# Features of Adaptive Phase Correction of Optical Wave Distortions under Conditions of Intensity Fluctuations

## Abstract

**:**

## 1. Introduction

## 2. The Influence of Amplitude Fluctuations on Phase Measurements during the Propagation of Optical Waves in Random Media

## 3. Numerical Experiments on Adaptive Correction of Turbulent Distortions

- (1)
- an ideal correction system “on reception” of the signal;
- (2)
- a correction system “on reception” of only the “potential” part of phase aberrations;
- (3)
- an ideal PC system operating “for transmission” using a reference source;
- (4)
- an adaptive phase correction system that implements the PC algorithm and works “for transmission”, which corrects only the “potential” part of the phase aberrations.

_{o}was chosen as the transverse scale of the problem [15], and the wave diffraction length at the coherence radius L

_{T}= kr

_{o}

^{2}was chosen as the longitudinal scale, here r

_{o}is the Fried radius (or parameter) [15,16], and k is the wave number of radiation.

_{T}, normalized to the wave diffraction length at the coherence radius, and normalized to the coherence scale, the aperture diameter D/r

_{o}. According to Rytov’s theory [15], the level of intensity fluctuations in an optical wave can be characterized through the so-called scintillation index for a plane wave β

_{int}

^{2}. For a power-law turbulence spectrum [15], this parameter turns out to be uniquely related to the normalized optical path length:

_{int}

^{2}can be used as a problem parameter along with the ratio of the path length to the longitudinal scale L/L

_{T}.

_{o}= 10, 20, 30. As a result, each of the system operation schemes corresponds to a family of three curves.

**SR**for a circuit operating “to receive” radiation on the flicker index in the received optical wave. From Figure 1 it is clear that with phase correction of only the “irrotational” part of the phase (scheme 2), the correction efficiency decreases quite significantly with increasing scintillation index (this can be seen from three curves corresponding to values D/r

_{o}= 10, 20, 30). A twofold reduction in the value of the SR ratio was already achieved at values on the order of 1.5. A further increase in intensity fluctuations leads to the fact that the Strehl parameter

**SR**tends to the uncorrected value. A decrease in the correction efficiency by an order of magnitude occurs at β

_{int}

^{2}= 3, which approximately corresponds to the fact that the path length L turns out to be approximately equal to L

_{T}.

_{T}. It should be emphasized that, as shown in [19], the efficiency of correction even with ideal phase conjugation depends on the magnitude of intensity fluctuations. However, we note that, as shown in [2], this dependence is not as strong as might be expected. This is due to the fact that in the phase conjugation system the correction is performed at the entrance to the turbulent atmosphere; in fact, there is a preliminary distortion of the original optical wave. Moreover, this preliminary distortion is calculated based on phase measurements in a reference source, which propagates towards the original wave.

_{o}= 10, 20, 30, already at a parameter value of β

_{int}

^{2}= 1, the value of the Strehl parameter

**SR**decreases to a value of 0.8 and practically weakly depends on the normalized aperture diameter.

## 4. Comparison of Calculations with the Lincoln Laboratory Experiment

## 5. Operation of the Phase Correction System under Conditions of Weak Fluctuations

## 6. Comparison of Phase Adaptive Correction in Areas of Weak and Strong Intensity Fluctuations

- The efficiency of phase correction of turbulent distortions decreases approximately by half as the normalized dispersion of intensity fluctuations (five) increases from zero to unity. In this range of intensity fluctuation dispersion values, the correction efficiency is practically independent of the relationship between the aperture diameter and the coherence radius. With a further increase in intensity fluctuations, the dependence of the correction efficiency on the aperture diameter begins to appear. An increase in the dispersion of intensity fluctuations ${\beta}_{\mathrm{int}}^{2}$ to three leads to a drop in the efficiency of correction by an order of magnitude or more, and the Strehl parameter SR tends to the value obtained in the system without correction.
- Since level ${\beta}_{\mathrm{int}}^{2}$ = 1 approximately corresponds to the limit of applicability of the smooth perturbation method (SPM) [15], we can assume that the applicability of calculations in the SPM approximation is possible only in the region where there are no phase dislocations.
- The decrease in the efficiency of adaptive correction of the “irrotational” phase with increasing dispersion of intensity fluctuations occurs approximately equally both in the “receive” phase compensation scheme and in the “transmit” phase conjugation scheme. The differences between a plane wave and a beam are also insignificant, as follows from a comparison with the Lincoln Laboratory experiment [24].

**SR**to the uncorrected value of this parameter—

_{c}**SR**for a compensation scheme for only the “potential” phase for the case when the dispersion of intensity fluctuations for a plane wave ${\beta}_{\mathrm{int}}^{2}$ = 3.

_{0}**D/r**values from 10 to 30, the corrected value of the Strehl parameter

_{o}**SR**is approximately four times greater than the uncorrected

_{c}**SR**.

_{0}## 7. Study of the Influence of Amplitude Fluctuations on Phase Measurements

## 8. Phase Measurements in the Region of Strong Fluctuations

_{1}—the number of subapertures in which the illumination at the maximum of the diffraction pattern is below the threshold, N

_{0}—the total number of fully illuminated subapertures in the exit pupil of the telescope. Thus, it turns out that in individual frames the light level “fades” to almost zero. Such subapertures should be excluded from analysis when operating the sensor. This in turn leads to a loss of accuracy in its operation.

## 9. Experiments with WFS

- The influence of the central shielding of the input aperture of the WFS does not significantly affect the quality of wavefront reconstruction; only the spherical aberration is underestimated, which may require modification of the adaptive correction algorithm when operating AO systems.
- When the pupil of the entrance aperture of the WFS was vignetted, differences arose in the magnitude of aberrations of the reconstructed wavefront, which can be considered quantitatively significant for the aberrations coma and astigmatism. Moreover, it was found that the magnitude of these aberrations turned out to be higher than without vignetting.
- As a result, it can be expected that the distortion of WFS data associated with the non-use of part of the subapertures when reconstructing the wavefront caused by the action of turbulence will affect the quality of operation of the AO system when correcting laser radiation on the atmospheric path.
- In our experiments with the Shack–Hartmann sensor, it was discovered that there was a dependence of phase systems on the level of intensity fluctuations, which manifests itself, first of all, in a change in the brightness (flickering) of focal spots in the Hartmann matrix. In this case, the phase measurements become incorrect, or rather, errors appear in the assessment of certain mode components.
- Summarizing the results of these experiments, it should be noted that such a simulation of operation associated with data loss in some subapertures of the Shack–Hartmann wavefront sensor practically simulates cases of complete loss of information from some part of the receiving aperture of the sensor. At the same time, the influence of intensity fluctuations apparently occurs somewhat differently.

## 10. Discussion

## 11. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Dependence of the Strehl ratio SR on the scintillation index during phase correction of fluctuations in the potential phase of the “receiving” signal (according to scheme 2).

**Figure 2.**The same as in Figure 1 for a phase conjugation system operating “for transmission” according to scheme 4.

**Figure 4.**Spectral filtering functions for correlation functions of phase and intensity fluctuations, calculated using Formula (9).

**Figure 5.**Spectral filter in Figure 5. Spectral filtering functions for the structure functions of phase and intensity, calculated using Formula (10).

**Figure 6.**Appearance of a pattern of focal spots in the Shack–Hartmann sensor with weak (

**left**) and strong (

**right**) intensity fluctuations in the optical wave.

**Figure 7.**Illumination distribution of focal spots across the aperture of the wavefront sensor at frame with number 369.

**Figure 8.**Illumination distribution of focal spots across the aperture of the wavefront sensor at frame with number 306.

**Figure 9.**Time dependence of the normalized number of subapertures that form images with maximum illumination below the threshold (this threshold of illumination value is 1.5 times greater than background).

**Figure 12.**Simultaneous measurements of the mode components of the phase front: WFS1 with a central shielding coefficient of 13% and WFS2—without shielding.

**Table 1.**Strehl parameters for the corrected field in the region of “strong” intensity fluctuations.

D/r_{o} | SR_{0} | SR_{c} | SR_{c}/SR_{0} |
---|---|---|---|

10 | 0.0324 | 0.129 | 3.98 |

20 | 0.0106 | 0.038 | 3.58 |

30 | 0.0051 | 0.025 | 4.90 |

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Lukin, V.
Features of Adaptive Phase Correction of Optical Wave Distortions under Conditions of Intensity Fluctuations. *Photonics* **2024**, *11*, 460.
https://doi.org/10.3390/photonics11050460

**AMA Style**

Lukin V.
Features of Adaptive Phase Correction of Optical Wave Distortions under Conditions of Intensity Fluctuations. *Photonics*. 2024; 11(5):460.
https://doi.org/10.3390/photonics11050460

**Chicago/Turabian Style**

Lukin, Vladimir.
2024. "Features of Adaptive Phase Correction of Optical Wave Distortions under Conditions of Intensity Fluctuations" *Photonics* 11, no. 5: 460.
https://doi.org/10.3390/photonics11050460