1. Introduction

Topological insulators have garnered significant attention due to their unique ability to excite topologically protected edge states (TES) [1,2]. These states have shown robustness against disturbances and can enhance light-matter interactions, making them applicable in various fields [3]. Recently, nanophotonic devices such as optical waveguides, nanoparticles and one-dimensional photonic crystal (1D-PhC) structures have been investigated for their potential to excite these edge states [4,5,6]. Here, 1D-PhC structures being particularly promising due to their design flexibility and ease of fabrication and characterization [7–9]. Considering two 1D-PhCs having overlapped photonic bandgaps and opposite Zak phase results in band inversion at a considered operating wavelength [10]. This facilitates the excitation of TES at the inversion wavelength. Recently, this concept has been utilized to excite Tamm plasmon polaritons (TPP) modes and Fano resonance [11,12]. Based on this, a 1D-PhC-based TES is utilized to design a refractive index sensor having 254.5 nm/RIU sensitivity [13], which can be improved to 616 nm/RIU using functional materials [14]. Furthermore, a polarization independent sensor having 60°/RIU (70°/RIU) for TM (TE) polarized light is also reported [15]. This concept has also been utilized to develop a low threshold optical bistability [16].

The performance of these structures can further be improved by considering the gradient refractive index profile. The quest for cutting-edge and precise sensing technologies has led to the development of innovative sensors based on graded index photonic crystal (GPhC). These 1D-GPhC structures are designed considering multi-layers having gradual refractive index variation along the fixed thickness or thickness gradient at a constant refractive index [17,18]. The existing research on 1D-GPhCs has primarily focused on bandgap engineering for linear, sawtooth and exponential-graded index materials [19,20]. But very few literatures have been available on the advancement of 1D-GPhC structures that integrate materials with hyperbolic graded index. The hyperbolic variations of refractive indices in PhC structures hold great potential for integrating optical devices, offering novel and intriguing effects. The gradual change in refractive index, combines the benefits of other graded configurations. This provides precise control over mode confinement, significantly reduced interface losses and dispersion management. Nonetheless, it’s worth noting that the topological characteristics of hyperbolic graded photonic crystals (HGPhCs) remain unexplored till lately. Leveraging the HGPhC structure to excite topologically protected edge states (TES), holds the promise of significantly enhancing sensing performance due to the combination of graded refractive index and topological properties. This offers improved control over dispersion characteristics and enhance the electric field intensity at the interface, leading to more efficient interactions between light and matter. Further investigation and validation studies are necessary to fully understand the potential of these techniques.

To address this gap, the current study presents a novel approach using a hyperbolic-graded one-dimensional photonic crystal (1D-PhC) to excite the TES. The study optimizes two distinct HGPhC structures made from porous silicon, specifically aligning their optical thicknesses to create overlapping photonic bandgaps (PBG) with opposing topological characteristics, specifically the Zak Phase. As a result, TES are excited, exhibiting significantly higher electric field intensity at an operational wavelength of 1521 nm. Additionally, the research explores the impact of varying interface layer thicknesses on the excitation of TES. Finally, the research demonstrates the structure’s potential as a refractive index sensor, employing cholesterol as an analyte. The analytical results showcase an average sensitivity of 852.14 nm/RIU, a quality factor of 4019.23, and a figure of merit (FOM) of 1277.13 RIU⁻¹. Importantly, the proposed structure offers a 489% increase in sensitivity, a remarkable 28166% improvement in FOM, and a substantial 16358% enhancement in quality factor when compared to previous approaches [21].

2. Theoretical analysis and design

The schematic of the proposed 1D-GPhC structure with hyperbolic grading is shown in Fig. 1. Here, two 1D-PhC structures PhC1 (substrate $\left| {{\left( {n_{A1},\,n_{{\rm B1}}} \right)}^7} \right|$ Air) and PhC2 (substrate $\left| {{\left( {n_{B2},\,n_{{\rm A2}}} \right)}^7} \right|$ Air) made of the same material of silicon (having RI (3.45)) are considered and represented in Fig. 1(a) and Fig. 1(b).

 

Fig. 1. Schematic representation of proposed (a) Graded index PhC1 structure, (b) Graded index PhC2 structure, and (c) Corresponding photonic band gap.

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A BK7 prism of refractive index 1.515 is used as a substrate to couple the incident light [22]. The porosity is introduced to obtain the desired refractive index profile in different layers. The layer thicknesses are optimized to have overlapped photonic bandgaps (PBG). The PhC1 comprises step index material ‘B1’ having low refractive index (n_B1) of 1.6 (with 80% porosity), while material ‘A1’ is considered as a hyperbolic graded high refractive index (n_A1) ranging from 1.6 (with 80% porosity) to 3.45 (with 0% porosity) as shown in Fig. 1(a). Similarly, PhC2 comprises step index material ‘B2’ having low refractive index (n_B2) of 1.8 (with 70% porosity), while material ‘A2’ is considered as a hyperbolic graded high refractive index (n_A2) ranging from 1.8 (with 70% porosity) to 3.45 (with 0% porosity) as shown in Fig. 1(b).

The graded index profile can be obtained by introducing the porosity within the material. This also enables effective infiltration of analytes. A layer’s porosity (P) can be analytically calculated using Eq. (1) [23].

In Eq. (1), the refractive indices n_p, n_a, and n_ds represent the refractive indices of the porous material, air/analytes, and dense material, respectively. The effective RI of high-index material in PhC1 is considered the average RI (1.6 to 3.45), while the effective RI of high-index material in PhC2 is considered the average RI (1.8 to 3.45). The hyperbolic refractive index of material ‘A’ along the layer thickness from the starting point (d(0)) to the final point (d(y)) is calculated using $\textrm{}{n_H}(y )= \frac{{{n_i}}}{{1 - \alpha y}}$ [24]. Here, the hyperbolic grading parameter ‘$\alpha$’ is determined as $\alpha = [({n_f} - {n_{i\textrm{}}})/({n_f}.d)]\textrm{}$ having ‘n_i’ as the initial refractive index, ‘n_f’ as the final refractive index and ‘D’ is the physical thickness of the layer. The electric field distribution along the plane perpendicular to the hyperbolic grading layer’s surface is calculated using Eq. (2) to Eq. (4).

2.1 Topological effect and sensing application

Topological resonating states can be realized at junction of two 1D-PhC structures exhibiting overlapping photonic bandgaps, each possessing distinct topological properties or Zak phases. The optimized physical thicknesses D_A1 (153 nm), D_B1 (242 nm), D_A2 (148 nm), and D_B2 (215 nm) results in an overlapped PBG of width 556 nm (from 1270 nm to 1826nm) which is shown in Fig. 1(c). Moreover, the contrasting Zak phase nature further increases the probability of exciting a topological resonating state at the interface of structure. The hyperbolic graded topological 1D-PhC structure made of combined PhC1 and PhC2 structures to excite the TES is shown in Fig. 2(a).

 

Fig. 2. (a) Schematics of proposed hyperbolic graded topological structure, and (b) Corresponding reflectance spectrum having TES and TRS.

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The Zak phase typically takes quantized values of 0 or $\pi$ and can be computed using the eigen-frequency band of the 1D-PhC, as given in Eq. (5) [25]. In this context, the dielectric function is represented as $\varepsilon (z ),\; $ the wave vector is indicated as ‘k’ and the eigenfunction of the n_th band Bloch electric field is denoted as ${u_{n,k}}(z )$. Hence, computing the Zak phase for each individual photonic band gap (PBG) imparts topological characteristics. This results in excitation of TES at 1521 nm, which is shown by black dotted lines in Fig. 2(b). The robustness of this TES stems from its intrinsic excitement due to the topological characteristics of the structure, and it remains unaffected by minor perturbations in the surrounding environment. Initially, the impact of incidences angle on electric field confinement at the interface is analyzed and is shown in Fig. 3. The structure shows the maximum electric field confinement of around 0.807 × 10⁶ V/m at normal incidence, which increases to 2.16 × 10⁶ V/m at 20-degree incidence angle.

 

Fig. 3. Normalized electric field distribution along the length of the device.

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High electric field confinement observed in topological structures has versatile applications, such as in laser cavities, waveguides, and optical switches. While the TES remains unaffected by its surrounding perturbations, the combination of two adjacent interface layers leads to the creation of a hyperbolic graded topological resonator. This topological resonator structure is subsequently employed for sensing applications. The enhanced E-field intensity within the resonator facilitates superior light-matter interaction, leading to improved sensitivity [26,27]. Furthermore, the E-field intensity remains nearly constant across different infiltrated analyte concentrations in the topological cavity structure, highlighting the robustness of the excited topological resonance state. The topological cavity is formed by combining two interface layers. Initially, the cavity thickness (DL) is set at 457 nm, which is the combined value of D_B1 (242 nm) and D_B2 (215 nm) and the considered refractive index is the average of two interface layers (n_D = 1.6941). This results in the excitation of topological resonating state (TRS) at 1766nm, which is shown in Fig. 2(b).

The excited resonance mode is highly dependent on cavity thickness. Thus, the analysis is extended to see the impact of cavity thickness on the excitation characteristics of the TRS, which is shown in Fig. 4. Increasing the cavity thickness from 1.0DL to 4.0DL leads to the excitation of resonating TRS within wavelength range of 1600 nm to 2300 nm. The structure shows resonating TRS excitation at 1766nm, 2057nm, 1710nm, 1919nm, 1681 nm, 1837nm, and 1663 nm operating wavelengths for corresponding defect layer thicknesses of 1.0DL, 1.5DL, 2.0DL, 2.5DL, 3.0DL, 3.5DL, and 4.0DL, respectively, as shown in Fig. 4. Additionally, the proposed structure shows a very narrow FWHM of less than 0.5 nm, thus showing its potential in tunable sensing and filter applications [28]. This provides an added advantage of increased defect layer thickness to enhance light-matter interaction hence sensitivity.

For sensing applications, the cavity is infiltrated with the chosen analyte, which, in this research, is cholesterol. The experimentally verified refractive index of cholesterol, which varies with concentration, is utilized for the sensing analysis [29]. By gradually increasing the cholesterol concentration in 20 mg/dl increments, the refractive index spans from the normal range (200 mg/dl) to the high-risk range (280 mg/dl), and the resulting shift in the resonant wavelength is examined. This give the average refractive index of around 2.49, 2.62, 2.80, 3.06, 3.23, and 3.47 for corresponding cholesterol concentration of 200 mg/dl, 220 mg/dl, 240 mg/dl, 260 mg/dl, and 280 mg/dl, respectively. The structure exhibits the excitation of a topological resonating state at an operating wavelength of 2352 nm when the infiltration contains 200 mg/dl (within the normal range) of cholesterol concentration. This increases to 2901 nm for 280 mg/dl cholesterol concentration as depicted in Fig. 5(a)

 

Fig. 4. Cavity thickness dependent resonance wavelength variations in hyperbolic graded topological resonator.

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Fig. 5. Reflectance response of hyperbolic graded topological structure “Substrate|(n_A1, n_B1)⁶ | n_A1|D|n_A2|(n_B2, n_A2)⁶|Air” for DL defect layer thickness at varying incidence angles of (a) 0-degree, and (b) 20-degree.

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Increasing the incidence angle results in blue shift in TRS wavelength. For 20-degree incidence angle, the proposed structure excitation of TRS at 2313 nm and 2871 nm operating wavelength for 200 mg/dl and 280 mg/dl cholesterol concentration as shown in Fig. 5(b). This results in average sensitivity ($s = \Delta \lambda /\Delta N$) of around 740 nm/RIU and 755 nm/RIU at 0-degree and 20-degree incidence angle for cavity thickness of DL [30]. It’s worth noting that the sensitivity is also influenced by the thickness of the cavity layer [31]. Therefore, the analysis is further extended, and the sensor performance is compared with varying the cavity layer width from 1.0DL to 4.0 DL at the angle of incidence of 0-degree and 20-degree and corresponding sensitivity is recorded. Figure 6, shows the comparative sensitivity analysis at both the incidence angles and four different cavity thicknesses. At a 0-degree incidence angle, the defect layers with thicknesses of 1.0DL, 2.0DL, 3.0DL and 4.0DL exhibit average sensitivities of approximately 740 nm/RIU, 781 nm/RIU, 812 nm/RIU and 839 nm/RIU respectively. When the incidence angle is increased to 20 degrees, the average sensitivities for the same defect layer thicknesses improve to 755 nm/RIU, 802 nm/RIU, 829 nm/RIU and 852 nm/RIU, respectively. This gives the maximum quality Factor ($Q = \lambda R/FWHM$) of 4019.23 and a figure of merit ($FOM = S/FWHM$) of 1277.13 RIU⁻¹ at 20-degree incidence angle for 4.0DL cavity thickness. Finally, a proposed structure’s sensing performance is compared with the recently reported values and is documented in Table 1. This demonstrates that the performance of the proposed biosensor surpasses that of current biosensing research endeavors, as indicated by its superior average sensitivity, FOM, and quality factor. Moreover, the structure can easily be fabricated using stain-etching and anodization techniques [32–34].

Table 1. Comparative performance analysis with recently reported results

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Fig. 6. Comparative sensitivity analysis of four different cavity thicknesses at 0-degree and 20-degree incidence angle.

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3. Conclusions

This paper focuses on exciting topological protected edge states using hyperbolic-graded photonic crystal structure. The proposed structure utilizes two connected photonic crystal structures having overlapping bandgaps and opposite Zak phases. By incorporating a hyperbolic graded refractive index profile, the dispersion characteristics of the device are effectively modified. The combination of two interface layer results in excitation of a topological resonance state within the photonic bandgap. The reflectance spectrum is analyzed using the finite element method. Remarkably, the proposed graded topological structure exhibits a sensitivity of 852 nm/RIU and an impressive FOM of 1277 RIU⁻¹. Overall, the proposed device holds great promise for highly efficient and accurate detection and sensing of cholesterol concentration.