1. Introduction

Surface plasmon polariton (SPP), the collective electromagnetic excitation of free electrons on a metal surface, is sensitive to the local refractive index change of the dielectric environment, due to the intrinsic strong confinement and localization property of SPPs on the metal-dielectric interface [1]. This property has led to the development of label-free plasmonic sensors for diverse applications in areas such as medical diagnosis, environment monitoring, food safety screening, and threat detection. Most commercially available plasmonic sensors are surface plasmon resonance (SPR) sensors, such as the prism-coupler sensors based on attenuated total reflection (ATR) [2]. Owning to the resonant photo-SPP coupling, these prism-based SPR sensors can reach an extremely small detection limit of 10⁻⁵ refractive index unit (RIU), which can be further improved by using phase sensitive interferometric scheme [3] or measuring the Goos-Hänchen shift [4]. However, such SPR sensors are typically bulky and expensive and require a large amount of sample solution, which are not suitable for small-volume, high-throughput, and chip-based detections. Nowadays, it is demanded to develop compact and low-cost sensors for robust, portable, rapid, and multiplexed measurements [5]. Various plasmonic nanosensors have been exploited for these aims, such as metallic nanoparticles with localized SPR and metallic nanohole arrays with extraordinary optical transmission [5,6], although their sensing response are usually much lower than that of prism-based SPR sensors [2]. Efforts have been taken to enhance the sensitivity and resolution of plasmonic nanosensors. For example, some recently emerged nanostructures, such as the plasmonic vertical Mach-Zehnder interferometer based on subwavelength double slits [7,8] and the nanorod metamaterials [9], show almost one order of magnitude sensitivity enhancement. Some novel physical mechanisms, such as electromagnetically induced transparency [10], diffraction coupling between metal nanoparticles [11], and cross polarization detection scheme [12], have been proposed to narrow the resonance linewidth and thus increase its resolution.

Interferometry is one of the most sensitive optical interrogation methods and various interferometric schemes have been utilized in refractometric sensors, such as Mach-Zehnder interferometer [7,8,13], Young interferometer [14], dual-polarization interferometer [15], porous interferometer [16], and back-scattering interferometer [17]. Recently, it was reported that light transmission through a slit milled in a metal film can be enhanced or suppressed as a result of interference of the directly transmitted light with the SPPs launched by a nearby parallel groove [18,19]. The interference is sensitive to the characteristics of SPPs supported by the metal-dielectric interface, such as the effective refractive index and propagation loss of the SPPs [18,19]. This interferometric scheme has been applied in all-optical plasmonic switches and modulators [20,21] and is believed to be promising for sensitive label-free sensing applications [20].

In this paper, we propose a non-spectroscopic refractometric sensing scheme based on a plasmonic interferometer integrating a metallic groove array and a tilted nanoslit, as shown in Fig. 1 . Under normal incidence, SPPs can be excited by the groove array and propagate to the nanoslit. Due to the varied distance between the tilted nanoslit and the groove array at different x, the SPP-mediated transmission through the slit interferes constructively or destructively with the directly transmitted light, generating an intensity interference fringe along the nanoslit, which can be detected by, e.g., a charge coupled device (CCD). The variation of refractive index of the cover medium, which changes the wave vector (and therefore the accumulated phase shift) of the SPPs launched from the groove array, can be detected by inspecting the shift of the interference fringe.

 

Fig. 1 (a) Schematic of the proposed tilted nanoslit-groove interferometer. X is the auxiliary coordinate axis by rotating the x axis with a small angle α. (b) Cross-section view of the interferometer. (c) SEM image of a fabricated sample with N = 10, L₀ = 15.8 μm, and α = 4°.

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Previous slit-groove interferometers [18–20] work in spectral domain by inspecting the shift of wavelength- or frequency-dependent interference curve, which requires the illumination of a broadband light source [20]. Therefore, high-contrast interference cannot be readily obtained due to the varied excitation strength of SPPs at different wavelength, making the strengths of the two interference arms not always the same. In contrast, our sensing scheme is based on the inspection of an intensity interference fringe under the illumination of a monochromatic light source, which simplifies the measurement setup (using a CCD instead of a spectrometer) as well as the post-detection signal processing (cosine-shaped interference curve with a uniform period). The strengths of the two arms of the interferometer can be well balanced to produce high-contrast interference pattern. Therefore, this non-spectroscopic sensing scheme has potential for low-cost, robust, portable, and chip-based detections provided that its sensing performance (sensitivity, FOM, etc.) is also good enough.

2. Principle of the plasmonic interferometer

The proposed interferometer is schematically depicted in Fig. 1, in which an array of N shallow grooves with period p and a subwavelength nanoslit of width w are patterned in a 250 nm thick gold film deposited on a fused silica substrate with a 15 nm thick titanium adhesion layer in between. L₀ is the distance between the center of the groove array and the nanoslit center at x = 0 (see Appendix A for the selection of L₀). The small tilt angle of the nanoslit with respect to the grooves is denoted as α. A TM polarized light (whose magnetic field vector is parallel with the grooves) of wavelength λ = 1064 nm impinges on the structure from the upper side at normal incidence, for which the permittivities of gold and silica are −52.02 + 3.87i and 1.45², respectively. Since in experiment the incident light may deviate slightly from the normal direction, we denote the small angle between the incident wave vector k₀ and the z axis in the yz plane as γ. For convenience of analysis, an auxiliary coordinate axis X is introduced, which is parallel with the nanoslit so that x = Xcosα. Due to the interference, the transmitted intensity I(X) along the nanoslit can be derived as (see Appendix B for details)

The structural parameters are optimized with the goal of keeping the intensities of the two arms as close as possible to each other so as to get the best contrast of interference. To begin with, we consider a simple case with air (n = 1.00) as the upper dielectric. The groove period p is chosen equal to the SPP wavelength (λ_spp = 1054 nm) according to the phase-matching condition of SPP excitation by a grating at normal incidence. For simplicity, the grooves are in rectangular profile with a depth of 50 nm and a duty cycle of 0.5. The other parameters are numerically optimized, with a commercial finite-element-method software COMSOL Multiphysics 3.5a, to achieve the best matched intensities of the two interference arms.

We first simulate the transmission through a single nanoslit (without the groove array nearby), as shown in Fig. 2(a) . The transmitted intensity varies slightly with respect to the change of slit width w (when w > 70 nm). Thus we simply choose w = 100 nm. Then we consider a slit-groove structure with L₀ = 16 μm by illuminating only the groove array, in which case only the SPPs launched from the grooves contribute to the transmission. The calculated transmitted intensity from the nanoslit in terms of the number of grooves N is shown as the dashed curve in Fig. 2(a), which is almost equal in strength to the directly transmitted light when N = 11. Therefore, we choose N = 11 and w = 100 nm and simulate the total transmission intensity by illuminating the whole slit-groove structure, as shown in Fig. 2(b). As anticipated, the transmitted intensity shows a cosine-shaped interference curve with respect to L. The field distributions in Figs. 2(c1) and 2(c2) correspond to the minimum I_min and the maximum I_max points marked as c1 (with L = 16.58 μm) and c2 (L = 17.08 μm) in Fig. 2(b), respectively, in which the destructive and constructive interferences are evidently observed. We define an extinction ratio r = 10lg(I_max/I_min) to indicate the contrast of the interference curve. According to Fig. 2(b), we can derive r = 44.57 dB, which indeed shows very high contrast.

 

Fig. 2 Simulated transmission from the tilted nanoslit-groove interferometer with air as the upper dielectric. (a) Normalized transmitted intensity from the nanoslit by illuminating only a single nanoslit with varied w (the solid line) or illuminating only the groove array in a slit-groove structure with w = 100 nm, p = 1054 nm, and different N (the dashed line). (b) Normalized total transmitted intensity from the nanoslit by illuminating the whole slit-groove structure with w = 100 nm, p = 1054 nm, and N = 11. (c1) and (c2) show the distributions of magnetic field component H_z in the slit-groove structure when L = 16.58 μm and 17.08 μm as indicated by arrows c1 and c2 in (b), respectively.

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To validate the feasibility of the interferometer, a number of samples have been fabricated by focused ion beam milling (with FEI Nova 200 Nanolab), where the groove array and the nanoslit were milled in a 250 nm thick gold film deposited on a fused silica substrate with 15nm thick Ti adhesion layer in between. Figure 1(c) shows the scanning electronic microscope (SEM) image of a typical sample with N = 10, α = 4°, and L₀ = 15.8 μm. In our experiment, as shown in Fig. 3 , the optical setup operates in a collinear transmission mode, i.e., the light source, the sample, and the detector are aligned along the same optical axis. A 1064 nm 200 mW solid state laser with spot size of 2 mm is used as the light source. After passing through a tunable attenuator and a polarizer, the TM polarized laser beam is incident on the sample. The transmitted light of the interference fringe from the nanoslit is collected by a long working-distance microscope objective (Nikon, 40×, NA = 0.6) and imaged by a CCD camera with 1280 × 1024 resolution and pixel size of 6.7 μm. The tunable attenuator is used to control the intensity of the incident light so that the detected CCD image has the best contrast (but is not saturated) with high signal-to-noise ratio. Two reflectors are used to fold the light path. The sample is attached to a three-dimensional translation stage. We found that the samples with N = 10 (instead of N = 11 in simulation) show the best contrast, probably due to the deviation of practical parameters such as the refractive index of the deposited gold film and the depth and width of the etched grooves. Figure 4 shows the measured transmitted intensities of two samples with N = 10, p ≈λ_spp = 1054 nm, w = 100 nm, L₀ = 15.8 μm, nanoslit length of 60 μm, and α = 2° and 4°. The detected CCD images of the interference fringes can be clearly observed in Figs. 4(a) and 4(b). To retrieve the fringe period, we plot the intensity profiles along the central lines of the nanoslits and fit them with cosine functions according to Eq. (1), as shown in Fig. 4(c). The periods of the two interference curves are estimated to be 31.94 μm and 15.9 μm, which closely match the theoretical predictions of 30.19 μm and 15.10μm calculated by Eq. (2), respectively. The small discrepancies are probably caused by the small incident angle γ in our measurement, which are calculated to be 3.16° and 2.89° according to Eq. (2). This is possible in our measurement setup. With these experiments, the working principle of the proposed interferometer is proved.

 

Fig. 3 Schematic and experimental setup of the optical characterization.

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Fig. 4 Experimental characterization of the transmitted interference fringes in two samples with N = 10, p ≈λ_spp = 1054 nm, w = 100 nm, L₀ = 15.8 μm, and α = 2° or 4°. (a) Measured CCD image (false color image) of the interference fringe in a sample with α = 2°. (b) The same as (a) but for a sample with α = 4°. (c) Fringe profiles along the central lines of the nanoslits retrieved from (a) and (b) and fitted with Eq. (1).

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3. Refractometric sensing experiment

Then we investigate the use of the interferometer as a plasmonic refractometric sensor. According to Eq. (1), the transmitted intensity from the nanoslit is sensitive to the refractive index n of the upper dielectric. Therefore, the refractive index change can be detected, e.g., by inspecting the intensity variation at a fixed point X on the slit. But this detection scheme is liable to be disturbed by the intensity fluctuation of the light source [22]. Alternatively, we can inspect the shift of the whole interference fringe ∆X with respect to the cover refractive index change ∆n. Since we avoid the absolute intensity measurement, the scheme is robust to the source disturbance and allows high-precision detection. From Eq. (1), we can derive the sensitivity of the interferometer S = −∆X/∆n (see Appendix C for details). If we neglect the small influence by φ₀ and γ, the sensitivity can be written in an elegant form

Equation (3) shows that the sensitivity S can be improved by increasing the slit-groove spacing L₀ or decreasing the tilt angle α. In principle, we can get infinitely large S by taking α close to 0. However, in practice this is not realistic. According to Eq. (2), smaller α leads to larger fringe period d, which makes the detection resolution (i.e., the minimum detectable fringe shift δX) worse. In other words, there is a trade-off between the sensitivity and the detection resolution, both of which are important for sensing applications. Therefore, to have a more meaningful measure of the practical sensing quality, we should also evaluate the figure of merit (FOM) of the device, which is defined as the ratio of sensitivity S to the full width at half maximum (FWHM) of the fringe [23]. Larger FOM means higher sensitivity for a fixed FWHM or narrower fringe for a fixed sensitivity. For our cosine-shaped interference fringe, the FWHM is half of the fringe period d. Thus, the theoretical FOM can be derived from Eqs. (2) and (3) as (if we still neglect the influence of γ)

Figure 5 shows the measured interference fringes in two typical samples (Sample I with L₀ = 127.7 μm and α = 1°; and Sample II with L₀ = 127.7 μm and α = 2°) covered with analytes of different NaCl concentrations. For both samples, we can clearly see the spatial shifts of the interference fringes to the left (as indicated by the dashed lines) with the increase of the analyte concentration. In Figs. 5(c) and 5(d), the intensity profiles along the center of the nanoslits, which are retrieved from the CCD images in Figs. 5(a) and 5(b), are plotted and fitted with cosine functions I = Acos(BX+C)+D, where A, B, C, and D are fitting parameters. It is seen that, except for the small fluctuation of the measurement data caused by the small fabrication defects of the nanoslits [such as those indicated by the dashed circles in Figs. 5(a) and 5(c)], the fringes are well fitted with the cosine profiles. Note that some other methods such as the fringe pattern matching method [24] and the fast Fourier transform method [17] may also be used for the signal processing. However, since our interference fringe strictly follows the cosine function (as indicated in Section 2 and Appendix B), the simple cosine fitting is precise enough to determine the shift of the curves.

 

Fig. 5 Measured interference fringes of two samples with NaCl solution of different concentrations as the upper dielectric: (a) and (c) for Sample I with L₀ = 127.7 μm and α = 1°; and (b) and (d) for Sample II with L₀ = 127.7 μm and α = 2°. (a) and (b) are the CCD false color images of the transmitted interference fringes. (c) and (d) are the intensity profiles along the central nanoslits and their cosine fitting curves. The curves are vertically displaced by 230 for clarity of demonstration. The dashed lines are visual guides to the valley positions of the interference fringes. The dashed circles show the position of a defect in the nanoslit.

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The valley positions of the interference fringes in Fig. 5 are extracted and fitted in Fig. 6(a) . As can be seen, the fringe position and the refractive index of the analyte hold a good linear relationship. The actual refractive indices of the analytes were measured by an Abbe refractometer, as shown in the inset of Fig. 6(a), in which a linear fitting relation of n = 1.3332+0.1720C between the refractive index n and the concentration of the analyte C was obtained. According to Fig. 6(a), the sensing properties of the two samples are calculated and summarized in Table 1 .In Fig. 6(b), we compare the theoretical [calculated by Eqs. (3) and (4)] and measured sensitivities and FOM of eight samples with different L₀ (67.7 μm, 87.7 μm, 107.7 μm, and 127.7 μm) and α (1° and 2°), which show relatively good theory-experiment correspondence. Therefore, for L₀ much larger than the illuminating wavelength, the interference is indeed mainly contributed by the phase difference accumulated by the SPP propagation [20]. Actually, due to the small tilt angle α, the term Xsinα (< 1.047μm for α = 1° and < 2.095 μm for α = 2°) is significantly smaller than L₀. So it is appropriate to neglect the small influence by φ₀ and γ in Eqs. (3) and (4). The experimental results also show that the FOM of our interferometric sensors can reach 250.9 (for the samples with L₀ = 127.7 μm). Compared with some previous plasmonic refractometric sensors, such as a typical prism-based SPR sensor with FOM ~55 [25], a typical nanohole array sensor with FOM ~23 [26], and the nanoparticle sensors with FOM < 10 [5], our sensing scheme is among the ones with the highest FOM.

 

Fig. 6 (a) Valley positions of the interference curves extracted from Fig. 5 and their linear fitting for Sample I (red points and curves) and Sample II (blue points and curves). The inset shows the refractive indices of the analytes measured by an Abbe refractometer. (b) Theoretical (lines) and experimental (symbols) comparisons of the sensitivity S and FOM for different samples with varied L₀ (67.7 μm, 87.7 μm, 107.7 μm, and 127.7 μm) and α (1° and 2°).

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Table 1. Comparison of the Measured and Theoretically Estimated Sensing Performances of Two Samples

View Table

According to Fig. 6(b), the performance of the sensor mainly depends on the slit-groove distance L₀ and the tilt angle α. By increasing L₀, higher sensitivity and higher FOM can be achieved. However, in practice, this is limited by two factors. On one hand, the requirement for miniaturized nanosensors with small footprint for probing nanovolume analyte limits the increase of L₀. On the other hand, L₀ cannot be arbitrarily large due to the propagation loss of SPPs. For example, in our sensing experiment, the numerically optimized L₀ is 62.43μm, in which case the two arms have the same intensity (I_spp = I_dir) and the interference pattern has the best contrast V = 1. By increasing L₀, if we want to keep the high contrast, the number of grooves N must be increased so as to maintain the intensity balance of the two arms (as discussed in Section 2). However, if all the other parameters are fixed and only L₀ is increased, obviously the contrast will decrease due to the propagation loss of the SPPs. In other words, we may obtain higher sensitivity and higher FOM by paying the cost of losing the contrast of the interference fringe. In practical application, as long as the interference contrast is not too small and can still be resolved by the CCD, it is acceptable. We have used a tunable attenuator to control the incident light intensity to let the maximum fringe intensity always close to 256 (maximum pixel value on our 8-digit CCD camera). Then the minimum intensity of the fringe is 256−A, where A is the difference of the maximum and minimum intensities. For example, we may set up a constraint condition (which depends on the performance of the detection system): if the noise fluctuation of signal is less than A/4, the fringe signal can be resolved. According to our experimental results, the mean noise fluctuation of signal is around the magnitude of 25.43 [as estimated from the results in Fig. 5(c)]. Thus, with A/4 > 25.43 we can derive that the interference contrast should be V > 0.2479. Denote that L₀ is increased by ∆L. Then I_spp is decreased due to the SPP propagation loss by a factor of exp(−2k″_spp∆L), where k″_spp is the imaginary part of the wave number of the SPPs and can be calculated from the SPP dispersion relation. Since I_dir does not change, the interference contrast becomes

 

Fig. 7 Contrast of the interference fringes in the eight samples used in the sensing experiment with respect to the change of L₀

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Reducing α is another effective way of enhancing the sensitivity, as seen from Eq. (3) and Fig. 6, where almost one-fold sensitivity enhancement can be achieved by changing α from 2° to 1° with the FOM almost unchanged. However, higher sensitivity makes the line width of the interference fringe broader (see Fig. 5), which degrades the detection resolution. Therefore, L₀ and α should be properly chosen so as to make a reasonable compromise among the sensitivity S, FOM, and the detection resolution in practical sensing applications.

4. Conclusion

To conclude, we have proposed and experimentally demonstrated a non-spectroscopic refractometric sensing scheme based on a plasmonic interferometer integrating a metallic groove array and a tilted nanoslit. Owing to the interference of the directly transmitted light from the nanoslit and that mediated by the SPPs launched from the groove array, high-contrast intensity interference fringe with uniform cosine-shaped pattern can be generated under the illumination of monochromatic light. By inspecting the spatial shift of interference fringe, the refractive index change of the cover medium can be detected. Our sensing experiments with NaCl solutions as the analytes show that the sensitivity of the interferometric sensors can reach 5674 μm/RIU and the FOM is up to 250, which are among the highest performance of the reported plasmonic refractometric sensors.

The proposed sensing scheme exhibits some new features. First, compared with most existing plasmonic sensing schemes that typically rely on the spectral interrogation requiring the use of expensive spectrometers and broadband light sources, our refractometric sensor is based on the imaging of an intensity interference pattern under the illumination of a monochromatic light source. The shift of the interference fringe is measured instead of its absolute intensity, which simplifies the measurement system and also makes the detection robust to the light source fluctuation. Second, by balancing the strengths of two interference arms, high-contrast interference fringe with uniform cosine-shaped profile can be produced, so that we can utilize all the measured data points for signal processing, which is more beneficial than monitoring a single peak or valley (such as that in previous SPR sensors). Third, the footprint of the proposed sensor is typically less than 0.01 mm² so that one can integrate more than 10⁴ sensing units on a single 1 cm² chip based on microfluidic platform, so that the multi-analyte and self-reference detections on the same chip are possible. Therefore, the proposed non-spectroscopic sensing scheme has the potential for low-cost, robust, portable, and chip-based detections in chemical and biosensing applications.

Appendix A: On the selection of L₀

As shown in Fig. 1, an array of N shallow grooves is used for SPP excitation. Since each groove can excite SPPs but has different distance with the nanoslit, the accumulated phase delay of the SPPs propagating from each groove to the nanoslit is also different, which yields different sensitivity according to Eq. (3). That is, the groove farthest from the nanoslit has the highest sensitivity, while the groove closest to the nanoslit has the lowest sensitivity. In our work, we selected L₀ as the distance from the center of the groove array to the nanoslit. To show that this is a reasonable solution, we treat each groove as a SPP source and the groove period is chosen equal to the SPP wavelength at n = 1.333. In Fig. 8 , we show the calculated interference fringes along the nanoslit by considering the respective contributions of each grooves (the blue curves) as well as their superposition (the red curve). It’s seen that the superposed fringe is always in the same position as that contributed by the groove in the middle for the groove array. Therefore, our definition of L₀ is reasonable, which is also validated by the good correspondence between experiment and theory in our sensing experiments (see Fig. 6).

 

Fig. 8 Calculated interference fringes contributed by each grooves (the blue curves) and the superposed total interference fringe (the red curve) in a slit-groove structure with L₀ = 120μm and α = 1°.

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Appendix B: Intensity expression of the interference fringe

When a TM polarized plane wave illuminates the tilted slit-groove interferometer at an angle γ with respect to the z axis, the grooves excite a fraction of the incident light into SPPs towards the nanoslit (whose propagation direction is indicated by dashed arrows in Fig. 1(a)). The SPPs scattered at the nanoslit reradiate into transmitted light whose electric field amplitude is denoted as E_spp. The directly transmitted light through the nanoslit, whose amplitude is denoted as E_dir, would interfere with the SPP-mediated transmitted light. Then the total transmitted intensity I(X) of the interference pattern along the nanoslit can be derived as

(B1)$$I\left(X\right)=E_{\text{spp}}^2+E_{\text{dir}}^2+2E_{\text{spp}}^{}{E}_{\text{dir}}^{}\cos [{\phi }_{\text{spp}}(X)+{\phi }_{\text{inc}}(X)+{\phi }_{\text{0}}],$$

where

(B2)$${\phi }_{\text{spp}}\left(X\right)=k_{\text{spp}}L\left(X\right)=k_0{\left(\frac{{{\epsilon }^{\prime }}_m{n}^2}{{{\epsilon }^{\prime }}_m+n^2}\right)}^{\frac{1}{2}}L\left(X\right),$$

is the phase delay of SPPs when propagating from the groove array to the slit,

(B3)$${\phi }_{\text{inc}}\left(X\right)=k_{0}L\left(X\right)\sin \gamma ,$$

is the phase difference induced by the non-normal incident angle γ, and φ₀ is the additional initial phase of SPPs excited by the grooves.

Appendix C: Derivation of the sensitivity

The sensitivity of the interferometer is defined as S = −∆X/∆n, which can be obtained by putting the phase term in Eq. (B1) as a constant, i.e.,

(C1)$${\phi }_{\text{spp}}(X)+{\phi }_{\text{inc}}(X)+{\phi }_{\text{0}}=\text{const,}$$

and taking its derivative with respect to n. Therefore, with Eqs. (B2), (B3), and (C1), we have

(C2)$$S=\frac{-dX}{dn}=\frac{{\left(\frac{{{\epsilon }^{\prime }}_m}{{{\epsilon }^{\prime }}_m+n^2}\right)}^{\frac{3}{2}}\left(L_0+X\sin \alpha \right)+\frac{d{\phi }_{\text{0}}}{dn}}{\left[{\left(\frac{{{\epsilon }^{\prime }}_m{n}^2}{{{\epsilon }^{\prime }}_m+n^2}\right)}^{\frac{1}{2}}+\sin \gamma \right]\sin \alpha },$$

If we neglect the small influence by φ₀ and γ, the sensitivity can be expressed in an elegant form as

(C3)$$S=\frac{{{\epsilon }^{\prime }}_m}{{{\epsilon }^{\prime }}_{m}n+n_{}^3}\left(\frac{L_0}{\sin \alpha }+X\right),$$

Since L₀/sinα >> X, Eq. (C3) can be further simplified as

(C4)$$S\approx \frac{{{\epsilon }^{\prime }}_m}{{{\epsilon }^{\prime }}_{m}n+n_{}^3}\frac{L_0}{\sin \alpha }.$$

Acknowledgments

We thank Mr. Hao Zhu for measuring the actual refractive indices of the analytes with an Abbe refractometer. We acknowledge the support by the National Basic Research Program of China (Project No. 2007CB935303), the National Natural Science Foundation of China (Project No. 11004119), and the Academy of Finland (Project No. 128420).

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