A quartz plate, 3 cm in diameter and 3 mm in thickness, is taken as the substrate. Both surfaces of the substrate are mechanically polished with a nanodiamond suspension up to the flatness better than *λ*/30. Special precautions are made to avoid the formation of surface damage layer that may destabilize subsequent etching. A multi-level stepped microrelief, imitating the continuous helicoidal profile, is fabricated using precise sequential etchings of the surface through a photoresist mask in a mixture based on hydrofluoric acid. At every stage, a level pattern is formed in the photoresist layer using a method of deep UV photoli‐ thography. The temperature during etching is stabilized with an accuracy of ±0.1ºC. The 16 and 32-level spiral phase plates at *m*=1 and 2 have been fabricated. As the calculations show, such stepped plates and an ideal plate with an exactly helicoidal surface give practically the same optical vortices in the far field. As contrasted to the examples of spiral phase plates [140], the plates [142] have a very high laser damage threshold and a working diameter of 2 cm that is larger by an order of magnitude. The high laser damage resistance of such plates allows their use in experiments with powerful laser beams [52-54]. The general view of the

The 3D image of the central part of a 32-level spiral phase plate designed for *λ*=0.633 µ, *m*=1 is shown in Figure 13. As it is seen from Figure 13, upon motion around the plate axis along a circle (perpendicular to the propeller bosses), the change of the etched profile altitude is linear with a rather high accuracy. The total break height of the microrelief on the plate sur‐ face of 1317.5 nm agrees with the calculated value *λ*/(*n*-1)=1339.6 nm, where *n* is the sub‐ strate refraction index, with an accuracy about of one and a half percent. As the measurements show, the roughness of the etched and non-etched surfaces (including the deepest one) is approximately the same. The roughness rms of each step surface equals 1-1.5

1

165

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

tical vortices.

minimizes the laser beam distortions on passing.

spiral phase plate is shown in Figure 12.

nm, amounting to 2-3% of the height of one step of 43.2 nm.

**Figure 11.** The intensity of the lateral diffraction order in the far field and interference pattern with the plane wave in the case of computer-synthesized amplitude diffraction grating.

Phase transparencies can be used to generate optical vortices. Application of the phase mod‐ ulator results in phase changes and, after that, in amplitude changes with deep intensity modulation and the advent of zeros. In [132] the optical schematic was described, in which the wave carrying the optical vortex is recorded on thick film (Bregg's hologram) that is used to reproduce the vortex beam. The diffraction efficiency in this schematic is about 99%. A relatively thin transparency with thickness varied gradually in one of the half planes is used in the other method [133]. The efficiency of this method is greater than 90%. A similar method was proposed in [134], but a dielectric wedge was used as the phase modulator. In general, a chain of several vortices is formed as the product of this process. The shape defor‐ mation of each vortex depends on the wedge angle and on the diameter of the beam waist on the wedge surface. Varying the waist radius, one can obtain the required number of vor‐ tices (even a single vortex).

One more method of the optical vortex generation was proposed in [95]. In this method a phase transparency is used, which immediately adds the artificial vortex component into the phase profile. One such phase modulator is a transparent plate, one surface of which has a helical profile, repeating the singular phase distribution. To obtain Laguerre-Gaussian mode the depth of break onto the surface should be equal to *mλ*/(*n*1-*n*2), where *n*<sup>1</sup> is the plate index of refraction, and *n*2 is the medium index of refraction. If these conditions are met, the opti‐ cal vortex appears in the far field. The main difficulty of this method lies in the problems of fabricating such a transparency. A special mask is used in the manufacture of such plates, which is made negative relative to the spiral phase plate to be formed [135]. The mask is made of brass and checked by the control interferometric system of high precision. After fabrication the mask is filled with a polymer substance and covered by glass. The spiral phase plate is formed on the glass as a result of polymerization. The transfer coefficient of such a plate is 0.98. Publications have appeared recently concerning the generation of phase transparencies using liquid crystals [136, 137].

We note that the manufacture of phase modulators is a special branch of optics called kino‐ form optics. At the heart of this branch lays the possibility to realize the phase control of ra‐ diation by a step-like change of the thickness or the refraction index of some structures [138]. Light weight, small size, and low cost are the most attractive features of kinoform phase elements, when compared with lenses, prisms, mirrors, and other optical devices. The kinoforms can be described as optical elements performing phase modulation with a depth not greater than the wavelength of light. This aim is realized by jumps of the optical path length not less than the even number of half wavelengths. These jumps form the lines divid‐ ing the kinoform into several zones. In boundaries of each zone the optical path length can be constant (there are two levels of binary phase elements), they can change discretely (*n*level phase elements), or they can change more continuously (in an ideal phase element *n* approaches infinity). With an increase of *n* the phase efficiency of the element increases as well as its ability to control light properly. The application of kinoforms facilitates a reduc‐ tion in the number of optical elements in the system by combining optical properties of sev‐ eral elements into one kinoform. Thus, these optical methods offer broad potential for anyone who wants to obtain beams with desired properties and to generate beams with op‐ tical vortices.

**Figure 11.** The intensity of the lateral diffraction order in the far field and interference pattern with the plane wave in

Phase transparencies can be used to generate optical vortices. Application of the phase mod‐ ulator results in phase changes and, after that, in amplitude changes with deep intensity modulation and the advent of zeros. In [132] the optical schematic was described, in which the wave carrying the optical vortex is recorded on thick film (Bregg's hologram) that is used to reproduce the vortex beam. The diffraction efficiency in this schematic is about 99%. A relatively thin transparency with thickness varied gradually in one of the half planes is used in the other method [133]. The efficiency of this method is greater than 90%. A similar method was proposed in [134], but a dielectric wedge was used as the phase modulator. In general, a chain of several vortices is formed as the product of this process. The shape defor‐ mation of each vortex depends on the wedge angle and on the diameter of the beam waist on the wedge surface. Varying the waist radius, one can obtain the required number of vor‐

One more method of the optical vortex generation was proposed in [95]. In this method a phase transparency is used, which immediately adds the artificial vortex component into the phase profile. One such phase modulator is a transparent plate, one surface of which has a helical profile, repeating the singular phase distribution. To obtain Laguerre-Gaussian mode the depth of break onto the surface should be equal to *mλ*/(*n*1-*n*2), where *n*<sup>1</sup> is the plate index of refraction, and *n*2 is the medium index of refraction. If these conditions are met, the opti‐ cal vortex appears in the far field. The main difficulty of this method lies in the problems of fabricating such a transparency. A special mask is used in the manufacture of such plates, which is made negative relative to the spiral phase plate to be formed [135]. The mask is made of brass and checked by the control interferometric system of high precision. After fabrication the mask is filled with a polymer substance and covered by glass. The spiral phase plate is formed on the glass as a result of polymerization. The transfer coefficient of such a plate is 0.98. Publications have appeared recently concerning the generation of phase

We note that the manufacture of phase modulators is a special branch of optics called kino‐ form optics. At the heart of this branch lays the possibility to realize the phase control of ra‐ diation by a step-like change of the thickness or the refraction index of some structures [138]. Light weight, small size, and low cost are the most attractive features of kinoform phase elements, when compared with lenses, prisms, mirrors, and other optical devices. The kinoforms can be described as optical elements performing phase modulation with a depth

the case of computer-synthesized amplitude diffraction grating.

tices (even a single vortex).

164 Adaptive Optics Progress

transparencies using liquid crystals [136, 137].

The fabrication of spiral (or helicoidal) phase plates techniques has progressed in recent years [139, 140, 141]. We will describe the generation of a doughnut Laguerre-Gaussian *LG*<sup>0</sup> 1 mode with the help of a spiral phase plate [129, 130, 142, 143, 144] manufactured with etch‐ ing of the fused quartz substrate using kinoform technology. Quartz displays a high damage threshold at *λ*=0.3-1.3 *μ*, high uniformity of chemical composition and refractive index *n* that minimizes the laser beam distortions on passing.

The fabrication of a kinoform spiral phase plate of fused quartz is performed as follows [142]. A quartz plate, 3 cm in diameter and 3 mm in thickness, is taken as the substrate. Both surfaces of the substrate are mechanically polished with a nanodiamond suspension up to the flatness better than *λ*/30. Special precautions are made to avoid the formation of surface damage layer that may destabilize subsequent etching. A multi-level stepped microrelief, imitating the continuous helicoidal profile, is fabricated using precise sequential etchings of the surface through a photoresist mask in a mixture based on hydrofluoric acid. At every stage, a level pattern is formed in the photoresist layer using a method of deep UV photoli‐ thography. The temperature during etching is stabilized with an accuracy of ±0.1ºC. The 16 and 32-level spiral phase plates at *m*=1 and 2 have been fabricated. As the calculations show, such stepped plates and an ideal plate with an exactly helicoidal surface give practically the same optical vortices in the far field. As contrasted to the examples of spiral phase plates [140], the plates [142] have a very high laser damage threshold and a working diameter of 2 cm that is larger by an order of magnitude. The high laser damage resistance of such plates allows their use in experiments with powerful laser beams [52-54]. The general view of the spiral phase plate is shown in Figure 12.

The 3D image of the central part of a 32-level spiral phase plate designed for *λ*=0.633 µ, *m*=1 is shown in Figure 13. As it is seen from Figure 13, upon motion around the plate axis along a circle (perpendicular to the propeller bosses), the change of the etched profile altitude is linear with a rather high accuracy. The total break height of the microrelief on the plate sur‐ face of 1317.5 nm agrees with the calculated value *λ*/(*n*-1)=1339.6 nm, where *n* is the sub‐ strate refraction index, with an accuracy about of one and a half percent. As the measurements show, the roughness of the etched and non-etched surfaces (including the deepest one) is approximately the same. The roughness rms of each step surface equals 1-1.5 nm, amounting to 2-3% of the height of one step of 43.2 nm.

that the beam intensity distribution has a true doughnut-like shape. The wavefront singular‐ ity appears, as before, by fringe branching in the beam center with the forming of a "fork"

Adaptive Optics and Optical Vortices http://dx.doi.org/10.5772/53328 167

**Figure 14.** Experimental distribution of intensity of a vortex beam in far field and its interference pattern with obli‐

The experimental data are in good agreement with the results of numerical simulation of the optical system, taking into account the stepped structure of spiral phase plate. The results barely differ from the distribution shown in Figure 2. It should be noted that the vortex

ral phase plate throughout its area. This circumstance gives us grounds to believe that the vortex wave front to be reconstructed by the Hartmann-Shack sensor has to be close to the

The problem of phase reconstruction using the Shack-Hartmann technique was successfully solved for optical fields with smooth wavefronts [145, 146, 147]. In the simplest case, to ob‐ tain the phase *S*(**r**) using the results of measurement of phase gradient projection ∇⊥ *Sm*(**r**) on the transverse plane it is possible to employ the numeric integration of the gradient over

mode) is very good, caused by the high surface quality of the spi‐

∇⊥*Sm*(*ρ*)*dρ*, (9)

quely incident reference plane wave in an experiment with a kinoform spiral phase plate.

*S*(*r*)=*S*0(*r*) + *∫*

*Γ*

where r={*x*, *y*}. Since for the ordinary wave fields the phase distribution is the potential func‐ tion, the values of *S*(r) do not depend formally on the configuration of the integration path. In the actual experiments, however, some errors are always present, so the potentiality of phase is violated and the results of phase reconstruction depend on the integration path [147]. To reduce the noise influence on the results it was proposed to consider the phase re‐

1

**5. Wavefront sensing of optical vortex**

quality (similarity to *LG*<sup>0</sup>

wave front.

1

ideal *LG*<sup>0</sup>

a contour Γ:

typical for screw dislocation with unity topological charge.

**Figure 12.** The photo-image of the 32-level spiral phase plate.

**Figure 13.** The image of surface in the near-axis region of the 32-level phase plate designed for λ=0.633 μ and its profile shape under motion along the circular line.

It should be noted that a laser beam in the form of a principal Gaussian mode with a plane wavefront that passes through a spiral phase plate maximally resembles the *LG*<sup>0</sup> <sup>1</sup> mode in the focal plane of a lens (in far field). The part of this mode in the beam exceeds more than 90%; the residual energy is confined in general in the higher Laguerre-Gaussian modes *LGn* 1 . It should be noted that the proper intensity modulation of the beam incident to the plate can additionally enhance the portion of the *LG*<sup>0</sup> 1 mode.

To generate a vortex beam with the help of a spiral phase plate, the experimental setup shown in Figure 10 is used. The spiral phase plate is installed into the scheme instead of the amplitude diffraction grating. In this case a vortex is formed in the 0th diffraction order in far field. Figure 14 demonstrates the experimental distributions of laser intensity in the far field and the pattern of interference of this beam with a obliquely reference plane wave. It is seen that the beam intensity distribution has a true doughnut-like shape. The wavefront singular‐ ity appears, as before, by fringe branching in the beam center with the forming of a "fork" typical for screw dislocation with unity topological charge.

**Figure 14.** Experimental distribution of intensity of a vortex beam in far field and its interference pattern with obli‐ quely incident reference plane wave in an experiment with a kinoform spiral phase plate.

The experimental data are in good agreement with the results of numerical simulation of the optical system, taking into account the stepped structure of spiral phase plate. The results barely differ from the distribution shown in Figure 2. It should be noted that the vortex quality (similarity to *LG*<sup>0</sup> 1 mode) is very good, caused by the high surface quality of the spi‐ ral phase plate throughout its area. This circumstance gives us grounds to believe that the vortex wave front to be reconstructed by the Hartmann-Shack sensor has to be close to the ideal *LG*<sup>0</sup> 1 wave front.
