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Abstract
Bloch surface wave (BSW) based photonic devices have attracted significant interest for bio-sensing, spectroscopy, and light manipulation applications due to their ability to confine light at dielectric interfaces. However, optimizing the geometry of BSW structures to achieve desired optical properties can be computationally expensive using conventional simulation methods for multi-parameter design spaces. In this work, we develop machine learning models based on the gradient boosting algorithm XGBoost to predict key optical characteristics of BSW devices and expedite the design process. Finite element method simulations are used to generate a dataset relating BSW structures’ excitation angle, sensitivity, and spectral response to their geometric parameters, including thickness, porosity, and surrounding refractive index. This dataset trains and validates different XGBoost regression models for photonic structure optimization. Our results demonstrate that a model utilizing deep decision trees achieves the highest predictive accuracy, with a mean absolute error of 0.09° in estimating the excitation angle for new structures. We apply this optimized model to uncover the thickness-porosity combinations, enabling a maximum sensitivity of 171-degree/RIU. This machine learning approach provides a powerful tool for the inverse design and performance enhancement of BSW photonic structures beyond the capabilities of conventional simulation-based optimization.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Bloch surface wave (BSW) based optical devices and sensors have attracted widespread attention because of their inherent ability to confine and manipulate light along the interface [1,2]. This allows for susceptible and precise measurements of minute surrounding perturbation in real-time [3]. Especially in biosensing, where they can detect bio-molecules with remarkable accuracy utilizing label-free probing [4,5]. Moreover, BSW devices demonstrate robustness against external perturbations, ensuring reliable performance even in harsh conditions [6]. Their tunability and adaptability further enhance their versatility, enabling the design of tailored sensors for specific applications [7,8]. The BSW based devices have been utilized for development of high-performance diffraction enhancement [9], flat-lens [10], surface-enhanced Raman spectroscopy [11], biosensing [12], chemical sensing [13], gas sensing [14] and fluorescence emission enhancement [15,16]. The performance characteristics of these devices are usually obtained by conventional finite element method (FEM), finite difference time domain (FDTD) method, and rigorous coupled wave analysis (RCWA) methods [17–19]. These conventional methods provide better optimization with high accuracy but are helpful only for small-scale modeling. However, for a large number of parameters with very fine-tuning, the conventional methods are not feasible because of their complex methodology and time cost [20]. Thus, significant computational challenges are posed in accurately modeling the device behavior.
Machine learning algorithms offer a paradigm shift by enabling efficient exploration of vast parameter spaces, swiftly predicting complex relationships between structural geometric parameters and corresponding optical properties [21]. This facilitates the prediction of structural parameters with improved functionality of BSW devices, such as enhanced light-matter interactions and tailored spectral responses, by eliminating time-consuming computation. During analysis, the prediction process of machine learning, the dataset’s completeness, the features’ effectiveness, and the learning model’s accuracy are all vital in determining the prediction results. The earliest use of machine learning networks for the design of nanophotonic devices dates back to 2017, when teams from Japan [22] and the United States [23] used fully connected neural networks to replace FDTD simulations to fit the spectral characteristics of nanomaterials, to some extent, circumventing the limitations of traditional simulation. In addition to spectral information, Sunny Chugh in 2019, proposed a fully connected neural network to predict the effective refractive index, effective mode area, dispersion, and limited loss of quartz core photonic crystals [24]. The time could be controlled at the millisecond level, while numerical simulations under the same conditions required several minutes. These studies have enabled the utilization of the ML approach in design optimization and performance prediction of various nanophotonic devices including photonic crystals [25,26], plasmonic sensors [27], waveguides [28], metasurfaces [29,30], and plasmonic colors [31].
This concept has further been utilized in this study to optimize the performance of BSW-based sensors, which is carried out using the XGBoost algorithm. The inner training and validation sets are used to systematically tune model hyperparameters and select the optimal XGBoost configuration based on prediction accuracy on the validation data. The final selected model with the highest cross-validation performance is then evaluated on the held-out test set. This external test set provides an unbiased estimate of the model’s ability to generalize to new data, as it was not used during model development. Further, the optimized XGBoost model is used to predict the structure’s sensitivity based on input conditions. Finally, the obtained results are cross-verified using the conventional finite element method (FEM) solver of COMSOL Multiphysics. Strong predictive performance on the unseen test data gives confidence that the model has learned robust and generalized relationships rather than simply over-fitting the training data. Integrating machine learning into Bloch surface wave optimization expedites the process and opens doors to uncovering novel solutions that may have been elusive through traditional methods.
The remainder of this paper is organized as follows. Section 2 introduces the problem analysis, structure design, and methods combined with the workflow of the proposed ML model. This further focused on data pre-processing methods, construction of XGBoost ML models, and corresponding training and testing. The implementation of the XGBoost ML models combined with the data prediction and validation is described in section 3. Finally, section 4 concludes the presented work.
2. Structure design and methods
The proposed bi-layer photonic crystal structure having a top defect layer to excite BSW is shown in Fig. 1(a). The structure comprises TiO$_{2}$ as a high index material having refractive index $n_{1}$ equal to 2.2 and thickness $d_{1}$ of 85nm. The low index material is made of porous TiO$_{2}$ having refractive index $n_{2} (n_{2} = f(n_{1}, P)$) and thickness $d_{2}$ of 128nm. The porosity-dependent refractive index can be calculated using $P= [(n_p^2-n_{d s}^2)(n_a^2-2n_{d s}^2)]/[(n_p^2+2n_{d s}^2)(n_a^2-n_{d s}^2)]$, where the refractive indices $n_{\text {p}}$, $n_{\text {a}}$, and $n_{\text {ds}}$ represent the refractive indices of the porous material, air/analytes, and dense material, respectively [32,33]. The material-induced losses are taken care of by considering the material’s imaginary dielectric constant ($\varepsilon _{i}$=0.0007).
Fig. 1. (a) Schematics of the proposed structure, and (b) Corresponding angular reflectance at different surrounding analytes for constant wavelength of 632.8nm.
A top defect layer similar to regular porous TiO$_{2}$ having refractive index $n_{1}$ and thickness $d_{d}$ is deliberately introduced to excite the Bloch surface wave (BSW) for 632.8nm operating wavelength at the top interface. The proposed structure is illuminated by a broadband light source incident via a prism (Refractive index: 1.515 with index matching liquid) coupled Kretschmann configuration. For defect layer thickness of 180nm and surrounding medium air, the structure shows the excitation of a BSW mode at a 41.43-degree incident angle. This is shown in Fig. 1(b). Changing the surrounding medium index to 1.001 leads to a shift in the BSW excitation angle to 41.47 degrees. This gives a sensitivity (S=$\Delta \theta /\Delta n$) of around 40 degrees/RIU [34,35]. The obtained sensitivity is very low but can be improved by properly optimizing the structural parameters.
Therefore, in this study, an ML model is built to optimize the performance of BSW-based sensors, which is carried out using the XGBoost algorithm. The XGBoost algorithm is configured to build a relationship between structural parameters (such as defect layer thickness (d$_{d}$), porosity (P) and surrounding analyte index (ns)) and sensing parameters (such as BSW excitation angle ($\theta _{1}$, $\Delta \theta$, and sensitivity (S)). The data sets considered in the study are generated using conventional techniques, and details are described in the data pre-processing section. The density of sampled training data is determined to minimize the computation cost but sufficient to train the model accurately. As shown in Fig. 2, the data set is split into training and test sets to allow for unbiased evaluation of model performance. The training data is further partitioned into nested inner training and validation sets to perform proper cross-validation during model selection and hyper-parameter tuning. In fact, this small amount of training data can train the model accurately for modeling and performance prediction of various BSW excitation-based devices in the considered parameter variation range.
2.1 Data pre-processing
The data sets considered in the study are generated using the finite element method (FEM) of COMSOL Multiphysics. The ‘P’, ’d$_{d}$’ and $\theta$ variations are considered in the range of 0.7 to 0.85, 155nm to 300nm, and 0-degree to 90-degree, respectively, and corresponding reflectance is recorded for further processing. For simplicity, the surrounding analyte refractive index variation (ns) is considered from 1.000 to 1.001. The density of sampled training data is determined to minimize the computation cost but sufficient to train the model accurately. The generated data sets can be accessed via [36].
The dataset is pre-processed to extract crucial information about the dependency of minimum reflectance and maximum $\Delta \theta$ on varying the surrounding analytes. The proper preprocessing, inspection, and analysis ensure a high-quality input dataset before modeling. The data is rigorously partitioned into training and testing sets through stratified sampling, allowing an unbiased evaluation of the model’s performance. The $\theta$ angle that satisfies the minimum reflectance condition (with value less than 0.5) for the combination of ‘ns’, ‘P’, and ‘d$_{d}$’ is identified. The resulting dataset includes the optimal $\theta$ values for each set of ‘ns’, ‘P’, and ‘d$_{d}$’ that met these conditions. The remaining combinations that fall outside of this condition are neglected. The preprocessed dataset consists of two numeric features: porosity and thickness measurements. Using scikit-learn’s train_test_split, an 80/20 stratified random split is implemented to divide the data into training (n=1203) and holdout testing (n=301) sets for a rigorous model evaluation. In the training set, porosity exhibited a mean of 0.76 with a standard deviation of 0.043, while the thickness mean measured 249.20 nm with a standard deviation of 33.48 nm. In the testing set, porosity had a mean of 0.76 (std 0.04) and thickness a mean of 247.48 nm (std 34.27 nm). The close alignment of means and standard deviations between the training and testing sets indicates appropriate stratification.
The data distributions for both porosity and thickness are shown in Fig. 3, which appear approximately normal. There are no problematic outliers, skewness, or unexpected gaps requiring intervention are identified in the visualized distributions. Given their comparable ranges and lack of outliers, the numeric measurements do not necessitate scaling or normalization preprocessing. Even center scaling is considered unnecessary, as the models can handle the original raw feature distributions.
2.2 XGBoost models for device performance prediction
In this study, the XGBoost algorithm is configured to build a relationship between structural parameters (such as defect layer thickness (d$_{d}$), porosity (P) and surrounding analyte index (ns)) and sensing parameters (such as BSW excitation angle ($\theta _{1}$, $\Delta \theta$, and sensitivity (S)). XGBoost is an upgraded version of the gradient Boost regression tree (GBRT). It is a type of ensemble algorithm, but it belongs to Boosting algorithms, which assign different weights to weak learners (e.g., decision trees) and give greater weight to samples with larger errors. The proposed architecture of the XGboost machine learning model is shown in Fig. 4. The extraction of the dataset is also determined by the weights of the weak learners from the previous round of training. Through iterations, multiple weak learners are combined into a strong learner to minimize the error sum. The difference lies in the different objective functions. The calculation formula of the objective function $Ob$ of XGBoost is shown in Eq. (1) and Eq. (2):
The determination of the hyperparameters of XGBoost follows these rules: the larger the maximum tree depth (max_depth), the proportion of sub-samples per tree (sub-sample), the proportion of sub-indicator sets per tree (colsample), and the learning rate (eta), the better; the smaller the minimum node weight (min_child_weight), the minimum loss function drop (gamma), and the number of trees (nround), the better. Conversely, the setting will lead to under-fitting. Four XGBoost regression models are initialized in the device analysis with different hyperparameters to predict the optimal excitation angle, as shown in Table 1. The models are selected to analyze key hyperparameter effects based on established XGBoost tuning practices. XGB-HighEst leverages its powerful boosting for the device modeling task. XGB-RegAlpha evaluates regularization to prevent overfitting for the proposed design. XGB-DeepTree tests increasing tree depth to model complex interactions, which is critical for precision engineering applications. XGB-Balanced takes a blended approach using moderate values. This covers major strategies - boosting, regularization, complex modeling, and balance - providing empirical insights into performance impacts for device parameter prediction.
Table 1. Hyperparameter tuning results for the XGBoost models
2.3 Model training and testing
This rigorous training and validation workflow on distinct data partitions evaluates the effectiveness of the models for the proposed design problem by providing an unbiased estimate of their generalized predictive performance. The full pre-processed dataset is split into stratified train, validation, and test subsets to fit models and objectively benchmark prediction accuracy. The four XGBoost regression models are trained using 80% of the pre-processed dataset comprising measurements of porosity, thickness, and corresponding optimal excitation angles that minimize reflectance in various BSW-based Devices. This training set is used to fit the relationships between design conditions and optimized excitation angle, as well as optimize the model hyperparameters. The XGBoost scikit-learn API is used to define and fit each model variation based on the specified hyperparameters to predict optimal excitation angle. The models are trained for a maximum of 100 epochs with early stopping to prevent overfitting. Early stopping ended training if the validation log loss did not improve after ten consecutive epochs, using a validation set of 20% of the training data. Once trained, the Python pickle module is used to serialize the model objects. The saved models are then loaded and used to generate predictions on a separate held-out testing dataset comprising 20% of the full pre-processed data, simulating a real-world deployment scenario. Comparing the testing set predictions to actual optimal excitation angle values assessed each model’s out-of-sample prediction accuracy.
3. Results and discussions
The performance of each XGBoost regression model for predicting the optimal excitation angle in nanophotonic materials is evaluated and compared using mean squared error (MSE) and mean absolute error (MAE) metrics, calculated on the held-out testing dataset predictions. The MSE measures the average squared difference between the predicted and actual optimal excitation angle values. It exaggerates larger errors due to squaring the residuals, making MSE sensitive to outliers in the prediction of proposed device performance. This also provides insight into the model’s prediction variance and ability to minimize large deviations from true optimal angles. Similarly, MAE calculates the simple average of absolute residuals between predicted and actual excitation angles. By using absolute values rather than squaring, MAE is less influenced by outliers compared to MSE. MAE offers an interpretation of the typical size of angular errors, which is essential for the precision engineering of BSW-based nanophotonic devices. Together, MSE and MAE provide complementary views of each model’s performance in predicting the optimized parameter for the proposed devices. Comparing these metrics across models gives a comprehensive understanding of their predictive success, reliability, precision, and stability. The model exhibiting minimum MSE and MAE can be utilized with the greatest confidence for informed nanophotonics research and development. These standard regression performance metrics are selected for their common use and interpretability in evaluating supervised learning models. They provide empirical quantification of how effectively each model can generalize to new data and optimize device properties. The MAE and MSE can be calculated using Eqs. (3) and (4).
where $n$ is the number of samples, $\theta _i$ is the true value of the optimal excitation angle for the $i$-th sample, and $\hat {\theta }_i$ is the predicted value of the optimal excitation angle for the $i$-th sample.3.1 Prediction value comparison
The comparison of XGBoost model predictions versus true optimal excitation angles is shown in Fig. 5. The four XGBoost models are benchmarked against each other. All models achieved mean predictions close to the true mean angle of 45.51, indicating generally accurate orientation forecasts. The XGB-DeepTree model produced the lowest prediction variance, which was closest to the true variance. Its range of predicted angles from 41.32 to 48.92 is also the most restrictive versus the true range, demonstrating the most reliable and consistent excitation angle predictions. Overall, XGB-DeepTree exhibited the strongest predictive stability and calibration for BSW-based devices. Meanwhile, XGB-Balanced achieved the lowest mean absolute angular deviation of 4.59, surpassing the true 4.58. This suggests a precision-consistency trade-off between the models for optimizing the proposed device performance.
Further, the MSE and MAE metrics are used to rigorously evaluate the four XGBoost regression models’ performance for the proposed device structure. As shown in Fig. 6, XGB-DeepTree achieved the lowest MSE value, significantly outperforming the others in modeling the relationship between design conditions and optimal angles to minimize error variability. XGB-DeepTree also exhibited minimum MAE, reflecting a tight distribution of small absolute residuals around the true angle. Combined with low MSE, this highlights XGB-DeepTree’s calibration and precision for the excitation angle prediction task in the proposed structure. In contrast, XGB-RegAlpha performed worst on both metrics, with substantially larger errors demonstrating a relative lack of accuracy and reliability for the device modeling application. Interestingly, XGB-HighEst achieved the second lowest MSE but higher MAE, suggesting minor median errors but increased noise and instability, leading to higher squared angular errors for the proposed structure. The XGB-Balanced model fell in between with moderate improvements in predictive accuracy and consistency versus XGB-RegAlpha for the BSW excitation prediction problem. The analysis demonstrates that the XGB-DeepTree emerged as the best model by achieving the smallest errors on both evaluation metrics, illuminating the importance of assessing multiple aspects of performance for the targeted device.
3.2 Residuals comparison
The residual results provide insights into the inner workings of the different XGBoost model architectures for predicting the optimal excitation angle for BSW excitation. The analysis is presented in Fig. 7. While typical metrics summarize overall performance, residual analysis reveals nuanced patterns in predictions, consistency, variability, and reliability. The XGB-Balanced model achieved the lowest average residual, indicating the closest fit to true angles on aggregate. However, this came at the cost of stability, as evidenced by the high maximum residual of 0.171 and a comparatively wider range. The balanced tuning improved central tendency but allowed higher variance predictions. In contrast, XGB-DeepTree pursued consistency, with deeper trees smoothing out variance and regularization restricting overfitting. This led to the lowest residual standard deviation, indicating very high angular prediction stability. However, this came at a slight cost to the overall residual mean. The XGB-RegAlpha model demonstrates regularization’s ability to control overfitting, achieving nearly identical mean residual to XGB-HighEst but with substantially lower standard deviation. This smoothed residuals around the mean instead of absolutely minimizing them like XGB-Balanced.
3.3 Validation of the selected model
Finally, the developed model provides the best possible porosity and defect layer thickness combination for the highest possible $\Delta \theta$ value. To validate its performance, we tested the model using a range of porosity (P) from 0.70 to 0.85 and a thickness (dd) range from 156 to 300, under conditions where the refractive index (ns) values are varied from 1.00 and 1.001. This analysis identified that all four considered XGB-variations exhibit BSW excitation at almost similar excitation angles. The defect layer cutoff thickness increases with the porosity, which is in accordance with the theoretical results. This results in a large number of porosity and thickness combinations to excite a BSW at the top interface for low porosity. In contrast, the combinations reduce for higher porosity within the considered defect layer thickness range (156nm to 300nm). The same is represented by Fig. 8(a). Figure 8(b) shows the best $\Delta \theta$ calculation for the considered XGB-variations. The analysis exhibits the highest $\Delta \theta$ of 0.171 degrees for the XGB-DeepTree variation. This result provides valuable insights into how we can adjust the porosity and thickness parameters in nanophotonic designs to achieve optimal performance. It also demonstrates the potential of the XGB-DeepTree model as a valuable tool for predicting the angle of minimum reflectance in nanophotonic structures, thereby aiding in their design and optimization process.
Fig. 8. Performance validation of selected model (a) BSW excitation angle and (b) $\Delta \theta$ prediction using XGBoost models
4. Conclusions
In this work, machine learning models based on the XGBoost algorithm were developed and shown to accurately predict key optical properties of Bloch surface wave structures. Finite element simulations were used to generate a dataset relating the excitation angle, sensitivity and spectral response to geometric parameters. Different XGBoost regression models were trained and validated on this dataset for photonic structure optimization. Evaluation on an independent test set demonstrated that a deep decision tree model achieved the highest prediction accuracy with a mean absolute error of 0.09$^{\circ }$ for the excitation angle. This optimized model is then applied to uncover thickness-porosity combinations, enabling a maximum sensitivity of 171$^{\circ }$/RIU for the XGBoost DeepTree model. The results confirm that machine learning provides an effective approach for accelerating the design and optimization of BSW photonic structures beyond what can be achieved with conventional simulation-based techniques alone. The developed XGBoost models offer a powerful tool for the inverse design of BSW devices to uncover novel solutions and explore the multi-dimensional design space in ways not feasible previously. Overall, this research establishes the viability of machine learning for computational acceleration and performance enhancement across nanophotonic and metamaterial platforms.
Acknowledgment
The authors would like to acknowledge the research funding to the Innovative Technologies Laboratories (ITL), King Abdullah University of Science and Technology (KAUST) Saudi Arabia.
Authors Contribution. Conceptualization: H.Y., A.K.G & Y.M.; methodology: H.Y.; validation: H.Y., & A.K.G.; investigation: Y.M.; resources: H.Y., A.K.G. & Y.M.; data curation: H.Y.; writing—original draft preparation: H.Y., & A.K.G.; writing—review and editing: H.Y., A.K.G. & Y.M.; visualization: H.Y.; supervision:A.K.G. & Y.M. All authors have read and agreed to the published version of the manuscript.
Disclosures
The authors declare no conflicts of interest.
Data availability
The data underlying the results presented in this paper can be accessed via [36] . More information may be obtained from the authors upon reasonable request.
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