# Reconstructing the Spatial Parameters of a Laser Beam Using the Transport-of-Intensity Equation

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Methods and Approaches

_{00}Gaussian beam, ${M}^{2}=1$, and for a real beam ${M}^{2}>1$.

#### 2.1. ISO Method

#### 2.2. TIE Method (Proposed Method)

## 3. Experimental Part

## 4. Results and Discussion

#### 4.1. ISO Method

^{2}, $b=-1.307\times {10}^{-3}$ mm, $c=7.866\times {10}^{-6}$ for plane X0Z and $a=0.059$ mm

^{2}, $b=-1.458\times {10}^{-3}$ mm, $c=9.506\times {10}^{-6}$ for plane Y0Z. Then, these coefficients were used to calculate the spatial parameters of the beam, the relationship of which is given in Table 1. The numerical results are presented in Table 3.

#### 4.2. TIE Method (Proposed Method)

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Schematic of the experimental measurements and algorithm for calculating the spatial parameters of the laser beam.

**Figure 5.**Numerical calculations of (

**a**) Radius of curvature $R\left({z}_{2}\right)$ depending on the longitudinal coordinate; (

**b**) Rayleigh length ${z}_{R}$; (

**c**) Waist radius ${w}_{0}$; (

**d**) Parameter ${M}^{2}$.

**Figure 3.**Experimental scheme consisting of a focusing lens and a CMOS-camera. The planes (

**a**–

**c**) show the cross-sections of the intensity distributions captured along the propagation axis. The plane (

**d**) marks the longitudinal section of the intensity distribution.

**Figure 4.**Cross-sections of the intensity distribution in planes with (

**a**) Z = 22 mm; (

**b**) Z = 82 mm; (

**c**) Z = 140 mm; (

**d**) Reconstructed longitudinal intensity distribution.

Parameters | Equations |
---|---|

Radius of the laser beam waist | ${w}_{0}=\frac{\sqrt{4ac-{b}^{2}}}{2\sqrt{c}}$ |

Position of the beam waist relative to the selected reference plane (taking into account the sign rule adopted in optics) | ${S}_{0}=-\frac{b}{2c}$ |

Rayleigh length | ${z}_{R}=\frac{\sqrt{4ac-{b}^{2}}}{2c}$ |

Angular divergence of the beam | $2\theta =2\sqrt{c}$ |

BPP | $BPP=\frac{\sqrt{4ac-{b}^{2}}}{2}$ |

Parameters | X0Z | Y0Z | Comment |
---|---|---|---|

Central wavelength, (nm) | 1031.2 | ||

Beam diameter, (mm) | 1.627 | 1.576 | at 60 mm from exit |

Beam ellipticity, (%) | 3 | at 60 mm from exit | |

Quality parameter ${M}^{2}$ | 1.075 | 1.042 | |

Astigmatism (%) | 0.2 | ||

Beam divergence, (mrad) | 1.424 | 1.383 | full angle |

Parameters | TIE (X0Z) | TIE (Y0Z) | ISO 11146 (X0Z) | ISO 11146 (Y0Z) |
---|---|---|---|---|

Radius of the laser beam waist (mm) | 0.064 | 0.059 | 0.061 | 0.053 |

Beam waist position (mm) | 83.087 | 77.014 | 83.306 | 76.686 |

Rayleigh length (mm) | 21.606 | 17.576 | 21.785 | 17.251 |

Quality parameter ${M}^{2}$ | 1.066 | 1.051 | 1.061 | 1.047 |

Angular divergence of the beam (mrad) | 0.55 | 0.58 | 0.482 | 0.485 |

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**MDPI and ACS Style**

Kovalev, M.; Gritsenko, I.; Stsepuro, N.; Nosov, P.; Krasin, G.; Kudryashov, S.
Reconstructing the Spatial Parameters of a Laser Beam Using the Transport-of-Intensity Equation. *Sensors* **2022**, *22*, 1765.
https://doi.org/10.3390/s22051765

**AMA Style**

Kovalev M, Gritsenko I, Stsepuro N, Nosov P, Krasin G, Kudryashov S.
Reconstructing the Spatial Parameters of a Laser Beam Using the Transport-of-Intensity Equation. *Sensors*. 2022; 22(5):1765.
https://doi.org/10.3390/s22051765

**Chicago/Turabian Style**

Kovalev, Michael, Iliya Gritsenko, Nikita Stsepuro, Pavel Nosov, George Krasin, and Sergey Kudryashov.
2022. "Reconstructing the Spatial Parameters of a Laser Beam Using the Transport-of-Intensity Equation" *Sensors* 22, no. 5: 1765.
https://doi.org/10.3390/s22051765