2.1 Working principle of PCF-SPR biosensor

Fig 2 illustrates the working principle of an optical fiber-based sensing system that relies on evanescent field interaction for detection. A broadband light source injects light into the core of a specially designed optical fiber, where it propagates through total internal reflection. At each reflection point, a portion of the light extends beyond the core into the cladding region as an evanescent wave. A syringe injection pump introduces the target sample (such as a chemical or biomolecule solution) into a flow chamber that surrounds the sensitive region of the fiber. When the evanescent field interacts with the sample, any change in the local RI, absorption, or scattering alters the transmitted light’s properties. This modified optical signal is collected at the output end and sent to an optical spectrum analyzer (OSA), which measures variations in the transmission spectrum (e.g., wavelength shift or intensity change). The data is then processed and analyzed using computer software to detect and quantify the presence of the analyte, making the system highly suitable for label-free and real-time sensing applications.

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Fig 2. Schematic diagram of a Fiber-Optic SPR Biosensor system for biochemical detection.

https://doi.org/10.1371/journal.pone.0330944.g002

The proposed PCF-SPR biosensor is engineered for effective real-time sensing in both medical diagnostics and environmental monitoring. In the medical domain, it is capable of detecting a wide range of biomarkers, including cancer indicators, hormones, proteins, and pathogens such as viruses and bacteria in aqueous biofluids like blood and serum. Its high sensitivity and broad n_a detection range (1.31–1.42) make it particularly suitable for early-stage disease diagnosis and continuous health monitoring. This RI range covers most biological analytes, typically found in environments with n_a values between 1.33 and 1.40, while the inclusion of the 1.31 to 1.32 range, though less common in practical biosensing, allows for theoretical analysis of sensor behavior near the lower detection threshold. This supports a more comprehensive understanding of sensitivity performance and helps evaluate the design’s robustness under edge-case or simulation-driven conditions. In environmental applications, the sensor can be employed to identify chemical contaminants, solvents, and pollutants in water or air samples. Its fast response time, compatibility with microfluidic systems, and adaptability to portable and point-of-care platforms make it highly suitable for on-site detection, addressing critical challenges in both clinical and environmental settings.

2.2 Sensor cross-sectional design

The proposed biosensor optimization approach involves simulation with defined geometric features, incorporating components like gold layers, a fused silica substrate, air cavities, and an analyte zone. Fig 3 presents the cross-sectional structure of a PCF based SPR biosensor, developed for precise biochemical detection. The design with a smaller number of air holes reduces fabrication complexity. In our design, we have utilized only 11 air holes, a significantly smaller number compared to previous redesigns. The fiber consists of a core surrounded by a structured cladding, where periodically arranged air holes (d1, d2) help guide light through the fiber. The spacing between these air holes is defined by the pitch (p), ensuring proper light propagation. Key structural distances include D1 = 2.6 × p, D2 = 3 × p, and D3 = 3.5 × p, which represent the radial placement of different regions in the fiber structure. D1 extends from the fiber’s center to the gold layer, the region where SPR takes place. D2 reaches the analyte layer, where interactions between biomolecules and the sensing surface occur. D3 extends further to the PML, which absorbs unwanted radiation and minimizes reflection. These structural parameters are essential for optimizing light transmission, improving sensor performance, and enhancing detection sensitivity. The color-coded regions represent different fiber components.

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Fig 3. Cross-sectional diagram of a PCF-Based SPR biosensor.

https://doi.org/10.1371/journal.pone.0330944.g003

2.3 Methodologies

The performance of a PCF-SPR biosensor depends on several key parameters [27]. N_eff ensures proper light propagation and phase matching. While CL measures optical power leakage, lower values indicate better efficiency. S_A enhances detection by quantifying intensity changes, and S_λ detects small RI variations by measuring resonance wavelength shifts. Resolution determines the smallest detectable RI change, with higher resolutions offering more precision. The FOM balances sensitivity and resonance sharpness, ensuring accurate detection. These parameters collectively evaluate sensor performance for biomedical, chemical, and environmental applications. Moreover, the surface plasmon polariton (SPP) in PCF SPR biosensors states electromagnetic waves that move over the metal-dielectric interface and are produced when light combines with free electrons on the metal surface. The phenomenon that generates SPP is called SPR. These waves are highly sensitive to changes in the n_a, which makes them crucial in PCF-SPR biosensors for analyte detection. Conversely, the fundamental mode directs light through the PCF, facilitating its propagation within the sensor. Here we are giving the brief overview of different optical parameters for the PCF-SPR biosensor.

The Sellmeier equation defines the calculation of the RI of fused silica, based upon the wavelength of light [28]. The equation is stated as follows:

(1)

The Sellmeier model utilizes specific constants to characterize the wavelength-dependent RI of materials. The constants used in this context are: B1 = 0.69616300, B2 = 0.407942600, B3 = 0.897479400, and C1 = 0.00467914826, C2 = 0.0135120631, C3 = 97.9340025.

The CL is denoted by the symbol α(dB/cm) [28]. CL in the sensor is computed according to the following relation, using the imaginary part of the N_eff:

(2)

Where k₀ = 2π/λ represents the free-space wave number, and Im(N_eff) is the imaginary component of the N_eff. This expression quantifies optical attenuation resulting from imperfect mode confinement, which plays a crucial role in sensor characterization.

Amplitude sensitivity (S_A) is another vital property that indicates the sensor’s response with the variations of n_a by observing changes in transmitted light intensity [29]. The numerical formulation is expressed as:

(3)

Here, n_a indicates the RI of the analyte, and ∂na represents the difference between the n_a of two neighboring analytes. The function α(λ, n_a) denotes the CL, while ∂α(λ, n_a) presents the variation in CL as the n_a changes. This formula evaluates the effect of n_a fluctuations on the amplitude of light waves through the SPR sensor.

Another critical performance metric is wavelength sensitivity (S_λ), which reflects how the sensor output responds to changes in the incident light’s wavelength [29]. It is defined by the following relation:

(4)

Where Δλ_peak represents the peak wavelength variation, and Δn_a is the variation in n_a between two adjacent analytes.

The calculation of sensor resolution is essential to assess the detection capability of the proposed sensor, as defined by the equation below [30]:

(5)

Where Δn_a represents the change in n_a, and Δλ_min = 0.1 nm denotes the smallest detectable wavelength shift. This equation helps quantify the sensor’s ability to distinguish small variations in analyte concentration.

The Figure of Merit (FOM) is an important parameter used to evaluate the performance of an SPR-based biosensor. It is defined as the ratio of S_λ to the full width at half maximum (FWHM) of the resonance curve [30]. The formula for FOM is given by:

(6)

Where, S_λ (nm/RIU) indicates the shift in resonance wavelength per unit change in n_a, and FWHM (nm) representing the spectral width of the resonance dip.

For a high-performance PCF-SPR biosensor, higher values of FOM, S_λ, and S_A contribute to improved detection accuracy, enhanced signal response, and better analyte sensitivity. In contrast, lower CL improves light propagation, minimizes signal attenuation, and enhances overall sensor efficiency. The mapping of optical parameters and their impact on biosensor performance is given below.

• Higher FOM → Sharper resonance peak → Improved detection accuracy and resolution
• Higher S_A → Greater intensity variations → More precise signal detection
• Higher S_λ → Larger resonance wavelength shift → Enhanced sensitivity to analyte changes
• Lower CL → Better light propagation → Reduced signal attenuation and improved efficiency

2.4 Dataset generation

In this study, we have used COMSOL Multiphysics software to design and simulate the sensor models and evaluated these models using the finite element method (FEM) [31]. The dataset is derived from simulation outputs and includes critical structural and optical features such as core diameters (d1, d2), spacing between air holes pitch (p), thickness of the gold coating (t_g), n_a, operational wavelength (wl), calculated N_eff, CL, and S_A. In addition to these primary features, we derived further optical performance metrics such as S_λ, resolution, and FOM. This comprehensive dataset was also utilized to train ML models aimed at predicting key optical responses of the proposed sensor across various structural configurations, thereby facilitating the identification of high-performance design combinations.

To evaluate each single-mode calculation simulation took around 2 minutes. The amount of time varies according to the number of degrees of freedom resolved. The number of degrees of freedom solved for our mode analysis is 242881. This number depends on the structural design of the sensor, particularly the size and arrangement of the air holes. We used a workstation with a 2.3 GHz Intel Core i3 processor and 12 GB of RAM to perform the calculations.

The simulation process generally follows these steps:

• Build the biosensor design according to the plan.
• Choose suitable electromagnetic interfaces for the analysis.
• Define the materials used for the core and cladding.
• Create a mesh to ensure precise estimates.
• Choose the input settings.
• Use a suitable solver to run the exercise.
• Evaluate the results and explain what the data means.
• Change the settings if necessary and improve the model.
• Gather the results and create a dataset for more research.

Here, Fig 4 represents different states of the PCF-SPR biosensor model, depending on its design and field distributions. Here, Fig 4(a) shows the meshed model of our biosensor model. That divides the computational domain into small, finite elements to solve partial differential equations (PDEs) for light propagation using the FEM [32]. The accuracy and efficiency of the simulation depend on the quality of the mesh. That depends on the air hole arrangement and other design parameters of the sensor structure. Again, Fig 4(b) illustrates the electric field distribution of the core mode for the y-polarized mode. The bright regions indicate areas of strong electric field intensity, revealing how light interacts within the fiber. Fig 4(c) displays the power flow distribution of the SPP mode for the y-polarized mode. The energy is mostly gathered near the plasmonic interface. This shows that the core mode and the plasmonic mode are coupled, which is important for sensing purposes.

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Fig 4. PCF-SPR biosensor: (a) meshed model, (b) core mode electric field distribution, and (c) SPP mode power flow distribution for y-polarized mode.

https://doi.org/10.1371/journal.pone.0330944.g004

This dataset employs parameters such as d1, d2, p, t_g, n_a, and wl as predictor variables. The outputs or target attributes include the N_eff, CL, S_A, and S_λ. From the simulation result, we have got N_eff and then numerically calculated the optical properties. The data provides insights into how structural modifications influence light propagation and resonance behavior of the sensor. This dataset provides an important insight for PCF-SPR biosensor performance optimization. Table 1 presents an overview of the main variables included in the dataset used for analyzing the PCF-SPR biosensor.

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Table 1. Description of dataset variables used in PCF-SPR biosensor modeling.

https://doi.org/10.1371/journal.pone.0330944.t001

2.5 AI techniques for PCF-SPR sensor optimization

Prior studies [33] mentioned that SPR-based sensors are well-suited for label-free detection of cancer markers due to their ability to monitor RI variations resulting from specific molecular binding events. AI techniques, particularly ML, have emerged as a powerful tool in the design and optimization of PCF-SPR biosensors. These data-driven models facilitate rapid prediction of complex optical properties, such as N_eff, CL, and S_A, based on sensor design parameters. That significantly reduces the time-consuming simulation processes. ML models, including RF, SVM, GB, and Bayesian regularization neural networks (BRANNs), have shown potential in modeling nonlinear relationships between structural features and sensor performance. Such approaches enable efficient inverse design, structural tuning, and enhanced prediction fidelity, making them ideal for biosensing applications. Furthermore, integrating ML with XAI enhances both optimization speed and interpretability. Based on other studies [34], different AI techniques can be used for sensor performance optimization as follows:

BRANNs are capable of learning from small, noisy datasets and capturing complex nonlinear dependencies between input parameters and output targets. The method reduces overfitting and enhances generalization through regularization, making it ideal for high-accuracy, real-time applications and inverse sensor design. Classical ML models such as SVM, DT, and ensemble methods like RF, AdaBoost (AB), and GB have been used for classifying optical modes and predicting sensor behavior. These models are particularly useful in early-stage design to efficiently evaluate large design spaces and estimate key performance indicators. Furthermore, advanced deep learning architectures, including feedforward neural networks (FNNs), convolutional neural networks (CNNs), and generative models such as variational autoencoders (VAEs) and conditional generative adversarial networks (GAN), are now being applied for precise resonance prediction, analyte classification, and automated inverse design of PCF structures. CNNs can analyze mode profile images, while autoencoders and 1D-CNNs help in denoising experimental data. Deep reinforcement learning (DRL) offers an adaptive framework for multi-objective optimization under fabrication and material constraints. It enables the sensor to learn from operational history and adjust design parameters in real-time, thereby supporting self-calibrating and self-optimizing systems. Adaptive sensing and environmental compensation AI techniques can also be employed to monitor environmental drift and sensor degradation. By learning from prior data, these models can dynamically adjust sensing parameters to maintain consistent accuracy over extended periods, which is especially critical in remote or in vivo diagnostics. So it is clear that the integration of these AI techniques not only enhances predictive accuracy but also reduces design time and computational cost, thus enabling the rapid development of next-generation PCF-SPR biosensors with enhanced sensitivity, robustness, and adaptability.

2.6 ML algorithms

Since our target variables are continuous, regression models were the appropriate choice for prediction tasks. We applied various regression algorithms to analyze the dataset and assess how design and other factors impact the outcomes. Below is an overview of the ML regression techniques employed:

Decision Tree Regression (DTR) operates by dividing the input feature space into distinct regions and making predictions based on the average target value within each region [35]. This supervised, non-parametric approach is intuitive and allows straightforward visualization of decision rules. However, if left unchecked, it can easily overfit the training data.

Random Forest Regression (RFR) builds an ensemble of decision trees by training each tree on a different random subset of the data and selecting random features at each split [36,37]. The final prediction is obtained by averaging the outputs of all individual trees. This approach enhances prediction stability and accuracy while effectively reducing the risk of overfitting, making it well-suited for handling complex and varied datasets.

Gradient Boosting Regression (GBR) sequentially constructs an ensemble of weak learners, where each subsequent model aims to correct the residual errors of its predecessors [38]. This iterative refinement enables GBR to model complex patterns effectively and achieve high prediction accuracy.

Extreme Gradient Boosting Regressor (XGBR) extends GB by optimizing both computational speed and predictive performance [39]. It combines a series of decision trees trained to minimize errors, excelling at handling large-scale data and intricate relationships within features.

BR improves model stability by training several base regressors on different randomly sampled subsets of data with replacement [40]. This ensemble method reduces variance and enhances prediction accuracy. The ensemble’s output is determined by combining the predictions of all base models through averaging, which helps minimize prediction variability and mitigates the risk of overfitting.

The integration of ML techniques plays a crucial role in predicting the optical properties of the proposed PCF-SPR biosensor, such as N_eff, CL, and S_A. These ML models effectively learn the nonlinear mapping between key design parameters (e.g., n_a, t_g, pitch, air-hole diameters) and corresponding optical properties. This not only reduces computational time from minutes to milliseconds but also allows for rapid screening of multiple design combinations. Among the tested models, RFR and XGBR provided the excellent trade-off between prediction accuracy and execution time. Furthermore, ML approaches aid in precise finalization of sensor parameters based on maximum sensitivity, minimal CL, and high FOM.

2.7 ML models evaluation metrics

ML regression model performance evaluation needs to assess using different performance metrics [41]. Here the metrics description will be described in short to understand the implication and numerical calculation process. Smaller MSE values indicate a more accurate model, as they reflect smaller differences between predicted and actual outcomes. MSE evaluates how well a model predicts by quantifying the average squared gap between actual outcomes and predicted values. It reflects how much the predictions deviate from the true values, with lower MSE indicating better model accuracy. The metric is calculated by averaging the squares of all prediction errors across the dataset [41].

(7)

Here, stands for the model’s forecasted value, while y denotes the actual observed value.

A MAE reflects higher predictive accuracy, as it captures the average magnitude of deviation between forecasts and actual outcomes. MAE is computed by averaging the absolute errors across all predictions, providing a straightforward indicator of prediction quality [41]. The equation (8) presents the formula for MAE calculation.

(8)

In this context, N denotes the total number of data points and y_i represents the observed value, while represents the predicted value.

The R² value quantifies the efficacy of a model in predicting the dependent variable’s outcome. The range is from 0 to 1, with 0 indicating no explanation of the variance in the dependent variable and 1 indicating a complete explanation of the relationships. An R² value of 1 indicates that the model accounts for 100% of the variance in the target variable, while a score of 0.5 suggests that the model explains 50% of the variance [41].

We use the following formula to compute the R² value:

(9)

In this expression, the numerator ∑(yᵢ − ŷᵢ)² represents the total squared error between the predicted values ŷᵢ and the actual observations yᵢ. The denominator ∑(yᵢ − ȳ)² captures the total variation in the actual data around the mean value ȳ.

2.8 Explainable AI (XAI)

XAI that makes AI models easier to understand and interpret. It focuses on developing methods that make the decisions and processes of AI systems transparent, so users can make sense of how outcomes are reached. This is crucial in areas like healthcare, finance, and automotive industries, where the accuracy and reliability of AI predictions are critical [42–44]. SHAP is one of the most widely used methods for model interpretability. It provides both global explanations that capture the overall behavior of the model and local explanations that detail how individual features influence specific predictions. SHAP assigns each feature a value representing its contribution to the prediction. These values are consistent and additive, ensuring that the sum of all feature contributions equals the difference between the predicted value and the model’s average output, thereby offering a fair and transparent measure of feature importance [45].

To enhance the transparency and trustworthiness of the ML model used for predicting optical properties, SHAP was employed as an XAI technique. SHAP provides detailed insights into how each input parameter, such as air hole diameters, pitch, t_g, and n_a contributes to model outputs like N_eff, CL, and S_A. By visualizing SHAP values, we were able to identify the most influential design features, enabling more informed and efficient adjustments to the PCF-SPR biosensor structure. While SHAP does not directly enhance the sensor’s physical sensing performance, it plays a crucial role in interpreting the ML model’s predictions. This interpretability supports targeted design optimization, reduces the dependency on exhaustive simulations, and fosters the decision-making process for sensor performance optimization.

3.1 Analysis of simulation results

Fig 5 displays the coupling point where the surface SPP mode intersects with the core mode. The structure features air hole diameters of 0.6 μm for d1 and 1 μm for d2, with a t_g of 40 nm. We have used the COMSOL Multiphysics simulation to evaluate N_eff real and imaginary values along the y-axis that were used to calculate various propagation properties. Fig 5 displays the coupling point where the surface SPP mode intersects with the core mode under the conditions: n_a of 1.37, core diameters d1 = 0.6 μm and d2 = 1 μm, and a t_g = 40 nm. The figure demonstrates how loss increases with the decrease in wavelengths and then decreases as they increase. The figure also shows how N_eff varies with wavelength, indicating a decrease in N_eff as the wavelength increases. At a wavelength of 0.67 µm, the real parts of N_eff for the y-polarized core and SPP modes coincide, marking the phase matching condition. This point is significant as SPR occurs here. At this wavelength, CL reaches its maximum, and N_eff is 1.451. The graphs in Fig 6 illustrate how changes in d1 and d2 size influence CL variation with wavelength, where p = 2 µm, t_g = 40 nm, and n_a = 1.37. In Fig 6(a), where d2 is fixed at 1 µm, d1 increases gradually from 0.6 µm to 1.6 µm. With the increase of d1, CL also increases gradually. The lowest loss is observed when d1 is 0.6 µm, while larger values of d1 result in higher loss. Similarly, graph in Fig 6(b), where d1 is fixed at 1.2 µm and d2 changes from 0.6 µm to 1.2 µm. In this case, the highest CL is seen at d2 = 0.6 µm, and the loss goes down as d2 increases.

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Fig 5. N_eff core and SPP mode phase matching at peak CL.

https://doi.org/10.1371/journal.pone.0330944.g005

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Fig 6. Effect of d1and d2 variation on CL (a) wavelength vs. CL with different d1 for fixed d2 = 1µm (b) wavelength vs. CL with different d2 for fixed d1 = 1.2 µm.

https://doi.org/10.1371/journal.pone.0330944.g006

Furthermore, Fig 7 illustrates the relationship between S_A and wavelength for different air hole sizes. We kept the other parameters fixed at p = 2µm, t_g = 40 nm, and n_a = 1.37. From the graphs, it is clear that the variation in air hole dimensions influences the sensor’s response, which causes changes in S_A. In Fig 7(a), S_A is plotted for different d1 values, keeping d2 = 1 µm as a fixed value. From the graph, we get that as d1 increases from 0.6 µm to 1.2 µm, the S_A curve shifts to a higher value. The highest S_A is achieved at d1 = 1.2 µm, indicating that a larger air hole in this design enhances plasmonic interaction and improves sensitivity. Fig 7(b) presents the effect of d2 variation, keeping d1 = 1.2 µm as constant. The graph shows that as d2 increases from 0.6 µm to 1.2 µm, S_A also increases. This suggests that an increase in d2 makes the core-plasmon coupling stronger. Overall, the analysis demonstrates that optimizing d1 and d2 plays a crucial role in achieving high CL and S_A, which impacts the sensor’s detection capabilities.

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Fig 7. Effect of air hole diameter variation on S_A (a) S_A vs. wavelength with the changes of d1, (b) S_A vs. wavelength with the changes of d2.

https://doi.org/10.1371/journal.pone.0330944.g007

Additionally, from the experiment, we depict that the thickness of t_g has a significant effect on sensor performance. We varied t_g from 30nm to 60nm while keeping d1 = 0.6µm, d2 = 1µm, and p = 2µm constant. Fig 8(a) shows how CL changes with wavelength for different values of t_g. The graph in Fig 8(a) indicates that as the t_g increases, the CL decreases. For t_g= 30nm (blue curve), the highest CL is observed, with a peak value of 350 dB/cm 0.65µm. When the thickness of t_g increases to 40 nm (green curve), the peak CL shifts slightly toward higher wavelengths and reduces intensity. For t_g= 50nm (red curve), the CL continues to decline, and the peak becomes broader. At t_g= 60nm (cyan curve), the loss is at its lowest value, with a less pronounced peak, indicating a weaker plasmonic effect. This pattern suggests that a thinner gold layer enhances plasmonic resonance, leading to higher CL, while a thicker layer reduces it.

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Fig 8. (a) Wavelength vs. CL with variation of t_g (b) wavelength vs. S_A with different t_g.

https://doi.org/10.1371/journal.pone.0330944.g008

Again, Fig 8(b) presents the impact of different t_g on S_A across varying wavelengths. The graph indicates that as t_g increases, S_A tends to decrease. When t_g = 30 nm (blue curve), the most significant negative S_A is observed 0.66 µm. Increasing the thickness to 40 nm (green curve) results in a slight reduction in sensitivity and a shift in the peak toward a longer wavelength. With t_g = 50 nm (red curve), the sensitivity declines further, and the peak becomes broader. At t_g = 60 nm (cyan curve), the lowest sensitivity is recorded, with a less prominent peak, indicating weaker plasmonic interaction. These findings suggest that a thicker gold layer minimizes CL and improves sensitivity, while a thinner layer reduces the sensor’s overall performance.

Pitch size also plays a crucial role in determining the performance of the sensor. In this study, the pitch was varied from 2 µm to 3 µm while keeping n_a = 1.37, d1 = 0.6 µm, d2 = 1 µm, and t_g = 40 nm constant. Fig 9(a) illustrates how CL changes with wavelength for different pitch values. The graph in Fig 9(a) shows that as the pitch increases, CL decreases. At pitch, p = 2 µm (blue curve), the highest CL is observed at wavelength 0.66 µm with high intensity. When the pitch increases to 2.5 µm (green curve), the peak shifts slightly toward a longer wavelength, and the loss intensity reduces. With a pitch of 3 µm (red curve), CL continues to decline, and the peak becomes broader, indicating a weaker plasmonic effect. This suggests that a smaller pitch leads to stronger light confinement and higher loss, while a larger pitch allows better mode propagation and reduces CL. Again, Fig 9(b) examines the effect of pitch variation on S_A. A similar trend is observed, where a smaller pitch results in higher sensitivity, with a pitch of 2 µm showing the strongest negative S_A at wavelength 0.66 µm. As the pitch increases to 2.5 µm, the sensitivity decreases slightly, shifting toward a longer wavelength. When the pitch reaches 3 µm, the S_A further weakens, leading to a broader response. These findings indicate that pitch size must be optimized based on the sensor’s intended application. We should carefully adjust the pitch choice to balance sensitivity and stability in sensor design.

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Fig 9. (a) Wavelength vs. CL with variation of pitch (b) wavelength vs. S_A based on pitch variation.

https://doi.org/10.1371/journal.pone.0330944.g009

Fig 10 illustrates the variation in CL across wl for different analyte refractive indices, keeping d1 = 0.6 µm, d2 = 1 µm, p = 3 µm, and gold thickness at 60 nm constant. This analysis provides insights into how n_a influences the optical behaviour of the sensor. In Fig 10(a), CL rises with wavelength, displaying distinct peaks for n_a values between 1.31 and 1.41. As n_a increases, the resonance peak shifts to longer wavelengths, and the peak intensity gradually increasing. This shift reflects stronger plasmonic coupling at higher n_a values, resulting in greater CL. The trend underscores the strong dependence of CL on n_a, highlighting its importance in sensor performance. Fig 10(b) exhibits a similar pattern but extends the wavelength range up to 2.5 µm and includes an additional n_a value of 1.42. As n_a increases, the resonance peaks continue shifting toward longer wavelengths, with the highest CL observed for n_a = 1.42. The peaks become more prominent, and loss values rise significantly, especially at higher n_a values. These results emphasize the significant impact of n_a on CL, showing that increasing n_a shifts resonance to longer wavelengths while amplifying loss. This behaviour is critical for designing sensors with enhanced sensitivity and precision in detecting variations in n_a.

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Fig 10. CL vs. wavelength (a) n_a = 1.31-1.41, (b) n_a = 1.31-1.42.

https://doi.org/10.1371/journal.pone.0330944.g010

Again, Fig 11 displays how S_A varies with wavelength across different analyte refractive indices, using fixed parameters: d1 = 0.6 µm, d2 = 1 µm, p = 3 µm, and a gold thickness of 60 nm. The variations in S_A across different wl indicate the influence of n_a on plasmonic interactions. As n_a increases from 1.31 to 1.41, the sensitivity peaks shift towards longer wl. For n_a = 1.31–1.32 (blue curve), the first significant drop occurs around 0.65 µm, whereas for n_a = 1.40–1.41 (red curve), the minimum sensitivity shifts beyond 0.9 µm. The negative peaks indicate strong plasmonic coupling, highlighting the sensor’s capability to detect small changes in n_a. These variations suggest that the optimal sensing wavelength shifts as n_a increases, making precise wl selection crucial for achieving high sensitivity. Overall, the results emphasize the impact of n_a on S_A and the importance of choosing the appropriate wl for effective sensing. The shift in resonance peaks with increasing n_a highlights the need for wl tuning to maximize detection accuracy.

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Fig 11. S_A vs. wavelength for n_a = 1.31-1.41.

https://doi.org/10.1371/journal.pone.0330944.g011

The Table 2 presents different characteristics of the SPR biosensor across various parameter values of n_a, with constant d1 = 0.6 µm, d2 = 1 µm, p = 3 µm, and t_g = 60 nm. It presents CL, peak resonance wavelength (λ_peak), S_A, S_λ, resolution, and FOM corresponding to n_a values. As n_a increases, the resonance wavelength shifts to longer values. We observe that the peak loss found at 590 nm wavelength for n_a = 1.31, and it reaches 2210nm for n_a = 1.42. S_A remains low for smaller n_a values but increases significantly beyond n_a = 1.36. At n_a = 1.31, S_A is 10 (RIU^-1), but it jumps to 50 (RIU^-1) at n_a = 1.38 and reaches 1250 (RIU^-1) at n_a = 1.41, showing improved sensor response for higher na. S_λ also increases as n_a rises. It reaches a maximum of 125,000 nm/RIU at n_a = 1.41. The resolution improves with the increase of n_a, reaching 8.00 × 10⁻⁷ RIU at n_a = 1.41, which employs the sensor’s ability to detect small changes in n_a. Furthermore, FOM also rises steadily with the increase of n_a. It remains below 50 for lower n_a values but increases beyond n_a = 1.36 and highest at 1402.28 at n_a = 1.41. These results show that the sensor performs improves with higher n_a values, because CL, S_A, S_λ, and FOM reach their highest values.

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Table 2. Optical properties of the proposed SPR biosensor for different n_a.

https://doi.org/10.1371/journal.pone.0330944.t002

Table 3 presents the performance evaluation of the PCF-SPR biosensor based on various design parameters and n_a. The design parameters are small and large air hole diameters (d1, d2), pitch (p), t_g, and the optical properties, such as λ_peak, CL, S_λ, resolution, FOM, and S_A. With the increase of n_a from 1.31 to 1.41, the λ_peak shifts to higher values that vary from 0.57 µm to 0.95 µm. Maximum CL varies significantly, reaching its highest value of 445.37 dB/cm at n_a = 1.4. Similarly, S_λ increases and reaches a peak value of 105000 nm/RIU at n_a = 1.41. Resolution also improves as n_a rises, reaching 9.52 × 10 ⁻ ⁷ RIU at n_a= 1.41. Furthermore, the FOM shows a rapid increase and shows the highest value at 2112.15 RIU^-1 for n_a = 1.41, which confirms superior sensing capability. Again S_A also rises with increasing n_a, where the highest value is 95.28 RIU^-1 at n_a = 1.41. These results suggest that the sensor performs with optimized parameters such as d1 = 0.6 µm, d2 = 1.0 µm, p = 2.5 µm, and t_g = 40 nm, contributing to maximum sensitivity and resolution. From this table, we get the results and insights with parameter variation on biosensor efficiency that help to select optimal design configurations for the best performing sensor.

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Table 3. Optical properties for different n_a with best performing design parameters.

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Table 4 presents the maximum absolute S_A values for different n_a in a PCF-SPR biosensor. The listed parameters include the structural dimensions: d1, d2, p, t_g along with the operating wavelength (wl). The S_A (RIU^-1) values represent the sensor’s ability to detect changes in the surrounding RI, making them crucial for evaluating the sensor’s performance. As n_a increases, S_A values generally increase in magnitude, indicating enhanced sensitivity. The highest negative S_A value of −1422.34 RIU^-1 is observed at n_a = 1.39, highlighting the sensor’s peak detection capability at this n_a.

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Table 4. Maximum absolute S_A values for different n_a.

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3.2 ML-based performance analysis

Using ML, we have used the simulation-based dataset to explore other design combinations and to explore better-performing designs. The dataset “Dataset 1” contains 5252 samples with eight features, where d1, d2, t_g, n_a, and wl serve as inputs, while N_eff, CL, and S_A are designated as outputs. We have created two subsets, “Dataset 2” and “Dataset 3”, from the initial primary “Dataset 1”. “Dataset 2” contains the combined samples for N_eff and CL, while “Dataset 3” includes samples for S_A. Using the train-test split method using the Scikit-learn library, we partitioned the dataset into training and testing sets with a 75% and 25% ratio. After preprocessing and splitting the data, each model was trained on the training set, and its performance was evaluated on the testing set. We implemented various ML algorithms employing ensemble techniques and subsequently applied XAI methods to these models to improve interpretability and transparency.

The polynomial equation y = 40.15x² − 105.93x + 70 shown in Fig 12 describes how the resonance wavelength shifts as n_a changes. In this equation, x represents n_a, and y is the resonance wavelength in micrometers. The quadratic term (40.15x²) means that the wavelength shift increases more rapidly as n_a increases. The linear term (−105.93x) shows the direct impact of n_a on resonance wavelength. The negative coefficient −105.93 means that for smaller values of n_a the resonance wavelength (µm) initially decreases. However, as n_a increases, the positive quadratic term 40.15x² starts to dominate linear term (−105.93x), causing the resonance wavelength to rise sharply. The constant term (70.45) is the estimated resonance wavelength when n_a is zero, though this value is theoretical and does not hold physical significance. The equation follows a curved trend rather than a straight-line change, meaning that at higher n_a values, small shifts cause more noticeable wavelength changes. This pattern is useful in biosensor applications, where detecting small n_a variations is important with high sensitivity and accuracy.

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Fig 12. Polynomial regression analysis of resonance wavelength with the change of n_a.

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Fig 13 compares the actual and predicted N_eff values using the RFR model. The actual values are marked in purple, while the predicted values are in orange. The graph shows that the RFR model predicts N_eff accurately, with minor deviations in some areas. Additionally, Fig 14 presents the validation of different ML models for predicting N_eff. We have plotted together the actual data alongside the predicted data using the DTR, RFR, GBR, XGBR, and BR models. The close match between actual and predicted values implies that all models perform well, showing minimal variations. The validation confirms that these models generalize effectively, maintaining consistency across different wavelength values.

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Fig 13. Actual vs. predicted N_eff using RFR model.

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Fig 14. Validation of N_eff predictions across different ML models.

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Table 5 provides performance metrics such as R², MAE, and MSE for both training and test datasets for different ML models. Here, DTR achieves the highest train R² (0.999969) but has slightly lower test accuracy. RFR, BR, and XGBR show strong generalization, with high R² values and low errors. KNN records the lowest test R² (0.992828) and the highest test MAE. (0.007912), indicating lower accuracy. GBR performs well, achieving a high test R² (0.997951) and the lowest test MSE (0.000053). The validation results suggest that ensemble models such as RFR, GBR, and XGBR provide the most reliable predictions for N_eff with minimal error. In terms of execution time, the DTR is the fastest, with the lowest training time (0.020 sec) and test time (0.000099 sec). However, it is less accurate compared to ensemble models. While XGBR offers better efficiency, with a lower training time (0.238 sec) compared to RFR (0.618 sec), RFR remains a strong performer due to its consistent performance. Although GBR is slightly more accurate, it requires more computational resources. Overall, XGBR and RFR emerge as the best performers, offering strong predictive accuracy with a balance of execution efficiency.

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Table 5. Comparative performance metrics for N_eff across different ML models.

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Furthermore, we have calculated the SHAP values using the RFR model, to assess feature importance and interpretability in the analysis. SHAP values quantify how much each feature of the input data influences the model’s prediction, relative to the average prediction. Positive SHAP values indicate an increase in the target value N_eff, while negative values mean a decrease. It also helps to understand how much each input attribute contributes to N_eff’s final prediction. Fig 15 illustrates this by combining feature importance with feature effects. Each point on the summary plot represents a Shapley value for a feature and an instance, with the y-axis indicating the feature and the x-axis the Shapley value. The color gradient from blue to red represents the feature’s value from low to high. Notably, wl has a significant impact on N_eff, with higher values (in red) decreasing and lower values (in blue) increasing. This also shows that wl has a significant negative impact on N_eff, underscoring its crucial role in determining N_eff, while other variables have a less significant influence on N_eff value prediction.

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Fig 15. SHAP summary plot for N_eff prediction.

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Again, the SHAP waterfall plot in Fig 16 shows how different input features contribute to predicting N_eff, starting from a base value E[f(x)] = 1.453 and going to a final predicted value f(x) = 1.454. The plot depicts that each input feature either increases or decreases N_eff, impacting the final predicted value. The blue bars represent negative contributions, while the red bars indicate positive contributions. It reveals that wavelength (wl = 0.63 µm) has the most significant impact, which increases N_eff by 0.00187. Pitch (p = 2 µm) decreases N_eff by 0.00064, while d1 (1.2 µm) slightly decreases it by 0.00012. The influence of d2 (1 µm) is a small negative contribution of 0.00008. The n_a = 1.35 has a minor positive effect of 0.00003, whereas the t_g = 0.04 µm has no noticeable impact. This SHAP plot provides insight into how input parameters influence the predicted N_eff value. We observe that wl is the most dominant factor, followed by p, while other features have a much smaller effect on the final prediction.

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Fig 16. SHAP waterfall plot for N_eff prediction.

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3.2.1 Confinement loss (CL).

The graph in Fig 17 shows how the predicted CL values compare to the actual ones using the RFR model. The actual data points, represented in purple, are plotted alongside the predicted values in orange. The pattern suggests that the model accurately follows the trend of CL, although slight differences are noticeable in higher ranges. Overall, the predictions align well with actual values across different wavelengths, demonstrating the model’s effectiveness. A further evaluation of ML models in predicting CL is displayed in Fig 18. The actual data is plotted alongside predicted values using DTR, RFR, GBR, XGBR, and BR models. The results indicate that most of the models match the actual data closely, showing minimal variation, which confirms their reliability for generalization.

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Fig 17. Actual vs. predicted CL using RFR model.

https://doi.org/10.1371/journal.pone.0330944.g017

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Fig 18. Validation of different ML models for CL prediction against actual data.

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Additionally, to analyze performance, Table 6 provides a comparison of performance metrics such as R², MAE, and MSE for both training and test datasets. Where DTR has a high train R² of 0.999444, its lower test R² of 0.927159 suggests reduced accuracy when handling new data. RFR and BR exhibit strong predictive capabilities, with test R² values of 0.96148 and 0.961493, respectively, along with lower error rates. XGBR also performs well, achieving a test R² of 0.941124 and a test MSE of 0.000935. KNN and GBR show slightly lower test R² values (0.909516 and 0.912935) and higher MAE, indicating relatively lower accuracy compared to ensemble models. For CL prediction, DTR is the fastest, with a training time of 0.021 seconds and a test time of 0.0003 seconds, but it has lower test accuracy. RFR and BR achieve the highest test R² but require longer training times (0.627 and 0.655 seconds). XGBR provides a strong balance, with a high test R², a low test MSE, and moderate training time (0.261 seconds). KNN and GBR have the lowest test R² and higher errors, making them less suitable. Among all, RFR and BR demonstrate the most reliable CL predictions with minimal error.

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Table 6. Performance metrics of ML models for CL prediction.

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Like N_eff analysis, we also used SHAP to explore the impact of different input parameters on CL. SHAP values were computed using the RFR model that is one of the most effective models in this study. The summary plot in Fig 19 presents the distribution of SHAP values for various features for CL prediction. Each point represents an individual prediction instance, with the y-axis displaying the features and the x-axis showing their corresponding SHAP values. The color gradient from blue to red reflects the feature values, where blue denotes lower values and red represents higher ones. Among all parameters, wl emerges as the most influential, showing both high and low values leading to significant variations in the model’s output. The n_a and t_g also play a crucial role, generally increasing CL with higher values. Other parameters, such as p, d1, and the large air hole d2, have a smaller but still noticeable effect. The variations in SHAP values suggest complex feature interactions, which are essential for refining the model’s interpretability. A SHAP waterfall plot in Fig 20 shows the change from the baseline E[f(X)] =47.157 to the final prediction f(x)= 77.28. This shows how these features have an effect. The n_a parameter makes the most substantial contribution, increasing CL by 45.61283 units. In contrast, wl reduces the prediction by −35.02253 units, while t_g contributes an additional 21.55977 units. Similarly, d2 leads to a decrease of 2.51656 units, whereas p has a minimal positive and d2 negative impact. These changes demonstrate how different input variables influence CL, with wavelength standing out as the dominant factor.

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Fig 19. SHAP summary plot for CL prediction.

https://doi.org/10.1371/journal.pone.0330944.g019

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Fig 20. SHAP waterfall plot for CL prediction.

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3.2.2 Amplitude sensitivity (S_A).

In Fig 21, actual and predicted S_A values are compared, where actual values are shown in purple and predicted values are marked in orange. The results indicate that the model has excellent predictions, but some deviations appear, particularly at lower sensitivity values. Fig 22 presents a detailed evaluation to predict S_A using ML models. The black line represents the actual data, while various colors display the predictions from DTR, RFR, XGBR, GBR, and BR. Most models follow the expected trend, but RFR and BR exhibit the closest match to actual values, indicating their strong predictive accuracy.

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Fig 21. Actual vs. predicted result of S_A using RFR model.

https://doi.org/10.1371/journal.pone.0330944.g021

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Fig 22. Validation of S_A predictions from various ML models against experimental data.

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Again, Table 7 presents key performance metrics such as R², MAE, and MSE for both training and test datasets for predicting S_A. Where, DTR shows an R² of 1.0 in training but drops to 0.788216 in testing, indicating overfitting. Its test MAE (0.010117) and MSE (0.001413) are relatively high, suggesting weaker generalization. DTR is the fastest, with a training time of 0.018 seconds and a test time of 0.0005 seconds, but has a lower test R² (0.788216). RFR and BR achieve the highest test R² (0.868116 and 0.867602) with a test MAE around 0.00924 but require longer training times (0.474 and 0.470 seconds). XGBR provides a strong balance, with a test R² of 0.813183, low test MSE (0.001356), and moderate training time (0.218 seconds). KNN and GBR perform weaker, with KNN showing the lowest test R² (0.50024) and highest test MAE (0.023273). Overall, RFR and BR offer the best accuracy, their balance of high accuracy and low error makes them well-suited for S_A prediction.

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Table 7. S_A Prediction performance metrics for different ML models.

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The SHAP summary plot in Fig 23 makes it very clear that the feature wl has the biggest effect on S_A. This highlights how important it is for optimal sensor model performance. Additionally, the feature also demonstrates a substantial impact, underscoring its importance in predicting S_A. The SHAP summary plot also depicts that t_g has a moderate effect, indicating its notable impact on predictions. In contrast, d1 and d2 show minimal impacts, suggesting that these parameters have less influence on S_A compared to wl, n_a, and t_g. Again, Fig 24 illustrates a SHAP waterfall model with a specific s_ample. It shows how various features contribute to the model’s final prediction, starting with a baseline value of E [f(X)] = −32.067 and reaching an output of f(x) = −36.866. Here, the feature wl has the most significant negative impact, decreases the prediction by −9.11925 units. The n_a feature also adds a substantial 2.01146 units to the prediction. In contrast, the feature t_g increases the prediction by 1.09693 units. The d1 and d2 features contribute small negative adjustments. This demonstrates the sensitivity of the model’s output to changes in each feature, with wl and n_a being the most influential in driving the prediction upward.

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Fig 23. SHAP summary plot of S_A prediction.

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Fig 24. SHAP waterfall plot of S_A prediction.

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3.3 Comparison with previous works

Table 8 compares the performance of our proposed PCF-SPR biosensor with recent works, highlighting significant improvements in optical properties. The proposed design outperforms previous models in terms of CL, S_A, S_λ, resolution, and FOM, demonstrating superior sensing capabilities. In terms of S_λ, the proposed biosensor achieves 125,000nm/RIU, which is higher than all the works. The S_A of −1422.34 RIU ⁻ ¹ remains competitive, close to 7220 RIU ⁻ ¹ [46] and −1971.30 RIU ⁻ ¹ [47], while excelling in other performance areas. Additionally, the proposed biosensor achieves an exceptionally low resolution of 8 × 10 ⁻ ⁷ RIU, which is superior to previous models [20,21,47–52]. Among all the properties, the FOM of 2112.15 RIU ⁻ ¹ is significantly higher than those reported in other studies [17,22,49,50,52] indicating a better balance between sensitivity and resolution. In our study, we employed six ML models (DTR, RFR, KNN, GBR, XGBR, BR), along with XAI techniques, specifically SHAP, to gain insights into the model’s decision-making process. The XGBR model demonstrated excellent performance with an R² value of 0.9976 for the prediction of N_eff, outperforming other models in the literature. The use of XAI methods like SHAP made the results more interpretable, providing valuable information about how different design parameters impact the sensor’s performance. This transparency not only enhanced our understanding of the system but also allowed for more informed decisions during the optimization process. The ML models showed lower error rates and higher accuracy compared to previous works, achieving an MAE of 0.00221 and an MSE of 0.00059, which are significantly lower than most of the other studies. This demonstrates the robustness of the ML models in predicting sensor performance. Additionally, integrating XAI with ML optimization significantly reduced computational time and cost, making it more efficient than conventional methods for exploring optimal design combinations. Overall, the proposed PCF-SPR biosensor, with its combination of ML and XAI techniques, shows substantial improvements in S_A, S_λ, resolution, and FOM compared to previous designs. This makes it a promising candidate for high-precision applications in various sensing domains, while the inclusion of XAI offers valuable insights into the underlying sensor behavior.

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Table 8. Comparison with previous work.

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Furthermore, compared to traditional PCF-SPR sensor optimization approaches that rely solely on parametric sweeps using FEM, our hybrid methodology incorporating ML (e.g., RFR, XGB) and SHAP-based XAI significantly enhances computational efficiency and interpretability. FEM alone often demands substantial time and computing power for iterative tuning of design parameters. In contrast, our work reduces the time for predicting key optical parameters such as N_eff, CL, and S_A from minutes to milliseconds, enabling rapid exploration of a broader design space. Recent sensor architectures have demonstrated high performance using complex geometries. Such as IMD-EMD merged PCF [53] and quasi D-shaped waveguides with external metal deposition [54], offering high sensitivities for detecting heavy metals, oils [55], and cancer biomarkers [56]. However, these structures typically require higher fabrication and simulation costs. Our proposed sensor, in comparison, achieves high performance metrics (S_λ = 125,000 nm/RIU and FOM = 2112.15) using a simpler geometry with the help of an intelligent data-driven approach. By combining ML and XAI with efficient sensor design, this framework supports rapid development of adaptive, intelligent biosensing platforms for diverse real-world applications.