**6. Phase correction of optical vortex**

Next we consider the possibility to transform the wavefronts of the vortex beam by means of the closed-loop adaptive optical system with a wavefront sensor and a flexible deforma‐ ble wavefront corrector. We can use the bimorph [166] as well as pusher-type [167, 168] pie‐ zoceramic-based adaptive mirrors as a wavefront corrector. In the experiments a flexible bimorph mirror [166] and the Hartmann-Shack wavefront sensor with a new reconstruction algorithm [157-159] are employed. An attempt is made to correct the laser beam carrying the optical vortex (namely, the Laguerre-Gaussian *LG*<sup>0</sup> 1 mode), i.e., to remove its singularity. The dynamic effects are not considered, the goal is the estimation of the ability of the bi‐ morph mirror to govern the spatial features of the optical vortex. It is very interesting to de‐ termine whether the phase correction leads to full elimination of the singularity [165].

intended to compensate for the beam defocusing if need be. The second piezoplate destined to transform the vortex phase surface is glued to the first one. The 5x5=25 electrodes are pat‐ terned on the surface of the second piezoplate in the check geometry (close square packing). Each electrode has the shape of a square, with each side measuring 8.5 mm. The full thick‐ ness of the adaptive mirror is 4.5 mm. The wavefront corrector is fixed in a metal mounting with a square 45x45 mm window. The surface deformation of the adaptive mirror under the

Adaptive Optics and Optical Vortices http://dx.doi.org/10.5772/53328 175

**Figure 21.** The deformable bimorph mirror and the scheme of arrangement of control elements on the second piezo‐

The radiation beam reflected from the adaptive mirror 7 (see Figure 20) is directed by a plane mirror 8 through a reducing telescope 10 to a Hartmann-Shack sensor including a lenslet array 11 with *d*=0.2 mm, *f*=15 mm and a CCD camera 12. At the field size of 3.2 mm there are 16x16 spots in the hartmannogram on the CCD camera screen. The planes of adap‐ tive mirror and lenslet array are optically conjugated so that the wavefront sensor recon‐

A beam part is derived by a dividing plate 9 to a CCD camera 18 for additional characteriza‐ tion (see Figure 20). In addition, the wavefront corrector 7, plates 4, 8 and rear mirror 16 form a Mach-Zehnder interferometer. On blocking the reference beam from the mirror 16, the CCD cameras 12 and 18 simultaneously register, respectively, the hartmannogram and intensity picture of the beam going from the adaptive mirror. Upon admission of the refer‐ ence beam from the mirror 16, the CCD camera 18 registers the interference pattern of the beam going from the adaptive mirror with an obliquely incident reference beam. Screen of CCD camera 18 is situated at a focal distance from the lens 17 or in a plane of the adaptive mirror image (like the lenslet array) thus registering the intensity/interferogram of the beam

The wavefront has no singularity upon removal of the spiral phase plate 5 from the scheme in the Figure 20 and when switching off the wavefront corrector. The reference beam phase surface in the corrector plane is shown in Figure 22a. It is not an ideal plane (PV=0.33 µ) but it is certainly regular. Therefore the picture of diffraction at the square diaphragm 3 (see

maximal voltage ±300 V applied to any one electrode reaches ±1.5 µm.

structs in fact the phase surface of the beam just in the corrector plane.

plate.

in far or near field, respectively.

Figure 20) roughly takes place in far field in Figure 23a.

A closed-loop adaptive system intended for performance of the necessary correction of vor‐ tex wavefront is shown in Figure 20 [169]. A reference laser beam is formed using a He-Ne laser 1, a collimator 2, and a square pinhole 3, which restricts the beam aperture to a size of 10×10 mm2 . Next the laser beam passes through a 32-level spiral phase plate 5 of a diameter of 2 cm, a fourfold telescope 6 and comes to an adaptive deformable mirror 7. It should be noted that the laser beam with the plane phase front that passes through the spiral phase plate maximally resembles the Laguerre-Gaussian *LG*<sup>0</sup> 1 mode in the far field. In Figure 20 the wavefront corrector is situated in near rather than far field but at a relatively large dis‐ tance from the spiral phase plate so that the proper vortex structure of the phase distribution is already formed in the wavefront corrector plane.

**Figure 20.** The close-loop adaptive system for optical vortex correction:1 – He-Ne laser; 2 – collimator; 3 – pinhole 10×10 mm; 4, 8, 9 – optical plates; 5 -spiral phase plate; 6, 10, 15 – telescopes; 7 – deformable adaptive mirror; 11 – lenslet array; 12, 18– CCD cameras; 13 –computer; 14 – control unit of adaptive mirror, 16 – plane mirror; 17 – lens.

The wavefront corrector (the bimorph adaptive mirror) 7 [166] is shown in Figure 21. It is composed of a substrate of LK-105 glass with reflecting coating and two foursquare piezo‐ ceramic plates, each measuring 45x45 mm and 0.4 mm thick. The first piezoplate is rigidly glued to rear side of the substrate. It is complete, meaning it serves as one electrode, and is intended to compensate for the beam defocusing if need be. The second piezoplate destined to transform the vortex phase surface is glued to the first one. The 5x5=25 electrodes are pat‐ terned on the surface of the second piezoplate in the check geometry (close square packing). Each electrode has the shape of a square, with each side measuring 8.5 mm. The full thick‐ ness of the adaptive mirror is 4.5 mm. The wavefront corrector is fixed in a metal mounting with a square 45x45 mm window. The surface deformation of the adaptive mirror under the maximal voltage ±300 V applied to any one electrode reaches ±1.5 µm.

**6. Phase correction of optical vortex**

optical vortex (namely, the Laguerre-Gaussian *LG*<sup>0</sup>

plate maximally resembles the Laguerre-Gaussian *LG*<sup>0</sup>

is already formed in the wavefront corrector plane.

10×10 mm2

174 Adaptive Optics Progress

Next we consider the possibility to transform the wavefronts of the vortex beam by means of the closed-loop adaptive optical system with a wavefront sensor and a flexible deforma‐ ble wavefront corrector. We can use the bimorph [166] as well as pusher-type [167, 168] pie‐ zoceramic-based adaptive mirrors as a wavefront corrector. In the experiments a flexible bimorph mirror [166] and the Hartmann-Shack wavefront sensor with a new reconstruction algorithm [157-159] are employed. An attempt is made to correct the laser beam carrying the

The dynamic effects are not considered, the goal is the estimation of the ability of the bi‐ morph mirror to govern the spatial features of the optical vortex. It is very interesting to de‐ termine whether the phase correction leads to full elimination of the singularity [165].

A closed-loop adaptive system intended for performance of the necessary correction of vor‐ tex wavefront is shown in Figure 20 [169]. A reference laser beam is formed using a He-Ne laser 1, a collimator 2, and a square pinhole 3, which restricts the beam aperture to a size of

of 2 cm, a fourfold telescope 6 and comes to an adaptive deformable mirror 7. It should be noted that the laser beam with the plane phase front that passes through the spiral phase

the wavefront corrector is situated in near rather than far field but at a relatively large dis‐ tance from the spiral phase plate so that the proper vortex structure of the phase distribution

**Figure 20.** The close-loop adaptive system for optical vortex correction:1 – He-Ne laser; 2 – collimator; 3 – pinhole 10×10 mm; 4, 8, 9 – optical plates; 5 -spiral phase plate; 6, 10, 15 – telescopes; 7 – deformable adaptive mirror; 11 – lenslet array; 12, 18– CCD cameras; 13 –computer; 14 – control unit of adaptive mirror, 16 – plane mirror; 17 – lens.

The wavefront corrector (the bimorph adaptive mirror) 7 [166] is shown in Figure 21. It is composed of a substrate of LK-105 glass with reflecting coating and two foursquare piezo‐ ceramic plates, each measuring 45x45 mm and 0.4 mm thick. The first piezoplate is rigidly glued to rear side of the substrate. It is complete, meaning it serves as one electrode, and is

1

. Next the laser beam passes through a 32-level spiral phase plate 5 of a diameter

1

mode), i.e., to remove its singularity.

mode in the far field. In Figure 20

**Figure 21.** The deformable bimorph mirror and the scheme of arrangement of control elements on the second piezo‐ plate.

The radiation beam reflected from the adaptive mirror 7 (see Figure 20) is directed by a plane mirror 8 through a reducing telescope 10 to a Hartmann-Shack sensor including a lenslet array 11 with *d*=0.2 mm, *f*=15 mm and a CCD camera 12. At the field size of 3.2 mm there are 16x16 spots in the hartmannogram on the CCD camera screen. The planes of adap‐ tive mirror and lenslet array are optically conjugated so that the wavefront sensor recon‐ structs in fact the phase surface of the beam just in the corrector plane.

A beam part is derived by a dividing plate 9 to a CCD camera 18 for additional characteriza‐ tion (see Figure 20). In addition, the wavefront corrector 7, plates 4, 8 and rear mirror 16 form a Mach-Zehnder interferometer. On blocking the reference beam from the mirror 16, the CCD cameras 12 and 18 simultaneously register, respectively, the hartmannogram and intensity picture of the beam going from the adaptive mirror. Upon admission of the refer‐ ence beam from the mirror 16, the CCD camera 18 registers the interference pattern of the beam going from the adaptive mirror with an obliquely incident reference beam. Screen of CCD camera 18 is situated at a focal distance from the lens 17 or in a plane of the adaptive mirror image (like the lenslet array) thus registering the intensity/interferogram of the beam in far or near field, respectively.

The wavefront has no singularity upon removal of the spiral phase plate 5 from the scheme in the Figure 20 and when switching off the wavefront corrector. The reference beam phase surface in the corrector plane is shown in Figure 22a. It is not an ideal plane (PV=0.33 µ) but it is certainly regular. Therefore the picture of diffraction at the square diaphragm 3 (see Figure 20) roughly takes place in far field in Figure 23a.

face after the correction in Figure 22c is close to the reference one (Figure 22a) except for a narrow region at the break line (PV=0.5 µ). As the radiation from this part of the beam is scattered to larger angles and its portion in the beam is relatively small, the far field intensi‐ ty picture after the correction in Figure 23c is much closer to the reference beam (Figure 23a) rather than the vortex before correction (Figure 23b). Thus, the doughnut-like vortex beam is focused into a beam with a bright axial spot and weaker background that radically increases

The beam interferograms in near field before and after correction are shown in Figure 24. Unlike the former, the latter contains no resolved singularities (at least, under the given fringe density). The vortices, however, may appear under beam propagation from the adap‐ tive mirror plane as it was in the case of combined propagation of the vortex beam with a

from the actual deformed adaptive mirror surface) interferograms of a corrected beam in far field are shown in Figure 25. Two off-axis vortices (denoted by light circles) of opposite topological charge are seen here. The first of them is initial vortex shifted from the axis whereas the second arises in the process of beam propagation from the far periphery of the

**Figure 24.** Experimental pattern of interference of the beam with an obliquely incident regular wave in the near field

**Figure 25.** The pattern of interference of the corrected beam with an obliquely incident regular wave in the far field in

1 mode

Adaptive Optics and Optical Vortices http://dx.doi.org/10.5772/53328 177

regular beam [170]. The experimental and calculated (at the reflection of an ideal *LG*<sup>0</sup>

beam (in fact from infinity, according to the terminology of [170]).

the Strehl ratio and resolution of the optical system.

(left) before and (right) after correction.

(left) calculation and (right) experiment.

**Figure 22.** Experimental phase surface in near field: (a) reference beam and beam (b) before and (c) after correction.

**Figure 23.** Experimental far field intensity: (a) reference beam and beam (b) before and (c) after correction.

After inserting the spiral phase plate 5 and when switching off the adaptive mirror, the wavefront in near field in Figure 22b acquires the spiral form with *λ*-break (PV=0.63 µ) so the far-field intensity has a doughnut form (see Figure 23b). Note that the wavefront correc‐ tor software in the computer 13 is based on the singular reconstruction technique [157-159]; the conventional least-squares approach fails here. The vortex in far field is the *LG*<sup>0</sup> 1 mode distorted by presence of other modes mainly because of the phase surface imperfection of the reference beam. Note that for the task of adjusting the vortex wavefront sensing techni‐ que it was necessary to form a close-to-ideal *LG*<sup>0</sup> 1 mode as a "reference" optical vortex with maximally predetermined phase surface, to determine the sensing algorithm accuracy. Here, for the correction task, it is even more attractive to work with a distorted vortex.

In order to correct the vortex wavefront in the closed loop, the recovered phase surface in Figure 22b is decomposed on the response functions of control elements of the deformable mirror. The response function of a control element is the changing of the shape of the de‐ formable mirror surface upon the energizing of this control element with zero voltages ap‐ plied to the others actuators. The expansion coefficients on response functions are proportional to voltages to be applied from control unit 14 to appropriate elements of the deformable mirror. When applying control voltages to the adaptive mirror its surface is de‐ formed to reproduce the measured vortex wavefront maximally and thus to obtain a wave‐ front close to a plane one upon reflection from the corrector. However, each superposition of the response functions of a flexible wavefront corrector is a smooth function, and the correc‐ tor is not able to exactly reproduce the phase discontinuity of a depth of 2*π*. The phase sur‐ face after the correction in Figure 22c is close to the reference one (Figure 22a) except for a narrow region at the break line (PV=0.5 µ). As the radiation from this part of the beam is scattered to larger angles and its portion in the beam is relatively small, the far field intensi‐ ty picture after the correction in Figure 23c is much closer to the reference beam (Figure 23a) rather than the vortex before correction (Figure 23b). Thus, the doughnut-like vortex beam is focused into a beam with a bright axial spot and weaker background that radically increases the Strehl ratio and resolution of the optical system.

The beam interferograms in near field before and after correction are shown in Figure 24. Unlike the former, the latter contains no resolved singularities (at least, under the given fringe density). The vortices, however, may appear under beam propagation from the adap‐ tive mirror plane as it was in the case of combined propagation of the vortex beam with a regular beam [170]. The experimental and calculated (at the reflection of an ideal *LG*<sup>0</sup> 1 mode from the actual deformed adaptive mirror surface) interferograms of a corrected beam in far field are shown in Figure 25. Two off-axis vortices (denoted by light circles) of opposite topological charge are seen here. The first of them is initial vortex shifted from the axis whereas the second arises in the process of beam propagation from the far periphery of the beam (in fact from infinity, according to the terminology of [170]).

**Figure 22.** Experimental phase surface in near field: (a) reference beam and beam (b) before and (c) after correction.

**Figure 23.** Experimental far field intensity: (a) reference beam and beam (b) before and (c) after correction.

the conventional least-squares approach fails here. The vortex in far field is the *LG*<sup>0</sup>

for the correction task, it is even more attractive to work with a distorted vortex.

que it was necessary to form a close-to-ideal *LG*<sup>0</sup>

176 Adaptive Optics Progress

After inserting the spiral phase plate 5 and when switching off the adaptive mirror, the wavefront in near field in Figure 22b acquires the spiral form with *λ*-break (PV=0.63 µ) so the far-field intensity has a doughnut form (see Figure 23b). Note that the wavefront correc‐ tor software in the computer 13 is based on the singular reconstruction technique [157-159];

distorted by presence of other modes mainly because of the phase surface imperfection of the reference beam. Note that for the task of adjusting the vortex wavefront sensing techni‐

maximally predetermined phase surface, to determine the sensing algorithm accuracy. Here,

In order to correct the vortex wavefront in the closed loop, the recovered phase surface in Figure 22b is decomposed on the response functions of control elements of the deformable mirror. The response function of a control element is the changing of the shape of the de‐ formable mirror surface upon the energizing of this control element with zero voltages ap‐ plied to the others actuators. The expansion coefficients on response functions are proportional to voltages to be applied from control unit 14 to appropriate elements of the deformable mirror. When applying control voltages to the adaptive mirror its surface is de‐ formed to reproduce the measured vortex wavefront maximally and thus to obtain a wave‐ front close to a plane one upon reflection from the corrector. However, each superposition of the response functions of a flexible wavefront corrector is a smooth function, and the correc‐ tor is not able to exactly reproduce the phase discontinuity of a depth of 2*π*. The phase sur‐

1

1 mode

mode as a "reference" optical vortex with

**Figure 24.** Experimental pattern of interference of the beam with an obliquely incident regular wave in the near field (left) before and (right) after correction.

**Figure 25.** The pattern of interference of the corrected beam with an obliquely incident regular wave in the far field in (left) calculation and (right) experiment.

Thus, the phase surface of the distorted *LG*<sup>0</sup> 1 mode is corrected in the closed-loop adaptive optical system, including the bimorph piezoceramic mirror and the Hartmann-Shack wave‐ front sensor with the singular reconstruction technique. Experiments demonstrate the abili‐ ty of the bimorph mirror to correct the optical vortex in a practical sense, namely, to focus the doughnut-like beam into a beam with a bright axial spot that considerably increases the Strehl ratio and optical system resolution. Since the phase break is not reproduced exactly on the flexible corrector surface, the off-axis vortices can appear in far field at the beam periphery.

the Hartmann-Shack sensor, the reconstruction of the "reference" vortex phase surface has

including a Hartmann-Shack wavefront sensor with singular reconstruction technique and a flexible bimorph piezoceramic mirror with 5х5 actuators allocated in the check geometry. The mirror has high laser damage resistance meaning it can operate with powerful laser beams. The purpose of the correction is to eliminate the singularity of the beam to the high‐ est degree possible. Experiments have demonstrated the ability of the bimorph mirror to correct the optical vortex in a practical sense. As a result of phase correction, the doughnutlike beam is focused into a beam with a bright axial spot that considerably increases the Strehl ratio and is important for practical applications. However, since the wavefront break cannot be reproduced exactly by a mirror with a flexible surface, the residual off-axis vorti‐

The investigations described above consolidate the actual birth of the experimental field of

and Yu. I. Malakhov2\*

1 Russian Federal Nuclear Center –VNIIEF, Institute of Laser Physics Research, Russia

[1] Vorontsov, M. A., & Shmalgauzen, V. I. (1985). Principles of adaptive optics. *Moscow:*

[2] Roggemann, M. C., & Welsh, B. M. (1996). Imaging through turbulence. *Boca Raton,*

[4] Hardy, J. W. (1998). Adaptive optics for astronomical telescopes. *New York: Oxford*

[5] Roddier, F. (1999). Adaptive optics in astronomy. *Cambridge: Cambridge University*

[3] Tyson, R. K. (1998). Principles of adaptive optics. *Boston, MA: Academic*.

mode) is corrected in the closed-loop adaptive system

Adaptive Optics and Optical Vortices http://dx.doi.org/10.5772/53328 179

1

been carried out to a high degree of accuracy.

ces can appear in far field at the beam periphery.

novel scientific branch – singular adaptive optics.

\*Address all correspondence to: malakhov@istc.ru

2 International Science and Technology Center, Russia

**Author details**

**References**

*Nauka*.

*Press*.

*FL: CRC Press*.

*University Press*.

S. G. Garanin1\*, F. A. Starikov1

The vortex laser beam (distorted *LG*<sup>0</sup>
