# Atmospheric Turbulence with Kolmogorov Spectra: Software Simulation, Real-Time Reconstruction and Compensation by Means of Adaptive Optical System with Bimorph and Stacked-Actuator Deformable Mirrors

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Experimental Setup

#### 2.2. Stacked-Actuator Deformable Mirror

#### 2.3. Bimorph Deformable Mirror

#### 2.4. Shack–Hartmann Wavefront Sensor

_{x}and S

_{y}) from the center of the sub-aperture in proportion to the tip-tilt value. In other words, if we measure these displacements, S

_{x}and S

_{y}, of the focal spot per the X and Y axis, we will correspondingly obtain the values of the partial derivatives ∂W/∂x and ∂W/∂y of the wavefront W within each sub-aperture. On the other hand, to describe and visualize the wavefront surface analytically, one can use the polynomial approximation, for example, B-Splines [62] or Zernike polynomials [63,64,65,66], which are commonly used in optics. Thus, the partial wavefront derivatives ∂W/∂x and ∂W/∂y can be defined analytically using Zernike polynomials. They can also be calculated from the measured displacements S

_{x}and S

_{y}of the focal spots on the Shack–Hartmann sensor. Finally, we determined the overdetermined system of linear equations with the unknown coefficient’s ${a}_{i}$. By solving the least squares problem [67], we obtain the coefficient’s ${a}_{i}$. From here on, the wavefront can be analytically described and analyzed.

#### 2.5. Algorithm of Phase Screens Simulation

- As we had the Zernike approximation of the simulated phase screen, we could calculate the values of the wavefront derivatives in each sub-aperture of the wavefront sensor.
- Knowing the values of the wavefront derivatives, we could calculate the displacements of the focal spots corresponding to these derivatives.
- Thus, knowing the focal spot shifts associated with the mirror response functions and the focal spot displacements corresponding to the wavefront to be reproduced, we could solve the overdetermined system of linear equations using the least squares method and calculate the vector of voltages that had to be applied to the mirror actuators.

#### 2.6. Algorithm of Phase Screen Compensation

- The wavefront of the laser beam reflected from the stacked-actuator and bimorph mirrors was analyzed on the Shack–Hartmann wavefront sensor.
- Having, on the one hand, the matrix of displacements of the focal spots on the Shack–Hartmann sensor $\left\{\begin{array}{c}{S}_{x}^{k}\\ {S}_{y}^{k}\end{array}\right\}$, corresponding to the wavefront of the reconstructed phase screen, and, on the other hand, the matrix of values of the bimorph mirror response functions RF, also consisting of the focal spots shifts, we obtained an overdetermined system of linear equations for unknown coefficients, which were the values of the voltages at the mirror electrodes. To solve this system of equations, the least squares method was used.
- The calculated voltages were applied to the electrodes of the bimorph mirror.
- The residual wavefront was measured by means of the Shack–Hartmann sensor.

## 3. Results and Discussion

- Calculation of the control voltages to be applied to the actuators of the stacked-actuator deformable mirror [73].
- Reconstruction of the simulated phase screens using the stacked-actuator mirror.
- Measurement of the introduced wavefront distortions using the Shack–Hartmann wavefront sensor.
- Calculation of the control voltages to be applied to the electrodes of the bimorph deformable mirror in order to compensate for the wavefront distortions.
- Compensation of the reconstructed phase screens by the bimorph mirror.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Correction Statement

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**Figure 1.**Optical scheme of the adaptive system for atmospheric turbulence simulation and compensation.

**Figure 2.**(

**a**) The drawing of the stacked-actuator mirror, (

**b**) the photo of the manufactured stacked-actuator mirror and (

**c**) actuators’ layout scheme.

**Figure 3.**(

**a**) Electrodes’ layout scheme, (

**b**) the photo of the manufactured bimorph mirror and (

**c**) the principal scheme of the bimorph mirror construction.

**Figure 4.**Results of the approximation of the phase screen using Zernike polynomials. (

**a**) Fringes map, (

**b**) phase map, (

**c**) calculated far field intensity distribution—for the approximated phase screen, (

**d**) Zernike decomposition and results of the reconstruction of the phase screen by means of stacked-actuator mirror, (

**e**) fringes map, (

**f**) phase map, (

**g**) calculated far field intensity distribution—for the approximated phase screen, (

**h**) Zernike decomposition.

**Figure 5.**Actuator’s voltages calculated to reconstruct the phase screen. (

**a**) Schematical representation of the actuators of the stacked-actuator mirror where each color from the palette corresponds to the voltage value, (

**b**) table of absolute voltages values on each actuator, (

**c**) bar diagram of the voltages.

**Figure 6.**Residual error of phase screen reconstruction by means of the stacked-actuator mirror: (

**a**) fringes map, (

**b**) phase map, (

**c**) calculated far field intensity distribution, (

**d**) Zernike decomposition.

**Figure 7.**Residual error of phase screen correction by means of the bimorph mirror: (

**a**) fringes map, (

**b**) phase map, (

**c**) calculated far field intensity distribution, (

**d**) Zernike decomposition.

**Figure 8.**Electrode voltages calculated to compensate for the induced wavefront distortions: (

**a**) schematical representation of the bimorph mirror electrodes where each color from the palette corresponds to the voltage value, (

**b**) table of absolute voltages values on each electrode, (

**c**) bar diagram of the voltages.

Parameter | Value |
---|---|

Substrate aperture | 40 mm |

Clear aperture | 35 mm |

Substrate material | glass |

No. of control actuators | 19 |

Type of actuators | PZT |

Actuators geometry | Hexagonal |

Maximum input voltage | −30…+130 V |

Parameter | Value |
---|---|

Substrate aperture | 35 mm |

Clear aperture | 30 mm |

Substrate material | glass |

No. of PZT | 2 |

No. of control electrodes | 32 |

Type of actuators | PZT discs |

Actuators geometry | sectorial |

Maximum input voltage | −200…+300 V |

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## Share and Cite

**MDPI and ACS Style**

Galaktionov, I.; Sheldakova, J.; Samarkin, V.; Toporovsky, V.; Kudryashov, A.
Atmospheric Turbulence with Kolmogorov Spectra: Software Simulation, Real-Time Reconstruction and Compensation by Means of Adaptive Optical System with Bimorph and Stacked-Actuator Deformable Mirrors. *Photonics* **2023**, *10*, 1147.
https://doi.org/10.3390/photonics10101147

**AMA Style**

Galaktionov I, Sheldakova J, Samarkin V, Toporovsky V, Kudryashov A.
Atmospheric Turbulence with Kolmogorov Spectra: Software Simulation, Real-Time Reconstruction and Compensation by Means of Adaptive Optical System with Bimorph and Stacked-Actuator Deformable Mirrors. *Photonics*. 2023; 10(10):1147.
https://doi.org/10.3390/photonics10101147

**Chicago/Turabian Style**

Galaktionov, Ilya, Julia Sheldakova, Vadim Samarkin, Vladimir Toporovsky, and Alexis Kudryashov.
2023. "Atmospheric Turbulence with Kolmogorov Spectra: Software Simulation, Real-Time Reconstruction and Compensation by Means of Adaptive Optical System with Bimorph and Stacked-Actuator Deformable Mirrors" *Photonics* 10, no. 10: 1147.
https://doi.org/10.3390/photonics10101147