# Design of a High Uniformity Laser Sheet Optical System for Particle Image Velocimetry

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## Abstract

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## 1. Introduction

## 2. Basic Knowledge of the Laser Sheet Optical System

^{N}steps can be obtained after N times overlay fabrication. The multi-step structure with phase depth $\mathsf{\pi}$ is generated in the first fabrication. The multi-step structure with phase depth $\mathsf{\pi}/2$ is generated in the second fabrication. The multi-step structure with phase depth $\mathsf{\pi}/4$ is generated in the third fabrication. So on, the multi-step structure with phase depth $\mathsf{\pi}/{2}^{\mathrm{N}-1}$ will be generated in the N th fabrication. For example, a DOE with 8 steps can be obtained through 3 times overlay fabrication, and the phase depths are 0, $\mathsf{\pi}/4$, $\mathsf{\pi}/2$, $3\mathsf{\pi}/4$, $\mathsf{\pi}$, $5\mathsf{\pi}/4$, $3\mathsf{\pi}/2,$ and $7\mathsf{\pi}/4$, respectively. The range of phase depth is $0\le \mathsf{\phi}<2\mathsf{\pi}$, and the relationship between substrate depth d and phase depth $\mathsf{\phi}$ can be expressed as $\mathrm{follows}$:

## 3. Design Algorithm of DOE

- Randomly select the initial phase distribution $\mathsf{\phi}\left({\mathrm{x}}_{\mathrm{i}},{\mathrm{y}}_{\mathrm{i}}\right)$ and estimate the initial temperature ${\mathrm{T}}_{0}$ of the optimization system. According to Metropolis criterion, ${\mathrm{T}}_{0}=-\frac{\overline{\mathsf{\Delta}\mathrm{E}}}{{\mathrm{lnP}}_{0}}$. The initial acceptance probability ${\mathrm{P}}_{0}=0.95$. $\overline{\mathsf{\Delta}\mathrm{E}}=\frac{{{\displaystyle \sum}}_{\mathrm{K}}\left({\mathrm{E}}_{1}-{\mathrm{E}}_{0}\right)}{\mathrm{K}}$. ${\mathrm{E}}_{0}$ is the evaluation function value in the initial state. ${\mathrm{E}}_{1}$ is the evaluation function value which is calculated once by the hybrid algorithm. $\overline{\mathsf{\Delta}\mathrm{E}}$ is the average of K cycles. Thus, the initial temperature ${\mathrm{T}}_{0}$ is obtained.
- Optimize the initial phase distribution with the iterative algorithm which is shown in Figure 5. After this calculation, a local optimal solution ${\mathsf{\phi}}_{1}\left({\mathrm{x}}_{\mathrm{i}},{\mathrm{y}}_{\mathrm{i}}\right)$ and its evaluation function ${\mathrm{E}}_{1}$ can be found.
- Change the phase distribution ${\mathsf{\phi}}_{1}\left({\mathrm{x}}_{\mathrm{i}},{\mathrm{y}}_{\mathrm{i}}\right)$ as follow:$${\mathsf{\phi}}^{\prime}\left({\mathrm{x}}_{\mathrm{i}},{\mathrm{y}}_{\mathrm{i}}\right)={\mathsf{\phi}}_{1}\left({\mathrm{x}}_{\mathrm{i}},{\mathrm{y}}_{\mathrm{i}}\right)+\mathsf{\beta}\mathsf{\epsilon}\left({\mathrm{x}}_{\mathrm{i}},{\mathrm{y}}_{\mathrm{i}}\right)$$
- Then make $\mathsf{\phi}{}^{\prime}\left({\mathrm{x}}_{\mathrm{i}},{\mathrm{y}}_{\mathrm{i}}\right)$ as initial phase distribution for iteration (see Figure 5) and calculate another local optimal solution ${\mathsf{\phi}}_{2}\left({\mathrm{x}}_{\mathrm{i}},{\mathrm{y}}_{\mathrm{i}}\right)$ and its evaluation function ${\mathrm{E}}_{2}$.
- If ${\mathrm{E}}_{2}<{\mathrm{E}}_{1}$, accept ${\mathsf{\phi}}_{2}\left({\mathrm{x}}_{\mathrm{i}},{\mathrm{y}}_{\mathrm{i}}\right)$, and replace ${\mathsf{\phi}}_{1}\left({\mathrm{x}}_{\mathrm{i}},{\mathrm{y}}_{\mathrm{i}}\right)$ with ${\mathsf{\phi}}_{2}\left({\mathrm{x}}_{\mathrm{i}},{\mathrm{y}}_{\mathrm{i}}\right).$ If ${\mathrm{E}}_{2}>{\mathrm{E}}_{1}$, calculate $\mathrm{P}=\mathrm{exp}\left[-\left({\mathrm{E}}_{2}-{\mathrm{E}}_{1}\right)/\mathrm{T}\right]$, if P is greater than or equal to some random number between 0 and 1, accept ${\mathsf{\phi}}_{2}\left({\mathrm{x}}_{\mathrm{i}},{\mathrm{y}}_{\mathrm{i}}\right)$, and replace ${\mathsf{\phi}}_{1}\left({\mathrm{x}}_{\mathrm{i}},{\mathrm{y}}_{\mathrm{i}}\right)$ with ${\mathsf{\phi}}_{2}\left({\mathrm{x}}_{\mathrm{i}},{\mathrm{y}}_{\mathrm{i}}\right)$, otherwise, refuse ${\mathsf{\phi}}_{2}\left({\mathrm{x}}_{\mathrm{i}},{\mathrm{y}}_{\mathrm{i}}\right)$. This step allows the search process to change from a better solution to a worse solution with a certain probability to avoid the trap of local optimization. This is the biggest difference between global optimization algorithm and local optimization algorithm.
- Go to step 3. In order to reach thermal equilibrium at temperature T, the number of cycles should be more than 50 times.
- Decrease T gradually. If the system temperature is higher than the minimum temperature ${\mathrm{T}}_{\mathrm{min}}$, go back to step 3 to continue the calculation. If the temperature T is less than the minimum temperature ${\mathrm{T}}_{\mathrm{min}}$, this program should be ended and output final phase distribution.

## 4. Design Results

## 5. Analysis of the Sheet Light Quality in Measurement Area

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 8.**Intensity distribution of the sheet light at Z = 2000 mm. (

**a**) Intensity distribution on the image plane; (

**b**) intensity distribution along the X axis; (

**c**) intensity distribution along the Y axis.

**Figure 10.**Intensity distribution at Z = 1900 mm and Z = 2100 mm. (

**a**) Intensity distribution along X axis at Z = 1900 mm; (

**b**) intensity distribution along Y axis at Z = 1900 mm; (

**c**) intensity distribution along X axis at Z = 2100 mm; (

**d**) intensity distribution along Y axis at Z = 2100 mm.

Design Parameter | Data |
---|---|

Incident light | Φ5 mm collimated Gaussian beam |

Wavelength | λ = 532 nm |

DOE size | $5\text{}\mathrm{mm}\times 5\text{}\mathrm{mm}$ |

The number of sampling points | 2000 × 2000 |

The size of each cell | $2.5\text{}\mathsf{\mu}\mathrm{m}\times 2.5\text{}\mathsf{\mu}\mathrm{m}$ |

Focal length of the lens | F = 2000 mm |

The size of the sheet light | $400\text{}\mathrm{mm}\times 1\text{}\mathrm{mm}$ |

Algorithms | Diffractive Efficiency η (%) | Non-Uniformity σ (%) |
---|---|---|

Simulated annealing algorithm | 88.68 | 3.13 |

G-S algorithm | 96.22 | 2.37 |

The hybrid algorithm | 97.77 | 0.03 |

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**MDPI and ACS Style**

Yin, K.; Zhang, J.; Chen, S.
Design of a High Uniformity Laser Sheet Optical System for Particle Image Velocimetry. *Aerospace* **2021**, *8*, 393.
https://doi.org/10.3390/aerospace8120393

**AMA Style**

Yin K, Zhang J, Chen S.
Design of a High Uniformity Laser Sheet Optical System for Particle Image Velocimetry. *Aerospace*. 2021; 8(12):393.
https://doi.org/10.3390/aerospace8120393

**Chicago/Turabian Style**

Yin, Kewei, Jun Zhang, and Shuang Chen.
2021. "Design of a High Uniformity Laser Sheet Optical System for Particle Image Velocimetry" *Aerospace* 8, no. 12: 393.
https://doi.org/10.3390/aerospace8120393