# Inverse Design for Coating Parameters in Nano-Film Growth Based on Deep Learning Neural Network and Particle Swarm Optimization Algorithm

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Research Methods

_{i}is the propagation matrix of the i-th layer, T

_{i,j}is the transition matrix of the interface between layers i and j, which can be given as follows:

_{i}and d

_{i}are the refractive index and thickness of layer i, respectively, and the reflectivity of the k-layer structure can be given by ${R}_{k}={m}_{21}^{k}/{m}_{11}^{k}$.

^{−6}. The maximum number of epochs of NN training is 1000, the additional momentum factor is 0.9, and all other parameters are set to their default values [25,26]. Here, the network has 3 hidden layers (each layer contains 50 neurons), 5 inputs, and 131 outputs.

^{−5}, while all other parameters take their default values [33]. The CPU we used in this paper for the nano thin-film simulations is a Intel(R) Core (TM) i5-8500 processor with an NVIDIA GeForce GTX 1050 Ti GPU acceleration card. The software we used for this study are MATLAB and Python for the TMM and NN/PSO simulations, respectively.

## 3. Results and Discussion

^{−2}%, as shown in Figure 3 for the NN-PSO optimization process.

## 4. Experiment Case

#### 4.1. The Anti-Reflection (AR) Coating Case

_{2}and Ta

_{2}O

_{5}, whose refractive indices were measured individually in their thin-film format, as shown in Figure 6b. In the experiment, the thicknesses were set to 255.88, 24.33, 28.85, 74.90, and 98.58 nm for the five layers (denoted by D1 to D5), respectively. Then, we grew the film by the e-beam heating of their solid sources such that SiO

_{2}and Ta

_{2}O

_{5}molecules could evaporate and escape from the source to be deposited on the substrate held at the top section of the coating machine, which rotated under a constant speed to ensure the uniformity of the deposition. The quartz crystal monitor was used to check the layer thickness, and the growth process was held under a low pressure of 10

^{−6}Torr and a constant temperature of 150 °C. After growth, the thin-film was tested for its reflection spectrum, as shown in Figure 7a.

^{−6}, and the error distribution is shown in Figure 7b.

^{−6}, as in Figure 8b. Here, the dash-asterisk lines are the results of D1 to D5, whose average values are 243.58, 18.17, 36.27, 76.58, and 98.99 nm, respectively. Then, the relative error of the predicted results as compared to the preset values (as marked by the solid lines) were 5.05, 33.90, 20.46, 2.19, and 0.41 (%), respectively. Here, D1 had the largest absolute error, and D2 and D3 had larger REs of the predicted results than the preset values. This may be because D1 was closest to the substrate and it may have had a larger error due to film monitoring during the initial growth, and D2 and D3’s larger REs were due to their relatively thinner thicknesses. The MSE distribution histogram of the NN-PSO predicted spectra is shown in Figure 8c, which shows that all of the MSEs of the spectra were less than 6 × 10

^{−5}, with most of them being less than 2 × 10

^{−5}. Therefore, it can be considered that the inverse result is accurate.

#### 4.2. Bragg Case

_{2}and Ta

_{2}O

_{5}on top of the BK7 substrate. For periodic even (odd) layers, we can use only two thickness parameters to characterize the film, i.e., D1 for the SiO

_{2}layers and D2 for the Ta

_{2}O

_{5}layers. The output parameter of the network is the transmission spectrum, whose wavelength range is 400~1500 nm, with a total number of 1101 sampling points for 1 nm spacing. The refractive indices of the materials used in the experiment were the same as in the AR film, as shown in Figure 6b, whose film thicknesses for growth were preset to 179.57 nm (SiO

_{2}) and 126.43 nm (Ta

_{2}O

_{5}), respectively. Then, we could take [179.57 − 30, 179.57 + 30] nm (SiO

_{2}) and [126.43 − 30, 126.43 + 30] (Ta

_{2}O

_{5}) as the input parameter ranges to generate transmission spectra for the training database of our deep learning neural network.

^{−4}, as verified against the test set.

^{−4}. The prediction results and error distributions of the two NNs are shown in Figure 10a,b, respectively.

_{2}and 127.73 nm for Ta

_{2}O

_{5}, which were the mean values of the samples close to the growth values of 179.57 nm (SiO

_{2}) and 126.43 nm (Ta

_{2}O

_{5}), i.e., with REs of 0.77% (SiO

_{2}) and 1.03% (Ta

_{2}O

_{5}) for NN1-PSO. For NN2, the number of PSO particles was 60, the maximum number of iterations was 50, the repeated time of inverse designs was 25, and the distribution of the results is shown in Figure 11b. It can be seen that the results of D3 and D4 were close to 178.66 and 127.14 nm (with REs of 0.51% and 0.56%), while the other four parameters had large fluctuations. This may be because D3 and D4 represent the majority of the thickness of the Bragg layers except for the top and bottom ones. Thus, these two parameters affect the spectrum more than the others. Moreover, this indicates that the Bragg structure is relatively robust against variations in the individual layers, such that even if the top and bottom layers fluctuate heavily, the spectrum still keeps its main shape.

^{−4}, and the MSE of NN2-PSO was 65.0205 × 10

^{−4}, which is still of high accuracy, considering the large fluctuations in the curves.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**(

**a**) The reflection spectrum generated by TMM and NN, and (

**b**) the error distribution of the NN prediction against the test dataset.

**Figure 4.**(

**a**) Reflection spectrum verification and (

**b**) the MSE distribution histogram of NN-PSO predicted spectra.

**Figure 5.**The reflection spectra for various structures inversely designed by NN-PSO for the same 1400~1500 nm wavelength range.

**Figure 6.**(

**a**) The ion-beam-assisted coating system and (

**b**) the indices of the materials used in the thin-film growth.

**Figure 7.**(

**a**) The predicted reflection spectrum by NN compared to TMM and (

**b**) the error distributions for NN in the five-layer AR case.

**Figure 8.**(

**a**) The reflection curves as compared to the experiment, TMM and NN-PSO, (

**b**) the distribution of the NN-PSO inversely designed layer thicknesses, and (

**c**) the MSE distribution histogram of NN-PSO predicted spectra.

**Figure 9.**The reflection spectra of various structures inversely designed for the 550~650 nm wavelength range.

**Figure 10.**(

**a**) The predicted spectra compared to the original one that was randomly selected from the training dataset and (

**b**) the error distributions for the two NNs.

**Figure 11.**(

**a**,

**b**) The result distribution statistics for the two NNs, and (

**c**) the transmission spectra from the experimental measurement, from TMM simulation for the preset grown structure, and from NN1(/2)-PSO’s predictions.

Parameters | D1 (nm) | D2 (nm) | D3 (nm) | D4 (nm) | D5 (nm) |
---|---|---|---|---|---|

Original | 189.22 | 185.79 | 388.76 | 247.13 | 41.48 |

Mean (RE) | 188.92 (0.16%) | 185.87 (0.04%) | 388.86 (0.03%) | 248.71 (0.64%) | 40.05 (3.45%) |

STD | 0.33 | 0.22 | 0.45 | 1.63 | 1.38 |

Min. | 187.88 | 185.27 | 388.07 | 246.12 | 36.42 |

Max. | 189.90 | 186.35 | 390.16 | 253.42 | 42.19 |

**Table 2.**The structures obtained by NN-PSO for AR coating in Figure 5.

Parameters | D1 (nm) | D2 (nm) | D3 (nm) | D4 (nm) | D5 (nm) |
---|---|---|---|---|---|

Original | 189.22 | 185.79 | 388.76 | 247.13 | 41.48 |

S1 | 187.07 | 182.88 | 441.05 | 278.13 | 36.78 |

S2 | 232.96 | 203.52 | 428.88 | 264.52 | 11.22 |

S3 | 183.02 | 184.97 | 419.23 | 266.13 | 33.76 |

S4 | 276.47 | 219.01 | 426.10 | 272.36 | 20.65 |

Parameters | D1 (nm) | D2 (nm) | D3 (nm) | D4 (nm) | D5 (nm) |
---|---|---|---|---|---|

Original | 255.88 | 24.33 | 28.85 | 74.90 | 98.58 |

S1 | 258.41 | 15.62 | 28.32 | 100.86 | 90.62 |

S2 | 284.86 | 23.91 | 26.59 | 77.28 | 99.63 |

S3 | 264.31 | 19.41 | 45.08 | 48.84 | 111.64 |

S4 | 227.79 | 18.58 | 34.12 | 93.87 | 94.20 |

**Table 4.**The structures obtained by NN-PSO for the Bragg case in Figure 12.

Parameters | D1 (nm) | D2 (nm) |
---|---|---|

Original | 179.57 | 126.43 |

S1 | 182.13 | 125.80 |

S2 | 175.31 | 130.64 |

PSO range | 142.57~202.57 | 96.43~156.43 |

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**MDPI and ACS Style**

Guo, X.; Lu, J.; Li, Y.; Li, J.; Huang, W.
Inverse Design for Coating Parameters in Nano-Film Growth Based on Deep Learning Neural Network and Particle Swarm Optimization Algorithm. *Photonics* **2022**, *9*, 513.
https://doi.org/10.3390/photonics9080513

**AMA Style**

Guo X, Lu J, Li Y, Li J, Huang W.
Inverse Design for Coating Parameters in Nano-Film Growth Based on Deep Learning Neural Network and Particle Swarm Optimization Algorithm. *Photonics*. 2022; 9(8):513.
https://doi.org/10.3390/photonics9080513

**Chicago/Turabian Style**

Guo, Xiaohan, Jinsu Lu, Yu Li, Jianhong Li, and Weiping Huang.
2022. "Inverse Design for Coating Parameters in Nano-Film Growth Based on Deep Learning Neural Network and Particle Swarm Optimization Algorithm" *Photonics* 9, no. 8: 513.
https://doi.org/10.3390/photonics9080513