# Interferometric Wavefront Sensing System Based on Deep Learning

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## Abstract

**:**

## 1. Introduction

## 2. Method

#### 2.1. Wavefront Detecting System

#### 2.2. Wavefront Analysis Neural Network

#### 2.2.1. Models

#### 2.2.2. Training Data Generated

#### 2.2.3. Training Networks

## 3. Experiment

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Phase-shifting interferometer. ND: neutral density filter; B: beam expander and collimator; BS: beam splitter; M1, M2: mirror; M3: deformable mirror; PLS: precision linear stage.

**Figure 2.**The wavefront analysis neural networks. (

**a**) The overall structure and process of the deep learning wavefront analysis system, including two neural network modules net1 and net2, and the relationship between the two modules. (

**b**) The structure of net1. (

**c**) The structure of net2.

**Figure 3.**The smooth process of the output of net1. (

**a**) Wrapped phase; (

**b**) wrap count estimated by net1; (

**c**) unwrapped phase of net1; (

**d**) individual points corrected by smooth method; (

**e**) regions corrected by smooth method; (

**f**) 3D image of the corrected wavefront.

**Figure 4.**Samples of the training dataset. (

**a**) true phase, (

**b**) wrapped phase, (

**c**) wrap count, (

**d**) reconstructed wavefront using Zernike polynomials.

**Figure 7.**Zernike coefficients fitting results. (

**a**) Distorted wavefront, (

**b**) the fitting result of the Zernike coefficient using different methods.

**Figure 10.**Output results of the real experiments. (

**a**) Wrapped phase obtained by the four-step phase-shifting method, (

**b**) unwrapped phase calculated by net1, (

**c**) unwrapped phase calculated by Goldstein’s branch cut algorithm, (

**d**) the difference of the two unwrapped phases, (

**e**) Zernike coefficients fitted by net2, (

**f**) reconstructed wavefront, (

**g**) errors of the reconstructed wavefront.

**Table 1.**Accuracies (RMSEs) and time consumption of different phase-unwrapping algorithms (SNR = 9).

Net1 | Branch Cut | Quality-Guided | |
---|---|---|---|

RMSE | 0.07 | 0.06 | 0.22 |

Time | 1.04 s | 1.94 s | 12.48 s |

**Table 2.**Accuracies (RMSEs) and time consumption of different phase-unwrapping algorithms (SNR = 3).

Net1 | Branch Cut | Quality-Guided | |
---|---|---|---|

RMSE | 0.37 | 1.48 | 1.02 |

Time | 1.06 s | 1.97 s | 12.73 s |

${a}_{1}$ | ${a}_{2}$ | ${a}_{3}$ | ${a}_{4}$ | ${a}_{5}$ | ${a}_{6}$ |

1.7944 | 0.6126 | 0.27979 | −0.96737 | −1.01558 | −0.44856 |

${a}_{7}$ | ${a}_{8}$ | ${a}_{9}$ | ${a}_{10}$ | ${a}_{11}$ | ${a}_{12}$ |

−0.00485 | 0.11478 | −0.02006 | −0.14852 | −0.07189 | −0.09368 |

${a}_{13}$ | ${a}_{14}$ | ${a}_{15}$ | ${a}_{16}$ | ${a}_{17}$ | ${a}_{18}$ |

−0.11007 | −0.07977 | −0.06551 | 0.04326 | −0.11056 | −0.09216 |

${a}_{19}$ | ${a}_{20}$ | ${a}_{21}$ | ${a}_{22}$ | ${a}_{23}$ | ${a}_{24}$ |

−0.14109 | 0.04773 | 0.09532 | 0.5545 | 0.06727 | 0.10555 |

${a}_{25}$ | ${a}_{26}$ | ${a}_{27}$ | ${a}_{28}$ | ${a}_{29}$ | ${a}_{30}$ |

0.05787 | −0.01177 | −0.06569 | −0.01927 | 0.05708 | 0.00327 |

${a}_{31}$ | ${a}_{32}$ | ${a}_{33}$ | ${a}_{34}$ | ${a}_{35}$ | ${a}_{36}$ |

−0.03634 | −0.04515 | −0.04231 | 0.02874 | 0.05204 | −0.0082 |

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**MDPI and ACS Style**

Niu, Y.; Gao, Z.; Gao, C.; Zhao, J.; Wang, X.
Interferometric Wavefront Sensing System Based on Deep Learning. *Appl. Sci.* **2020**, *10*, 8460.
https://doi.org/10.3390/app10238460

**AMA Style**

Niu Y, Gao Z, Gao C, Zhao J, Wang X.
Interferometric Wavefront Sensing System Based on Deep Learning. *Applied Sciences*. 2020; 10(23):8460.
https://doi.org/10.3390/app10238460

**Chicago/Turabian Style**

Niu, Yuhao, Zhan Gao, Chenjia Gao, Jieming Zhao, and Xu Wang.
2020. "Interferometric Wavefront Sensing System Based on Deep Learning" *Applied Sciences* 10, no. 23: 8460.
https://doi.org/10.3390/app10238460