High-Sensitivity and Temperature-Robust Gas Sensor Based on Magnetically Induced Differential Mode Splitting in InSb Photonic Crystals

1. Introduction

The accurate identification of dangerous gases is a crucial issue in chemical engineering and environmental safety [1,2]. Specifically, common hazardous industrial gases such as methane (CH₄), carbon monoxide (CO), and nitrogen oxides (NO_x) exhibit extremely low refractive indices, typically ranging from 1.0003 to 1.0008 at standard atmospheric pressure. The release or accumulation of these flammable, explosive, or poisonous volatile organic compounds presents significant hazards to industrial infrastructure and public health. However, the fluctuation in gas concentration is inherently linked to minute changes in the refractive index of the medium, often on the magnitude of 10^-4 to 10^-5 RIU. This necessitates sensing devices with an exceptionally low Limit of Detection (LOD) to distinguish target analytes from the background environment. Consequently, optical refractive index sensing has emerged as an indispensable technology in chemical analysis [3,4], biosensing [5,6], new energy monitoring [7,8], and environmental governance [9,10]. Utilizing benefits such as non-contact measurement, rapid response, high precision, and resistance to electromagnetic interference, optical sensors have emerged as the predominant method for detection [11,12,13,14]. A variety of sensing designs has been developed, including total internal reflection techniques [15,16], surface plasmon resonance [17,18], interferometry [19,20], and fiber-optic sensing [21,22,23]. Notwithstanding the compactness and stability of these conventional designs, numerous contemporary systems continue to be hindered by restricted resolution and poor quality factors (Q) [24,25]. This constraint necessarily results in inadequate sensitivity and a heightened chance of false negatives or overlooked detection in practical applications [26,27]. With the progression of the Internet of things and intelligent manufacturing [28], the necessity for sensors that overcome performance limitations has become critical.

To tackle these issues, one-dimensional photonic crystals (1D PCs) [29,30,31,32] have attracted considerable interest as a viable sensing platform owing to their great integration ease and relatively straightforward fabrication methods. Recent improvements have enhanced the performance of PCs significantly. In 2024, Qi et al. [33] attained angle-insensitive ultra-high-Q resonances by integrating bound states in the continuum, developing a hypersensitive environmental refractive index sensor and a temperature sensor with maximum sensitivities of 8.67 × 10⁵ μm/RIU and 2.8 × 10⁶ μm/℃, respectively. While this work achieved an impressive detection limit approximately 10^-5 RIU within the gaseous refractive index range, it primarily relies on wavelength interrogation, which inherently necessitates high-resolution spectrometers, thereby increasing the system complexity and cost. In 2025, Li et al. [34] introduced a Ω-shaped sensor utilizing Gold Nanoparticles/Polydimethylsiloxane (AuNPs/PDMS) core-shell structures for the concurrent measurement of refractive index and temperature. Despite these advancements in spectral detection, angular interrogation techniques have emerged as a robust alternative for constructing compact and cost-effective sensing systems. By fixing the excitation frequency and monitoring the angular shift of resonance modes, angular sensors eliminate the need for bulky spectral analyzers. Currently, high-performance angular sensing is predominantly realized through Surface Plasmon Resonance (SPR) or defect-mode PCs. However, most existing angular sensors employ passive measurement schemes that track absolute angular shifts, leaving them susceptible to source fluctuations and environmental noise. Therefore, developing an active angular sensing architecture that incorporates differential detection capabilities is critical for further improving measurement stability and resolution. To enhance sensing capabilities from passive detection to active tunable sensing, semiconductor materials are progressively incorporated into dielectric stacks. Indium antimonide (InSb) is a remarkable candidate for the terahertz (THz) spectrum [35,36]. InSb is distinguished by its exceptionally low effective electron mass and elevated carrier mobility, resulting in a significant magneto-optical effect [37]. In the presence of an external magnetic field, permittivity transforms into a tensor facilitating the non-reciprocal manipulation of light, which is a capability absent in passive dielectrics [38]. To further optimize the structural performance, porous silica (porous SiO₂) is utilized as the low-refractive-index material [39,40]. Porous SiO₂ represents the most mature and technologically established solution for achieving extremely low refractive indices, which is essential for maximizing the refractive index contrast and enhancing the optical field confinement within the device.

The design of high-performance magneto-optical devices constitutes a complicated multiparametric challenge. Conventional design methodologies frequently depend on trial-and-error or empirical intuition, resulting in inefficiency and a tendency to achieve only local optima. The recent surge in Artificial Intelligence (AI) research has facilitated the integration of intelligent algorithms with nanophotonics, leading to novel opportunities for device optimization [41,42]. In the realm of optical sensing and perception, meta-heuristic algorithms such as Genetic Algorithms and Particle Swarm Optimization have been extensively applied to solve inverse design problems, enabling the precise tailoring of geometric parameters for enhanced light-matter interaction. For instance, in 2023, Teng et al. [43] significantly enhanced the performance of surface plasmon resonance sensors in a Kretschmann structure through genetic algorithms, exhibiting anti-crossing behavior and achieving an unprecedented sensitivity of 1364 °/RIU. However, traditional algorithms often struggle with premature convergence when navigating the high-dimensional search spaces of complex multilayer structures. To execute the inverse design of the sensor architecture in this study, the Multi-Objective Dragonfly Algorithm (MODA) was employed, which was first proposed by Mirjalili in 2015 [44]. Distinct from standard evolutionary algorithms, MODA is characterized by its unique ability to switch between static (exploitation) and dynamic (exploration) swarming behaviors. The core inspiration stems from the natural hunting and migration mechanisms of dragonflies, by dynamically adjusting weights for separation, alignment, and cohesion, MODA effectively maintains population diversity to avoid local optima while ensuring rapid convergence towards the Pareto front. This capability makes it uniquely suited for simultaneously optimizing conflicting objectives, such as maximizing the Rectangular Coefficient (RC) [45] while maintaining fabrication feasibility. Through this computational optimization of layer thicknesses, a resonance mode exhibiting an exceptionally high RC is attained. This guarantees a spectral profile characterized by a steep and virtually vertical band edge, establishing the physical basis for high-sensitivity detection that manual tuning cannot readily achieve.

Notwithstanding these breakthroughs, a significant gap persists in current sensing methodologies. Most existing refractive index sensors rely heavily on wavelength interrogation, which necessitates bulky and expensive spectral analyzers, or depend on the absolute angular shift of a single resonance peak, leaving them vulnerable to source fluctuations and environmental noise. To address these restrictions, this work introduces a novel Magneto-Optical Differential Photonic Crystals Sensor (MO-DPCS) based on a precise angular interrogation architecture. Unlike intricate lithographic structures, the approach preserves a straightforward multilayered arrangement while implementing a differential angular detection strategy [46,47,48]. The sensing mechanism is fundamentally rooted in the magneto-optical non-reciprocity of the InSb material. Under an external magnetic field, the gyrotropic permittivity of InSb breaks the time-reversal symmetry, causing the dispersion relations of Transverse Electric (TE) and Transverse Magnetic (TM) modes to split. Consequently, at a fixed THz frequency, the photonic bandgap edges of the two modes exhibit distinct angular cutoff characteristics. By utilizing the angular difference (Δθ) between the TE and TM modes as the sensing signal, this method not only doubles the interrogation efficiency but also efficiently mitigates common-mode noise such as thermal drift and mechanical vibrations. This greatly improves the resolution and stability of the sensor in intricate sensing situations.

2. Configuration and Methods

The proposed MO-DPCS is engineered as a one-dimensional multilayered structure specifically for the detection of hazardous gases with low refractive indices, such as CH₄, CO, and sulfur dioxide (SO₂). The design focuses on a refractive index spectrum ranging from 1.000 to 1.100. This interval is strategically selected to strictly encompass the refractive indices of vacuum (n = 1.0), atmospheric air (n ≈ 1.0003), and the majority of industrial volatile gases, ensuring comprehensive coverage for gas leakage monitoring. This apparatus attains enhanced sensitivity (S) via a differential sensing methodology. Figure 1 demonstrates that the structure operates as a magneto-optical Fabry-Pérot cavity [49], consisting of a core host structure situated between two symmetrical anti-reflection structures (AFSs) [50]. The fundamental host structure comprises a singular layer of InSb with an optimized thickness d_H of 433.7464 μm, leveraging the epsilon-near-zero (ENZ) [51,52] property to stimulate a designated angular transmission window (ATW). To reduce signal attenuation caused by thin-film resonance effects at energy transition points, the host structure is interposed between two identical AFSs designed using impedance matching to minimize reflection losses [53,54]. Each AFS consists of 7 periods of alternating layers of InSb, with a thickness of d_a = 7.3047 μm, and porous SiO₂ layers, with a thickness of d_b = 1.442 μm. The porous SiO₂ is chosen for its steady low refractive index and a regulated dielectric constant of 1.50. A key aspect of this design is the synchronized modulation of the InSb layers within both the host structure and the AFSs. The device is exposed to an incoming electromagnetic wave (EW) characterized by a wave vector k, which propagates in the xoz-plane (the plane of incidence) at an incidence angle θ. An external static magnetic field B is oriented along the positive +y-axis, maintaining a direction perpendicular to the plane of incidence to establish the Voigt configuration [55]. Under this geometric arrangement, the incident light is decomposed into two orthogonal polarization modes. In the TE wave, the electric field vector E oscillates parallel to the +y-axis and is thus aligned with B. Conversely, in the TM wave, the magnetic field vector H oscillates parallel to the +y-axis, which constrains the electric field vector E to lie within the xoz-plane and remain perpendicular to the direction of B. This configuration generates a significant non-reciprocal magneto-optical effect specifically for the TM mode while maintaining the TE mode as a quasi-static reference, thereby providing the MO-DPCS with enhanced anti-interference capabilities against environmental variations and improved detection resolution by effectively mitigating common-mode noise [56].

The permittivity of InSb can be expressed in tensor form [57]:

$${\epsilon }_i=\left[\begin{array}{ccc}{\epsilon }_x& 0& i{\epsilon }_{xz}\\ 0& {\epsilon }_y& 0\\ -i{\epsilon }_{xz}& 0& {\epsilon }_x\end{array}\right]$$

(1a)

$${\epsilon }_x={\epsilon }_{\infty }-{\epsilon }_{\infty }{\displaystyle \frac{{\omega }_p^2(\omega +i\gamma )}{\omega \left[{(\omega +i\gamma )}^2-{\omega }_c^2\right]}}$$

(1b)

$${\epsilon }_y={\epsilon }_{\infty }-{\epsilon }_{\infty }{\displaystyle \frac{{\omega }_p^2}{\omega (\omega +i\gamma )}}$$

(1c)

$${\epsilon }_{xz}={\epsilon }_{\infty }{\displaystyle \frac{{\omega }_p^2{\omega }_c}{\omega \left[{(\omega +i\gamma )}^2-{\omega }_c^2\right]}}$$

(1d)

In this context, ε_∞ denotes the high-frequency dielectric constant of InSb, γ signifies the carrier collision frequency [57], m and e correspond to the mass and charge of an electron, respectively, and m* equals 0.015m, indicating the effective mass of the carriers. ω_p is the plasma frequency, defined as (Ne/ε₀ε_∞m*)^1/2. Here, ε₀ represents vacuum permittivity, while ambient temperature is denoted by T₀. N is specified as [57]:

$$N\left(c{m}^{-3}\right)=5.76\times {10}^{14}{T}_0^{1.5}\times e^{\left({\displaystyle \frac{-0.26}{2\times 8.625\times {10}^{-5}\times T_0}}\right)}$$

(2)

The effective dielectric constant for an incident plane electromagnetic wave in TE mode can be expressed as [57]:

$${\epsilon }_{TE}={\epsilon }_{\infty }\left(1-{\displaystyle \frac{{\omega }_p^2}{\omega (\omega +i\gamma )}}\right)$$

(3)

In the Voigt configuration, an external magnetic field B is applied perpendicularly to the wave propagation direction, resulting in an anisotropic dielectric response of InSb. In TE mode, the electric field oscillates parallel to B, resulting in carrier motion being unaffected by the Lorentz force. Thus, the effective refractive index for TE mode is isotropic, represented as n_a(TE) = (ε_TE)^1/2 [58]. Conversely, the TM mode exhibits an electric field orthogonal to B, resulting in robust coupling between the EWs and free carriers influenced by the Lorentz force. This interaction is defined by the cyclotron frequency ω_c = eB/m* [57], and the effective refractive index in this mode is given by n_a(TM) = [(ε_x - ε_xz)/ε_x]^1/2 [59].

Owing to these distinct dielectric responses, the TE mode remains isotropic, whereas the TM mode becomes magneto-optically anisotropic under the influence of the Lorentz force. This divergence in propagation characteristics dictates that, within a periodically layered medium, the characteristic transfer matrix of each layer must be formulated independently for the TM and TE waves, which are respectively expressed as[60,61]:

$$M_n=\left[\begin{array}{cc}\cos {\delta }_n& -i{p}_n^{-1}\sin {\delta }_n\\ -i{p}_n\sin {\delta }_n& \cos {\delta }_n\end{array}\right]$$

(4a)

$$M_{TM}=\left[\begin{array}{cc}\cos \left(k_{1z}{d}_n\right)+{\displaystyle \frac{k_{1x}{\epsilon }_{xz}}{k_{1z}{\epsilon }_{xx}}}\sin \left(k_{1z}{d}_n\right)& -i{p}_1^{-1}\sin \left(k_{1z}{d}_n\right)\\ -i{p}_1\sin \left(k_{1z}{d}_n\right)& \cos \left(k_{1z}{d}_n\right)-{\displaystyle \frac{k_{1x}{\epsilon }_{xz}}{k_{1z}{\epsilon }_{xx}}}\sin \left(k_{1z}{d}_n\right)\end{array}\right]$$

(4b)

$$M_{TE}=\left[\begin{array}{cc}\cos \left(k_{2z}{d}_n\right)& -i{p}_2^{-1}\sin \left(k_{2z}{d}_n\right)\\ -i{p}_2\sin \left(k_{2z}{d}_n\right)& \cos \left(k_{2z}{d}_n\right)\end{array}\right]$$

(4c)

where k_iz = k_icosθ_i = (ω/c)n_icosθ_i , k_1x = k₁sinθ_i , p_i = (ε₀/μ₀)^1/2n_icosθ_i , ε₀ = 8.8542×10^-12 is the vacuum dielectric constant, and μ₀ = 4π×10^-7 is the permeability of vacuum [59]. To thoroughly examine the optical transmission characteristics of the proposed MO-DPCS, the equations of Maxwell were resolved utilising the transfer matrix method (TMM) [60,61] under the continuity boundary conditions for tangential electric and magnetic fields. Let M_a represent the characteristic matrix of the InSb material within the AFSs structure, M_b represent the characteristic matrix of the porous SiO₂ within the AFSs structure, and M_H represent the characteristic matrix of the host structure. The TMM can be employed to calculate energy transfer between dielectric layers.

$$M=\left[\begin{array}{cc}{M}_{11}& M_{12}\\ M_{21}& M_{22}\end{array}\right]={\left(M_a{M}_b\right)}^7{M}_H{\left(M_b{M}_a\right)}^7$$

(5)

The transmission coefficient t can be articulated using the subsequent formula [62]:

$$t=\left|{\displaystyle \frac{2p_0}{\left(M_{11}+M_{12}{p}_0\right)p_0+\left(M_{21}+M_{22}{p}_0\right)}}\right|$$

(6)

The transmittance T can be articulated as [62]:

$$T=|t{|}^2$$

(7)

3. Results

3.1. Spectral Characteristics and Algorithm-Driven Optimization of Magneto-Optical Polarization

The optical transmission properties of the proposed MO-DPCS were initially examined to confirm its efficacy as a high-performance angular filter. Figure 2 illustrates that the periodic stratification of the structure results in a pronounced photonic band gap (PBG) [63,64] attributable to Bragg interference [65]. The MO-DPCS demonstrates an exceptional ATW for both TE and TM waves inside the passband. The transmission efficiency typically surpasses 0.9 across the operational angular range, guaranteeing adequate signal intensity for reliable detection even in lossy experimental conditions. The steepness of the transmission band edge is a crucial characteristic for ensuring excellent detection resolution [45]. A steeper edge indicates a more acute angular transition from the passband to the stopband, hence reducing the ambiguity in identifying the cutoff angle θ_cutoff and improving the LOD. The RC is employed as a statistic to quantify the sharpness of the band edge. The RC is delineated as [45]:

$$RC={\displaystyle \frac{\Delta {\theta }_{-3\mathrm{dB}}}{\Delta {\theta }_{-30\mathrm{dB}}}}$$

(8)

Δθ_-3dB and Δθ_-30dB represent the angular widths recorded at -3 dB with T = 0.5 and at -30 dB with T = 0.001, respectively. The rectangular coefficient varies from 0 to 1, with values approaching unity signifying a sharper band edge, therefore indicating more angular selectivity and greater measurement precision.

Figure 3 presents a quantitative evaluation regarding the spectral quality of the proposed sensor. Observations indicate that the RC values for both TE and TM modes remain consistently above 0.975 across the complete refractive index detection range from 1.000 to 1.100. Simultaneously, the device exhibits exceptional optical throughput, with the peak transmittance of the Angular Transmission Window (ATW) exceeding 99%. This persistently high RC indicates a nearly vertical band-edge profile, which is mathematically correlated with a minimized width of the transition region. Physically, this "flat-top and steep-edge" spectral characteristic offers significant advantages, the high transmission efficiency markedly improves the Signal-to-Noise Ratio (SNR) of the detection system, ensuring that the cutoff point remains distinct even under weak signal conditions, while the high RC directly constrains the Full Width at Half Maximum (FWHM) of the resonance edge [25]. As a critical parameter for defining spectral resolution, the minimization of FWHM is the primary mechanism for maximizing the Figure of Merit (FOM). Consequently, the FOM—where higher values indicate superior quality—serves as the most equitable benchmark for evaluating the comprehensive performance of the sensor in practical applications. Although high-Q transmission is crucial, the implementation of differential detection requires divergent optical characteristics for TE and TM modes. In absence of an external magnetic field (B = 0 T), InSb functions as an isotropic material. Under these circumstances, the transmission spectra for TE and TM modes are essentially indistinguishable, with the cutoff angles θ_TE and θ_TM approximately coinciding. Thus, any environmental fluctuation would cause simultaneous shifts in both modes, leading to an insignificant differential signal where Δθ ≈ 0. To address this issue, a static magnetic field of 0.033 T is employed in the Voigt configuration. This field effectively disrupts the time-reversal symmetry of the system and induces gyrotropic anisotropy in the InSb layers. Crucially, the Lorentz force exerts a selective influence, it specifically modifies the effective refractive index of the TM mode, where the electric field vector E is orthogonal to the magnetic field vector B. In contrast, the TE mode, with E aligned parallel to B, remains predominantly unaffected. This magneto-optical polarization splitting generates two distinct signal channels within a single physical structure, creating the necessary bias for the ensuing differential detection technique.

The MODA, a bio-inspired metaheuristic first proposed by Mirjalili in 2015 [44], was utilized to accurately ascertain the ideal magnetic field strength B that maximizes the polarization splitting effect while maintaining a high transmission efficiency of the TM mode. MODA epitomizes a leading edge in contemporary artificial intelligence research, distinguished by its unique mechanism that mimics the static and dynamic swarming behaviors of dragonflies. In the static phase (exploration), dragonflies form sub-groups to fly over different areas, which allows the algorithm to effectively avoid local optima. Conversely, in the dynamic phase (exploitation), the swarm migrates in a common direction, facilitating rapid convergence toward the global optimum. The core of this algorithm is to calculate the step vectors and position vectors of dragonflies, which are shown as follows, representing their movement directions ΔX_i and positions X_i [44]. The detailed variable explanations are provided in Table 1 pseudo-code.

$$\Delta X_{i+1}=\left(s{S}_i+a{A}_i+c{C}_i+f{F}_i+e{E}_i\right)+w\Delta X_i$$

(9a)

$$X_{i+1}=X_i+\Delta X_{i+1}$$

(9b)

This exceptional capacity to balance exploration and exploitation makes MODA particularly suitable for addressing intricate, multi-variable engineering challenges with conflicting aims. The efficacy of such AI-driven optimization techniques has been extensively validated in the design of high-performance nanophotonic devices. For instance, Genetic Algorithms have been employed to develop ultra-sensitive SPR sensors, including a generic optimization technique that identified a series of sensors with sensitivity improvements of approximately 100%. Notably, a dual-mode SPR structure coupling Surface Plasmon Polaritons and waveguide modes in germanium dioxide was achieved, exhibiting anti-crossing behavior and an unprecedented sensitivity of 1364 °/RIU [43]. Other GA-optimized designs include bimetallic Al/Ag structures sandwiched in hBN (578 °/RIU at 633 nm) and hybrid hBN/MoS₂/hBN structures (676 °/RIU at 785 nm). Furthermore, deep learning approaches such as Convolutional Neural Networks (CNNs) have proven superior to traditional methods in the inverse design of Photonic Crystal Waveguides (PCWs). Hybrid frameworks combining the Grey Wolf Optimizer (GWO) with CNNs have also been utilized to optimize complex D-shaped photonic crystal fiber polarization filters, achieving a crosstalk of 965.3 dB and high confinement losses [42]. These cases underscore the powerful potential of intelligent algorithms in pushing the performance boundaries of photonic sensors. In this study, the optimization focused on a highly refined search space, as the requisite magnetic field adjustments are extremely subtle yet decisive for the resolution of the sensor. The algorithm was assigned the duty of navigating this delicate trade-off to determine a specific magnetic field intensity that maximizes the divergence between TE and TM modes while specifically maintaining a high transmission rate. Through rigorous iterative optimization, a magnetic field of 0.032 T was determined to be the global optimum. Figure 4 illustrates the optical response of the structure under optimal conditions, showcasing significant polarization separation while maintaining TM passband transmission above 0.9 (with a maximum transmissivity reaching 0.99). The detailed implementation logic and the mathematical foundation of the optimization technique are encapsulated in the pseudo-code provided in Table 1.

To comprehensively execute the inverse design and ensure the rigorous coupling between the meta-heuristic optimization and the transfer matrix method calculations, the initial parameters of the MODA were meticulously configured. The optimization framework is multifaceted, encompassing global algorithmic settings, dynamically adaptive swarming coefficients, and strict electrodynamic constraints. Notably, the swarming parameters, such as separation, alignment, and cohesion weights, are non-static and adaptively transition from the exploration phase to the exploitation phase as iterations progress. Furthermore, the objective functions are evaluated under high-resolution angular interrogations and strictly bounded by intrinsic physical properties. The complete architectural parameters and boundary conditions governing this complex optimization process are systematically detailed in Table 2.

3.2. Refractive Index Sensing Response and Sensitivity Analysis

The primary sensing mechanism of the proposed MO-DPCS relies on the dependence of PBG edge on the dielectric surroundings. Variations in the target gas concentration cause fluctuations in the background refractive index n₀, which subsequently modify the effective optical path length within the crystal layers. This alteration in the Bragg resonance condition results in a notable angular displacement of the transmission window edge. To clarify the precise effect of magnetic modulation, the response without an external magnetic field B = 0T was initially analyzed to establish a baseline. Specifically, Figure 5(a) and Figure 5(b) illustrate the transmission spectra for the TE and TM modes, respectively, demonstrating a concurrent trend towards reduced incidence angles as the refractive index increases from 1.000 to 1.100. While this synchronous blue-shift is characteristic of the unmagnetized state, the application of the optimal external magnetic field induces a distinct magneto-optical phenomenon. Under these magnetized conditions, the two orthogonal polarizations demonstrate contrasting shifting tendencies. The cutoff angle of the TE mode progressively decreases, whereas the cutoff angle of the TM mode shifts in the opposite direction and increases.

The sensitivity, denoted as S, was computed for both modes and is fundamentally defined as the slope of the angular shift in relation to the change in refractive index. As depicted in Figure 6(a), in the unmagnetized state, both modes exhibit a highly similar synchronous response. The TE mode follows the linear equation θ_cutoff = -26.42n₀ + 54.03 ° with a correlation coefficient (R²) of 0.99910, yielding a sensitivity θ_TE of -26.42 °/RIU. Concurrently, the TM mode is described by θ_cutoff = -27.22n₀ - 55.34 ° (R² = 0.99901), resulting in a sensitivity θ_TM of -27.22 °/RIU. Upon the application of the optimized external magnetic field, as shown in Figure 6(b), the two orthogonal polarizations demonstrate contrasting shifting tendencies. The TE mode, remaining unaffected by the magnetic field, maintains its negative angular shift trend with θ_TE = -26.42 °/RIU. Conversely, the TM mode experiences strong Lorentz force coupling, causing its cutoff angle to progressively increase. This transition is mathematically defined by the linear fitting equation θ_cutoff = 4.42n₀ - 2.84 ° (R² = 0.88498), which corresponds to a positive sensitivity θ_TM of +4.42 °/RIU. Both polarizations exhibit exceptional linearity, with correlation coefficients nearing unity, signifying that the measurement outcomes are highly dependable and consistent over the detection range. Although the absolute values of the sensitivities vary, the essential discovery is their contrasting signals. This unique differentiation in directional response establishes the physical basis for the ensuing differential detection technique, which is employed to enhance the overall sensor responsiveness and mitigate common-mode noise.

3.3. Differential Detection Strategy and Comprehensive Performance Evaluation

To enhance the sensing capability of the proposed structure, a differential detection scheme was adopted to quantify the sensing signal. Unlike conventional detection methods where the sensor response relies on a single resonance feature that is susceptible to environmental noise, the response in this system is determined by the absolute difference between the cutoff angles of the two orthogonal polarization modes. The differential angular shift, denoted as Δθ_diff, represents the separation between the TE and TM cutoff edges. Consequently, the total differential sensitivity S_diff is theoretically defined as the absolute value of the disparity between the individual sensitivities.

$$S_{diff}=\left|S_{TE}-S_{TM}\right|$$

(10)

The effectiveness of this differential method relies significantly on the directional shifting patterns of the orthogonal modes. As illustrated in Figure 7, the differential angular shift exhibits completely distinct behaviors depending on the external magnetic field configuration. In the absence of an external magnetic field (B = 0T), both modes undergo synchronous blue-shifts. Because the algebraic signs of the individual sensitivities S_TE and S_TM are congruent, the differential computation inherently results in the subtraction of their magnitudes. Quantitative linear fitting verifies this limitation, yielding the equation Δθ = -0.8n₀ - 1.31 ° with R² of 0.9935. This corresponds to a negligible differential sensitivity of merely 0.8 °/RIU, offering minimal advantage over conventional single-mode detection. Conversely, the application of the external magnetic field (B = 0.033T) exploits the unique reverse-motion characteristic, wherein the individual sensitivities exhibit opposing signs. Under this condition, subtracting the negative S_TE from the positive S_TM results in a constructive superposition of their absolute values. The linear fitting analysis for this magnetized state defines the relationship as Δθ = -30.84n₀ + 56.87 ° (R² = 0.9941), which confirms a markedly enhanced differential sensitivity of 30.84 °/RIU. This exceptional value significantly surpasses the sensitivity attainable in the unmagnetized state, conclusively demonstrating that the magneto-optical effect is the principal driver for the overall performance enhancement of the proposed sensor.

A comprehensive evaluation of the realistic resolution limit was conducted by carefully assessing the FWHM, FOM, and 、LOD of the sensor. As delineated in Section 3.1, the elevated RC of the structure guarantees an almost vertical band edge, which immediately correlates with a reduced FWHM of the derivative spectrum. This spectral acuity is essential for performance enhancement. The evaluation is fundamentally grounded in the optimized differential sensitivity, which demonstrates a robust linear response characterized by the equation Δθ = -49.090n₀ + 75.554 ° with a correlation coefficient of R² = 0.9979. The FOM is defined as the ratio of sensitivity to FWHM. The proposed sensor attains remarkable figures of merit due to this enhanced differential sensitivity combined with the reduced FWHM. Figure 8(a) illustrates that the differential mode displays the highest FOM throughout the full detection range, markedly exceeding the individual TE and TM modes. This superior performance is quantitatively defined by the linear fitting equation y = -2.575n₀ + 15.171 /RIU with R² = 0.8500. Furthermore, the LOD is inversely related to the product of sensitivity and the quality factor, mathematically represented as the FWHM divided by twenty times the sensitivity. Figure 8(b) demonstrates that the differential mode achieves the optimal resolution capability. The linear regression for the differential LOD is described by y = 0.000854n₀ - 0.000478 RIU with an exceptional linearity of R² = 0.9993, ultimately reaching a minimal detection limit of roughly 4.18 × 10^-4 RIU. This superior performance metric indicates that the sensor is capable of resolving extremely minute variations in refractive index, thereby validating the suitability of the proposed device for high-precision biosensing and gas detection applications.

To further highlight the practical superiority of the proposed MO-DPCS, Table 3 presents a comprehensive comparison with recently reported optical refractive index gas sensors. Conventional sensing designs often exhibit highly restricted detection ranges, thereby limiting their applicability to a narrow selection of target analytes. In contrast, the differential sensing architecture developed in this study provides an exceptionally broad detection range spanning from 1.000 to 1.100 RIU. Coupled with the ultra-low Limit of Detection established in the preceding analysis, this expansive range empowers the proposed sensor to effectively monitor a highly diverse array of critical industrial and hazardous gases. As detailed in the table, these include methane, carbon monoxide, sulfur dioxide, and various nitrogen oxides, whose refractive indices are notoriously close to that of atmospheric air. This comparison conclusively validates the versatility and engineering potential of the MO-DPCS for complex environmental and industrial monitoring scenarios.

3.4. Temperature Cross-Sensitivity and Environmental Stability Analysis

A major obstacle in the implementation of semiconductor-based photonic sensors is the inherent heat sensitivity of the materials involved. The permittivity of InSb is very responsive to temperature fluctuations, resulting in a refractive index alteration that may be indistinguishable from the signal produced by the target analyte. This effect, termed temperature cross-sensitivity, significantly undermines detection accuracy in real settings where thermal fluctuations are unavoidable, potentially resulting in false positive results. The spectral response was examined throughout a temperature range of 274 K to 275 K to assess the robustness of the proposed device against external noise.

Figure 9(a) specifically illustrates the limitations of traditional single-mode detection. With a temperature rise of just 1 K, the cutoff angles for both TE and TM modes demonstrate a significant synchronous shift, transitioning from roughly +11° to -12°. The significant angular deviation over 20° suggests that even little thermal fluctuations may be misconstrued as considerable alterations in the refractive index of the target gas, hence compromising the reliability of single-mode measurements without stringent temperature stabilizing measures.

In contrast, the differential detection strategy effectively mitigates this issue by exploiting the common-mode nature of the thermal response. Since the temperature variation induces a similar kinematic shift in both polarization modes, the subtraction of the two cutoff angles inherently cancels out the majority of the thermal drift. The stability of the differential signal is presented in Figure 9(b). It is observed that the magnitude of the differential angular shift is confined within a narrow range of approximately 0.35° over the same temperature interval. Compared to the single-mode drift, the temperature cross-sensitivity is suppressed by nearly two orders of magnitude. This result confirms that the proposed differential sensor possesses a self-referencing capability, allowing it to distinguish the true refractive index signal from thermal noise and ensuring high measurement reliability in complex experimental conditions.

4. Conclusion

The results presented in this study demonstrate that the integration of the magneto-optical effect with 1D PCs provides a robust mechanism for enhancing refractive index sensing performance, validating the working hypothesis that the non-reciprocal nature of magnetized InSb can be manipulated to induce opposing spectral shifts in orthogonal polarization modes. Crucially, the structural efficacy was maximized through the implementation of the MODA, which enabled the precise inverse design of layer thicknesses to achieve an optimal RC that manual tuning could not easily attain. While previous studies have predominantly focused on simple geometric optimization to narrow resonance linewidths, the approach utilized here synergizes algorithmic design with the external magnetic field as a new degree of freedom, enabling a transition from absolute angular detection to a differential scheme that amplifies sensitivity from a negligible 0.8 °/RIU to a substantial 30.8 °/RIU. Interpreting these findings in a broader context, the proposed MO-DPCS addresses the critical challenge of environmental instability inherent to semiconductor-based sensing. Specifically, the differential strategy effectively converts the high thermal cross-sensitivity of InSb into a common-mode noise factor that is mathematically cancelled out, thereby ensuring high measurement reliability. Future research directions should focus on the experimental realization of the proposed structure via precision growth techniques such as molecular beam epitaxy, and potentially extending the concept to intelligent sensing systems where machine learning algorithms could further enhance real-time data demodulation and pattern recognition in miniaturized on-chip sensor arrays.