# Identification of Model Particle Mixtures Using Machine-Learning-Assisted Laser Diffraction

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## Abstract

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## 1. Introduction

## 2. Experimental Methods

#### 2.1. Setup and Data Acquisition

Algorithm 1: Pseudo-code of experimental data collection. |

#### 2.2. Neural Network Architecture and Processing

## 3. Results and Discussion

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

DMD | Digital Micromirror Device |

SLS | Static Light Scattering |

DLS | Dynamic Light Scattering |

STA | Scattering Tracking Analysis |

LD | Laser Diffraction |

NN | Neural Network |

ML | Machine Learning |

CCD | Charge-coupled Device |

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**Figure 1.**Experimental Setup. A 405 nm laser beam, spatially filtered with two lenses (L1 and L2) and a pinhole (P), is expanded with a telescope system (L3 and L4) to illuminate a Digital Micromirror Device (DMD). The power of the illumination beam can be controlled using a Half-wave Plate (HWP) followed by a Linear Polarizer (LP). The light reflected by the DMD passes through a Fourier transform lens (L5), and the diffraction pattern is collected by a CCD camera at the focal plane of the Fourier lens L5. The power of the signal after L5 was measured using a beam splitter (BS) and a power meter (PM).

**Figure 2.**Machine-learning-assisted particle mixture identification. (

**a**) Flowchart of the machine learning algorithms used for extracting information from intensity-only measurements of the diffraction patterns. (

**b**) Image down-sampling process. The intensity signal extracted from the CCD camera is presented in false color for the sake of clarity of the presentation. (

**c**) Flow diagram of the neural networks used in each phase of the experiments described in the main text.

**Figure 3.**Diffraction patterns measured in experiments. (

**a**,

**d**,

**g**) show examples of the objects generated with the Digital Micromirror Device (DMD). (

**b**,

**e**,

**h**) are the theoretically predicted diffraction patterns created by the objects depicted in the leftmost column. (

**c**,

**f**,

**i**) are the experimentally measured diffraction pattern. Note that the images are normalized, so that the integrated signal over the detection area adds up to unity. In all cases, the overlap parameter $\mathsf{\Omega}$ is found to be larger than $0.9$.

**Figure 4.**Identification accuracy as a function of the feature matrix dimension. Error bars indicate standard deviations.

**Figure 5.**Confusion matrices that summarize the performance of machine-learning algorithms for particle mixture identification. The top row shows the confusion matrices containing information about the correct and incorrect predictions for (

**a**) geometry (shape), (

**b**) model particle characteristic size, and (

**c**) number of objects of the Experiment #1. The bottom row presents the confusion matrices for (

**d**) geometry (shape), (

**e**) number of model particles, and (

**f**,

**g**) dominant geometry (shape) of the Experiment #2. (

**f**,

**g**) matrices correspond to the case in which the number of objects is even and odd, respectively. In all cases, the diagonal elements of the matrices represent successful recognition, i.e., true-positives and true-negatives, whereas off-diagonal elements represent failed attempts, false-negatives, and false-positives.

Number of Pixels | Power [$\mathsf{\mu}$W] |
---|---|

≤500 | 200 |

501–1000 | 100 |

1001–1500 | 50 |

1501–3000 | 25 |

≥3001 | 15 |

**Table 2.**Overall accuracy, number of hidden layers, and number of neurons in each layer for the neural networks implemented in the described experiments.

Experiment | Neural Network | Accuracy | Number of Hidden Layers | Number of Neurons by Layer |
---|---|---|---|---|

1 | Geometry | 99% | 1 | 5 |

Object Size | 99% | 1 | 5 | |

Object Number | 93% | 2 | Layer 1 = 20; Layer 2 = 5 | |

2 | Geometry | 94% | 2 | Layer 1 = 30; Layer 2 = 20 |

Object Number | 92% | 2 | Layer 1 = 80; Layer 2 = 50 | |

Dominant geometry (even) | 95% | 2 | Layer 1 = 30; Layer 2 = 20 | |

Dominant geometry (odd) | 98% | 2 | Layer 1 = 30; Layer 2 = 20 |

Experiment | Neural Network | Training [s] | Test [s] |
---|---|---|---|

1 | Geometry | $2.308\times {10}^{-1}$ ± $2.7\times {10}^{-3}$ | $7.1\times {10}^{-3}$ ± $6.7\times {10}^{-7}$ |

Object Size | $2.491\times {10}^{-1}$ ± $4.7\times {10}^{-3}$ | $7\times {10}^{-3}$ ± $5.2\times {10}^{-7}$ | |

Object Number | $3.905\times {10}^{-1}$ ± $9.9\times {10}^{-3}$ | $7.2\times {10}^{-3}$ ± $2\times {10}^{-7}$ | |

2 | Geometry | 4.7212 ± $7.471\times {10}^{-1}$ | $1.2\times {10}^{-2}$ ± $1.4\times {10}^{-6}$ |

Object Number | $1.57444\times {10}^{1}$± 4.6636 | $1.82\times {10}^{-2}$ ± $2.4\times {10}^{-6}$ | |

Dominant geometry (even) | 1.045 ± $6.1\times {10}^{-2}$ | $2.7\times {10}^{-5}$ ± $9.9\times {10}^{-10}$ | |

Dominant geometry (odd) | $9.513\times {10}^{1}$ ± $6.5\times {10}^{-2}$ | $2.1\times {10}^{-5}$ ± $2.7\times {10}^{-9}$ |

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## Share and Cite

**MDPI and ACS Style**

Villegas, A.; Quiroz-Juárez, M.A.; U’Ren, A.B.; Torres, J.P.; León-Montiel, R.d.J.
Identification of Model Particle Mixtures Using Machine-Learning-Assisted Laser Diffraction. *Photonics* **2022**, *9*, 74.
https://doi.org/10.3390/photonics9020074

**AMA Style**

Villegas A, Quiroz-Juárez MA, U’Ren AB, Torres JP, León-Montiel RdJ.
Identification of Model Particle Mixtures Using Machine-Learning-Assisted Laser Diffraction. *Photonics*. 2022; 9(2):74.
https://doi.org/10.3390/photonics9020074

**Chicago/Turabian Style**

Villegas, Arturo, Mario A. Quiroz-Juárez, Alfred B. U’Ren, Juan P. Torres, and Roberto de J. León-Montiel.
2022. "Identification of Model Particle Mixtures Using Machine-Learning-Assisted Laser Diffraction" *Photonics* 9, no. 2: 74.
https://doi.org/10.3390/photonics9020074