Ex) Article Title, Author, Keywords
Current Optics
and Photonics
Ex) Article Title, Author, Keywords
Curr. Opt. Photon. 2024; 8(3): 230-238
Published online June 25, 2024 https://doi.org/10.3807/COPP.2024.8.3.230
Copyright © Optical Society of Korea.
Seok Jin Hong1, Jung Hee Lee2, Devarajulu Gelija1, Woon Jin Chung1
Corresponding author: *wjin@kongju.ac.kr, ORCID 0000-0002-1523-338X
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/4.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
The refractive index is a key material-design parameter, especially for high-refractive-index glasses, which are used for precision optics and devices. Increased demand for high-precision optical lenses produced by the glass-mold-press (GMP) process has spurred extensive studies of proper glass materials. B2O3, SiO2, and multiple heavy-metal oxides such as Ta2O5, Nb2O5, La2O3, and Gd2O3 mostly compose the high-refractive-index glasses for GMP. However, due to many oxides including up to 10 components, it is hard to predict the refractivity solely from the composition of the glass. In this study, the refractive index of optical glasses based on the B2O3-La2O3-Ta2O5-SiO2 system is predicted using machine learning (ML) and compared to experimental data. A dataset comprising up to 271 glasses with 10 components is collected and used for training. Various ML algorithms (linear-regression, Bayesian-ridge-regression, nearest-neighbor, and random-forest models) are employed to train the data. Along with composition, the polarizability and density of the glasses are also considered independent parameters to predict the refractive index. After obtaining the best-fitting model by R2 value, the trained model is examined alongside the experimentally obtained refractive indices of B2O3-La2O3-Ta2O5-SiO2 quaternary glasses.
Keywords: B2O3-La2O3-Ta2O5-SiO2, Glass compositions, Glass mold press, Machine learning, Refractive index
OCIS codes: (120.4610) Optical fabrication; (160.2750) Glass and other amorphous materials; (230.0230) Optical devices
The refractive index (RI) is an important property of a material, and an essential characteristic for selecting a suitable material for various scientific and engineering applications, including the design of lenses, prisms, and other optical devices. Recent development in various precision-optical applications such as cameras, telecommunications, and integrated photonic devices have extended demand for suitable precision-optical lenses with high RI (nd ≥ 1.7), which are mostly produced by the glass-mold-press (GMP) process. The glasses for GMP optical lenses thus need to be carefully designed, in terms of their RI and thermal properties, for the GMP process. To achieve high RI via compositional design, multicomponent glasses based on heavy-metal oxides such as Nb2O5, Ta2O5, La2O3, and Gd2O3, along with B2O3 or SiO2 as a glass former, were mostly used [1–3]. For example, borate glasses containing Nb2O5, Ta2O5, La2O3, and Gd2O3 could modify the structural changes and improve the RI by altering the content like Ta2O5 and La2O3 [1–6].
Since the RI of a glass mostly depends on its composition, prediction of RI based on glass composition is highly important and useful for the design of optical glasses. Although estimation of RI from the composition has been suggested [7], this was mostly based on SiO2-rich glasses and required characteristic parameters for each component, which required complicated calculations. However, most high-RI GMP glasses have low concentrations of SiO2 and require multiple components (up to 10) to adjust the RI, thermal properties, and glass stability at the same time [1–3, 7], making it difficult to estimate the RI correctly. Meanwhile, recent computational methods based on machine learning (ML) and semiempirical methods can provide a more efficient way to determine RI than traditional methods.
Machine learning is classified into supervised learning, unsupervised learning, and reinforcement learning according to the application method [8, 9]. In addition, classification, regression, and clustering models are produced by applying various algorithms such as linear regression (LR), Bayesian ridge regression (BRR), nearest-neighbor regression (NN), and random-forest regression (RFR). The selection of the appropriate regression model depends on the nature of the data, the assumptions about the relationships between variables, and the specific goals of the analysis [8, 9]. After proper training of the data with a suitable algorithm, it can be used to predict various glass properties such as density, glass transition temperature, Young’s modulus, and atomic structures [10–14]. For example, Bhattoo et al. [15] reported the role of each input component and their control effects on electrical, mechanical, and physical properties of inorganic glasses. Alsaif et al. [16] reported the prediction of the density of B2O3-ZnO-BaO-PtO2 glasses, and proved that random-forest regression was best, with an R2 value of 0.945. However, literature survey reveals that there is still a lack of ML-model studies focusing on prediction of high-RI (≥1.7) glasses based on multi-component borate glasses.
In this study, GMP glass compositions based on SiO2, B2O3, Ta2O5, and La2O3 content are collected with their RI data from a literature survey, including patents [17–32]. The polarizability and density are considered additional variables affecting the RI according to the glass composition [33–37]. Advanced data SCiEnce toolkit for non-data scientists (ASCENDS) [10, 38–40] is employed as a machine-learning tool, and four algorithms (LR, BRR, NN, and RF) are applied to train the model and predict the RI. The accuracy of the model is estimated by R2 value, to identify the optimal one for RI prediction. The trained model is examined and evaluated using the experimental data obtained for B2O3-La2O3-Ta2O5-SiO2 quaternary glass.
A dataset comprising composition and RI data is prepared by collecting 271 high-RI optical glasses with ten components: SiO2, Ta2O5, Nb2O5, BaO, ZnO, La2O3, ZrO2, GeO2, B2O3, and Gd2O3 based on [17–32]. Ten components are used as variables, and the corresponding RI (nd) is used as the target property. Since RI can be described by the polarizability and density of the glass [41], these are used for each glass composition as additional independent variables for RI prediction. The polarizability of the glass is estimated by the sum of each oxide’s polarizability, considering the molar ratio of the component. After the construction of a dataset, it is trained with a machine learning tool using ASCENDS [10, 38–40] and varying the supervised-training model among LR, BRR, NN, and RF. The R2 value of each model’s predicted data examines its accuracy compared to the actual data.
To ensure the robustness of our ML model for RI prediction, we conduct a thorough validation process. We use glasses from the B2O3-La2O3-Ta2O5-SiO2 (BLTS) quaternary system, which had been previously reported [42]. A total of 26 quaternary glasses are synthesized, and their RIs meticulously inspected with an Abbe refractometer (NAR-2T; ATAGO Co., LTD., Tokyo, Japan) [42]. The measured values are then compared to the ones predicted by our ML model, with accuracy further examined by the R2 value. This rigorous validation process provides confidence in the reliability of our present study.
As described above, composition and RI data for high-RI optical glasses based on ten components (SiO2, Ta2O5, Nb2O5, BaO, ZnO, La2O3, ZrO2, GeO2, B2O3, and Gd2O3) are collected to compose a dataset [17–32]. After labeling, the dataset is trained with four models (LR, NN, BRR, and RF). After training the data, the accuracy of the trained model is examined according to the linear relationship between the predicted value and the trained data. Since RI can be affected by composition, polarizability, and density, we apply composition, polarizability, and density as variables in different combinations to find an optimal combination of variables for RI prediction.
As a first trial of ML training, the data are trained with 10 components as variables and the corresponding RI as the target property. The goodness of fit for each trained model is shown in Fig. 1, along with the linear relationship between the predicted values and actual data for each. As seen in Fig. 1, a high R2 value of 0.979 is obtained for LR and BRR, while 0.935 and 0.956 are obtained for the NN and RF models respectively. The results show that the RI can be trained and predicted with conventional supervised-ML models with relatively low statistical error. The results also imply that only composition information can give a meaningful RI value. It should be noted that among the four algorithms, LR and BRR produced more reliable data, suggesting that the regression algorithms can be more suitable for establishing the relationship between composition and the resultant RI.
The verification of the trained model is examined with the measured RI for 26 BLTS quaternary-component glasses. Table 1 shows the measured and predicted RI is for varying variable combinations, and Fig. 2 shows the linear correlation between the measured RIs and the predicted RIs for a 10-component dataset. Unlike the high fitness of the literature data, the trained model with ten components shows a relatively low R2 value of 0.732 for the actual synthesized quaternary glasses. The result suggests that some of the quaternary glasses can be out of the composition range of the pre-trained data, and further parameters are required to improve the accuracy of the trained model.
TABLE 1 Linear-regression prediction of refractive index (nd) of B2O3-La2O3-Ta2O5-SiO2 glasses, varying dataset variables
Glass Code | Glass Composition (mol%) | nd | |||||||
---|---|---|---|---|---|---|---|---|---|
B2O3 | La2O3 | Ta2O5 | SiO2 | Actual | Predicted with 10-component | Predicted with 10-component + Polarizability + Density | Predicted with 9-component | Predicted with 9-component + Polarizability + Density | |
BLTS-59 | 70 | 25 | - | 5 | 1.7342 | 1.7275 | 1.7139 | 1.7275 | 1.7350 |
BLTS-1 | 65 | 25 | 5 | 5 | 1.7826 | 1.7720 | 1.7606 | 1.7720 | 1.7789 |
BLTS-7 | 65 | 20 | 10 | 5 | 1.7946 | 1.7798 | 1.7739 | 1.7798 | 1.7864 |
BLTS-2 | 60 | 25 | 10 | 5 | 1.8189 | 1.8166 | 1.8066 | 1.8166 | 1.8197 |
BLTS-6 | 60 | 20 | 15 | 5 | 1.8312 | 1.8244 | 1.8197 | 1.8244 | 1.8261 |
BLTS-4 | 55 | 30 | 10 | 5 | 1.8415 | 1.8534 | 1.8391 | 1.8534 | 1.8517 |
BLTS-5 | 55 | 25 | 15 | 5 | 1.8541 | 1.8611 | 1.8521 | 1.8611 | 1.8575 |
BLTS-9 | 50 | 30 | 15 | 5 | 1.8815 | 1.8979 | 1.8842 | 1.8979 | 1.8879 |
BLTS-60 | 65 | 25 | - | 10 | 1.7370 | 1.7264 | 1.7138 | 1.7264 | 1.7342 |
BLTS-70 | 65 | 20 | 5 | 10 | 1.7702 | 1.7342 | 1.7274 | 1.7342 | 1.7428 |
BLTS-10 | 60 | 25 | 5 | 10 | 1.7936 | 1.7710 | 1.7605 | 1.7710 | 1.7783 |
BLTS-16 | 60 | 20 | 10 | 10 | 1.8032 | 1.7788 | 1.7739 | 1.7788 | 1.7858 |
BLTS-11 | 55 | 25 | 10 | 10 | 1.8301 | 1.8155 | 1.8066 | 1.8155 | 1.8192 |
BLTS-13 | 50 | 30 | 10 | 10 | 1.8453 | 1.8523 | 1.8391 | 1.8523 | 1.8514 |
BLTS-14 | 50 | 25 | 15 | 10 | 1.8559 | 1.8601 | 1.8521 | 1.8601 | 1.8572 |
BLTS-18 | 45 | 30 | 15 | 10 | 1.8792 | 1.8969 | 1.8843 | 1.8969 | 1.8877 |
BLTS-61 | 60 | 25 | - | 15 | 1.7375 | 1.7254 | 1.7138 | 1.7254 | 1.7334 |
BLTS-75 | 60 | 20 | 5 | 15 | 1.7712 | 1.7332 | 1.7273 | 1.7332 | 1.7421 |
BLTS-19 | 55 | 25 | 5 | 15 | 1.7937 | 1.7699 | 1.7605 | 1.7699 | 1.7777 |
BLTS-25 | 55 | 20 | 10 | 15 | 1.8133 | 1.7777 | 1.7738 | 1.7777 | 1.7852 |
BLTS-20 | 50 | 25 | 10 | 15 | 1.8208 | 1.8145 | 1.8066 | 1.8145 | 1.8188 |
BLTS-22 | 45 | 30 | 10 | 15 | 1.8465 | 1.8512 | 1.8391 | 1.8512 | 1.8511 |
BLTS-23 | 45 | 25 | 15 | 15 | 1.8663 | 1.8590 | 1.8521 | 1.8590 | 1.8569 |
BLTS-27 | 40 | 30 | 15 | 15 | 1.8807 | 1.8958 | 1.8843 | 1.8958 | 1.8875 |
BLTS-41 | 40 | 25 | 20 | 15 | 1.8919 | 1.9036 | 1.8972 | 1.9036 | 1.8925 |
BLTS-42 | 35 | 30 | 20 | 15 | 1.9133 | 1.9404 | 1.9291 | 1.9404 | 1.9215 |
As discussed earlier, the polarizability of a material is related to its RI through the Lorentz-Lorenz equation, as follows [36, 37]:
where n is the RI, α is the polarizability, NA is the Avogadro number, M is the molar weight, and ρ is the density of the glass. The equation indicates that the RI increases with the polarizability and density of the glass. Thus, the polarizability and density of the glass are added to the dataset as additional independent variables to improve the RI-prediction accuracy. Polarizability values are calculated based on the mole fraction of each glass component [37, 38], and the density values are calculated by applying the density formula with the packing density [41].
Four ML models, LR, BRR, NN, and RF, are also applied to train and predict RI using the dataset including 10 components along with polarizability and density as variables. As displayed in Fig. 3, the LR and BRR models show high accuracy, with R2 value of 0.979, which is the same as the previous results without polarizability and density. The NN and RF models also show similar fitting accuracy, with R2 values of 0.935 and 0.958 respectively. While almost no change in RI prediction is observed with polarizability and density when trained with the literature dataset, a slight improvement in accuracy is seen when RIs of the BLTS quaternary glasses are predicted using the trained LR model and fitted with the measured values (Fig. 4). The predicted RI values are summarized in Table 1. The R2 value between the predicted and measured values increased to 0.758, which is slightly improved compared to the previous model (R2 = 0.732) with the 10-component dataset only. This suggests the possible role of polarizability and density of glass in predicting experimentally prepared glasses.
However, it should be noted that the predicted RI value of the quaternary system still has a weak correlation with the experimental results, with a relatively low R2 value of 0.758. It should also be noted that the BLTS quaternary system is based on SiO2 with B2O3 as a glass former and does not contain GeO2, which also acts as a glass former and has a stronger effect on RI than SiO2 and B2O3, with its higher polarizability and molecular weight. Considering that most optical glasses with high RI are also based on silicate or borate glass systems with little contribution of GeO2, to improve accuracy, a new dataset based on [18–32] is composed of glasses having up to nine components, removing glasses with GeO2.
With the new 9-component dataset, four ML models are also employed to train and predict the RI of the glasses. As shown in Fig. 5, among the 4 models LR and BRR models again show the highest fitness when R2 value is 0.988, which is improved from the result for the 10-component dataset. In contrast, the NN and RF models present relatively lower accuracy (with R2 values of 0.922 and 0.952 respectively) than those for the 10-component dataset. The new dataset with 9 components is also validated using the trained LR model and BLTS glass system. The predicted RI values using the LR model with a 9-component dataset are summarized in Table 1. As found in Fig. 6, using a 9-component dataset, the accuracy of the predicted RI is highly improved in terms of R2, which increases to 0.846. This result indicates that the confined compositional range in the dataset gives more appropriate ML training and prediction, by removing data with less correlation to the quaternary system.
Further enhancement in fitting accuracy for the experimental results of the quaternary system is also attempted, with the introduction of polarizability and density as additional variables along with 9 components. After composing a new dataset with 9 components along with polarizability and density, it is also trained with four models and their predicted RI results correlate with the data, as exhibited in Fig. 7. Per Fig. 7, a significant correlation between the predicted RI and data is obtained when R2 value is 0.991 in LR and BRR method. NN and RF models produce relatively low accuracy, with R2 values of 0.928 and 0.948 respectively, further demonstrating that LR or BRR is more suitable for the RI prediction of optical glasses. Using the trained LR model, the RI of the BLTS glass system is predicted and compared to the experimental data for validation. As depicted in Fig. 8, highly improved accuracy is obtained when R2 value is 0.925. This is a significantly enhanced value compared to the previous R2 values of 0.732, 0.758, and 0.846 (using a dataset with only ten components, ten components with polarizability and density, and only nine components). In addition, it clearly demonstrates that the LR model trained with a dataset based on nine components, polarizability, and density as variables is suitable for predicting RI for the BLTS quaternary glass system.
The high accuracy asserts that the RI of optical glass can be predicted based on the glass composition in a simple way using ML, compared to the traditional calculation methods that require complicated parameters for each composition. An efficient design of glass compositions to adjust RI can thus be anticipated. However, it should also be noted that the present results are focused on the BLTS quaternary system. At the same time, the suitable ML model and dataset can be varied depending on the target compositions. Thus, further study is required to expand the available dataset and reduce possible errors and noisy data, to extend the approach’s feasibility to various glass systems.
A dataset consisting of 271 glass compositions and refractive indices, including ten components (SiO2, Ta2O5, Nb2O5, BaO, ZnO, La2O3, ZrO2, GeO2, B2O3, and Gd2O3) was prepared by literature survey, to predict the RI of optical glass for GMP with high RI using ML. The dataset was trained with various models (LR, BRR, NN, and RF). When the trained models were examined according to the correlation accuracy, using the R2 value between the predicted RI and actual data values, LR and BRR showed high accuracy, proving their suitability for RI prediction. However, the trained LR model resulted in low accuracy, with R2 of 0.732, when verified with the experimentally prepared and measured B2O3-La2O3-Ta2O5-SiO2 quaternary glass system. Polarizability and density were added to the dataset as additional variables and showed meaningful improvement. However, significant enhancement in the prediction accuracy was obtained when the dataset was constructed with nine components (excluding GeO2) along with polarizability and density. A highly improved correlation between the predicted RI value using the trained LR model and the experimental data was successfully obtained, with R2 value of 0.925 for the present quaternary glass system. The result successfully demonstrated the RI prediction of B2O3-La2O3-Ta2O5-SiO2 quaternary glasses based on composition using ML.
The authors are grateful to the Technology Innovation Program named High Refractive Optical Glass for GMP (Grant no. 20011325), and Korea Evaluation Institute of Industrial Technology (KEIT) (Grant no. G012001132504). This work was also supported by Korea Institute for Advancement of Technology (KIAT) grant (Grant no. G02P13780002112) funded by the Korea Government (MOTIE) Human Resource Development Program for Industrial Innovation (Grant no. P0017012).
The Technology Innovation Program (Grant no. 20011325, High refractive optical glass for GMP); Korea Evaluation Institute of Industrial Technology (KEIT) (Grant no. G012001132504); Korea Institute for Advancement of Technology (KIAT) (Grant no. G02P13780002112); The Human Resource Development Program for Industrial Innovation (Grant no. P0017012).
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
The data used to support the findings of this study are available from the corresponding author upon request.
Curr. Opt. Photon. 2024; 8(3): 230-238
Published online June 25, 2024 https://doi.org/10.3807/COPP.2024.8.3.230
Copyright © Optical Society of Korea.
Seok Jin Hong1, Jung Hee Lee2, Devarajulu Gelija1, Woon Jin Chung1
1Institute for Rare Metals and Division of Advanced Materials Engineering, Kongju National University, Cheonan 31080, Korea
2Taihan Fiber Optics Co., Ansan 15601, Korea
Correspondence to:*wjin@kongju.ac.kr, ORCID 0000-0002-1523-338X
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/4.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
The refractive index is a key material-design parameter, especially for high-refractive-index glasses, which are used for precision optics and devices. Increased demand for high-precision optical lenses produced by the glass-mold-press (GMP) process has spurred extensive studies of proper glass materials. B2O3, SiO2, and multiple heavy-metal oxides such as Ta2O5, Nb2O5, La2O3, and Gd2O3 mostly compose the high-refractive-index glasses for GMP. However, due to many oxides including up to 10 components, it is hard to predict the refractivity solely from the composition of the glass. In this study, the refractive index of optical glasses based on the B2O3-La2O3-Ta2O5-SiO2 system is predicted using machine learning (ML) and compared to experimental data. A dataset comprising up to 271 glasses with 10 components is collected and used for training. Various ML algorithms (linear-regression, Bayesian-ridge-regression, nearest-neighbor, and random-forest models) are employed to train the data. Along with composition, the polarizability and density of the glasses are also considered independent parameters to predict the refractive index. After obtaining the best-fitting model by R2 value, the trained model is examined alongside the experimentally obtained refractive indices of B2O3-La2O3-Ta2O5-SiO2 quaternary glasses.
Keywords: B2O3-La2O3-Ta2O5-SiO2, Glass compositions, Glass mold press, Machine learning, Refractive index
The refractive index (RI) is an important property of a material, and an essential characteristic for selecting a suitable material for various scientific and engineering applications, including the design of lenses, prisms, and other optical devices. Recent development in various precision-optical applications such as cameras, telecommunications, and integrated photonic devices have extended demand for suitable precision-optical lenses with high RI (nd ≥ 1.7), which are mostly produced by the glass-mold-press (GMP) process. The glasses for GMP optical lenses thus need to be carefully designed, in terms of their RI and thermal properties, for the GMP process. To achieve high RI via compositional design, multicomponent glasses based on heavy-metal oxides such as Nb2O5, Ta2O5, La2O3, and Gd2O3, along with B2O3 or SiO2 as a glass former, were mostly used [1–3]. For example, borate glasses containing Nb2O5, Ta2O5, La2O3, and Gd2O3 could modify the structural changes and improve the RI by altering the content like Ta2O5 and La2O3 [1–6].
Since the RI of a glass mostly depends on its composition, prediction of RI based on glass composition is highly important and useful for the design of optical glasses. Although estimation of RI from the composition has been suggested [7], this was mostly based on SiO2-rich glasses and required characteristic parameters for each component, which required complicated calculations. However, most high-RI GMP glasses have low concentrations of SiO2 and require multiple components (up to 10) to adjust the RI, thermal properties, and glass stability at the same time [1–3, 7], making it difficult to estimate the RI correctly. Meanwhile, recent computational methods based on machine learning (ML) and semiempirical methods can provide a more efficient way to determine RI than traditional methods.
Machine learning is classified into supervised learning, unsupervised learning, and reinforcement learning according to the application method [8, 9]. In addition, classification, regression, and clustering models are produced by applying various algorithms such as linear regression (LR), Bayesian ridge regression (BRR), nearest-neighbor regression (NN), and random-forest regression (RFR). The selection of the appropriate regression model depends on the nature of the data, the assumptions about the relationships between variables, and the specific goals of the analysis [8, 9]. After proper training of the data with a suitable algorithm, it can be used to predict various glass properties such as density, glass transition temperature, Young’s modulus, and atomic structures [10–14]. For example, Bhattoo et al. [15] reported the role of each input component and their control effects on electrical, mechanical, and physical properties of inorganic glasses. Alsaif et al. [16] reported the prediction of the density of B2O3-ZnO-BaO-PtO2 glasses, and proved that random-forest regression was best, with an R2 value of 0.945. However, literature survey reveals that there is still a lack of ML-model studies focusing on prediction of high-RI (≥1.7) glasses based on multi-component borate glasses.
In this study, GMP glass compositions based on SiO2, B2O3, Ta2O5, and La2O3 content are collected with their RI data from a literature survey, including patents [17–32]. The polarizability and density are considered additional variables affecting the RI according to the glass composition [33–37]. Advanced data SCiEnce toolkit for non-data scientists (ASCENDS) [10, 38–40] is employed as a machine-learning tool, and four algorithms (LR, BRR, NN, and RF) are applied to train the model and predict the RI. The accuracy of the model is estimated by R2 value, to identify the optimal one for RI prediction. The trained model is examined and evaluated using the experimental data obtained for B2O3-La2O3-Ta2O5-SiO2 quaternary glass.
A dataset comprising composition and RI data is prepared by collecting 271 high-RI optical glasses with ten components: SiO2, Ta2O5, Nb2O5, BaO, ZnO, La2O3, ZrO2, GeO2, B2O3, and Gd2O3 based on [17–32]. Ten components are used as variables, and the corresponding RI (nd) is used as the target property. Since RI can be described by the polarizability and density of the glass [41], these are used for each glass composition as additional independent variables for RI prediction. The polarizability of the glass is estimated by the sum of each oxide’s polarizability, considering the molar ratio of the component. After the construction of a dataset, it is trained with a machine learning tool using ASCENDS [10, 38–40] and varying the supervised-training model among LR, BRR, NN, and RF. The R2 value of each model’s predicted data examines its accuracy compared to the actual data.
To ensure the robustness of our ML model for RI prediction, we conduct a thorough validation process. We use glasses from the B2O3-La2O3-Ta2O5-SiO2 (BLTS) quaternary system, which had been previously reported [42]. A total of 26 quaternary glasses are synthesized, and their RIs meticulously inspected with an Abbe refractometer (NAR-2T; ATAGO Co., LTD., Tokyo, Japan) [42]. The measured values are then compared to the ones predicted by our ML model, with accuracy further examined by the R2 value. This rigorous validation process provides confidence in the reliability of our present study.
As described above, composition and RI data for high-RI optical glasses based on ten components (SiO2, Ta2O5, Nb2O5, BaO, ZnO, La2O3, ZrO2, GeO2, B2O3, and Gd2O3) are collected to compose a dataset [17–32]. After labeling, the dataset is trained with four models (LR, NN, BRR, and RF). After training the data, the accuracy of the trained model is examined according to the linear relationship between the predicted value and the trained data. Since RI can be affected by composition, polarizability, and density, we apply composition, polarizability, and density as variables in different combinations to find an optimal combination of variables for RI prediction.
As a first trial of ML training, the data are trained with 10 components as variables and the corresponding RI as the target property. The goodness of fit for each trained model is shown in Fig. 1, along with the linear relationship between the predicted values and actual data for each. As seen in Fig. 1, a high R2 value of 0.979 is obtained for LR and BRR, while 0.935 and 0.956 are obtained for the NN and RF models respectively. The results show that the RI can be trained and predicted with conventional supervised-ML models with relatively low statistical error. The results also imply that only composition information can give a meaningful RI value. It should be noted that among the four algorithms, LR and BRR produced more reliable data, suggesting that the regression algorithms can be more suitable for establishing the relationship between composition and the resultant RI.
The verification of the trained model is examined with the measured RI for 26 BLTS quaternary-component glasses. Table 1 shows the measured and predicted RI is for varying variable combinations, and Fig. 2 shows the linear correlation between the measured RIs and the predicted RIs for a 10-component dataset. Unlike the high fitness of the literature data, the trained model with ten components shows a relatively low R2 value of 0.732 for the actual synthesized quaternary glasses. The result suggests that some of the quaternary glasses can be out of the composition range of the pre-trained data, and further parameters are required to improve the accuracy of the trained model.
TABLE 1. Linear-regression prediction of refractive index (nd) of B2O3-La2O3-Ta2O5-SiO2 glasses, varying dataset variables.
Glass Code | Glass Composition (mol%) | nd | |||||||
---|---|---|---|---|---|---|---|---|---|
B2O3 | La2O3 | Ta2O5 | SiO2 | Actual | Predicted with 10-component | Predicted with 10-component + Polarizability + Density | Predicted with 9-component | Predicted with 9-component + Polarizability + Density | |
BLTS-59 | 70 | 25 | - | 5 | 1.7342 | 1.7275 | 1.7139 | 1.7275 | 1.7350 |
BLTS-1 | 65 | 25 | 5 | 5 | 1.7826 | 1.7720 | 1.7606 | 1.7720 | 1.7789 |
BLTS-7 | 65 | 20 | 10 | 5 | 1.7946 | 1.7798 | 1.7739 | 1.7798 | 1.7864 |
BLTS-2 | 60 | 25 | 10 | 5 | 1.8189 | 1.8166 | 1.8066 | 1.8166 | 1.8197 |
BLTS-6 | 60 | 20 | 15 | 5 | 1.8312 | 1.8244 | 1.8197 | 1.8244 | 1.8261 |
BLTS-4 | 55 | 30 | 10 | 5 | 1.8415 | 1.8534 | 1.8391 | 1.8534 | 1.8517 |
BLTS-5 | 55 | 25 | 15 | 5 | 1.8541 | 1.8611 | 1.8521 | 1.8611 | 1.8575 |
BLTS-9 | 50 | 30 | 15 | 5 | 1.8815 | 1.8979 | 1.8842 | 1.8979 | 1.8879 |
BLTS-60 | 65 | 25 | - | 10 | 1.7370 | 1.7264 | 1.7138 | 1.7264 | 1.7342 |
BLTS-70 | 65 | 20 | 5 | 10 | 1.7702 | 1.7342 | 1.7274 | 1.7342 | 1.7428 |
BLTS-10 | 60 | 25 | 5 | 10 | 1.7936 | 1.7710 | 1.7605 | 1.7710 | 1.7783 |
BLTS-16 | 60 | 20 | 10 | 10 | 1.8032 | 1.7788 | 1.7739 | 1.7788 | 1.7858 |
BLTS-11 | 55 | 25 | 10 | 10 | 1.8301 | 1.8155 | 1.8066 | 1.8155 | 1.8192 |
BLTS-13 | 50 | 30 | 10 | 10 | 1.8453 | 1.8523 | 1.8391 | 1.8523 | 1.8514 |
BLTS-14 | 50 | 25 | 15 | 10 | 1.8559 | 1.8601 | 1.8521 | 1.8601 | 1.8572 |
BLTS-18 | 45 | 30 | 15 | 10 | 1.8792 | 1.8969 | 1.8843 | 1.8969 | 1.8877 |
BLTS-61 | 60 | 25 | - | 15 | 1.7375 | 1.7254 | 1.7138 | 1.7254 | 1.7334 |
BLTS-75 | 60 | 20 | 5 | 15 | 1.7712 | 1.7332 | 1.7273 | 1.7332 | 1.7421 |
BLTS-19 | 55 | 25 | 5 | 15 | 1.7937 | 1.7699 | 1.7605 | 1.7699 | 1.7777 |
BLTS-25 | 55 | 20 | 10 | 15 | 1.8133 | 1.7777 | 1.7738 | 1.7777 | 1.7852 |
BLTS-20 | 50 | 25 | 10 | 15 | 1.8208 | 1.8145 | 1.8066 | 1.8145 | 1.8188 |
BLTS-22 | 45 | 30 | 10 | 15 | 1.8465 | 1.8512 | 1.8391 | 1.8512 | 1.8511 |
BLTS-23 | 45 | 25 | 15 | 15 | 1.8663 | 1.8590 | 1.8521 | 1.8590 | 1.8569 |
BLTS-27 | 40 | 30 | 15 | 15 | 1.8807 | 1.8958 | 1.8843 | 1.8958 | 1.8875 |
BLTS-41 | 40 | 25 | 20 | 15 | 1.8919 | 1.9036 | 1.8972 | 1.9036 | 1.8925 |
BLTS-42 | 35 | 30 | 20 | 15 | 1.9133 | 1.9404 | 1.9291 | 1.9404 | 1.9215 |
As discussed earlier, the polarizability of a material is related to its RI through the Lorentz-Lorenz equation, as follows [36, 37]:
where n is the RI, α is the polarizability, NA is the Avogadro number, M is the molar weight, and ρ is the density of the glass. The equation indicates that the RI increases with the polarizability and density of the glass. Thus, the polarizability and density of the glass are added to the dataset as additional independent variables to improve the RI-prediction accuracy. Polarizability values are calculated based on the mole fraction of each glass component [37, 38], and the density values are calculated by applying the density formula with the packing density [41].
Four ML models, LR, BRR, NN, and RF, are also applied to train and predict RI using the dataset including 10 components along with polarizability and density as variables. As displayed in Fig. 3, the LR and BRR models show high accuracy, with R2 value of 0.979, which is the same as the previous results without polarizability and density. The NN and RF models also show similar fitting accuracy, with R2 values of 0.935 and 0.958 respectively. While almost no change in RI prediction is observed with polarizability and density when trained with the literature dataset, a slight improvement in accuracy is seen when RIs of the BLTS quaternary glasses are predicted using the trained LR model and fitted with the measured values (Fig. 4). The predicted RI values are summarized in Table 1. The R2 value between the predicted and measured values increased to 0.758, which is slightly improved compared to the previous model (R2 = 0.732) with the 10-component dataset only. This suggests the possible role of polarizability and density of glass in predicting experimentally prepared glasses.
However, it should be noted that the predicted RI value of the quaternary system still has a weak correlation with the experimental results, with a relatively low R2 value of 0.758. It should also be noted that the BLTS quaternary system is based on SiO2 with B2O3 as a glass former and does not contain GeO2, which also acts as a glass former and has a stronger effect on RI than SiO2 and B2O3, with its higher polarizability and molecular weight. Considering that most optical glasses with high RI are also based on silicate or borate glass systems with little contribution of GeO2, to improve accuracy, a new dataset based on [18–32] is composed of glasses having up to nine components, removing glasses with GeO2.
With the new 9-component dataset, four ML models are also employed to train and predict the RI of the glasses. As shown in Fig. 5, among the 4 models LR and BRR models again show the highest fitness when R2 value is 0.988, which is improved from the result for the 10-component dataset. In contrast, the NN and RF models present relatively lower accuracy (with R2 values of 0.922 and 0.952 respectively) than those for the 10-component dataset. The new dataset with 9 components is also validated using the trained LR model and BLTS glass system. The predicted RI values using the LR model with a 9-component dataset are summarized in Table 1. As found in Fig. 6, using a 9-component dataset, the accuracy of the predicted RI is highly improved in terms of R2, which increases to 0.846. This result indicates that the confined compositional range in the dataset gives more appropriate ML training and prediction, by removing data with less correlation to the quaternary system.
Further enhancement in fitting accuracy for the experimental results of the quaternary system is also attempted, with the introduction of polarizability and density as additional variables along with 9 components. After composing a new dataset with 9 components along with polarizability and density, it is also trained with four models and their predicted RI results correlate with the data, as exhibited in Fig. 7. Per Fig. 7, a significant correlation between the predicted RI and data is obtained when R2 value is 0.991 in LR and BRR method. NN and RF models produce relatively low accuracy, with R2 values of 0.928 and 0.948 respectively, further demonstrating that LR or BRR is more suitable for the RI prediction of optical glasses. Using the trained LR model, the RI of the BLTS glass system is predicted and compared to the experimental data for validation. As depicted in Fig. 8, highly improved accuracy is obtained when R2 value is 0.925. This is a significantly enhanced value compared to the previous R2 values of 0.732, 0.758, and 0.846 (using a dataset with only ten components, ten components with polarizability and density, and only nine components). In addition, it clearly demonstrates that the LR model trained with a dataset based on nine components, polarizability, and density as variables is suitable for predicting RI for the BLTS quaternary glass system.
The high accuracy asserts that the RI of optical glass can be predicted based on the glass composition in a simple way using ML, compared to the traditional calculation methods that require complicated parameters for each composition. An efficient design of glass compositions to adjust RI can thus be anticipated. However, it should also be noted that the present results are focused on the BLTS quaternary system. At the same time, the suitable ML model and dataset can be varied depending on the target compositions. Thus, further study is required to expand the available dataset and reduce possible errors and noisy data, to extend the approach’s feasibility to various glass systems.
A dataset consisting of 271 glass compositions and refractive indices, including ten components (SiO2, Ta2O5, Nb2O5, BaO, ZnO, La2O3, ZrO2, GeO2, B2O3, and Gd2O3) was prepared by literature survey, to predict the RI of optical glass for GMP with high RI using ML. The dataset was trained with various models (LR, BRR, NN, and RF). When the trained models were examined according to the correlation accuracy, using the R2 value between the predicted RI and actual data values, LR and BRR showed high accuracy, proving their suitability for RI prediction. However, the trained LR model resulted in low accuracy, with R2 of 0.732, when verified with the experimentally prepared and measured B2O3-La2O3-Ta2O5-SiO2 quaternary glass system. Polarizability and density were added to the dataset as additional variables and showed meaningful improvement. However, significant enhancement in the prediction accuracy was obtained when the dataset was constructed with nine components (excluding GeO2) along with polarizability and density. A highly improved correlation between the predicted RI value using the trained LR model and the experimental data was successfully obtained, with R2 value of 0.925 for the present quaternary glass system. The result successfully demonstrated the RI prediction of B2O3-La2O3-Ta2O5-SiO2 quaternary glasses based on composition using ML.
The authors are grateful to the Technology Innovation Program named High Refractive Optical Glass for GMP (Grant no. 20011325), and Korea Evaluation Institute of Industrial Technology (KEIT) (Grant no. G012001132504). This work was also supported by Korea Institute for Advancement of Technology (KIAT) grant (Grant no. G02P13780002112) funded by the Korea Government (MOTIE) Human Resource Development Program for Industrial Innovation (Grant no. P0017012).
The Technology Innovation Program (Grant no. 20011325, High refractive optical glass for GMP); Korea Evaluation Institute of Industrial Technology (KEIT) (Grant no. G012001132504); Korea Institute for Advancement of Technology (KIAT) (Grant no. G02P13780002112); The Human Resource Development Program for Industrial Innovation (Grant no. P0017012).
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
The data used to support the findings of this study are available from the corresponding author upon request.
TABLE 1 Linear-regression prediction of refractive index (nd) of B2O3-La2O3-Ta2O5-SiO2 glasses, varying dataset variables
Glass Code | Glass Composition (mol%) | nd | |||||||
---|---|---|---|---|---|---|---|---|---|
B2O3 | La2O3 | Ta2O5 | SiO2 | Actual | Predicted with 10-component | Predicted with 10-component + Polarizability + Density | Predicted with 9-component | Predicted with 9-component + Polarizability + Density | |
BLTS-59 | 70 | 25 | - | 5 | 1.7342 | 1.7275 | 1.7139 | 1.7275 | 1.7350 |
BLTS-1 | 65 | 25 | 5 | 5 | 1.7826 | 1.7720 | 1.7606 | 1.7720 | 1.7789 |
BLTS-7 | 65 | 20 | 10 | 5 | 1.7946 | 1.7798 | 1.7739 | 1.7798 | 1.7864 |
BLTS-2 | 60 | 25 | 10 | 5 | 1.8189 | 1.8166 | 1.8066 | 1.8166 | 1.8197 |
BLTS-6 | 60 | 20 | 15 | 5 | 1.8312 | 1.8244 | 1.8197 | 1.8244 | 1.8261 |
BLTS-4 | 55 | 30 | 10 | 5 | 1.8415 | 1.8534 | 1.8391 | 1.8534 | 1.8517 |
BLTS-5 | 55 | 25 | 15 | 5 | 1.8541 | 1.8611 | 1.8521 | 1.8611 | 1.8575 |
BLTS-9 | 50 | 30 | 15 | 5 | 1.8815 | 1.8979 | 1.8842 | 1.8979 | 1.8879 |
BLTS-60 | 65 | 25 | - | 10 | 1.7370 | 1.7264 | 1.7138 | 1.7264 | 1.7342 |
BLTS-70 | 65 | 20 | 5 | 10 | 1.7702 | 1.7342 | 1.7274 | 1.7342 | 1.7428 |
BLTS-10 | 60 | 25 | 5 | 10 | 1.7936 | 1.7710 | 1.7605 | 1.7710 | 1.7783 |
BLTS-16 | 60 | 20 | 10 | 10 | 1.8032 | 1.7788 | 1.7739 | 1.7788 | 1.7858 |
BLTS-11 | 55 | 25 | 10 | 10 | 1.8301 | 1.8155 | 1.8066 | 1.8155 | 1.8192 |
BLTS-13 | 50 | 30 | 10 | 10 | 1.8453 | 1.8523 | 1.8391 | 1.8523 | 1.8514 |
BLTS-14 | 50 | 25 | 15 | 10 | 1.8559 | 1.8601 | 1.8521 | 1.8601 | 1.8572 |
BLTS-18 | 45 | 30 | 15 | 10 | 1.8792 | 1.8969 | 1.8843 | 1.8969 | 1.8877 |
BLTS-61 | 60 | 25 | - | 15 | 1.7375 | 1.7254 | 1.7138 | 1.7254 | 1.7334 |
BLTS-75 | 60 | 20 | 5 | 15 | 1.7712 | 1.7332 | 1.7273 | 1.7332 | 1.7421 |
BLTS-19 | 55 | 25 | 5 | 15 | 1.7937 | 1.7699 | 1.7605 | 1.7699 | 1.7777 |
BLTS-25 | 55 | 20 | 10 | 15 | 1.8133 | 1.7777 | 1.7738 | 1.7777 | 1.7852 |
BLTS-20 | 50 | 25 | 10 | 15 | 1.8208 | 1.8145 | 1.8066 | 1.8145 | 1.8188 |
BLTS-22 | 45 | 30 | 10 | 15 | 1.8465 | 1.8512 | 1.8391 | 1.8512 | 1.8511 |
BLTS-23 | 45 | 25 | 15 | 15 | 1.8663 | 1.8590 | 1.8521 | 1.8590 | 1.8569 |
BLTS-27 | 40 | 30 | 15 | 15 | 1.8807 | 1.8958 | 1.8843 | 1.8958 | 1.8875 |
BLTS-41 | 40 | 25 | 20 | 15 | 1.8919 | 1.9036 | 1.8972 | 1.9036 | 1.8925 |
BLTS-42 | 35 | 30 | 20 | 15 | 1.9133 | 1.9404 | 1.9291 | 1.9404 | 1.9215 |