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Microring resonator based on polarization multiplexing for simultaneous sensing of refractive index and temperature on silicon platform

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Abstract

Silicon photonic integrated sensors based on microring resonators are a promising candidate to achieve high-performance on-chip sensing. In this work, a novel dual-parameters sensor based on polarization multiplexing on silicon-on-insulator (SOI) platform is proposed and demonstrated experimentally, simultaneously achieving refractive index (RI) and temperature sensing with high sensitivity and large detection range (DR). The experimental results show that the RI sensitivity and temperature sensitivity of the TM-operated sensor are 489.3 nm/RIU and 20.0 pm/°C, respectively, and that of the TE-operated sensor are 102.6 nm/RIU and 43.3 pm/°C, respectively. Moreover, the DR of the fabricated sensor is 0.0296 RIU, which is 4.2 times that of the conventional TM-operated sensor based on the microring resonator. The dual-parameters sensor based on polarization multiplexing can successfully realize the simultaneous measurement of the RI and the temperature, showing potential applications of silicon photonic on-chip sensors in reality.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Silicon photonic integrated sensors have attracted intensive attention due to their high sensitivity, compact footprint, low manufacturing cost, combined with high-volume production capability [13]. A variety of silicon photonic integrated sensors have been proposed before and are mainly divided into several types according to the configuration, including Mach–Zehnder interferometer (MZI) [4], Bragg grating [5], microring resonator [6], and photonic crystal cavity [7]. Among all various sensing configurations, the microring resonator featuring ultra-compact footprint, high quality factor (Q-factor), and multiplexed capabilities is particularly fascinating for on-chip optical sensing [610].

For silicon photonic sensors based on microring resonators, the refractive index variations of the analyte are transferred to the optical intensity variation or resonant wavelength shift. Typically, the resonant shift can be determined with high accuracy, while the intensity interrogation with high accuracy is difficult to be achieved without a highly stable measurement system [11]. Thus, it is more feasible to realize high-performance microring resonator-based sensors using wavelength interrogation. Considerable research for refractive index sensing has been reported and some of them show remarkable performance in sensitivity and detection limit [6,12]. However, the periodic transmission spectrum of the microring resonator leads to ambiguous detecting, since the resonance tracking is potentially disturbed by adjacent resonance peaks [13]. Consequently, the detection range (DR) of the microring resonator optical sensor is inversely proportional to its sensitivity, indicating a trade-off between DR and sensitivity for a typical microring resonator-based on-chip sensors. However, achieving microring resonator-based sensors with both high sensitivity and large DR are of great significance in practice. Several solutions have been proposed before, such as the sensor based on cascade rings [9] and the sensor patterned with a periodically arranged set of gold nanodisks [14]. Nevertheless, in practical biochemical sensing applications, the resonant wavelength of refractive index-based sensors is affected by the ambient temperature drift [1517]. Therefore, it is critical to illuminate the cross-talk caused by temperature drift or achieve temperature self-calibration for practical biomedical applications. There are mainly two approaches to overcome this limitation. One approach is to dynamically heat the sensor and compensate for the resonant wavelength shift, and the other is to clad the device with negative thermo-optic coefficient (TOC) materials. However, those two approaches are not appropriate for achieving temperature-independent optical sensors, since the former consumes extra power and the latter conflicts with the sensitivity enhancement of sensors. Therefore, the simultaneous measurement of the resonant wavelength shift in different optical modes has been proposed to illuminate the cross-talk of the temperature and obtain the variation of the temperature, as the resonant wavelength shifts of different optical modes is different to the identical variation of the analyte [18]. However, optical sensors utilizing this method reported before are mostly based on optical fibers [1820] and photonic crystals [21], while there are few reports focused on on-chip sensors based on microring resonators [22,23].

In this work, a novel dual-parameters sensor based on polarization multiplexing is proposed and demonstrated to achieve simultaneous measurement of refractive index and temperature with high sensitivity and a large DR on the silicon photonics platform. The proposed device integrates a polarization beam splitter (PBS) with two microring resonators based on transverse magnetic (TM) mode and transverse electric (TE) mode, respectively. Due to the different mode field distribution of the TE mode and the TM mode, the responses of the two resonators show different sensitivity to the RI and the temperature. The different RI sensitivities and temperature sensitivities of two microring resonators operated in different modes can be combined to construct a characteristic matrix, which is used to obtain information on the variations of refractive index and temperature. The high RI sensitivity of the TM-based sensor combined with the large DR of the TE-based sensor simultaneously achieves a high sensitivity of 489.3 nm/RIU and a wide detection range of 0.0296 RIU. The temperature sensitivity of TM-operated and TE-operated microrings are 20.0 pm/°C and 43.3 pm/°C, respectively. The proposed novel dual-parameters sensor based on polarization multiplexing shows the promising potential of silicon photonic on-chip sensor with simultaneous measurement of multi-analytes.

2. Working principle and device design

2.1 Waveguide analysis

The waveguide is one of the most important components of the optical sensing device. The difference between the strip waveguide based on TE mode and TM mode on the aspects of electric field (E) distribution and waveguide sensitivity (Swg) are analyzed in this section. The thickness and width of the silicon core channel are h = 220 nm and w = 500 nm, respectively. The finite difference eigenmode (FDE) method is adopted to characterize the strip waveguide in different polarization states. Here, the refractive indices of Si, SiO2, and the analyte are 3.45, 1.45, and 1.33, respectively.

The cross-sections and E distributions of the strip waveguide in TE mode and TM mode are shown in Fig. 1 (a) and (b). The E of the strip waveguide in TE mode is mainly confined in the core of the waveguide, resulting in a weak interaction between the light and the surrounding analyte. On the contrary, the E of the strip waveguide in TM mode is mainly distributed at the top and the bottom of the waveguide. The E intensity is enhanced at the interface of the waveguide and the cladding, since the electric displacement vector (D) at the interface of the waveguide and the cladding is continuous, while the refractive indices of the waveguide and cladding are not identical. The E distribution of the TM-operated waveguide clearly exhibits a significantly larger E field enhancement than that of the TE-operated waveguide. To precisely illustrate the characteristics of various waveguides, the evanescent field ratio (EFR) is adopted to express the ratio of the optical power in the surrounding analyte to the total optical power. EFR is given by Eq. (1) [24],

$${\Gamma _{\textrm{cladding}}} = \frac{{\smallint {\smallint _{\textrm{cladding}}}\textrm{Re}({\boldsymbol{E} \times {\boldsymbol{H}^\ast }} )\cdot {e_z}\textrm{dxdy}}}{{\smallint {\smallint _{\textrm{total}}}\textrm{Re}({\boldsymbol{E} \times {\boldsymbol{H}^\ast }} )\cdot {\boldsymbol{e}_z}\textrm{dxdy}}}, $$
where Re represents the real part of complex electric (E) and magnetic (H) fields; * denotes the complex conjugate and ez is the unit vector in the z-direction. The EFR of the TM-operated waveguide is 23.88%, which is higher than the 20.0% of the TE-operated strip waveguide.

 figure: Fig. 1.

Fig. 1. The cross-section E distribution of (a) TE-operated strip waveguide, and (b) TM-operated strip waveguide. The calculated waveguide sensitivity of (c) TE-operated strip waveguide, and (d) TM-operated waveguide.

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Swg is a key factor of the waveguide for sensing devices, which is described by the ratio between the effective refractive index change and the refractive index change (Δneff/Δnc) of analytes (homogenous sensing) [25], and proportional to EFR and the strength of E. Figure 1 (c) and (d) show the calculated Swg of the TE-operated strip waveguide and the TM-operated waveguide, respectively. From the lines of Fig. 1 (c) and (d), the waveguide sensitivity of the TE-operated strip waveguide is 0.150, while the waveguide sensitivity of the TM-operated waveguide is 0.502, which is much larger than that of the TE-operated strip waveguide. Consequently, the sensing sensitivity of the optical sensors based on the microring resonator is much higher when using the TM-operated waveguide. Furthermore, the large different TOCs of Si, SiO2, and analytes, combined with the difference in mode field distribution between TE-operated waveguides and TM-operated waveguides, result in different temperature waveguide sensitivities. Thus, the waveguide with different sensitivity can be adopted to realize microring resonator-based sensors with different sensitivity of RI and temperature.

2.2 Polarization beam splitter

To achieve polarization multiplexing on one chip, a polarization beam splitter (PBS) is needed to separate optical signals of different polarization states in the optical path [26,27]. Many optical structures can be used to realize PBS, including multimode interference (MMI) [28,29], Mach-Zehnder Interferometers (MZI) [30], gratings [31,32], and directional couplers (DC) [26]. Among them, the PBS based on DC has been considered as a preferred candidate to achieve a high extinction ratio and extraordinarily low insertion loss in a compact footprint due to its obvious birefringence effect. Moreover, the wavelength sensitivity of conventional symmetric DC is reduced by applying an asymmetric directional coupler (ADC), which makes PBS based on ADC has attracted great attention in recent years [3335].

The 3D sketch of PBS based on ADC is shown in Fig. 2 (a). Since the effective refractive index of TE mode and TM mode in the waveguide is different, the phase-matching conditions of the two polarization states in the ADC structure are different. Utilizing the difference in phase-matching, the waveguide of ADC was designed with one polarization mode that meets the phase-matching and the other is not. In the strip waveguide, the effective refractive index and mode field distribution characteristics of TE mode are mainly affected by the width of the waveguide, while the effective refractive index and mode field distribution of TM mode are less affected by the width of the waveguide. Therefore, the main structure of PBS based on ADC includes two ordinary strip waveguides of unequal width. The structure adopted in this work is cascaded ADC, which further improves the extinction ratio between the two modes, as shown in Fig. 2(b). When the light of two polarization modes passes through PBS, the light satisfying the phase-matching condition is coupled into the cross waveguide, while the polarized light not satisfying the condition of phase matching is output from the through port, thus realizing the function of polarization separation.

 figure: Fig. 2.

Fig. 2. The 3D sketch of (a) the ADC-based PBS; (b) the cascaded ADC-based PBS in this work.

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2.3 Microring resonator based on polarization multiplexing for sensing

According to the contradiction between sensitivity and detection range of the microring resonator-based optical sensors, a novel sensor configuration with high sensitivity and large detection range was proposed, which consists of two microrings operated in different modes and a PBS, as shown in Fig. 3. The fundamental mode light emitted from the fiber is coupled into the silicon waveguide through the edge coupler, and then the TM mode and TE mode will be separated through the PBS. The TM mode light enters the upper microring from the cross port, and the TE mode enters the lower microring from the through port. The transmission spectrum lines of the two ports can be obtained simultaneously through the optical spectrometer analyzer. Since the resonant wavelength of the microring resonator is periodic, the shift of the resonant peak can be described by (N + α)FSR, where N is an integer and α is a decimal between 0 and 1. When the shift of a resonant peak exceeds one or several FSR, it cannot be judged exactly. The waveguide sensitivity of the TM-operated waveguide is 3.35 times higher than that of the TE-operated waveguide, therefore the microring resonator-based sensor working on TM mode with high sensitivity can be used for precise measurement (α), while the microring resonator-based sensor working on TE mode with low sensitivity can be used to calibrate the shift range of resonance peak (N). Taking a clock as an analogy, the microring of TE mode is like an hour hand (N), and the micro ring of TM mode is like a minute hand (α). By cooperating, the two microrings can achieve a wide range of accurate measurements.

 figure: Fig. 3.

Fig. 3. The schematic of the proposed sensor based on polarization multiplexing

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In the sensing structure, the sensitivity matrix Sn, T composed of TE mode and TM mode to the refractive index and external temperature of the analyte is given by,

$${S_{n,T}} = \left[ {\begin{array}{cc} {{S_{n,TE}}}&{{S_{T,TE}}}\\ {{S_{n,TM}}}&{{S_{T,TM}}} \end{array}} \right], $$
where Sn, TE, ST, TE, Sn, TM, and ST, TM are RI sensitivity and the temperature sensitivity of the TE-based and TM-based sensors, respectively.

The sensing functions discussed in this work are achieved by detecting the shift of the resonant wavelength, which is positively correlated with the effective refractive index of the waveguide. Without considering the high-order dispersion, the sensitivity of the sensor is given by [36],

$${S_{{\mathop{\boldsymbol{var}}} }} = \frac{{\varDelta \lambda }}{{\varDelta {\mathop{\boldsymbol{var}}} }} = \frac{{\varDelta {n_{eff}}}}{{\varDelta {\mathop{\boldsymbol{var}}} }} \cdot \frac{\lambda }{{{n_g}}}, $$
where var represents external variables (In this work, variables are the refractive index of the analyte and the surrounding temperature).

According to the optical waveguide theory and perturbation theory, the change of effective refractive index of silicon waveguide can be expressed by Eq. (4) [37,38],

$$\varDelta {n_{eff}}\textrm{ = }{f_c} \cdot \frac{{{n_g}}}{{{n_c}}} \cdot \varDelta {n_c} + {f_{si}} \cdot \frac{{{n_g}}}{{{n_{si}}}} \cdot \varDelta {n_{si}}\textrm{ + }{f_{si{o_2}}} \cdot \frac{{{n_g}}}{{{n_{si{o_{_2}}}}}} \cdot \varDelta {n_{si{o_2}}}, $$
where ${\textrm{f}_\textrm{C}}$, ${\textrm{f}_{\textrm{Si}}}$, and ${\textrm{f}_{\textrm{Si}{\textrm{O}_\textrm{2}}}}$ represent the EFR of light in the analyte, silicon waveguide, and silicon dioxide, respectively. When the external refractive index nc changes, such as the concentration of gas or solution, only the first term in Eq. (4) will affect the effective refractive index of the waveguide, and Δnsi and $\varDelta {\textrm{n}_{\textrm{Si}{\textrm{O}_\textrm{2}}}}$ are zero. Although the change of analytes affects the distribution of light field and leads to little change of EFR, the RI in this work is a slight change, which has little effect on the change of EFR. However, when the temperature changes, the index of the analyte, silicon, and silicon dioxide will change simultaneously. Since the TOC of silicon dioxide is one order of magnitude lower than that of silicon, the effect of temperature on silicon dioxide can be ignored. Combined Eq. (3) with Eq. (4), the sensitivity of the refractive index and the temperature of can be obtained,
$$\left\{ {\begin{array}{c} {{S_n}\textrm{ = (}{f_c} \cdot \frac{{\varDelta {n_c}}}{{\varDelta n}}\textrm{)} \cdot \frac{\lambda }{{{n_c}}}}\\ {{S_T} = \textrm{(}{f_c} \cdot \frac{{\varDelta {n_c}}}{{\varDelta T}}\textrm{)} \cdot \frac{\lambda }{{{n_c}}} + \textrm{(}{f_{si}} \cdot \frac{{\varDelta {n_{si}}}}{{\varDelta T}}\textrm{)} \cdot \frac{\lambda }{{{n_{si}}}}\varDelta {n_{si}}} \end{array}} \right.. $$

Combining Eq. (2) with Eq. (5), the relationship between the changes of the RI and the temperature and wavelength shift of TE mode and TM mode can be described by a matrix equation,

$$\left[ {\begin{array}{c} {\varDelta {\lambda_{TE}}}\\ {\varDelta {\lambda_{TM}}} \end{array}} \right] = {S_{n,T}} \times \left[ {\begin{array}{c} {\varDelta n}\\ {\varDelta T} \end{array}} \right]. $$

When the rank of the matrix Sn, T is not zero, Δn and ΔT can be obtained by measuring the resonant wavelength shifts in different polarization modes simultaneously. Hence, a novel polarization-multiplexing-based sensor with high sensitivity and a large detection range is successfully constructed to achieve the detection of refractive index and temperature parameters simultaneously.

3. Fabrication and characterization

3.1 Fabrication

The silicon photonic integrated microring sensor based on polarization multiplexing for simultaneous sensing of refractive index and temperature was designed and fabricated on the SOI platform. As shown in Fig. 4(a), First, a 200 mm SOI wafer with 220 nm-thick top silicon and 3 µm-thick buried oxide was prepared. Then the deep ultraviolet (DUV) photolithography was adopted to define the waveguide, grating, and device patterns on the photoresist. Subsequently, double Inductively coupled plasma (ICP) dry etching processes were adopted to transfer the patterns from the photoresist layer to silicon, as shown in Fig. 4(b). After a 2 µm-thick SiO2 was deposited on top of the wafer by plasma-enhanced chemical vapor deposition (PECVD), the ICP etching process was used to open the sensing window, as shown in Fig. 4(c) and (d). For the avoidance of the surface damage during the dry etching process, the etching depth was set to 1.7 µm, followed by wet etching (buffered oxide etch (BOE) solution) to remove the residual 0.3 µm-thick SiO2, as shown in Fig. 4(e). Last, the photoresist was removed and the fabrication of the proposed device was completed. Figure 5 shows the optical microscope image of the fabricated device and the scanning electron microscope (SEM) images of the PBS, the TM-operated microring resonator, and the TE-operated microring resonator.

 figure: Fig. 4.

Fig. 4. The fabrication process flow of the silicon photonic integrated microring sensor based on polarization multiplexing. (Not to scale)

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 figure: Fig. 5.

Fig. 5. (a) The optical microscope image of the fabricated microring resonator based on polarization multiplexing for sensing; the SEM images of (b) PBS, (c) TM-operated microring resonator, and (d) the TE-operated microring resonator.

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3.2 Characterization

The PBS is a key component of the proposed sensing configuration, and the performance of the PBS is characterized first. Since the sensing characterization of the proposed device is performed near the wavelength of 1550 nm, only the spectral response of the fabricated PBS in the wavelength range of 1542 to 1557 nm is concerned. As shown in Fig. 6, for the TE polarization mode, the extinction ratio (ER) is higher than 17 dB in the wavelength range of 1542 to 1557 nm, and the loss at the TE-TE port almost approaches 0. For the TM polarization mode, the insertion loss at the TM-TM port is about 1.5 dB and the ER is higher than 25 dB. According to the test results, it can be concluded that the PBS can effectually achieve the function of polarization separation in the required wavelength range, which can be used in the proposed sensor structure.

 figure: Fig. 6.

Fig. 6. The transmission of the PBS.

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Then the sensing properties of the proposed sensor were characterized. In order to characterize the sensing performance of the proposed sensor, two identical fabricated devices were connected with two different grating couplers, one for TE input PBS and the other for TM input PBS. The light from the tunable laser was coupled into the waveguide through the grating coupler and propagated to the sensing ring, followed by the detection of the optical spectrometer analyzer. The refractive indices of the water-NaCl mixtures as a function of NaCl concentration were determined as 1.3331 + 0.00185C (C% is the NaCl concentration) [39]. Thus, NaCl solutions with different concentrations were filled in the sensing window to characterize the sensing performance by tracing the resonant wavelength of the sensing ring operated in different polarizations, respectively.

It can be observed from Fig. 7 (a) and (b) that the sensitivity of TE mode and TM mode are different. For the same refractive index variation, the shift of the resonant peak of the TM mode is obviously larger than that of the TE mode, which suggests that the sensitivity of the TM-operated sensor is much higher than that of the TE-operated sensor. Sensitivity is one of the most important performance indicators of sensors, but for microring resonator-based sensors, high sensitivity leads to a problem: a small detection range. As shown in Fig. 7 (a) and (b), although the sensing sensitivity of TE mode is small, the concentration range of the NaCl solution can be measured to exceed 10%, and its resonant peak drift exceeds one FSR when the solution concentration is about 15%. Nevertheless, when the concentration of NaCl solution is 5%, the resonant wavelength shift of the sensor in TM mode obviously exceeds one FSR. In Fig. 7 (a) and (b), the fluctuation of the output optical power is attributed to the error of the coupling efficiency from one time to the next of the manual testing platform. By measuring the variation of resonant wavelength of the transmission spectrum of the microring resonator in two polarization modes with different NaCl solution concentrations, a linear fitting of its offset was conducted, as shown in Fig. 7 (c). From the slope of the fitting curve, it can be obtained that the refractive index sensitivity of the TM-operated sensor is 489.3 nm/RIU, and that of the TE-operated sensor is 102.6 nm/RIU, showing the sensitivity of TM mode is 4.76 times that of TE mode. This ratio is higher than 3.35 times as analyzed in Section 2, since the local SiO2 beneath the TM-operated micro-ring resonator was removed partially during the wet etching process. The DR of the two kinds of polarization can be calculated by DR = FSR/S, resulting in that the DR of the TM-operated sensor is 0.0071 RIU, and that of the TE-operated sensor is 0.0296 RIU, which suggests the DR of the latter is 4.2 times that of the former.

 figure: Fig. 7.

Fig. 7. (a) TE mode and (b) TM mode transmission spectra with various NaCl solution concentrations; (c) Linear fitting of resonant wavelength shifts of TE mode and TM mode under different refractive indices; (d) Linear fitting of resonant wavelength shifts of TE and TM modes at different temperatures.

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Moreover, the responses of the sensor to changes in temperature were also measured. As shown in Fig. 7 (d), the positions of the resonant wavelength at various temperatures were measured and the temperature sensitivity was calculated from the shifts of the resonant wavelengths. The temperature sensitivity of TM and TE modes is 20.0 pm/°C and 43.3 pm/°C respectively. The electric field of the TM mode waveguide is mainly distributed in silicon dioxide and NaCl solution under test, while that of the TE mode waveguide is mainly distributed in silicon material. Since the TOC of silicon is larger than that of silicon dioxide and NaCl solution, the temperature sensitivity of the TM-operated sensor is much lower than that of the TE-operated sensor.

Through the above test results, the sensitivity matrix of the proposed structure was obtained,

$$\left[ {\begin{array}{c} {\varDelta {\lambda_{TE}}}\\ {\varDelta {\lambda_{TM}}} \end{array}} \right] = \left[ {\begin{array}{cc} {102.6\textrm{nm/RIU}}&{43.3\textrm{pm/}^\circ \textrm{C}}\\ {489.3\textrm{nm/RIU}}&{20.0\textrm{pm/}^\circ \textrm{C}} \end{array}} \right] \times \left[ {\begin{array}{c} {\varDelta n}\\ {\varDelta T} \end{array}} \right], $$
and its inverse can be calculated. By measuring the shifts of the resonant wavelength in two different polarization states simultaneously, and then these results are substituted into Eq. (7), the unique value of the refractive index and temperature change can be obtained. In this work, the fitting degree of the fitting lines is not particularly high, and more points need to be tested to improve the fitting degree of the fitting lines. Thus, the two-parameter sensing with high accuracy was achieved.

4. Conclusion

In this work, a novel dual-parameters sensor based on polarization multiplexing was proposed and demonstrated experimentally, which simultaneously achieves refractive index and temperature sensing with high sensitivity and a large DR. Firstly, according to the contradiction between the sensitivity and DR of the microring resonator, a polarization multiplexing sensing structure based on the different waveguide sensitivity of TE-operated waveguide and TM-operated waveguide was proposed. Moreover, due to the different mode field distribution characteristics of TE mode and TM mode in the waveguide, the shift of resonant wavelength caused by temperature change and refractive index change can form a two-dimensional matrix. Thus, by coordination between polarization modes, the sensing structure can detect the variations of the refractive index and temperature simultaneously. The experimental results show that the refractive index sensitivity and temperature sensitivity of TM microring are 489.3 nm/RIU and 20.0 pm/°C, respectively, and that of TE microring are 102.6 nm/RIU and 43.3 pm/°C, respectively. Moreover, the DR of the fabricated sensor is 0.0296 RIU, which is 4.2 times that of the conventional TM-operated sensor based on the microring resonator. The dual-parameters sensor based on polarization multiplexing show potential applications of silicon photonic on-chip sensors with multiparametric sensing capability.

Funding

National Natural Science Foundation of China (61904196).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (7)

Fig. 1.
Fig. 1. The cross-section E distribution of (a) TE-operated strip waveguide, and (b) TM-operated strip waveguide. The calculated waveguide sensitivity of (c) TE-operated strip waveguide, and (d) TM-operated waveguide.
Fig. 2.
Fig. 2. The 3D sketch of (a) the ADC-based PBS; (b) the cascaded ADC-based PBS in this work.
Fig. 3.
Fig. 3. The schematic of the proposed sensor based on polarization multiplexing
Fig. 4.
Fig. 4. The fabrication process flow of the silicon photonic integrated microring sensor based on polarization multiplexing. (Not to scale)
Fig. 5.
Fig. 5. (a) The optical microscope image of the fabricated microring resonator based on polarization multiplexing for sensing; the SEM images of (b) PBS, (c) TM-operated microring resonator, and (d) the TE-operated microring resonator.
Fig. 6.
Fig. 6. The transmission of the PBS.
Fig. 7.
Fig. 7. (a) TE mode and (b) TM mode transmission spectra with various NaCl solution concentrations; (c) Linear fitting of resonant wavelength shifts of TE mode and TM mode under different refractive indices; (d) Linear fitting of resonant wavelength shifts of TE and TM modes at different temperatures.

Equations (7)

Equations on this page are rendered with MathJax. Learn more.

Γ cladding = cladding Re ( E × H ) e z dxdy total Re ( E × H ) e z dxdy ,
S n , T = [ S n , T E S T , T E S n , T M S T , T M ] ,
S v a r = Δ λ Δ v a r = Δ n e f f Δ v a r λ n g ,
Δ n e f f  =  f c n g n c Δ n c + f s i n g n s i Δ n s i  +  f s i o 2 n g n s i o 2 Δ n s i o 2 ,
{ S n  = ( f c Δ n c Δ n ) λ n c S T = ( f c Δ n c Δ T ) λ n c + ( f s i Δ n s i Δ T ) λ n s i Δ n s i .
[ Δ λ T E Δ λ T M ] = S n , T × [ Δ n Δ T ] .
[ Δ λ T E Δ λ T M ] = [ 102.6 nm/RIU 43.3 pm/ C 489.3 nm/RIU 20.0 pm/ C ] × [ Δ n Δ T ] ,
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