Thermal Stability Analysis of Surface Wave Assisted Bio-Photonic Sensor

Abstract

In this paper, the thermal stability of a Bloch Surface Wave (BSW) assisted bio-photonic sensor is investigated. The structural analysis is carried out using the transfer matrix method (TMM). The design comprises a truncated one-dimensional photonic crystal (1D-PhC) structure along with a defective top layer. The structural parameters are optimized to excite a BSW at the top interface for an operating wavelength of 632.8 nm. The mode confinement is confirmed by using wavelength interrogation, angular interrogation and surface electric field profile. Further, the effect of thermal variation on BSW excitation angle and sensitivity is carried out. The analysis shows the average variations in excitation angle and sensitivity of about −0.00096 degree/$°\mathrm{C}$ and 0.01046 (degree/RIU)/$°\mathrm{C}$, respectively. Additionally, the analysis is also extended towards different lower wavelengths of 400 nm and 550 nm, which provides average variations in the excitation angles of about −0.0027 degree/$°\mathrm{C}$, and 0.0016 degree/$°\mathrm{C}$. This shows that the structural sensitivity response is more thermally stable at the lower wavelength range. Thus, showing its potential applications in designing thermally stable bio-photonic sensors.

1. Introduction

Surface wave based optical sensing devices are a prominent research area globally. The Surface Plasmon Polaritons (SPP) is a very prominent technique, which founds its applications in integrated photonic, healthcare monitoring, medical diagnostics, drug detection, and environmental monitoring [1,2,3,4]. It is also widely used as a label free biosensor to study molecular interactions along with their binding mechanism and wave-guiding applications. The most interesting and important features of SPR devices are their good sensitivity, instantaneous response, and label free detection [5,6,7,8]. However, its performance improvement is limited because of the higher absorption losses, hence lower sensitivity and figure-of-merit (FOM). The SPR can only be used to excite TM modes (longitudinal modes) with some specific momentum coupling technique (grating or prism). Moreover, operating wavelength tuning is also not possible in the SPR based devices. These problems can be alleviated by replacing the metallic top layer by a defect dielectric layer or 2D materials (i.e. Graphene, Transition-metal Dichalcogenides) on a truncated interface of a one-dimensional photonic crystal (1D-PhC) structure [9,10]. This leads to the excitation of plasmonic-like resonance mode called Bloch Surface Wave (BSW) or Optical Tamm States (OTS) at the interface of one-dimensional photonic crystals (1D-PhC) and a defect layer [11,12,13]. Excitation of Tamm Plasmon Polariton is not possible without a metallic layer; however, Optical Tamm state and Bloch surface mode can be excited using dielectric materials [12,14,15,16,17].

Recently, a number of optical sensors using surface waves have been proposed in the literature for biochemical sensing applications. Bijalwan et al. (2020) proposed a graphene and WSe${}_2$ (TMDC) material based structure and reported an average sensitivity of 163°/RIU (WSe${}_2$) and 160°/RIU (graphene), respectively [18]. The PtSe${}_2$ TMDC-based sensor is also explored for RI-based sensing applications [19]. The authors reported a gold and silver assisted sensitivity of around 165°/RIU and 162°/RIU, respectively. A purely dielectric structure-based 1D-PhC surface wave sensor structure having a top dielectric defect layer as a cover is introduced for sensing purposes [20]. The authors monitored the shift in coupling angle by infiltrating various concentrations of hemoglobin at a top air–dielectric interface and reported an average sensitivity of 69°/RIU. Recently, in 2021 Gryga et al. [21] also proposed a purely dielectric based 1D-PhC surface wave sensor and reported a sensitivity of around 1456 nm/RIU for an analyte refractive index of 1.000–1.005. Some other surface wave based integrated optics devices such as waveguides [22,23], lenses [24,25], resonators [26,27] and interferometers [28] have also been realized.

The characteristic parameters (such as excitation angle, mode wavelength and sensitivity, etc.) of these devices are very sensitive to the thermal variation [29]. Yaremchuk et al. [30] studied the impact of temperature variation on SPR based sensor performance and reported a 0.23 degree change in excitation angle for a corresponding thermal variation from 100 K to 800 K. Moreover, the spectrum is also broadened, which deteriorate the accuracy of the sensor. A grating coupled SPR based sensor structure has been evaluated [31]. The authors reported a sensitivity variation of 0.002 ((degree/RIU)/$°\mathrm{C}$) for gold metal in the temperature range of 300 K to 400 K. Similarly, an 88.7 pm/$°\mathrm{C}$ sensitivity variation has also been reported for an optical Tamm state based sensor structure [32]. Most of the reported thermal studies have been performed on SPR based sensor structures and confirm the adverse impact on device performance. Less attention has been paid to the study of thermal effect on BSW based devices for most of the mentioned applications. Since temperature exhibits its prominent impact on the device performance, a detailed analysis is required to measure its impact on BSW excitation and the corresponding sensing application, specifically, for the devices in which only dielectrics are employed.

In this paper, a thermal stability analysis of a designed BSW assisted bio-photonic sensor is carried out. The structure comprises alternate dielectric layers of titanium dioxide (TiO${}_2$) and silicon dioxide (SiO${}_2$), which possess a photonic band-gap (PBG) centered on a 632.8 nm wavelength. A top defect layer of SiO${}_2$ is deliberately introduced to excite Bloch Surface modes at the interface. The detailed analysis is carried out to investigate the effect of varying surrounding temperatures on the excitation angle of BSW at a 632.8 nm wavelength. Further, the impact of thermal variation is also studied at a comparatively lower wavelengths of 550 nm and 400 nm, respectively. Finally, detailed temperature dependent sensitivity analysis at all three wavelengths is also carried out. The analysis exhibits that for a SiO${}_2$ and TiO${}_2$ material based structure, the BSW excitation angle variation is smaller for a higher wavelength range, whereas sensitivity response is more thermally stable at the lower wavelength range. Thus, showing its major potential applications in developing thermally stable biophotonic sensors. Additionally, the analysis can also be extended to design other thermally stable BSW based devices such as waveguides, filters and resonators.

2. Structure Design and Methods

A multilayer PhC structure is designed by considering alternate layers of material ‘A’ (high refractive index) and material ‘B’ (low refractive index) on glass substrate as shown in Figure 1a. The structure possesses periodicity in the ’y’ direction with defect layer ’D’ at the end for the confinement of BSW. The structure is designed considering a quarter wavelength Bragg stack having ni × di $=\lambda o/4$ [33,34], where ni and di are the refractive index and physical thickness of ith layer, and $\lambda$o is the central reflected wavelength [35,36]. The phase thicknesses ($\delta$) of each layer is calculated by considering optical thickness (L), frequency (f) and angle of refraction ($\theta$) from each layer and is represented in Equation (1). The reflected electric field profile of the proposed design can be calculated by considering the summation of both forward and backward propagating fields. This can be represented by a Helmholtz equation as shown below in Equation (2) [11].

$$\delta =2\pi \frac{f}{f_0}LCos\left(\theta \right)$$

(1)

$$E(y,x)=A_n{e}^{i{k}_{n}y}+B_n{e}^{-i{k}_{n}y},$$

(2)

where A${}_n$ and B${}_n$ are the amplitudes of forward and backward propagating waves, ‘n’ is the layer number and k${}_n$ is the wavenumber. The structural parameters are optimized such that overall constructive interference occurs at the last layer.

This electric field is calculated for all the considered layers and the field components in reflected and transmitted waves are calculated. Since the Eigen value problem needs to be solved for a periodic structure, the Floquet theorem (Equation (3)) along with continuous boundary condition is imposed to formulate the Eigen value problem, which is given by Equation (4) [37,38].

$$E(y,x)=E\left(y\right)e^{iKy}{e}^{i\beta x}$$

(3)

$$e^{-iK\Lambda }\left(\begin{array}{c}{A}_n\\ B_n\end{array}\right)=\left(\begin{array}{cc}M1& M3\\ M2& M4\end{array}\right)\left(\begin{array}{c}{A}_n\\ B_n,\end{array}\right)$$

(4)

where constant ‘K’ is known as the Bloch wave number, $\beta$ is the phase propagation constant, Mn is the matrix parameter to formulate the ‘M’ matrix in the TMM method and $\Lambda$ is the period of one stack. The solution of Equation (4) provides Eigen value (e${}^{-iK\Lambda }$). Thus, the dispersion relation for the proposed surface wave structure can be calculated and represented by Equation (5).

$$K(\beta ,\omega )=\frac{1}{\Lambda }co{s}^{-1}\left(\frac{1}{2}(M1+M4)\right).$$

(5)

Here, the Bloch wave vector is a complex quantity, which results in both evanescent and propagating surface modes at the top interface. The structure optimizations and sensing analysis are carried-out using the transfer matrix method (TMM) [38,39]. The structural parameters such as layer thicknesses, refractive index contrast, and number of stacks are optimized to obtain the highest possible sensitivity at different central wavelengths of 632.8 nm, 550 nm and 400 nm, respectively. Suitable thermo-physical, experimentally verified values for $n(\lambda$, T) and $ϵ(\lambda$, T) have been considered for the analysis. This gives a thickness of SiO${}_2$ and TiO${}_2$ to be around 128 nm and 85 nm, respectively for corresponding refractive indices of 1.46 and 2.2. The imaginary parts of the dielectric constant are also considered to be 0.0007i and 0.0001i for TiO${}_2$ and SiO${}_2$, respectively for the consideration of the lossy nature of the dielectric material.

The defect layer thickness is also adjusted to generate optimized BSW. This results in the confinement of BSW at the excitation angle of 42.25 degrees, 46.48 degrees and 58.08 degrees at the corresponding operating wavelengths of 632.8 nm, 550 nm and 400 nm, respectively, as shown in Figure 1b. Here, the BSW has been excited for an optimized defect layer thickness of 165 nm for a six stacked 1D-PhC structure. The mode excitation has also been confirmed by wavelength interrogation and the surface electric filed profile. Here, the analysis is carried out to confirm the BSW excitation at 632.8 nm operating wavelength for Glass|(SiO${}_2$,TiO${}_2$)${}^6$|SiO${}_2$|air structure. Figure 2a represents the wavelength interrogation of excited BSW at an incident angle of 42.25 degrees. This clearly shows a strong dip in the reflection spectrum at 632.8 nm wavelength. The same has also been confirmed by seeing the electric field profile at the top interface. Figure 2b represents the spectral electric field distribution and mode confinement in the proposed structure for a defect layer thickness of 165 nm. Additionally, the confined modes are evanescent in nature, thus showing its potential applications in bio-photonic sensors.

Figure 3 represents the impact of thermal variations on the effective refractive index (n${}_{eff}$) of the considered structure. The n${}_{eff}$ is calculated by considering the device parameters such as periodicity ($\Lambda$), filling factor of each layer (F), and refractive indices of the low (n${}_l$) and the high index (n${}_h$) materials. Thus, n${}_{eff}$ is calculated by Equation (6) [40].

$$n_{eff}=\sqrt{F_l{\epsilon }_l+F_h{\epsilon }_h},$$

(6)

where $F_l$ and $F_h$ are the filling factor (F = d/$\Lambda$) of low and high refractive index layer along with ${\epsilon }_l$ and ${\epsilon }_h$ are the corresponding materials permittivity. The thermal variation will affect the material permittivity and thus results in change in the n${}_{eff}$ of the structure. For a central wavelength of 632.8 nm, the calculated n${}_{eff}$ comes to be around 1.82569, 1.82227, 1.82171, 1.82111 and 1.82055 for corresponding thermal variations of ($\Delta T$ =) 5$°$, 10$°$, 15$°$ and 20$°$, respectively. The analysis has also been carried out to measure the n${}_{eff}$ at lower wavelengths (550 nm and 400 nm) and the same trend has been observed. The comparative n${}_{eff}$ analysis is represented in Figure 3 and summarized in

Table 1. Effect of increasing temperature on excitation angle and sensitivity at different incident wavelengths of 400 nm, 550 nm and 632.8 nm.

Incident Wavelength	Temperature Variations ($\Delta \mathit{T}$)	Effective Refractive Index	Excitation Angle (Degree)	Sensitivity (Degree/RIU)	Excitation Angle Variations	Sensitivity Variation
400 nm	0 $°\mathrm{C}$	1.89546	58.08219	5.50975	−0.0027 (Degree/$°\mathrm{C}$)	0.00231 ((Degree/RIU)/$°\mathrm{C}$)
	5 $°\mathrm{C}$	1.89122	58.06899	5.52432
	10 $°\mathrm{C}$	1.89051	58.05579	5.53118
	15 $°\mathrm{C}$	1.88978	58.04199	5.54404
	20 $°\mathrm{C}$	1.88907	58.02819	5.55775
550 nm	0 $°\mathrm{C}$	1.84284	46.48411	23.90073	−0.0016 (Degree/$°\mathrm{C}$)	0.00389 ((Degree/RIU)/$°\mathrm{C}$)
	5 $°\mathrm{C}$	1.83921	46.47631	23.91959
	10 $°\mathrm{C}$	1.83861	46.46851	23.93587
	15 $°\mathrm{C}$	1.83798	46.46011	23.96159
	20 $°\mathrm{C}$	1.83738	46.45231	23.97702
632.8 nm	0 $°\mathrm{C}$	1.82569	42.24868	47.0266	−0.00096 (Degree/$°\mathrm{C}$)	0.01046 ((Degree/RIU)/$°\mathrm{C}$)
	5 $°\mathrm{C}$	1.82227	42.24388	47.0788
	10 $°\mathrm{C}$	1.82171	42.23908	47.132
	15 $°\mathrm{C}$	1.82111	42.23428	47.18174
	20 $°\mathrm{C}$	1.82055	42.22948	47.2366

.

3. Results and Discussions

In order to characterize the effect of thermal fluctuations on device performance, thermo-optic coefficient (TOC) and thermal expansion coefficient (TEC) of both SiO${}_2$ and TiO${}_2$ are considered. The TOC of titanium dioxide measured for coatings deposited by electron beam deposition is negative and equal to −1.7 × 10${}^{-4}$ K${}^{-1}$ between 18 $°\mathrm{C}$ and 120 $°\mathrm{C}$ and corresponds to −3.04 × 10${}^{-4}$ K${}^{-1}$ between 220 $°\mathrm{C}$ and 325 $°\mathrm{C}$ [41]. The TOC of silicon dioxide is positive with a value around 1 × 10${}^{-5}$ K${}^{-1}$ [42]. The positive TOC of SiO${}_2$ implies that the rise in temperature will cause an increase in effective refractive index of the device. Since the TOC of both the materials is different and opposite in sign, this results in an overall decrease in the effective refractive index of devices. Besides the increase in the refractive index, the device heating also changes the effective optical path length due to positive thermal expansion coefficient. All these factors contribute to the shift in the excitation angle of BSW confinement. However, the refractive index change due to the thermal expansion coefficient is expected to be one order less compared to that of TOC. Thus, in the analysis, only the thermo-optic coefficients of both SiO${}_2$ and TiO${}_2$ are considered.

Figure 4a represents the effect of temperature variation on the reflection spectrum of Glass|(SiO${}_2$,TiO${}_2$)${}^6$|SiO${}_2$|air structure. The increase in device temperature causes a blue shift in the excitation angle required to confine BSW at the same incident wavelength of 632.8 nm. This is because of the decrease in the overall effective refractive index of the device. This also affects the overall reflection intensity. Figure 4b represents the dependency of excitation angle and normalized reflectance Rn (=R/R0) as a function of $\Delta T$ (= T − Tamb). Here, R0 is the reflection intensity for an ambient temperature of 25 $°\mathrm{C}$ ($\Delta T$ = 0). This clearly indicates that the increase in device temperature leads to a decrease in both required excitation angle (at fixed incident wavelength) and reflection intensity. This is because of the decrease in effective refractive index, which leads to a change in the phase thickness. Thus, the phase matching condition would be satisfied at a lower excitation angle. From the linear fit, we obtain the slope $\partial Rn/\partial T$ = −0.000745/$°\mathrm{C}$ for normalized reflection and $\partial \theta /\partial T$ = −0.00096 degree/$°\mathrm{C}$ for excitation angle along with coefficients of determination R${}^2$ of around 0.9924 and 1, respectively.

Since incident wavelength also affects the device performance [20], therefore, thermal analysis is also carried out for lower incident wavelengths of 550 nm and 400 nm as well and shown in Figure 5. The decrease in incident wavelength also confines the BSW at the top interface but at different incident angles. Therefore, BSW excitation is observed at 46.48 degrees and 58.08 degrees for corresponding operating wavelengths of 550 nm and 400 nm, respectively. Additionally, this leads to higher thermal instability in the required excitation angle. The temperature dependent variation in excitation angle $\partial \theta /\partial T$ comes to be around −0.0027 and −0.0016 along with coefficients of determination R${}^2$ of around 0.9998 and 0.9997 for an incident wavelength of 400 nm and 500 nm, respectively. Finally, the effect of temperature variation on the sensitivity of the proposed structure is analysed at all incident wavelengths of 400 nm, 550 nm and 632.8 nm respectively. The RI of infiltrated analyte is varied from 1.00 to 1.1 with a step size of 0.02 and the corresponding reflection spectrum is measured.

Increasing the analyte refractive index leads to a red shift in excitation angle and the same trend has been observed in all configurations. This is because of the increased effective index of the device [43,44], which directly affects the phase thickness. Hence, a shift in the excitation angle is observed. Figure 6a represents the temperature dependent reflectivity spectrum for an incident wavelength on 400 nm at varying infiltrated analyte RI. It is evident from Figure 6a that increasing the infiltrated analyte RI leads to red shift in the required BSW excitation angle at all device temperature ranges. For $\Delta T$ = 0$°$, the excitation angle is red shifted to 58.64 degrees for the analyte refractive index (RI) of 1.1. Similarly, the same has been measured to be around 58.63 degrees, 58.61 degrees, 58.60 degrees and 58.586 degrees for corresponding temperature variations of ($\Delta T$ =) 5$°$, 10$°$, 15$°$ and 20$°$, respectively.

This sensitivity analysis is extended for the other two operating wavelengths of 550 nm and 632 nm as well. Figure 6b compares the temperature dependent sensitivity analysis at incident wavelengths of 400 nm, 550 nm and 632.8 nm, respectively. This shows that an increase in operating wavelengths improves structural sensitivity response. Moreover, the sensitivity variation in all three configurations is also almost linear. Figure 7 represents the temperature dependent sensitivity comparison at incident wavelengths of 400 nm, 550 nm and 632.8 nm. Further, it is also evident from the analysis that an increase in device temperature leads to comparatively higher sensitivity.

It is evident from Figure 7 that the higher the operating wavelength, the greater the thermal instability in the structural sensitivity performance. There is a maximum 0.1046 degrees/RIU change in sensitivity for every 10 $°\mathrm{C}$ variation in temperature at an incident wavelength of 632.8 nm. This sensitivity variation decreases to 0.0389 degrees/RIU and 0.0231 degrees/RIU for an incident wavelength of 550 nm and 400 nm, respectively, with coefficients of determination R${}^2$ of around 0.99271 and 0.98648. Additionally, the temperature dependent variation in excitation angle ($\partial S/\partial \theta$) comes to be around −0.00096 degrees/$°\mathrm{C}$, −0.0016 degrees/$°\mathrm{C}$ and −0.0027 degrees/$°\mathrm{C}$ for incident wavelengths of 632.8 nm, 550 nm and 400 nm, respectively. The obtained results are summarized in Table 1.

4. Conclusions

In this paper, thermal stability analysis is carried out for a proposed SiO${}_2$ and TiO${}_2$ material based Bloch Surface Wave assisted photonic crystal bio-photonic sensor. The structural parameters are optimized to excite BSW at the top interface and the same is confirmed by wavelength interrogation, angular interrogation and a surface electric filed profile. The detailed analysis reveals that the structure is highly sensitive and possesses more thermally stable excitation angle variations at higher operating wavelengths. The optimized structure exhibits average excitation angle variations of about −0.0027 degrees/$°\mathrm{C}$, −0.0016 degrees/$°\mathrm{C}$ and −0.00096 degrees/$°\mathrm{C}$ for incident wavelengths of 400 nm, 550 nm and 632.8 nm, respectively. Furthermore, the performed analysis also shows a 0.1046 degrees/RIU maximum change in sensitivity for every 10 $°\mathrm{C}$ change in temperature at a 632.8 nm wavelength.