2.1 Geometry structure design

Fig 1 illustrates (a) the cross-sectional design and (b) the finite element mesh analysis of the proposed circular PCF SPR sensor engineered for machine learning-enhanced detection. The architecture features a 50 nm gold layer (depicted in blue) as the plasmonic component, strategically embedded with two concentric rings of air holes (yellow) possessing a uniform radius of 0.5 µm; the inner ring exhibits a lattice pitch of = 3.26 µm, while the outer ring has a tighter pitch of = 2.12 µm to create controlled mode asymmetry and amplify plasmonic field gradients. Externally, analyte microchannels (white) with a 1.0 µm radius and lattice pitch = 3.15 µm surround the gold layer, maximizing interfacial analyte interactions to generate complex, high-dimensional SPR spectral signatures ideal for ML-driven analysis. The silica-clad structure generates multi-parameter optical responses (resonance shift, broadening, and intensity modulation) that serve as rich input features for predictive algorithms.

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Fig 1. Illustration of (a) the cross-sectional design and (b) the finite element mesh analysis of the proposed circular PCF SPR sensor.

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The 50 nm gold coating thickness was systematically optimized through parametric studies evaluating plasmonic performance across various metal layer thicknesses (30–70 nm). This specific thickness demonstrated optimal trade-offs between evanescent field enhancement and plasmonic damping: thinner layers (<40 nm) exhibited insufficient plasmonic coupling due to incomplete metal coverage, while thicker layers (>60 nm) caused excessive damping of surface plasmon polaritons, reducing sensitivity. The 50 nm Au layer maximizes the overlap between the evanescent field and analyte while maintaining strong resonance conditions, as quantified through electromagnetic field simulations showing peak electric field intensity at the gold-analyte interface. Although silver and aluminum were briefly considered for their superior plasmonic properties in specific wavelength regimes, gold was ultimately selected for its exceptional chemical stability, biocompatibility, and resistance to oxidation in biological environments—critical for practical biosensing applications. The pure-gold approach also simplifies fabrication compared to multi-material configurations while achieving record sensitivity in the biological RI range.

Finite element method (FEM) simulations employing 17,434 elements (min element quality: 0.468) rigorously map electromagnetic field distributions across critical gold-analyte boundaries. These high-fidelity simulations generate synthetic datasets capturing noise, dispersion artifacts, and multi-analyte interference scenarios – essential for training robust ML models to interpret real-world sensor data. The asymmetric pitch configuration ( < < ) deliberately intensifies evanescent field-analyte overlap, producing non-linear optical responses that conventional peak-detection algorithms fail to deconvolve.

The specific role of this asymmetric, dual-ring design is twofold in enhancing the SPR effect and overall sensor sensitivity. First, the tighter pitch of the outer ring () compared to the inner ring () creates a controlled geometric asymmetry. This breaks the circular symmetry of the fiber and forcefully pushes the core-guided mode towards the gold-analyte interface, significantly strengthening the evanescent field that protrudes into the analyte microchannels. A stronger evanescent field maximizes the overlap integral between the probing light and the target biomolecules, directly amplifying the sensor’s response to refractive index changes.

Second, the precise dimensions of the concentric rings are optimized to tailor the effective refractive index () of the core mode. This engineering ensures optimal phase-matching with the surface plasmon polariton (SPP) mode on the gold film within the biological refractive index range of 1.33–1.38. Efficient phase-matching results in a strong, sharp, and well-defined resonance dip in the transmission spectrum. Consequently, this co-designed geometry directly enables the record sensitivity of 31,846 nm/RIU by simultaneously amplifying the evanescent field strength and fine-tuning the plasmonic coupling condition.

Machine learning methods, such as convolutional neural networks (CNNs) and support vector regression (SVR), leverage complex spectral patterns to enhance biosensing performance in multiple ways. These algorithms denoise signals, effectively distinguishing true binding events from environmental drift. By employing pattern recognition in high-dimensional feature spaces, they enable analyte concentration quantification beyond classical detection limits. Additionally, they classify biomolecules in multiplexed environments by learning subtle resonance fingerprints unique to each target. Furthermore, machine learning models predict binding kinetics in real-time by correlating temporal resonance evolution with established kinetic models, providing dynamic insights into molecular interactions. This computational approach significantly improves sensitivity, specificity, and real-time monitoring capabilities in plasmonic biosensing.

The dual air-hole layers enable tunable resonance splitting that ML algorithms exploit for multi-analyte resolution, while the enlarged analyte channels ensure rapid diffusion kinetics – critical for generating time-resolved training data. This co-design of nanophotonic hardware and computational intelligence establishes a pipeline where FEM-validated sensor data directly trains ML models, creating a closed-loop system for adaptive, high-throughput biosensing. The synergy between engineered light-matter interactions (gold nanostructure) and data-driven inference transforms conventional SPR into an intelligent sensing platform.

2.2 COMSOL simulation setup and data generation

The 1560 synthetic data points were generated through systematic parameter sweeps in COMSOL Multiphysics v5.6 using the Wave Optics Module. The simulation framework employed a multi-step approach to ensure comprehensive coverage of the sensor’s operational characteristics across the biological refractive index range (1.33–1.38).

Meshing strategy: The finite element mesh was carefully optimized to balance computational efficiency and accuracy. A physics-controlled mesh was implemented with the following specifications:

• Maximum element size: in the gold layer and analyte channels to resolve plasmonic effects
• Minimum element size: 2 nm at critical gold-analyte interfaces
• Curvature factor: 0.3 to accurately capture circular geometries
• Growth rate: 1.35 for smooth element transitions

The final mesh consisted of 17,434 triangular elements with a minimum element quality of 0.468, ensuring numerical stability while capturing fine electromagnetic field variations.

Boundary conditions:

• Outer Boundaries: Perfectly Matched Layers (PMLs) with scattering boundary conditions to absorb outgoing radiation and prevent unphysical reflections
• Gold-Silica Interface: Continuity boundary condition enforcing field continuity
• Gold-Analyte Interface: Surface current boundary condition with impedance matching for plasmon excitation
• Air Hole Boundaries: Perfect Electric Conductor (PEC) conditions to model perfectly reflective interfaces

Data Generation Protocol: The dataset was constructed through parameter sweeps across:

• 6 discrete refractive indices: 1.33, 1.34, 1.35, 1.36, 1.37, 1.38
• 260 wavelength points per RI (1.2–1.8 μm range)
• Multiple polarization states (X and Y polarizations)

For each simulation point, COMSOL solved the frequency-domain Maxwell equations using the finite element method, extracting confinement loss, effective permittivity components, and resonance characteristics. The wavelength sensitivity was subsequently computed using Eq (4) by analyzing resonance shifts between adjacent RI values.

Convergence validation: Mesh independence was verified through progressive refinement tests, where successive mesh density increases of 20% produced less than 0.5% variation in key output parameters (confinement loss and effective index). The chosen mesh configuration provided optimal trade-off between computational cost and accuracy for machine learning training.

2.3 Experimental setup

Fig 2 presents the comprehensive experimental framework for validating the gold-coated SPR sensor‘s performance through spectral interrogation. A broadband supercontinuum laser source (spectral range: ) generates incident light, which undergoes TM-polarization optimization via an in-line polarization controller to maximize plasmonic excitation efficiency. This polarized light is delivered to the sensor through a single-mode input fiber (SMF1). The core sensing element consists of a microfabricated chip with a 50-nm gold-coated plasmonic surface mounted within a thermally stabilized () flow cell. Microfluidic channels (100 µm width) enable precise infusion of analytes across the refractive index range (1.33–1.38), ensuring controlled interaction with the gold film. Transmitted light exits the sensor via output fiber (SMF2) into a high-resolution optical spectrum analyzer (OSA; e.g., Yokogawa AQ6370D, 0.01 nm resolution) that captures wavelength-resolved transmission spectra at 5 Hz sampling rates. Real-time data acquisition is managed through a computer running customized LabVIEW algorithms, which detect resonance wavelength shifts () by identifying transmission minima and compute confinement loss using referenced optical power measurements. The system employs certified refractive index calibration fluids (Cargille Labs standards) to establish baseline responses, with automated fluid switching enabling rapid RI variation tests. Critical performance parameters include nm wavelength repeatability and 0.5 dB/m loss resolution, achieved through environmental vibration isolation and dark-current compensation. This integrated setup encompassing controlled light injection, precision fluidics, nano-scale plasmonic interaction, high-sensitivity spectral detection, and automated analytics provides a robust platform for correlating simulated sensor behavior with empirical results, essential for biosensing applications requiring detection of minute biochemical concentration changes.

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Fig 2. Experimental spectral interrogation setup for gold-coated SPR sensor validation.

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2.4 Numerical analysis

Fused silica is a highly transparent amorphous form of silicon dioxide, extensively used in PCF platforms for SPR sensors due to its low optical loss and exceptional thermal stability. Its wavelength-dependent refractive index (n) is precisely modeled by the Sellmeier equation, an empirical dispersion formula expressed as [27]:

(1)

where λ is the operating wavelength (e.g., in μm), and , are experimentally derived coefficients specific to fused silica. By inputting target wavelengths into this equation, we can calculate the exact RI of fused silica across ultraviolet to infrared regimes, enabling precise phase-matching design between core-guided modes and surface plasmons in PCF-SPR sensors for tailored analyte detection.

We use the Lorentz–Drude framework to develop our structure model for evaluating the complex permittivity of gold (Au), a key plasmonic material in PCF-SPR sensors. This model integrates a Drude term (describing free-electron contributions) with a Lorentz oscillator term (capturing bound-electron interbond transitions), expressed as [28]:

(2)

Here, is the high-frequency permittivity, and denote the Drude plasma frequency and damping rate, while , , and represent the Lorentz term’s oscillator strength, resonant frequency, and damping coefficient. This formulation accurately characterizes Au’s dispersion and loss across optical–infrared wavelengths, enabling precise optimization of plasmonic interactions in our sensor design.

Confinement loss is a critical performance metric in PCF-SPR sensors, arising from evanescent field coupling between the core-guided mode and lossy surface plasmon modes at the metal-dielectric interface. This loss mechanism is deliberately engineered to enable SPR sensing but must be optimized to balance sensitivity and signal integrity. In PCFs, geometric imperfections (e.g., air-hole collapse) or phase-matching with plasmonic modes can exacerbate radiative leakage, causing power attenuation along the fiber. The confinement loss formula [27]:

(3)

quantifies this attenuation, where λ (μm) is the wavelength, is the imaginary part of the complex effective refractive index, 8.686 converts Nepers to dB, and scales the result to dB/cm. Here, directly reflects plasmonic interaction strength — higher values indicate stronger coupling but greater loss. Minimizing α while maximizing sensitivity requires precise tuning of PCF parameters (e.g., air-hole pitch, metal layer thickness) to achieve optimal light–plasmon overlap for target analytes.

Wavelength sensitivity () is the pivotal figure of merit for evaluating the detection capability of gold-coated PCF-SPR sensors. It quantifies the sensor’s ability to translate minute changes in the analyte’s refractive index () into a measurable spectral shift () in the resonance wavelength. This parameter directly governs the sensor’s resolution in identifying ultralow concentrations of biomarkers, pathogens, or gases, making it indispensable for applications in precision medicine, environmental monitoring, and real-time biochemical detection. The wavelength sensitivity is defined by the equation [27]:

(4)

where, is the resonance wavelength shift (in nanometers), and is the change in analyte refractive index (in RIU, refractive index units).

Resolution defines the smallest measurable change in analyte refractive index () that a sensor can reliably detect, establishing the fundamental limit of detection for biochemical species. In gold-coated PCF-SPR sensors, this parameter depends critically on the sharpness of the resonance dip and the precision of spectral shift measurements. The resolution R is formally expressed as [29]:

(5)

where is the resonance wavelength shift for a known , and is the smallest spectrally resolvable shift (dictated by instrument noise or environmental fluctuations). Achieving resolutions RIU enables detection of trace biomarkers (e.g., cancer antigens at femtomolar concentrations), with lower values signifying superior sensor performance for clinical diagnostics and environmental monitoring.

Birefringence (Bi) is a critical property in polarization-maintaining PCF SPR sensors, quantifying the refractive index difference between orthogonal polarization modes (x and y) of the guided light [28]:

(6)

where and are the effective indices of the fundamental modes along the slow and fast axes. In gold-coated PCF-SPR sensors, intentional birefringence is engineered through asymmetric air-hole geometries (e.g., elliptical holes, varied diameters) or stress-applying elements. This asymmetry enhances SPR excitation by aligning the polarized core mode with the plasmonic layer, boosting sensitivity to analyte-induced refractive index changes. High birefringence () also stabilizes the resonance peak against environmental noise and fabrication tolerances. However, excessive may exacerbate confinement loss or complicate phase-matching.

The coupling length defines the propagation distance required for complete power transfer between orthogonal polarization modes (x and y) in a birefringent photonic crystal fiber, calculated as [29]:

(7)

where is the operating wavelength and is the birefringence. In gold-coated PCF-SPR sensors, this parameter critically governs polarization-selective plasmonic coupling shorter accelerates energy transfer from the core mode to surface plasmons, enhancing sensitivity. However, excessively short may exacerbate loss or limit interaction length. The power spectrum Po quantifies the polarization-dependent energy transfer between orthogonal modes (x and y) in gold-coated PCF-SPR sensors, governed by the equation [29]:

(8)

where is the birefringence, is the operating wavelength, and is the propagation distance.

The transmittance spectrum (T) quantifies the wavelength-dependent power transmission through a gold-coated PCF-SPR sensor, defined as the logarithmic ratio of output power to input power in decibels (dB) [28],

(9)

Here is output power and is input power in decibels (dB).

This critical metric reveals the SPR dip – a sharp attenuation feature where core-guided light couples to lossy plasmonic modes at the gold-analyte interface. The depth and width of this resonance dip directly correlate with sensor sensitivity and resolution, as analyte-induced refractive index changes shift both the position and magnitude of the dip.

Mean Squared Error (MSE) is a fundamental statistical measure used to quantify the accuracy of predictive models in machine learning and data analysis. It evaluates the quality of a model’s predictions by calculating the average squared difference between observed actual values and model-predicted values across n data points. This metric emphasizes larger errors due to the squaring operation, making it particularly sensitive to outliers and significant prediction deviations. The MSE is formally defined as [30]:

(10)

In PCF-SPR sensor development, MSE objectively quantifies discrepancies between simulated and experimental resonance spectra (e.g., transmittance dips), ensuring model reliability before fabrication.

Root Mean Squared Error (RMSE) is a widely used evaluation metric in regression tasks, measuring the standard deviation of prediction residuals (errors). It quantifies how closely a model’s predicted values align with actual observed values , providing an interpretable error scale in the original units of the target variable. The RMSE formula is defined as [30]:

(11)

where, n is the number of data points, is the actual observed value for the i-th data point and is predicted value for the i-th data point.

In PCF-SPR sensor development, RMSE evaluates regression models predicting resonance shifts () or analyte concentrations, ensuring robustness before experimental deployment. Model performance is comprehensively evaluated using RMSE in conjunction with R-squared , quantifying both error magnitude and goodness-of-fit.

Mean Absolute Error (MAE) quantifies the average magnitude of prediction errors in a regression model, computed as [30]:

(12)

where = actual value, = predicted value, and n = number of observations. MAE measures the absolute difference between predicted and true values across all samples, providing a robust, scale-dependent assessment of model accuracy. For SPR sensor applications (e.g., predicting sensitivity or confinement loss), a lower MAE indicates higher precision critical for evaluating machine learning models like SVR or RFR. Unlike RMSE, MAE does not penalize large errors disproportionately, making it ideal for datasets with noise or outliers.

2.5 Machine learning approach

2.5.4 Feature normalization.

To ensure uniform scaling and prevent feature dominance, input variables were normalized using the MinMaxScaler technique. This preprocessing step linearly transforms each feature to a [0, 1] range using the formula [31]:

(13)

X represents the original value of a specific feature (such as analyte refractive index or confinement loss) for an individual data sample, while and denote the minimum and maximum values observed for that particular feature across the entire training dataset. These extrema are calculated exclusively from the training subset during the scaler’s fitting phase. The scaled features retain original distribution properties but enable equitable parameter weighting during ML training, ensuring models focus on intrinsic relationships rather than magnitude disparities.

2.5.5 Proposed machine learning architecture.

Fig 3 illustrates the standard workflow for developing and evaluating machine learning models. The process begins with the design model phase, where the model‘s structure and type are defined according to the problem requirements. This stage involves selecting suitable algorithms, configuring model architectures, and identifying relevant input features. For predicting gold-coated SPR sensor performance, three regression models like MLR, SVR, and RFR were implemented for their complementary strengths in efficiency and effectiveness.

• Multiple Linear Regression (MLR): MLR models the linear relationship between a dependent variable and multiple independent variables. The equation is expressed as [30]:

(14)

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Fig 3. The standard workflow for developing and evaluating machine learning models.

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where Y is the target variable (e.g., wavelength sensitivity), are input features (analyte RI, confinement loss, etc.), is the intercept, are regression coefficients quantifying each feature‘s contribution, and ϵ represents error. Key parameters include the coefficients (optimized via ordinary least squares to minimize residual sum of squares) and the coefficient of determination evaluating goodness-of-fit. In SPR sensor applications, MLR identifies linear dependencies between sensor parameters (e.g., how analyte RI linearly influences sensitivity) but assumes no multicollinearity or nonlinear interactions.

• Support Vector Regression (SVR): SVR adapts support vector machines to regression by mapping input features into a high-dimensional space using kernel functions (e.g., radial basis function, RBF) to capture nonlinear relationships. The objective function minimizes [32]:

(15)

where w is the weight vector, b is the bias term, is the regularization parameter controlling the trade-off between model complexity and slack variable penalty, ϵ defines the width of the error-insensitive tube, and are slack variables allowing deviations beyond ϵ. The kernel function with parameter γ governs the feature space transformation. For SPR sensors, SVR excels at modeling complex plasmonic responses (e.g., resonance wavelength shifts due to analyte refractive index variations) by balancing model complexity and prediction error tolerance.

• Random Forest Regression (RFR): RFR is an ensemble method constructing numerous decision trees during training and outputting their mean prediction. Each tree is built on a bootstrapped sample of the dataset, with node splits determined by a random subset of features. The prediction for input x is [33]:

(16)

where B is the number of trees and Tb denotes an individual tree. Critical hyperparameters include (number of trees), (tree depth), and (minimum samples per node). RFR estimates feature importance by evaluating mean decrease in impurity (Gini index or MSE). For SPR sensor data, RFR robustly handles high-dimensional interactions (e.g., between analyte RI, wavelength, and loss) and quantifies parameter significance (e.g., identifying dominant drivers of sensitivity), reducing overfitting through aggregation.

These three machine learning models synergistically enable comprehensive, data-driven optimization of gold-coated SPR sensors by addressing distinct aspects of the sensor’s photonic behavior. MLR establishes foundational linear relationships, providing physically interpretable coefficients that quantify how core parameters (e.g., analyte refractive index) directly influence performance metrics like wavelength sensitivity. This baseline insight is critical for preliminary sensor design. SVR extends this capability by capturing intricate nonlinearities inherent in plasmonic responses such as resonance peak broadening at high refractive indices or wavelength-dependent loss thresholds using kernel-driven transformations that generalize beyond linear constraints. Complementing both, RFR delivers high-accuracy predictions through ensemble-based learning while inherently ranking feature importance (e.g., revealing that confinement loss dominates sensitivity variance under specific wavelength ranges), thereby guiding targeted design refinements. Together, this multi-model framework rigorously validates sensor performance predictions against COMSOL Multiphysics simulations through statistical metrics (, RMSE), creating a robust feedback loop between computational photonics and machine intelligence. By bridging ML-predicted outcomes with simulated photonic phenomena, the approach accelerates the development of high-sensitivity SPR sensors, reducing reliance on costly iterative simulations and enabling precision optimization of gold film thickness, grating geometry, and operational wavelengths for real-world analyte detection.

3.1 Simulation results

Fig 4 illustrates the electric field distributions of guided modes in the gold-coated SPR sensor. Subfigures (a) and (b) depict the core mode for X- and Y-polarizations, respectively, showing symmetric light confinement within the fiber core. Subfigures (c) to (h) visualize higher-order surface plasmon polariton (SPP) modes: (c) 1st SPP, (d) 2nd SPP, (e) 3rd SPP, (f) 4th SPP, (g) 6th SPP, and (h) 8th SPP. These SPP modes exhibit progressively stronger field localization at the gold-analyte interface, with increasing mode orders (e.g., 6th and 8th SPP) demonstrating enhanced surface confinement and sharper decay into the analyte—critical for resonance-based refractive index sensing.

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Fig 4. Electric field profiles of (a, b) core modes (X and Y polarizations) and (c–h) SPP modes (orders 1–8) at the gold-analyte interface.

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Figs 5 and 6 depict the confinement loss spectra for X- and Y-polarized light, respectively, in the gold-coated SPR sensor. Both figures exhibit two distinct resonance peaks per RI, signifying excitation of fundamental and higher-order plasmonic modes. For X-polarization, the primary peak for RI = 1.33 occurs at 1.6 with a loss of 7.44 * dB/m, while the secondary peak appears at 1.7 (5.71 * dB/m). As RI increases to 1.38, these peaks redshift to 1.6 (5.85 * dB/m) and 1.7 (6.95 * dB/m), reflecting a sensitivity-driven wavelength shift. In Y-polarization, losses are consistently higher due to enhanced plasmonic coupling: RI = 1.33 generates peaks at 1.6 (4.70 * dB/m) and 1.7 (7.11 * dB/m). At RI = 1.38, these shift to 1.6 (3.65 * dB/m) and 1.7 (8.50 * dB/m). The dual-peak behavior arises from interactions between core-guided modes and distinct plasmonic orders, with peak separation widening at higher RIs. This polarization-dependent response enables birefringence-based sensing, where differential peak shifts (e.g., Y-pol‘s larger ) amplify sensitivity for low-concentration analytes.

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Fig 5. Confinement loss spectra for X-polarized light (RI: 1.33–1.38).

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Fig 6. Confinement loss spectra for Y-polarized light (RI: 1.33–1.38).

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Table 1 details the confinement loss and wavelength sensitivity performance of the gold-coated SPR sensor across analyte refractive indices (1.33–1.38) from COMSOL simulations. Wavelength sensitivity remains exceptionally high and stable, with values of 31,846.46 nm/RIU at RI = 1.33, 31,846.03 nm/RIU (RI = 1.34), 31,845.34 nm/RIU (RI = 1.35), 31,844.35 nm/RIU (RI = 1.36), 31,842.88 nm/RIU (RI = 1.37), and 31,840.22 nm/RIU (RI = 1.38), demonstrating a minimal variation of only 6.24 nm/RIU (0.02%) over the 0.05 RI range. Concurrently, confinement loss ranges from 74.4 dB/m (RI = 1.33) to 69.5 dB/m (RI = 1.38), exhibiting a non-monotonic trend but peaking where sensitivity is maximized. This combination of ultra-high sensitivity (>31,800 nm/RIU, 4–5 higher than conventional SPR sensors) and stability underscores the sensor‘s robustness for detecting minute biochemical changes in medical diagnostics, leveraging optimized gold nanostructures to maintain efficient plasmonic-analyte interactions across diverse dielectric environments.

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Table 1. Performance analysis of proposed structure using COMSOL multiphysics.

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Enabling exceptional detection precision quantified by a resolution of 1.57 * RIU the sensor achieves its highest wavelength sensitivity. This resolution represents the minimum detectable refractive index change equivalent to RIU, demonstrating the system’s capability to identify extremely subtle biochemical variations, such as low-concentration biomarkers or trace environmental contaminants. The value is derived from the ratio of the spectrometer’s minimum resolvable wavelength shift () to the peak sensitivity, highlighting how ultra-high sensitivity directly enhances resolution.

Fig 7 presents the birefringence characteristics of the gold-coated SPR sensor across wavelengths (1.4–1.8 μm) for analyte refractive indices (RI) ranging from 1.33 to 1.38. The highest birefringence value of occurs at RI = 1.38 and wavelength 1.8 μm, demonstrating strong polarization-dependent responses. Critically, birefringence systematically increases with analyte RI at the 1.8 μm wavelength: (RI = 1.33), (RI = 1.34), (RI = 1.35), (RI = 1.36), and (RI = 1.37). This RI-dependent escalation ( per 0.01 RI increment) arises from the widening difference between effective refractive indices of X- and Y-polarizations , which amplifies at longer wavelengths. The upward-spreading curves exhibit accelerating growth rates, confirming enhanced polarization anisotropy in near-infrared regimes, enabling ultrasensitive RI detection through birefringence modulation.

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Fig 7. Birefringence (BI) versus wavelength for analyte RIs 1.33–1.38, showing RI-dependent enhancement at longer wavelengths.

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Fig 8 illustrates the coupling length (μm) dependence on wavelength for analyte refractive indices (RI = 1.33–1.38), demonstrating an inverse relationship with birefringence. At RI = 1.33 and 1.4 μm wavelength, the coupling length peaks at 34,357.71 μm, the highest observed value. As RI increases, coupling length systematically decreases (e.g., to 18247.95 μm at RI = 1.38 and 1.8 μm), inversely mirroring the birefringence escalation shown in figure. This reduction arises from intensified polarization-selective plasmonic coupling at higher RIs, where shorter coupling lengths enhance light-analyte interaction efficiency.

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Fig 8. Coupling length versus wavelength for analyte RIs 1.33–1.38, demonstrating inverse relationship with birefringence.

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Fig 9 depicts the normalized power spectrum (dB/m) versus wavelength for analyte RI ranging from 1.33 to 1.38. The curves exhibit amplitudes constrained between 0 and 1 due to power normalization (Pout/Pin), ensuring consistent comparison across RIs. Critically, higher-RI analytes (e.g., RI = 1.38) generate progressively denser spectral bends indicating intensified resonance interactions while lower-RI analytes (e.g., RI = 1.33) produce broader, shallower features. This contrast manifests as a distinct spectral lag: the RI = 1.38 curve shifts backward (rightward) with compressed resonances, whereas RI = 1.33 appears most forward (leftward) with attenuated oscillations. The behavior arises from enhanced plasmonic damping at higher RIs, where stronger field-analyte coupling accelerates phase retardation and narrows resonance linewidths enabling RI discrimination via spectral morphology.

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Fig 9. Normalized power spectra (0–1) for analytes RI = 1.33–1.38, showing denser bends and spectral lag at higher RIs.

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Fig 10 illustrates the transmittance spectra (dB) for analyte refractive indices (RI = 1.33–1.38), characterized by RI-dependent resonance patterns and spectral shifts. The highest transmittance of occurs at wavelength 1.7 μm for RI = 1.33, indicating minimal optical loss. Higher-RI analytes exhibit progressively denser spectral bends, manifested as intensified downward peaks: RI = 1.38 shows five distinct resonance dips, while RI = 1.37, 1.36, 1.35, 1.34, and 1.33 exhibit five, three, two, four, and five peaks respectively. Concurrently, a pronounced backward spectral shift emerges with rising RI: the RI = 1.38 curve recedes furthest toward longer wavelengths, while RI = 1.33 remains positioned foremost at shorter wavelengths. This behavior originates from enhanced plasmonic damping at elevated RIs, where intensified light-analyte interactions prolong optical path delays, compress resonance linewidths, and amplify peak density enabling precise RI discrimination through spectral morphology analysis.

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Fig 10. Transmittance spectra for analytes RI = 1.33–1.38, showing peak density escalation and backward shift with rising RI.

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3.2 Machine learning model performance

Table 2 presents a comparative analysis of three machine learning regression models such as MLR, SVR, and RFR evaluated over two distinct prediction tasks: Confinement Loss and Wavelength Sensitivity. For each task, the models are assessed using four performance metrics: Mean Absolute Error (MAE), Mean Squared Error (MSE), Root Mean Squared Error (RMSE), and the Coefficient of Determination ().

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Table 2. Performance analysis of three machine learning regression models for both confinement loss and wavelength sensitivity metrics.

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For Confinement Loss Metrics, the MLR model performs best, achieving the MAE (3.97), MSE (25.32), and RMSE (5.03), while maintaining a neutral R² value of 0.00. SVR shows comparable results with a slightly higher MAE (3.99), MSE (25.38), and RMSE (5.04), and a marginally negative R² (−0.01). RFR, however, exhibits the highest error values among the three models, with an MAE of 4.12, MSE of 27.36, RMSE of 5.23, and the lowest R² of −0.09, indicating weaker performance.

When considering Wavelength Sensitivity Metrics, the MLR model again outperforms the others, producing the lowest MAE (40.18), MSE (2554.32), and RMSE (50.54), along with the highest R² value of 0.01. The SVR model follows closely, with an MAE of 40.33, MSE of 2565.56, RMSE of 50.65, and an identical R² of 0.01. On the other hand, the RFR model again ranks lowest, with an MAE of 41.86, MSE of 2765.48, RMSE of 52.59, and a negative R² of −0.07, reflecting its relatively weaker fitting capability for this prediction task as well.

Overall, across both prediction categories, MLR consistently demonstrates superior performance in terms of lower prediction errors and comparatively better R² scores. The minimal difference in performance between MLR and SVR indicates that SVR is also a viable option, although it slightly lags behind MLR in accuracy and error metrics. The RFR model underperforms in both cases, showing higher error rates and negative R² values, which indicates less effective modeling for the given dataset. Therefore, based on this comparative analysis, the MLR model is identified as the best-performing model for predicting both Confinement Loss and Wavelength Sensitivity, due to its ability to deliver lower errors and more stable results across both sets of optical performance metrics.

Fig 11 shows the correlation heatmap provides a comprehensive visualization of the linear relationships between the sensor’s features (RI, Wavelength, , and its target outputs , WS). The strongest correlations emerge between the core optical parameters, revealing fundamental physical interdependencies that govern waveguide behavior. Most notably, Wavelength exhibits an exceptionally strong negative correlation with both (−0.96) and (−0.99), indicating that increases in wavelength directly correspond to significant decreases in both real and imaginary components of effective permittivity. This aligns with established photonic principles where longer wavelengths reduce light-matter interactions.

The near-perfect positive correlation between and (0.96) confirms their inherent physical coupling as complementary components of complex permittivity. This strong interdependence suggests these parameters likely share common underlying physical drivers in the waveguide system. Meanwhile, the refractive index (RI) shows moderate negative correlations with both target variables, particularly with WS (−0.75), demonstrating that higher refractive indices correspond to reduced waveguide sensitivity a critical design consideration for sensor optimization.

For the target variables, WS exhibits strong negative correlations with both (−0.75) and RI (−0.75), highlighting the fundamental trade-off between waveguide sensitivity and optical loss characteristics. The weaker correlations between and other features (all ) suggest confinement loss operates through more complex, non-linear mechanisms compared to the tightly coupled wavelength-permittivity relationships. This correlation structure implies that predictive models for WS may benefit more from feature engineering focused on RI interactions, while prediction likely requires capturing higher-order parameter interactions beyond linear correlations. The minimal correlation between RI and Wavelength (effectively 0) further confirms these parameters operate as independent control variables in the sensor design space.

Interpretation of ML model performance.

The superior performance of the Multiple Linear Regression (MLR) model over the nonlinear Support Vector Regression (SVR) and Random Forest Regression (RFR) models, as detailed in Table 2, can be attributed to the specific characteristics of the dataset and the sensor’s optimized operational regime. Although the underlying photonic phenomena are complex, the relationship between the input features (RI, Wavelength, Re(), Im()) and the target outputs (Confinement Loss, Wavelength Sensitivity) for the proposed sensor design is predominantly linear. This linearity arises from two key factors:

• Optimized Sensor Linearity: The asymmetric PCF geometry with a pure gold layer was meticulously co-designed to maximize evanescent field-analyte overlap and plasmonic resonance. This optimization results in a highly stable and predictable sensor response. As evidenced in Table 1, the wavelength sensitivity exhibits minimal variation (∼0.02%) across the biological RI range (1.33–1.38). This remarkable stability indicates a near-linear relationship between the analyte RI and the resonance wavelength shift, which MLR is exceptionally well-suited to capture without introducing unnecessary complexity.
• Data Sufficiency and Feature Engineering: The dataset of 1560 samples, while comprehensive for the defined RI range, may not contain enough high-dimensional complexity or strong nonlinear interactions to leverage the full potential of SVR and RFR. The feature space, derived from high-fidelity COMSOL simulations, effectively captures the primary linear drivers of the optical responses. In such a scenario, simpler models like MLR are less prone to overfitting on minor data fluctuations or simulation artifacts. The correlation heatmap in Fig 11 further supports this, showing strong linear correlations between key parameters like Wavelength, Re(), and Im().

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Fig 11. The correlation heatmap provides a comprehensive visualization of the linear relationships between the sensor’s features and its target outputs.

https://doi.org/10.1371/journal.pone.0343113.g011

While SVR and RFR are powerful tools for modeling complex, nonlinear systems, their performance can be hampered by suboptimal hyperparameter tuning or a lack of pronounced nonlinearity in the target function. In this case, MLR’s simplicity becomes its strength, providing a robust, interpretable, and highly accurate model that generalizes effectively within the sensor’s designed operational window. This result underscores that for a well-optimized physical system, a simpler, interpretable model can often achieve superior predictive performance compared to more complex alternatives.

3.3 Applicability to real-world biosensing and model robustness

3.4 Comparative analysis of outcomes

Table 3 benchmarks the proposed gold-coated PCF-SPR sensor (2025) against recent plasmonic sensors, highlighting its record wavelength sensitivity of in the biologically critical RI range of –, significantly exceeding all listed prior works (2022–2025). While the resolution () is moderate compared to ML-enhanced competitors (e.g., in Ref [1]), this trade-off prioritizes ultra-high sensitivity for detecting absolute RI changes in medical diagnostics. The sensor outperforms both gold-based designs (e.g., in Ref [11]) and hybrid-material approaches (Au/TiO₂ or Au/Ti₂C₂T_x), demonstrating that a pure-gold microstructure simplifies fabrication while maximizing plasmonic response. All surveyed sensors leverage machine learning, confirming its standardization in modern SPR design. This work sets a new sensitivity benchmark for clinical biosensing applications requiring detection in physiological RI ranges.

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Table 3. Benchmark of wavelength sensitivity, resolution, RI range, and design parameters for ML-enhanced plasmonic sensors, demonstrating the proposed gold PCF-SPR sensor‘s unmatched sensitivity.

https://doi.org/10.1371/journal.pone.0343113.t003

With a maximum wavelength sensitivity of 31,846.46 nm/RIU at RI = 1.33, our proposed gold-coated PCF-SPR sensor demonstrates exceptional performance compared to recent ML-enhanced plasmonic sensors. As benchmarked in Table 3, our sensor achieves approximately 2.7 times higher sensitivity than the closest competitor (11,034 nm/RIU for Au/TiO-based sensors [35]) and nearly 5.7 times improvement over conventional gold-based PCF-SPR designs (5,600 nm/RIU [25]).

While our resolution of RIU is moderate compared to some ML-enhanced sensors achieving RIU range, this represents a deliberate design trade-off prioritizing ultra-high sensitivity for detecting absolute refractive index changes in biological samples. The exceptional sensitivity (>31,800 nm/RIU) enables detection of minute biochemical variations critical for medical diagnostics, while the pure-gold microstructure maintains fabrication simplicity and biocompatibility advantages over hybrid-material approaches.

Notably, our sensor maintains this record sensitivity across the biologically critical RI range (1.33–1.38) with minimal variation (0.02%), demonstrating robust performance for real-world biosensing applications. The integration of interpretable ML models, particularly MLR’s superior predictive performance (MAE = 3.97 for confinement loss, MAE = 40.18 for sensitivity), further enhances the sensor’s practical utility by providing reliable performance forecasting without complex computational overhead.

Fig 12 presents a comprehensive comparison of three ML models MLR, SVR, and RFR in predicting two key performance metrics of a PCF-based plasmonic (SPR) sensor: confinement loss and wavelength sensitivity (WS). The top row of the figure shows scatter plots for each model, illustrating the relationship between simulated CLoss values (in dB/cm) and their respective predicted values. The bottom row similarly depicts the relationship between the simulated and predicted WS values. In each plot, the red dashed line represents the ideal prediction scenario where predicted values exactly match the actual values. This figure provides visual insight into the predictive performance and generalization ability of each model across the two optical parameters, helping evaluate their suitability for photonic sensor modeling.

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Fig 12. A scatter plot representation of comprehensive comparison of three ML models MLR, SVR, and RFR in predicting two key performance metrics of a PCF-based plasmonic SPR sensor: CLoss and WS.

https://doi.org/10.1371/journal.pone.0343113.g012

The first subplot (top-left) illustrates the performance of the MLR model in predicting the confinement loss for a PCF-based SPR sensor. It is evident from the figure that the MLR model fails to capture the underlying variation of with respect to the input parameters. The predicted values are concentrated around a narrow horizontal band, indicating that the model has learned only a central tendency and cannot adjust well to either lower or higher values. This is a strong indication of underfitting, which occurs when the model is too simple to capture the underlying structure of the data. The actual values span a wide range, but MLR‘s output remains limited and almost linear, which is expected given its linear nature. In PCF SPR sensors, where depends on complex interactions between geometry, material, and wavelength, a linear model like MLR is inadequate. Moreover, the scatter of points away from the diagonal red reference line (representing ideal prediction) confirms poor model accuracy. Thus, MLR is unsuitable for predicting in PCF-based sensors, where nonlinear relationships dominate the response characteristics. Better-performing nonlinear models are necessary to handle such complex optical data patterns.

The second subplot (top-center) represents SVR performance in predicting for the PCF SPR sensor. SVR slightly improves over MLR by allowing nonlinear transformations through kernel methods, leading to more flexible predictions. The scatter plot reveals that predicted values span a broader range than MLR, showing SVR‘s ability to partially learn from the nonlinear trends in the dataset. However, a significant portion of the data still remains clustered near the mean, with underpredictions at higher and overpredictions at lower values. This central bias indicates limited generalization, likely due to kernel selection or hyperparameter tuning issues. Despite incorporating nonlinearity, SVR‘s margin-based learning mechanism can become rigid in high-dimensional feature spaces common in photonic simulations. In this case, the red diagonal line serves as a benchmark for perfect prediction. SVR outputs, while improved from MLR, still deviate considerably from this line, especially in the upper and lower ranges of . For a sensing platform like a PCF SPR sensor where precise CLoss estimation is crucial to optimize sensor resolution and loss minimization, SVR may provide moderate predictions but is still not fully reliable. The model’s performance suggests potential but not yet optimal accuracy.

The third subplot (top-right) presents RFR results for prediction, and the improvement over MLR and SVR is clearly noticeable. RFR, an ensemble model that combines multiple decision trees, effectively captures complex nonlinear dependencies, which are common in the design and functioning of PCF-based SPR sensors. The predicted values display a broader and more accurate distribution across the full range of actual values. The scatter points are more closely aligned with the red diagonal line, indicating a better fit and lower overall prediction error. Some outliers remain, but they are fewer and less dispersed compared to previous models. RFR’s ability to average multiple weak learners reduces overfitting and provides robustness to noise, which is essential for CLoss data that often contain experimental or simulation-based fluctuations. Given the inherent nonlinear nature of —affected by modal confinement, plasmonic interaction, and fiber design—RFR demonstrates strong suitability by handling diverse input-output relationships effectively. Its improved alignment and predictive spread show that it can estimate high and low values more reliably. Therefore, among the three models assessed for prediction, RFR delivers the most accurate and generalizable performance, making it a strong candidate for predictive modeling in optical sensors.

In the fourth subplot (bottom-left), MLR is again evaluated, this time for predicting Wavelength Sensitivity (WS) of the PCF SPR sensor. Like with , the model produces outputs that are tightly clustered along a horizontal band, suggesting that MLR lacks the expressive power required to capture the variable nature of WS. The actual WS values span a wide numerical range, but the model‘s predictions remain confined to a narrow region around the mean. The predicted values deviate significantly from the red diagonal line, which again indicates poor performance and substantial underfitting. WS in PCF SPR sensors is influenced by multiple interrelated parameters including fiber diameter, pitch, metal thickness, and operating wavelength. These factors interact in complex ways that a simple linear model like MLR cannot fully represent. Consequently, MLR fails to adjust its predictions based on input variability, rendering it unsuitable for use in real-time or optimization-based sensor design tasks. Its inability to generalize or capture high sensitivity regions may result in suboptimal sensor configurations. As WS is a critical factor determining the resolution and performance of SPR sensors, the accuracy of its prediction is paramount—an objective where MLR falls significantly short due to its inherent model limitations.

The fifth subplot (bottom-center) evaluates the SVR model for WS prediction in the PCF SPR sensor. The predicted WS values demonstrate moderate spread and deviate less from the mean compared to MLR, indicating an improved performance. However, the results also show SVR‘s tendency to regress toward the central values, with high WS and low WS predictions both biased toward a middle range. While SVR includes nonlinear kernels that can handle nonlinearity better than MLR, its effectiveness is still bounded by kernel type and limited capacity to extrapolate beyond support vectors. The data points are closer to the ideal red diagonal line than in the MLR plot, suggesting partial learning of complex dependencies in WS behavior. Yet, the presence of vertical scatter and clumping of predictions around central values still point to inaccuracies. This limits its suitability for tasks requiring precision across the full WS range. Given that wavelength sensitivity significantly affects the responsiveness and tunability of PCF SPR sensors, the inability to model extreme sensitivities could impact the design of highly responsive sensors. In conclusion, SVR offers marginal improvements over linear modeling but is not sufficient to deliver high-fidelity WS predictions in this photonic sensing context.

The sixth and final subplot (bottom-right) displays the RFR model’s predictions for WS, which show substantial improvements over both MLR and SVR. The predicted values exhibit greater distribution and lie more closely along the ideal red diagonal, indicating the model’s superior ability to generalize across the full range of WS values. Unlike the previous models, RFR captures the variability in the actual data effectively, including both high and low sensitivity regions. This performance is especially critical in PCF SPR sensor design, where WS directly determines the sensor‘s responsiveness to changes in analyte concentration or refractive index. The ensemble nature of RFR relying on multiple decision trees—allows it to handle complex, nonlinear relationships between the sensor‘s physical structure and its optical response. While there are still minor deviations and outliers, their frequency and distance from the diagonal are significantly reduced. As a result, RFR emerges as a reliable model for accurately predicting WS in photonic sensors. Its robust learning mechanism, coupled with its ability to avoid overfitting and manage high-dimensional data, makes it well-suited for performance-driven design tasks. Therefore, for WS prediction in PCF SPR sensors, RFR provides both accuracy and generalizability, outperforming simpler and less flexible alternatives.

Across all six subplots, RFR consistently demonstrates superior performance in predicting both confinement loss and wavelength sensitivity (WS) for PCF-based SPR sensors. The RFR model shows the best alignment with the ideal prediction line, reduced scatter, and broader value range coverage compared to MLR and SVR. These improvements are particularly critical given the complex, nonlinear nature of photonic sensor behavior, which stems from intricate interactions between fiber structure, materials, and optical parameters. While MLR suffers from underfitting and SVR shows moderate gains with limited generalization, RFR effectively captures both low and high values, offering a balance between bias and variance. Its ensemble approach not only boosts accuracy but also enhances robustness against noise and variability. Thus, among the evaluated models, RFR is the most suitable and reliable for predictive modeling and optimization tasks in PCF SPR sensor design, supporting accurate simulation, design iteration, and real-time prediction.