1. Introduction

Precise detection of trace amounts of biomolecules is required for many applications including diagnostics, genomics, proteomics, and drug screening. Today optical biosensors are powerful analysis tools for such applications by means of fluorescence-based or label-free detection. In contrast to fluorescence-based techniques, label-free sensors allow for quantitative and kinetic measurement of molecular interaction [1]. While surface plasmon resonance (SPR) is commonly used in commercial sensing systems [2–4], recently silicon photonic wire waveguide evanescent field (PWEF) sensors have seen rapid development [5–7]. Due to the high refractive index of silicon, the optical modal electric field is strongly localized near the waveguide surface, giving rise to high sensitivity to surface adsorption. Long cavity resonator sensors can be made with small footprint, making them easily scalable to large arrays [8]. Silicon PWEF sensors also benefit from established silicon photonics technologies developed in the past decade, allowing low cost fabrication.

We have developed highly sensitive silicon-on-insulator (SOI) photonic wire sensors with folded path in ring resonator and Mach-Zehnder interferometer configurations, operating in the TM polarization [8–11]. With these structures, we have demonstrated the real-time detection of surface-adsorbed DNA and protein molecules with a sensitivity of ~0.2 pg/mm², an order of magnitude better than the SPR technique, and the sensors are scaled up to × 128 arrays. The temperature-induced resonance shift is cancelled out using reference resonators which is protected from the sensing medium by a SU8 cladding layer [10]. In these label-free sensors, however, the changes in the refractive index ∆n_c of the aqueous solution and the adsorbed molecular layer thickness ∆d_F on the waveguide surface cannot be distinguished directly, similar to the case for SPR. In our previous experiments, the refractive index of the buffer solution was usually kept constant, and the measured sensing responses were attributed to the surface adsorption only. In general, several solutions of different refractive indices may be required for the immobilization and rinsing steps in sensing procedures, so the response of the PWEF sensors comes from both ∆n_c and ∆d_F [12]. In commercial SPR instruments, the fluidic index changes are monitored using a separate fluidic channel, requiring more complex instrumentation or regent preparation [13]. It is also difficult to ensure that the response of the reference channel solely comes from the fluidic index change. It is desirable to differentiate these responses optically for the same fluid flow.

In this work, we demonstrate the feasibility of retrieving these two changes (∆n_c and ∆d_F) simultaneously using experimentally measurable resonance-wavelength shifts of two sensors through a single fluidic channel, operating in the quasi transverse electric (TE) and transverse magnetic (TM) fundamental modes, respectively. The main challenge in this procedure comes from the waveguide size fluctuations common in silicon photonic wire fabrication, which can cause large errors in the retrieved values. We propose and numerically demonstrate that the actual waveguide size can be determined with high precision using measured group indices. Using this information, ∆n_c and ∆d_F can be obtained with good accuracy.

2. Theoretical background

The silicon wire waveguide structure in our resonator sensors is shown in Fig. 1(a) . For comparison with our experimental work, the nominal waveguide size is chosen as 450 nm wide (W) and 260 nm high (H). The upper cladding is assumed to be an aqueous solution unless otherwise specified. Due to the different optical field distributions of the TE and TM modes, the sensing response to the variations in the cladding fluidic refractive index ∆n_c and adsorbed molecular-film thickness ∆d_F differ between the two polarizations. The correspondingly induced effective index changes for TE and TM modes are:

 

Fig. 1 (a) Cross-sectional view of a silicon photonic wire waveguide sensor. (b) Schematic layout of our spiral cavity resonator sensors for TE and TM polarizations.

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In experiments, we can obtain the values of the resonance wavelength shift ∆λ and the free-spectral-range (FSR) for TE and TM modes in real time. The effective index changes ∆N_eff can then be calculated from ∆λ using the relation ∆λ = λ(∆N_eff / N_g), where the group index N_g = λ²/(FSR⋅L), and λ and L are the resonance wavelength and the resonator cavity length, respectively. The values of ∂N_eff(TE,TM)/∂n_c and ∂N_eff(TE,TM)/∂d_F can be calculated through simulations, provided that the waveguide size is known. Here the sensors operating in the TE and TM polarizations are assumed to have identical waveguide dimensions. Since these sensors are generally placed within a few hundred micrometers of each other, this is a reasonable assumption based on experimental observations. For a given waveguide cross-section, simulations show that ∂N_eff(TE,TM)/∂n_c is a constant for a wide range of fluidic refractive index. The value of ∂N_eff(TE,TM)/∂d_F can also be closely approximated by a constant for ∆d_F < 10 nm, which is larger than the size of most proteins and DNA molecules encountered in biosensing and therefore adequately cover the range of interest. Using Eq. (1), it is straightforward to find the solutions for ∆n_c and ∆d_F as expressed using an inverse sensitivity matrix S ⁻¹:

3. Simulation results and discussion

It is well known that the optical properties of silicon wire waveguides strongly depend on their cross-sectional dimensions. Figures 2(a) and 2(b) show the dependence of the inverse sensitivity matrix elements on the waveguide width for a height of 260 nm and the waveguide height for a width of 450 nm, calculated using the film mode matching (FMM) method at a wavelength of 1.55 µm. As expected, the matrix elements for the TE polarization vary with W more strongly than those for TM which have a stronger dependence on the height H. That is due to the high index contrast which leads to the large surface electric field magnitude and discontinuity at the waveguide interface in the surface normal direction. It is also evident that the sensing response of the TM mode is higher than that of the TE mode for W > 400 nm. The maximum response for the TM mode to ∆n_c and ∆d_F occurs at H of 220 nm and 245 nm, respectively.

 

Fig. 2 Elements of the inverse sensitivity matrix versus (a)(c) waveguide width at a height of 260 nm and (b)(d) waveguide height at a width of 450 nm.

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In order to obtain valid solutions from Eq. (2), it is required that the matrix determinant det S ≠0. In other words, when the combined responses of the TE and TM modes to ∆n_c and ∆d_F are the same, the differentiation between these two contributions becomes impossible. The dependence of 1/det S on the waveguide dimensions is shown in Figs. 2(c) and 2(d). The values of 1/det S should be as small as possible for best numerical accuracies. The waveguide dimensions (W ~300 nm or H ~200 nm) for minimum 1/det S, however, leads to other trade-offs such as large waveguide bend radius.

Figures 3(a) and 3(b) show the overall coefficients for variables ∆N_eff(TE) and ∆N_eff(TM) in Eq. (2) as a function of W and H. For a given waveguide size, ∆n_c and ∆d_F can easily be calculated using these coefficients. These results also illustrate the high sensitivity to waveguide dimensions of the retrieved ∆n_c and ∆d_F using measured ∆N_eff(TE) and ∆N_eff(TM). Good structural tolerance is found for waveguides with e.g. W ~350 nm for H = 260 nm, or H ~200 nm for W = 450 nm, where the optical modes are rather delocalized for at least one polarization. On the other hand, the sensor waveguide choice also depends on requirements including the single mode operation, high sensing response and small bend radius to ensure compact footprint. Our nominal design of 450 nm × 260 nm represents a compromise between all these considerations.

 

Fig. 3 Overall elements of the inverse sensitivity matrix S ⁻¹ versus (a) waveguide width at a height of 260 nm and (b) waveguide height at a width of 450 nm.

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Using current fabrication technologies, the silicon thickness (H) uniformity on a wafer is typically ± 2 nm, while the variation between wafers is typically ± 5 nm. For state-of-the-art patterning techniques such as e-beam and 193 nm DUV lithography, the deviation of the waveguide width W is typically ± 10 nm, although a systematic bias can often be larger [14–16]. The precise sensor waveguide dimensions are generally not available for each chip. In the following, we evaluate the retrieval accuracies using the nominal waveguide dimension of 450 nm × 260 nm in the presence of size fluctuations.

As shown in Fig. 4 , the vertical axis is defined as the normalized difference between the original (∆n_c and ∆d_F) and the retrieved values (∆n_c^’ and ∆d_F^’), with corresponding deviations of (∆n_c − ∆n_c^’)/∆n_c or (∆d_F − ∆d_F^’)/∆d_F. Two cases are considered. In the first case the maximum value for ∆n_c is 0.003 which is comparable to the index difference between 2 × PBS (phosphate buffered saline) solution and water, and ∆d_F of 1.2 nm is comparable to 1/3 of monolayer of streptavidin protein. The corresponding changes in the effective index for the TM mode are ∆N_eff(∆n_c) ~0.0012 and ∆N_eff(∆d_F) ~0.001, respectively. In other words, the contribution to the sensing signal from the fluidic index change and surface adsorption are comparable: ∆N_eff(∆n_c):∆N_eff(∆d_F) ~1:1. A more stringent case is also evaluated where ∆N_eff(∆n_c):∆N_eff(∆d_F) ~10:1. First, the response in ∆N_eff(TE) and ∆N_eff(TM) is calculated from Eq. (1) to emulate experimentally measured values using the waveguide dimensions as indicated in the Fig. 3. Then the values of ∆n_c and ∆d_F are retrieved using Eqs. (2) and (3) assuming the nominal waveguide size of 450 nm × 260 nm. As the waveguide size deviates from the nominal dimensions, significant errors in the retrieved fluidic index change or the surface adsorption are introduced, particularly for the component with the smaller sensing response. For the case of ∆N_eff(∆n_c):∆N_eff(∆d_F) ~10:1, the maximum errors are as large as about + 150% and −100%, respectively, for ∆W~ ± 10 nm and ∆H ~ ± 5 nm. Clearly, this level of errors is unacceptable. It is imperative that the actual waveguide dimensions are accurately known.

 

Fig. 4 Evaluation of the retrieval accuracy as a function of waveguide dimension changes in (a) waveguide width and (b) waveguide height. Two cases were estimated: ∆N_eff(∆n_c): ∆N_eff(∆n_c) ~“10: 1” and “1: 1”.

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In the following, we demonstrate that the fabricated waveguide dimensions can be determined with high accuracy using the group index N_g for the TE and TM polarizations that are deduced from the optically measurable FSR data. The measurement accuracy of FSR depends on the magnitude of FSR, the resonator quality factor and the extinction ratio. Therefore here we choose air as the upper cladding to avoid the degradation in the quality factor caused by the attenuation from water. It should be noted that air upper cladding is generally available during the setup procedure of sensing operations. Figure 5 shows the group indices for TE and TM modes as a function of W and H, respectively. As can be seen, the changes in N_g(TE) almost depends solely on W, while the changes in N_g(TM) depends mainly on H, allowing W and H being determined separately. The color-contours are separated by a group index interval of 0.01 which is experimentally achieved, as shown below.

 

Fig. 5 Group indices as the functions of waveguide width and height for (a) TE mode and (b) TM mode. The upper cladding is water and the contour interval was set to be 0.01 in both figures.

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The transmission spectrum of a spiral cavity resonator (see Fig. 1(b)) in TM mode is shown in Fig. 6(a) , measured with a wavelength scan step of 1 pm. Here the cavity length L = 700 µm which result in a FSR of ~0.7 nm. The quality factor is of a moderate value of Q ~20,000. The deduced N_g as well as the deviations from the polynomial fit is shown in Fig. 6(b). Clearly the group index can be determined to within an accuracy of ± 0.005. The measurement accuracy in N_g improves with decreasing L, but we found that similar precision can be achieved for resonators with cavity length L < 1000 µm.

 

Fig. 6 (a) Measured spectrum for spiral cavity resonator with the cavity length of 700 µm, and (b) calculated group index from the measured spectrum, and it’s deviation from the polynomial fit.

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Within the range of waveguide dimensions considered, N_g is a monotonic function of both W and H that can be approximated by a 2nd order polynomial fit:

Figures 7(a) and 7(b) show the W and H matrix as a function of group indices for TE and TM mode which are calculated from Eq. (4) with the use of the fitting parameters listed in Table 1 . Here the color contours are separated by a waveguide width ∆W of ± 2 nm in Fig. 7(a) and by a waveguide height ∆H was of ± 1 nm from Fig. 7(b). We can determine the size of the waveguide with these numerical expressions. With the obtained level of accuracy in measuring N_g, we can determine the waveguide size to an accuracy of ± 2 nm for W and ± 1nm for H.

 

Fig. 7 Distribution of (a) waveguide width and (b) waveguide height as the functions of group indices for TE and TM mode with the color interval of 2 nm and 1 nm, respectively. The squares with white dashed line show the range of measurement accuracy for N_g.

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Table 1. Optimized fitting parameters for Eq. (4).

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Finally, in order to evaluate the differentiation method, we evaluated the retrieval accuracy for fluidic index and film thickness changes again using the determined actual waveguide dimensions, as shown in the shaded areas of Fig. 4. In the case of ratio of 1:1, the maximum retrieval errors are reduced from 30% to 7%. Even when the effective index variation due to liquid index change exceeds the variation due to film thickness change by a factor of 10, the maximum errors for the smaller component are reduced from 150% to less than 40%, i.e. the retrieval accuracy is improved by more than 3 times by using our procedure.

4. Conclusion

We have presented a robust method of simultaneous monitoring of the fluidic index change ∆n_c and surface adsorption ∆d_F using silicon photonic wire resonator sensors. The data retrieval is achieved by measuring the resonance wavelength shifts for the TE and TM modes at the same time, and correlating them to n_c and ∆d_F using calculated sensitivity coefficients. We also showed how to circumvent a critical challenge in ∆n_c and ∆d_F discrimination accuracy, namely the large dependence of the sensitivity coefficients on the cross-section dimensions of the sensor waveguide core. Waveguide size fluctuations commonly encountered in silicon photonic wire fabrication can introduce unacceptable retrieval errors. We propose a new method of accurately determining the size of fabricated waveguides using TE and TM mode group indices obtained from measured resonator FSR. Using this approach, the waveguide dimensions can be determined to a precision of ∆W ~ ± 2 nm and ∆H ~ ± 1 nm. By applying the actual waveguide size, the retrieval accuracies for ∆n_c and ∆d_F are improved by more than 3 times. The proposed method of determining waveguide dimensions indeed is not restricted to evanescent field waveguide sensors and it can also be used in other types of planar waveguide circuits.