Temperature dependence of resonance characteristics of silicon resonators and thermal stability improvement by differential operation method

The resonance wavelength shift of Si resonators by thermal change is given by

$$\begin{equation} \Delta\lambda_{\text{res}} = \frac{\Delta n_{\text{eff}}}{n_{\text{eff}}}\lambda_{\text{res}} + \frac{\Delta L}{L}\lambda_{\text{res}}, \end{equation} \tag{ 1 }$$

where λ_res, n_eff, and L are the resonance wavelength, effective refractive index, and size of the resonator, respectively. Δλ_res, Δn_eff, and ΔL are the changes in these parameters due to the temperature change. The right-hand first term indicates the effective refractive index (Si) effect and the latter term indicates the thermal deformation effect. This equation is easily derived from the equation λ_res = n_effL/m, where m denotes the integer. As the temperature increases, Si and SiO₂ thermally expand with different thermal expansion coefficients and the SOI wafer has a curvature shape as shown in Fig. 2. The radius of the curvature of the SOI substrate is given by²²⁾

$$\begin{equation} \frac{1}{R} = \cfrac{6(\alpha_{1} - \alpha_{2})\Delta T(1 + p)^{2}}{h\biggl[3(1 + p)^{2} + (1 + pq)\biggl(p^{2} + \cfrac{1}{pq}\biggr)\biggr]}, \end{equation} \tag{ 2 }$$

where h, α₁, and α₂ are the total thickness and thermal expansion coefficients of Si and SiO₂, respectively. Moreover, p = d_SiO2/d_Si (where d_SiO2 and d_Si are the thicknesses of SiO₂ and Si, respectively), q = Y_SiO2/Y_Si (where Y_SiO2 and Y_Si are Young's moduli of SiO₂ and Si, respectively), and ΔT is the temperature change. The change in resonator size is expressed as

$$\begin{equation} \Delta L \approx \frac{dL}{2R}, \end{equation} \tag{ 3 }$$

where d is the thickness of the SOI substrate. Here, d = d_Si + d_SiO2 and d_Si ≫ d_SiO2 is assumed.

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In the differential silicon ring resonator, one ring is considered as a detection ring, on which a biomaterial is adsorbed, and the other is considered as a reference ring, which is not exposed to the biomaterial. The phase of one of the outputs is changed by π and merged again. When a biomaterial is not adsorbed on the detection ring, the differential output is zero because two resonance curves completely overlap. When a biomaterial is adsorbed on the detection ring, its resonance curve shifts and an output signal appears. Here, we discuss the output spectrum of the differential ring resonator sensor. First, for the single-ring resonator the transmission spectrum T_s is given by^23,24)

$$\begin{align} T_{\text{s}} &= \frac{\sin K[1 - X\cos\phi(\lambda) + i\sin\phi(\lambda)]}{1 + X^{2} - 2X\cos\phi(\lambda)}, \end{align} \tag{ 4 }$$

$$\begin{align} X &= \cos^{2}K\exp\left(-\frac{\alpha L}{2}\right), \end{align} \tag{ 5 }$$

$$\begin{align} \phi(\lambda) &= \frac{2\pi n_{\text{eff}}L}{\lambda}, \end{align} \tag{ 6 }$$

where sin K is the coupling coefficient, α is the attenuation constant of the waveguide, L is the peripheral length of the resonator, and n_eff is the effective refractive index of the Si waveguide. The transmission spectrum of the differential Si ring resonator sensor T_DS is given by²⁰⁾

$$\begin{equation} T_{\text{DS}} = |T_{\text{s}} + e^{i\theta}T_{\text{s}}|^{2} + (2 + 2\cos\theta)|T_{\text{s}}|^{2}, \end{equation} \tag{ 7 }$$

where θ is the phase difference between the detection and reference rings, which is usually π.

The temperature dependence is calculated as

$$\begin{align} \theta &= \frac{2\pi n_{\text{eff}}\Delta l}{\lambda}, \end{align} \tag{ 8 }$$

$$\begin{align} n_{\text{eff}} &= n_{23} + \beta(T - 23), \end{align} \tag{ 9 }$$

where Δl is the length of the phase shifter, T (°C) is the temperature, n₂₃ is the effective refractive index at 23 °C, and β is the thermal coefficient of the core and various clad layers at the above temperature. n₂₃ and β are obtained as

$$\begin{align} n_{\text{eff}}\left(\frac{dn_{\text{eff}}}{dT}\right)& = \left(n_{1}\frac{dn_{1}}{dT}\Gamma_{1}\right)_{\text{core}} {}+ \left(n_{2}\frac{dn_{2}}{dT}\Gamma_{2}\right)_{\text{topclad}} {}\\ &\quad + \left(n_{3}\frac{dn_{3}}{dT}\Gamma_{3}\right)_{\text{bottomclad}}, \end{align} \tag{ 10 }$$

where n₁, n₂, and n₃ are the refractive indices of the core, top clad, and bottom clad, respectively. The thermal coefficients of the core, top clad, and bottom clad are dn₁/dT, dn₂/dT, and dn₃/dT, respectively. Γ₁, Γ₂, and Γ₃ are the electric field confinement factors of the core, top clad, and bottom clad, respectively, which were obtained by simulation using the Rsoft photonic CAD suite (Synopsys). Equation (9) is obtained using an optical simulator. The values of the bulk materials are obtained from Ref. 25 for oil and Ref. 26 for silicon, SiO₂, resist, and air. The actual numerical values of coefficient β are 1.16 × 10⁻⁴/°C for SiO₂, 1.14 × 10⁻⁶/°C for water, 1.09 × 10⁻⁴/°C for air, 1.15 × 10⁻⁴/°C for oil, and 1.15 × 10⁻⁴/°C for resist. The mechanical deformation effect is not included because it is negligible as discussed in Sect. 4.

In the differential mode, large-bandwidth input light can be used and the output is the integral of the light intensity within the input light bandwidth or the maximum peak height. Figure 3(a) shows the maximum differential output versus Δn_eff/n_eff or ΔL/L, which is the difference in the effective refractive index or circumferential length of the Si waveguide between the detection and reference resonators divided by the original ones, respectively. It is found that there is a linear region in the output versus Δn_eff/n_eff or ΔL/L curve. Δn_eff or ΔL induces Δλ_res, which is the difference in resonance wavelength between the two ring resonators. In the linear region even though the resonance wavelengths of both resonators are slightly different, the sensitivity defined as output/Δn_eff or output/Δλ_res becomes constant when the operation point is in the linear region.²⁰⁾ Figures 3(b) and 3(c) show the calculated temperature change versus maximum output change for various cladding layers (e.g., air, water, resist, oil, and SiO₂).

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To simulate the temperature dependence of the PhC cavity resonator sensor, we used Eqs. (1)–(3) and (10). The length L of the PhC cavity resonator is considered the maximum electric field distributed area. The field is distributed in the Si resonator as well as in the SiO₂ layer. The confinement factors Γ₁, Γ₂, and Γ₃ are for the inside of the resonator area, Si, and SiO₂, respectively. The simulated electric field distributions obtained using the Rsoft photonic CAD suites are 58, 22, and 20% of the inside of the resonator region, Si, and SiO₂, respectively.

The temperature dependences of the differential Si ring and PhC cavity resonators are shown in Figs. 7 and 8, respectively. In Fig. 9(a), the measured results for the differential Si ring resonator at different temperatures are shown together with the simulation results obtained by the method described in Sect. 2. It is found that the measured data well fit the simulation results and it is clearly observed that the Si refractive index change effect is marked. The mechanical deformation effect is negligible (Δλ_res/ΔT ∼ −10⁻⁴ nm/°C). The result for the PhC cavity resonator is also shown in Fig. 9(b), and the same conclusion as for the ring resonator is derived. For both types of devices, we performed simulation by using Eq. (10) and considering core, top clad, and bottom clad individually. The calculated field distributions in the core (Si), top clad (air), and bottom (SiO₂) clad are 80, 8, and 12% respectively.

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In Fig. 10, the maximum differential output change for ΔT = 4 °C from 25 to 29 °C for the Si ring resonators with different top cladding layers are plotted. In the differential detection, only the maximum differential output, which corresponds to the difference between the resonance wavelength shifts of two rings, is counted. The spectral shift of the differential output is not considered. Theoretically, the maximum differential output change for ΔT = 4 °C is calculated under three different top cladding layer conditions, and the simulated changes for resist, air, and oil are 0.012, 0.02, and 0.01% respectively, as indicated by symbol A. The measured results are also shown in Fig. 10, where the changes in maximum differential outputs for the differential Si ring resonators with different top cladding layers for ΔT = 4 °C are plotted. The measured scatter error for multiple measurements in the air clad is represented by symbol B (reproducibility under the same condition), and the measured data deviation with the temperature changes (from 25 to 29 °C) is expressed by symbol C. The theoretical calculation suggests that the differential operation is temperature independent.

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In our research, we targeted to detect a 1 ng/mL prostate specific antigen (PSA) as the practical level of sensitivity. We used various concentrations of sucrose solutions and converted the equivalent refractive index using Refs. 28 and 29. The sucrose solution of 10⁻²% concentration corresponds to 1 ng/mL PSA. The change in effective refractive index for 1 ng/mL PSA is equivalent to Δn_eff ≈ 1.7 × 10⁻⁵ (Ref. 20) and Δn_eff/n_eff is 6.5 × 10⁻⁶, which corresponds to the differential output of 0.186 from Fig. 3(a). This is in the linear region of this graph. As the differential output is in the linear region, output/Δλ_res could be constant even though the temperature increases.