**5. Wavefront sensing of optical vortex**

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

166 Adaptive Optics Progress

profile shape under motion along the circular line.

additionally enhance the portion of the *LG*<sup>0</sup>

**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

It should be noted that a laser beam in the form of a principal Gaussian mode with a plane

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*

It should be noted that the proper intensity modulation of the beam incident to the plate can

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

1 mode. mode in

1 .

wavefront that passes through a spiral phase plate maximally resembles the *LG*<sup>0</sup> <sup>1</sup>

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 a contour Γ:

$$\mathcal{S}\{\mathbf{r}\} = \mathcal{S}\_0(\mathbf{r}) + \int\_{\Gamma} \nabla\_{\perp} \mathcal{S}\_m(\mathbf{p}) d\boldsymbol{\rho}\_{\prime} \tag{9}$$

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‐ construction as the minimization of a certain functional. The most commonly used function‐ al is the criterion corresponding to the minimum of the weighted square of residual error of gradient of the reconstructed phase ∇⊥ *S*(**r**) and phase gradients ∇⊥ *Sm*(**r**), obtained in the measurements:

$$\int\_{D} (\mathbf{W}(\mathbf{r}) \cdot (\nabla\_{\perp} \mathbf{S}(\mathbf{r}) - \nabla\_{\perp} \mathbf{S}\_{m}(\mathbf{r}))^{2} \mathbf{d}\mathbf{r} \to \min,\tag{10}$$

Ñ =Ñ +Ñ ^^ ^ *SS S* (**r rr** ) *p c* ( ) ( ) (13)

Adaptive Optics and Optical Vortices http://dx.doi.org/10.5772/53328 169

/2 *c z* **H e** (14)

where ∇⊥ *Sp* is the gradient of potential phase component and ∇⊥ *Sc* is the gradient of vor‐ tex (solenoidal) component. By using only the ordinary methods of phase reconstruction it is possible to reproduce just the part of phase distribution that corresponds to potential com‐

However, if the quantity ∇⊥ *Sc* is considered as a rotor of vector potential **H**, namely, ∇⊥ *Sc*=∇×H, which is dependent only on the coordinates of optical vortices [68] then potential phase component in (13) can be found by the least mean square method [68, 147]. By means of a novel "hydrodynamic" approach to the properties of the vector filed of phase gradients a new group of methods was formed [72, 150, 151], employing the discovered coordinates of dislocations and reconstructing the potential phase component with the least mean square method. Within another technique [74, 152] reproduction of the scalar potential is also based on the least square method but the vortex component is calculated with Eq. (9) via a consis‐ tent rotor of vector potential. The method of matching the vortex component was proposed

( ( )) ( ) <sup>2</sup> Ñ Ñ = Ñ´ × *Rot S* - ^

where **e***z* is the unit vector of *z* axis and *Rot*−*<sup>π</sup>*/2(∇⊥*Sc*) is operation of the rotation of each vector on −*π*/2 angle. This relation is a Poisson equation which allows one to find compo‐ nents of consistent vectors of vortex phase gradient. Now it is possible to take Eq. (9) and obtain a vortex phase component, assuming that the consistent gradients of vortex phase are

The searching for dislocation located positions, which is required in algorithms of phase re‐ construction [72, 150, 151], is a sufficiently difficult problem. Because of the infinite phase gradients in the points of zero intensity, the application of methods based on solution of (13) [74, 152] is also not straightforward. Presently there is no such an algorithm, which guaran‐ tees the required fidelity of wavefront reconstruction in the presence of dislocations [64]. However, according to some estimations [154, 155, 156] the accurate detection of vortex co‐ ordinates and their topological charges insures the sensing of wavefronts with high preci‐ sion. Therefore we expect a future improvement in reconstruction algorithms by involving more sophisticated methods into the consideration of gradient fields, insuring more accurate

Analysis shows that from the point of view of experimental realization, of the considered approaches of wavefront reconstruction the algorithm of D. Fried [74] is one of the best algo‐ rithms (with respect to accuracy, effectiveness and resistance to measurement noises) of re‐ covery of phase surface *S*(*x*, *y*) from its measured gradient ∇*S*<sup>⊥</sup> distribution in the presence of optical vortices. Fried's algorithm (*a noise-variance-weighted complex exponential reconstruc‐ tor*) consists of three parts: reduction or simplification, solving, and reconstruction. The algo‐

p

detection of dislocation positions and their topological charges.

ponent in (13).

in [153] and is based on the following equation:

measured without errors.

where **W**(**r**)={*Wx*(*x*, *y*), *Wн*(*x*, *y*)} is the vector weighting function introduced to account for the reliability of ∇⊥ *Sm*(**r**) measurements. This method is known as the least mean square phase reconstruction.

The approaches to the solution of variation problem (10) are well known [146, 147] and ac‐ tually mean the solution of the Poisson equation written with partial derivatives. Allowing for the weighting function W(**r**), it acquires the following form:

$$W\_x(\mathbf{x}, \ y) \Big| \frac{\partial \mathcal{S}}{\partial x} - \frac{\partial \mathcal{S}\_m}{\partial x} \Big| + W\_y(\mathbf{x}, \ y) \Big| \frac{\partial \mathcal{S}}{\partial y} - \frac{\partial \mathcal{S}\_m}{\partial y} \Big| = 0,\tag{11}$$

where ∂*S*/∂*x* and ∂*S*/∂*y* are gradients of reconstructed phase, ∂*Sm*/∂*x* and ∂*Sm*/∂*y* are measured gradients of the phase.

There are a wide variety of methods [145, 146, 147, 148] which can be used to solve the dis‐ crete variants of equation (11). For example, one can use the representation of (11) as a sys‐ tem of algebraic equations, the fast Fourier transform, or the Gauss-Zeidel iteration method applied to the multi-grid algorithm. This group of methods is equally well adopted for the application of centroid coordinates measured by the Shack-Hartmann sensor as input data:

$$\left\{\nabla\_{\perp}S\_{m}(r,z)\right\}\_{I} = \frac{\iint V(\mathbf{r}-\mathbf{r}\_{0})I(\mathbf{r}\_{0})\nabla\_{\perp}S(\mathbf{r}\_{0})d\mathbf{r}\_{0}}{\iint V(\mathbf{r}-\mathbf{r}\_{0})I(\mathbf{r}\_{0})d\mathbf{r}\_{0}},\tag{12}$$

where the integration is performed over the square of the subaperture, *V* is a subaperture function, and *I*(**r**0) is intensity of the input beam.

The sensing of wavefront with screw phase dislocations by the least mean square method is not agreeable. With this technique (along with other methods based on the assumption that phase surface is a continuous function of coordinates) it is possible to reconstruct only a fraction of the entire phase function. As it turned out [68, 149], the differential properties of the vector field of phase gradients help to find some similarity between this field and the field of potential flow of a liquid penetrated by vortex strings. It is also possible to represent this vector field as a sum of potential and solenoid components:

$$
\nabla\_{\perp} S(\mathbf{r}) \ = \nabla\_{\perp} S\_{\rho} \begin{pmatrix} \mathbf{r} \\ \end{pmatrix} + \nabla\_{\perp} S\_{c} \begin{pmatrix} \mathbf{r} \\ \end{pmatrix} \tag{13}
$$

where ∇⊥ *Sp* is the gradient of potential phase component and ∇⊥ *Sc* is the gradient of vor‐ tex (solenoidal) component. By using only the ordinary methods of phase reconstruction it is possible to reproduce just the part of phase distribution that corresponds to potential com‐ ponent in (13).

construction as the minimization of a certain functional. The most commonly used function‐ al is the criterion corresponding to the minimum of the weighted square of residual error of gradient of the reconstructed phase ∇⊥ *S*(**r**) and phase gradients ∇⊥ *Sm*(**r**), obtained in the

where **W**(**r**)={*Wx*(*x*, *y*), *Wн*(*x*, *y*)} is the vector weighting function introduced to account for the reliability of ∇⊥ *Sm*(**r**) measurements. This method is known as the least mean square

The approaches to the solution of variation problem (10) are well known [146, 147] and ac‐ tually mean the solution of the Poisson equation written with partial derivatives. Allowing

<sup>∂</sup> *<sup>x</sup>* ) <sup>+</sup> *Wy*(*x*, *<sup>y</sup>*)( <sup>∂</sup>*<sup>S</sup>*

where ∂*S*/∂*x* and ∂*S*/∂*y* are gradients of reconstructed phase, ∂*Sm*/∂*x* and ∂*Sm*/∂*y* are measured

There are a wide variety of methods [145, 146, 147, 148] which can be used to solve the dis‐ crete variants of equation (11). For example, one can use the representation of (11) as a sys‐ tem of algebraic equations, the fast Fourier transform, or the Gauss-Zeidel iteration method applied to the multi-grid algorithm. This group of methods is equally well adopted for the application of centroid coordinates measured by the Shack-Hartmann sensor as input data:

*∫∫V* (*r* −*r*0)*I*(*r*0)∇⊥*S*(*r*0)*dr*<sup>0</sup>

*∫∫V* (*r* −*r*0)*I*(*r*0)*dr*<sup>0</sup>

where the integration is performed over the square of the subaperture, *V* is a subaperture

The sensing of wavefront with screw phase dislocations by the least mean square method is not agreeable. With this technique (along with other methods based on the assumption that phase surface is a continuous function of coordinates) it is possible to reconstruct only a fraction of the entire phase function. As it turned out [68, 149], the differential properties of the vector field of phase gradients help to find some similarity between this field and the field of potential flow of a liquid penetrated by vortex strings. It is also possible to represent

<sup>∂</sup> *<sup>y</sup>* <sup>−</sup>

∂*Sm*

<sup>∂</sup> *<sup>y</sup>* ) =0, (11)

, (12)

(*W* (*r*) ⋅ (∇⊥*S*(*r*)−∇⊥*Sm*(*r*))2d*r* →min, (10)

measurements:

168 Adaptive Optics Progress

phase reconstruction.

gradients of the phase.

*∫ D*

for the weighting function W(**r**), it acquires the following form:

<sup>∂</sup> *<sup>x</sup>* <sup>−</sup>

{∇⊥*Sm*(*r*, *z*)}*<sup>I</sup>* =

this vector field as a sum of potential and solenoid components:

function, and *I*(**r**0) is intensity of the input beam.

∂*Sm*

*Wx*(*x*, *<sup>y</sup>*)( <sup>∂</sup>*<sup>S</sup>*

However, if the quantity ∇⊥ *Sc* is considered as a rotor of vector potential **H**, namely, ∇⊥ *Sc*=∇×H, which is dependent only on the coordinates of optical vortices [68] then potential phase component in (13) can be found by the least mean square method [68, 147]. By means of a novel "hydrodynamic" approach to the properties of the vector filed of phase gradients a new group of methods was formed [72, 150, 151], employing the discovered coordinates of dislocations and reconstructing the potential phase component with the least mean square method. Within another technique [74, 152] reproduction of the scalar potential is also based on the least square method but the vortex component is calculated with Eq. (9) via a consis‐ tent rotor of vector potential. The method of matching the vortex component was proposed in [153] and is based on the following equation:

$$\nabla^2 \left( \operatorname{Rot}\_{-\pi/2} \left( \nabla\_{\perp} S\_c \right) \right) = \left( \nabla \times \mathbf{H} \right) \cdot \mathbf{e}\_z \tag{14}$$

where **e***z* is the unit vector of *z* axis and *Rot*−*<sup>π</sup>*/2(∇⊥*Sc*) is operation of the rotation of each vector on −*π*/2 angle. This relation is a Poisson equation which allows one to find compo‐ nents of consistent vectors of vortex phase gradient. Now it is possible to take Eq. (9) and obtain a vortex phase component, assuming that the consistent gradients of vortex phase are measured without errors.

The searching for dislocation located positions, which is required in algorithms of phase re‐ construction [72, 150, 151], is a sufficiently difficult problem. Because of the infinite phase gradients in the points of zero intensity, the application of methods based on solution of (13) [74, 152] is also not straightforward. Presently there is no such an algorithm, which guaran‐ tees the required fidelity of wavefront reconstruction in the presence of dislocations [64]. However, according to some estimations [154, 155, 156] the accurate detection of vortex co‐ ordinates and their topological charges insures the sensing of wavefronts with high preci‐ sion. Therefore we expect a future improvement in reconstruction algorithms by involving more sophisticated methods into the consideration of gradient fields, insuring more accurate detection of dislocation positions and their topological charges.

Analysis shows that from the point of view of experimental realization, of the considered approaches of wavefront reconstruction the algorithm of D. Fried [74] is one of the best algo‐ rithms (with respect to accuracy, effectiveness and resistance to measurement noises) of re‐ covery of phase surface *S*(*x*, *y*) from its measured gradient ∇*S*<sup>⊥</sup> distribution in the presence of optical vortices. Fried's algorithm (*a noise-variance-weighted complex exponential reconstruc‐ tor*) consists of three parts: reduction or simplification, solving, and reconstruction. The algo‐ rithm designed for work in Hadjin geometry reconstructs the phase in the nodes of a quadratic grid with the dimensions (2*<sup>N</sup>*+1)×(2*<sup>N</sup>*+1), using the phase differences between these nodes. Obviously, to employ the algorithm we need (2*<sup>N</sup>*+1)×2*<sup>N</sup>* and 2*<sup>N</sup>*×(2*<sup>N</sup>*+1) array of phase differences along *x* and *y* axes. The words "*complex exponential*" mean that the phase recon‐ struction problem is reformulated to a task of recovery of "phasors" *u* (the complex number with a unity absolute value and an argument that is equal to the phase of optical field) dis‐ tribution in transverse section of the beam. Here, the analysis and transformation of differ‐ ential complex vectors (differential phasors) Δ*xu*≡exp(*i*Δ*xφ*), Δ*yu*≡exp(*i*Δ*yφ*), corresponding to phase differences Δ*xφ*, Δ*yφ* between different nodes of the computational grid, are used. The words "*noise-variance-weighted*" mean that the algorithm takes into account the distinctions of measurement variance of individual differential phasors, i.e. the influence ("weight") of dif‐ ferential phasors on the recovery result is inversely proportional to their variance. This fea‐ ture of Fried's algorithm allows us to apply it to a computational grid of arbitrary dimension, not only to the (2*<sup>N</sup>*+1)×(2*<sup>N</sup>*+1) grid [74]; to take into account the average statistical inequality of measurement errors of phase gradient in different areas of the beam (for exam‐ ple, on the sub-apertures of the Hartmann-Shack sensor) if the repeated characterization of the same beam is performed; to consider a prior concept of the inequality of measurement errors of phase gradient in these areas if the measurement of the beam characteristics is sin‐ gle.

**Figure 15.** Experimental setup for wavefront sensing of optical vortex in far field: 1 – He-Ne laser; 2 – collimator; 3 – optical plate; 4 – plane mirror; 5 – spiral phase plate; 6 – the lens F=6 m; 7 – the objective; 8 and 8' - focal plane of lens

Adaptive Optics and Optical Vortices http://dx.doi.org/10.5772/53328 171

A technical feature of the Hartmann-Shack wavefront sensor used involves the employment of a raster of 8-level diffraction Fresnel lenses as the lenslet array (see Figure 16). The raster is fabricated from fused quartz by kinoform technology, similar to the aforesaid spiral phase plate, with the minimum size of microlens *d*=0.1 mm and diffraction efficiency up to 90% [162]. The accuracy of etching profile depth is not worse than 2%, the difference of the focal spot size from the theoretical size is ~1%. The spatial resolution of the wavefront sensor and its sensibility depend on the microlens geometry, the number of registered focal spots and

**Figure 16.** Photo image of a fragment of the lenslet array and image of surface profile of a microlens.

tilts of wave front on the sub-apertures of lenslet array are determined.

Under the registration of phase front the reference beam in the second arm of the interfer‐ ometer is blocked. In the beginning the wavefront sensor is calibrated by a reference beam with plane phase front (the spiral phase plate is removed from the scheme). Then the spiral phase plate is inserted, and the picture of focal spots correspondent to singular phase front is registered. From the values of displacement of focal spots from initial positions, the local

Experiments with a different number of registration spots on the hartmannogram have been carried out [160, 161]. When using a lenslet array with subaperture size *d*=0.3 mm, focal length *f*=25 mm and *d*=0.2 mm, *f*=15 mm, the picture from 8×8 and 16x16 focal spots on the CCD camera screen has been registered, respectively. The results of experimental measure‐ ments of wave front gradients are given in Figure 17 for measurement points 8×8, where the picture of displacements of focal spots in the hartmannogram is shown. The vortex center is situated between the sub-apertures of the array. Displacement of each spot is demonstrated by the arrow (line segment). The arrow origin corresponds to the reference spot position

6 and its optically conjugated plane, respectively; 9 – lenslet array; 10 – CCD camera.

their size with respect to the CCD camera pixel size.

In Fried's algorithm the differential phasors are unit vectors. The operation of normalization of a complex vector is applied to provide for this requirement. However, the amplitudes of differential phasors and phasors, obtained under reduction and reconstruction, contain in‐ formation about measurement errors of phase differences in the actual experiment. Based on this reason the algorithm in question has been modified [157, 158, 159]. The modification in‐ volves exclusion of the operation of complex vector normalization and allows an increase in algorithm accuracy.

The experimental setup for registration of an optical vortex wavefront consists of a system for formation of collimated laser beam, the Mach-Zehnder interferometer (as in the scheme in Figure 10), and the additionally induced the Hartmann-Shack wavefront sensor [160, 161]. It is shown in Figure 15. The system of formation of collimated beam includes a He-Ne laser 1 (*λ*=0.633 мкм) and collimator 2 composed of lenses with focal lengths 5 cm and 160 cm. The collimator forms the reference basic Gaussian beam with a diameter of 1 cm and the plane wave front. The Mach-Zehnder interferometer includes two optical plates 3 and two mirrors 4. The spiral phase plate 5 for formation of the optical vortex is interposed into one of the interferometer arms. The working surface of the phase plate, with a diameter of 2 cm com‐ pletely covers the beam. After passing through the spiral phase plate the Gaussian beam turns into an optical vortex (the *LG*<sup>0</sup> 1 mode) in the focal plane 8 of the lens 6, i.e. in far field, with high conversion coefficient. Focal plane 8 of the lens 6 with focal length 700 cm is transfer‐ red (for purposes of magnification) by an objective 8 in the optically conjugated plane 8'. The wavefront sensor consists of a lenslet array 9 situated in the plane 8'and a CCD camera 10.

rithm designed for work in Hadjin geometry reconstructs the phase in the nodes of a quadratic grid with the dimensions (2*<sup>N</sup>*+1)×(2*<sup>N</sup>*+1), using the phase differences between these nodes. Obviously, to employ the algorithm we need (2*<sup>N</sup>*+1)×2*<sup>N</sup>* and 2*<sup>N</sup>*×(2*<sup>N</sup>*+1) array of phase differences along *x* and *y* axes. The words "*complex exponential*" mean that the phase recon‐ struction problem is reformulated to a task of recovery of "phasors" *u* (the complex number with a unity absolute value and an argument that is equal to the phase of optical field) dis‐ tribution in transverse section of the beam. Here, the analysis and transformation of differ‐ ential complex vectors (differential phasors) Δ*xu*≡exp(*i*Δ*xφ*), Δ*yu*≡exp(*i*Δ*yφ*), corresponding to phase differences Δ*xφ*, Δ*yφ* between different nodes of the computational grid, are used. The words "*noise-variance-weighted*" mean that the algorithm takes into account the distinctions of measurement variance of individual differential phasors, i.e. the influence ("weight") of dif‐ ferential phasors on the recovery result is inversely proportional to their variance. This fea‐ ture of Fried's algorithm allows us to apply it to a computational grid of arbitrary dimension, not only to the (2*<sup>N</sup>*+1)×(2*<sup>N</sup>*+1) grid [74]; to take into account the average statistical inequality of measurement errors of phase gradient in different areas of the beam (for exam‐ ple, on the sub-apertures of the Hartmann-Shack sensor) if the repeated characterization of the same beam is performed; to consider a prior concept of the inequality of measurement errors of phase gradient in these areas if the measurement of the beam characteristics is sin‐

In Fried's algorithm the differential phasors are unit vectors. The operation of normalization of a complex vector is applied to provide for this requirement. However, the amplitudes of differential phasors and phasors, obtained under reduction and reconstruction, contain in‐ formation about measurement errors of phase differences in the actual experiment. Based on this reason the algorithm in question has been modified [157, 158, 159]. The modification in‐ volves exclusion of the operation of complex vector normalization and allows an increase in

The experimental setup for registration of an optical vortex wavefront consists of a system for formation of collimated laser beam, the Mach-Zehnder interferometer (as in the scheme in Figure 10), and the additionally induced the Hartmann-Shack wavefront sensor [160, 161]. It is shown in Figure 15. The system of formation of collimated beam includes a He-Ne laser 1 (*λ*=0.633 мкм) and collimator 2 composed of lenses with focal lengths 5 cm and 160 cm. The collimator forms the reference basic Gaussian beam with a diameter of 1 cm and the plane wave front. The Mach-Zehnder interferometer includes two optical plates 3 and two mirrors 4. The spiral phase plate 5 for formation of the optical vortex is interposed into one of the interferometer arms. The working surface of the phase plate, with a diameter of 2 cm com‐ pletely covers the beam. After passing through the spiral phase plate the Gaussian beam turns

high conversion coefficient. Focal plane 8 of the lens 6 with focal length 700 cm is transfer‐ red (for purposes of magnification) by an objective 8 in the optically conjugated plane 8'. The wavefront sensor consists of a lenslet array 9 situated in the plane 8'and a CCD camera 10.

mode) in the focal plane 8 of the lens 6, i.e. in far field, with

gle.

170 Adaptive Optics Progress

algorithm accuracy.

into an optical vortex (the *LG*<sup>0</sup>

1

**Figure 15.** Experimental setup for wavefront sensing of optical vortex in far field: 1 – He-Ne laser; 2 – collimator; 3 – optical plate; 4 – plane mirror; 5 – spiral phase plate; 6 – the lens F=6 m; 7 – the objective; 8 and 8' - focal plane of lens 6 and its optically conjugated plane, respectively; 9 – lenslet array; 10 – CCD camera.

A technical feature of the Hartmann-Shack wavefront sensor used involves the employment of a raster of 8-level diffraction Fresnel lenses as the lenslet array (see Figure 16). The raster is fabricated from fused quartz by kinoform technology, similar to the aforesaid spiral phase plate, with the minimum size of microlens *d*=0.1 mm and diffraction efficiency up to 90% [162]. The accuracy of etching profile depth is not worse than 2%, the difference of the focal spot size from the theoretical size is ~1%. The spatial resolution of the wavefront sensor and its sensibility depend on the microlens geometry, the number of registered focal spots and their size with respect to the CCD camera pixel size.

**Figure 16.** Photo image of a fragment of the lenslet array and image of surface profile of a microlens.

Under the registration of phase front the reference beam in the second arm of the interfer‐ ometer is blocked. In the beginning the wavefront sensor is calibrated by a reference beam with plane phase front (the spiral phase plate is removed from the scheme). Then the spiral phase plate is inserted, and the picture of focal spots correspondent to singular phase front is registered. From the values of displacement of focal spots from initial positions, the local tilts of wave front on the sub-apertures of lenslet array are determined.

Experiments with a different number of registration spots on the hartmannogram have been carried out [160, 161]. When using a lenslet array with subaperture size *d*=0.3 mm, focal length *f*=25 mm and *d*=0.2 mm, *f*=15 mm, the picture from 8×8 and 16x16 focal spots on the CCD camera screen has been registered, respectively. The results of experimental measure‐ ments of wave front gradients are given in Figure 17 for measurement points 8×8, where the picture of displacements of focal spots in the hartmannogram is shown. The vortex center is situated between the sub-apertures of the array. Displacement of each spot is demonstrated by the arrow (line segment). The arrow origin corresponds to the reference spot position whereas the arrow end corresponds to spot position after the insertion of a spiral phase plate in the experimental scheme. In Figure 17 the results are also shown, which are ob‐ tained in calculation and are correspondent to the ideal *LG*<sup>0</sup> 1 mode with high accuracy. It is seen from Figure 17 that the experimental and calculated pictures of spots' displacements agree with each other. Some local data difference is caused by the distinction between the phase and amplitude structure of the beam incident on the phase plate in the calculation and actual experiment, by the inaccurate location of the vortex in the optical axis assumed in calculations and by the inevitable noises of the measurement.

course, from the viewpoint of its proximity to the theoretical results) is not worse than *λ*/20. The accuracy of recovery of phase surface break increases at the measurement spots of 16х16. For comparison purposes in Figure 18 the result is demonstrated of the vortex wave‐ front reconstruction with the help of the standard least-squares restoration technique in the

**Figure 18.** Experimental vortex phase surface reconstructed using modified Fried's (left) and conventional least-

In Figure 19 we show the calculation results [165] of phase front reconstruction of the beam

propagation (see Figure 4). The modified Fried's algorithm embedded into the Hartmann-Shack sensor software correctly restores the complicated singular structure of the phase surface.

**Figure 19.** The phase surface fragment of the beam after the turbulent path reconstructed using the modified Fried's

2

=10-14 cm-2/3 after 1 km distance

Adaptive Optics and Optical Vortices http://dx.doi.org/10.5772/53328 173

passed through the turbulent atmosphere in the case of *Cn*

squares (right) procedure.

algorithm.

Hartmann-Shack sensor. It is seen that the conventional approach obviously fails.

**Figure 17.** The picture of displacements of focal spots of the hartmannogram in experiment (black arrows) and calcu‐ lation (grey arrows).

In Ref. [163] the vortex-like structure of displacements of spots in the hartmannogram was registered for the *LG*<sup>0</sup> 1 and higher-order modes. As the primary information, the spot dis‐ placements can be used for deriving the Poynting vector skews (in fact, wavefront tilts), as it was made in [163], as well as for wavefront sensing that is more nontrivial. In this chapter we simply consider the reconstruction of singular phase surface by the Hartmann-Shack sensor and describe the realization of this operation with the new reconstruction technique.

In Figure 18 we present the wave front surface of optical vortex reconstructed by the Hart‐ mann-Shack sensor [161, 164] with software incorporating the code of restoration of singular phase surfaces [157-159]. Comparison of experimental data with calculated results shows that the wave front surface is restored by the actual Hartmann Shack wavefront sensor with good quality despite the rather small size of the matrix of wave front tilts (spots in the hart‐ mannogram). The reconstructed wave front has the characteristic spiral form with a break of the surface about 2*π*. Analysis shows that the accuracy of wave front reconstruction (of course, from the viewpoint of its proximity to the theoretical results) is not worse than *λ*/20. The accuracy of recovery of phase surface break increases at the measurement spots of 16х16. For comparison purposes in Figure 18 the result is demonstrated of the vortex wave‐ front reconstruction with the help of the standard least-squares restoration technique in the Hartmann-Shack sensor. It is seen that the conventional approach obviously fails.

whereas the arrow end corresponds to spot position after the insertion of a spiral phase plate in the experimental scheme. In Figure 17 the results are also shown, which are ob‐

seen from Figure 17 that the experimental and calculated pictures of spots' displacements agree with each other. Some local data difference is caused by the distinction between the phase and amplitude structure of the beam incident on the phase plate in the calculation and actual experiment, by the inaccurate location of the vortex in the optical axis assumed in

**Figure 17.** The picture of displacements of focal spots of the hartmannogram in experiment (black arrows) and calcu‐

In Ref. [163] the vortex-like structure of displacements of spots in the hartmannogram was

placements can be used for deriving the Poynting vector skews (in fact, wavefront tilts), as it was made in [163], as well as for wavefront sensing that is more nontrivial. In this chapter we simply consider the reconstruction of singular phase surface by the Hartmann-Shack sensor and describe the realization of this operation with the new reconstruction technique. In Figure 18 we present the wave front surface of optical vortex reconstructed by the Hart‐ mann-Shack sensor [161, 164] with software incorporating the code of restoration of singular phase surfaces [157-159]. Comparison of experimental data with calculated results shows that the wave front surface is restored by the actual Hartmann Shack wavefront sensor with good quality despite the rather small size of the matrix of wave front tilts (spots in the hart‐ mannogram). The reconstructed wave front has the characteristic spiral form with a break of the surface about 2*π*. Analysis shows that the accuracy of wave front reconstruction (of

and higher-order modes. As the primary information, the spot dis‐

1

mode with high accuracy. It is

tained in calculation and are correspondent to the ideal *LG*<sup>0</sup>

calculations and by the inevitable noises of the measurement.

lation (grey arrows).

172 Adaptive Optics Progress

registered for the *LG*<sup>0</sup>

1

**Figure 18.** Experimental vortex phase surface reconstructed using modified Fried's (left) and conventional leastsquares (right) procedure.

In Figure 19 we show the calculation results [165] of phase front reconstruction of the beam passed through the turbulent atmosphere in the case of *Cn* 2 =10-14 cm-2/3 after 1 km distance propagation (see Figure 4). The modified Fried's algorithm embedded into the Hartmann-Shack sensor software correctly restores the complicated singular structure of the phase surface.

**Figure 19.** The phase surface fragment of the beam after the turbulent path reconstructed using the modified Fried's algorithm.
