Enhancing the Spectral Sensitivity of Prism-Based SPR Sensors: The Role of Analyte RI

Abstract

A theoretical approach is presented to significantly enhance the spectral sensitivity of prism-based SPR sensors. The spectral sensitivity of prism-based SPR sensors is derived based on the coupling conditions of SPR and might exceed 10⁵ nm/RIU for analytes with large RI values when other sensor parameters are carefully considered, including the RI of the prism, the angle of incidence, and the SPR active material. The spectral sensitivity could be markedly enhanced, reaching up to 10,000 nm/RIU by fine-tuning the effective RI of the incident light to be slightly larger, specifically 0.01~0.02 RIU, than the RI of the analyte, which is attributed to the large dielectric permittivity of the SPR active material, the key factor for achieving high sensitivity. The dynamic range is 0.040 RIU in the case of high sensitivity, which is sufficient in most applications. Moreover, the spectral sensitivity could be pushed even higher, into the range of 10⁶~10⁸ nm/RIU, by positioning the effective RI of the incident light closer to that of the analyte. However, it requires a careful balance between optimizing the sensitivity and maintaining an acceptable dynamic range.

1. Introduction

Surface plasmon resonance (SPR) sensors have received much attention for their significant applications in investigating surface interactions and sensing of gases and bio-materials. Among the three types of coupling configurations (grating-based, prism-based, and optical-waveguide-based coupling), prism-based coupling is the most widely used due to its high sensitivity and operational simplicity [1,2]. Numerous attempts have been made to improve the sensitivity and the detection limit over the last two decades [3,4], such as modifying the refractive index (RI) and the dispersion of the prism [5,6], SPR active material optimization [7,8], enhancing the sensitivity in the infrared wavelength [9,10], long-range SPR [11,12], guided-wave SPR by the addition of a thin dielectric layer or 2D materials on top of the metal layer [13,14,15,16,17], and additions of gratings on top of the metal layer [18,19]. However, the existing studies of spectral sensitivity are mostly from the view of the wavelength; investigations from the perspective of the RI of the analyte are scarce. The present work investigates the spectral sensitivity of the prism-based SPR sensor in terms of the RI of the analyte. The sensitivity-improved approach is proposed, and the dynamic range is discussed. It provides a convenient method to improve the spectral sensitivity based on the maximum RI of the analyte during the measurement of all the prism-based SPR sensors using the wavelength interrogation method.

2. Spectral Sensitivity

An optical wave couples to the SPW (surface plasmon wave) of the metal with a prism by the attenuated total reflection (ATR) method [20]. The optical wave is totally reflected at the interface between the prism and the metal film and then evanescently penetrates through the metal film and excites the SPW at the interface between the metal film and the analyte, as shown in Figure 1a. The SPR leads to a drop in the intensity of the reflected light, which could be measured as a valley in the wavelength spectrum. The RI of the analyte would be obtained by detecting the wavelength of the valley, which is known as the wavelength interrogation method, as shown in Figure 1b.

The optimum coupling could be achieved for a metal film thickness in the range of 10 nm to 70 nm [21]. To excite the SPW, the component along the metal film of the wave vector of the incident light must match the vector of the SPW, and then the coupling condition is expressed as

$$k_0\cdot n_p\cdot \sin \theta =k_0\cdot \sqrt{{\displaystyle \frac{{\epsilon }_m\cdot n_a^2}{{\epsilon }_m+n_a^2}}}$$

(1)

where k₀ = 2π/λ₀ and λ₀ are the propagation constant and wavelength of the incident light in the free space, respectively. n_p and n_a are the RIs of the prism and the analyte, respectively. θ is the incident angle of the light at the interface between the prism and the metal film, as shown in Figure 1a, and ε_m is the dielectric permittivity of the SPR active material (usually noble metals, such as gold and silver). For wavelength interrogation method, the sensitivity can be obtained by differentiating Equation (1) in λ and n_a [3]

$$S_{\lambda }={\displaystyle \frac{d\lambda }{d{n}_a}}={\displaystyle \frac{2{\epsilon }_m^2}{-n_a^3\cdot \frac{d{\epsilon }_m}{d\lambda }+2\frac{n_a}{n_p}\cdot {\epsilon }_m\left(n_a^2+{\epsilon }_m\right)\cdot \frac{d{n}_p}{d\lambda }}}$$

(2)

Accompanied with Equation (1), the correlation between S_λ and n_a is illustrated in Figure 2. The incident angle is set at 82°, and gold and fused silica act as the SPR active material and the prism, respectively. Drude and Sellmeier models are used to describe the dielectric permittivity of gold and RI of fused silica, respectively [22,23]. It can be seen that the sensitivity increases with the RI of the analyte and exhibits high value at large RI. The absolute value of dielectric permittivity of gold or RI of the prism should be increased for satisfying the coupling condition of SPR at the fixed incident angle in order to compensate for the rise in the right term of Equation (1), i.e., the propagation constant of SPW, due to the increase in the analyte RI. A large dielectric permittivity of SPR active material or a large RI of the prism is the reason for the rising of the sensitivity according to Equation (2) (the second part of the denominator of Equation (2) is negative). There are three other parameters influencing the sensitivity in accordance with Equation (2), which are the RI of the analyte, the derivations of the dielectric permittivity of SPR active material, and the RI of the prism to the wavelength. As the material dispersion of the commonly used prism is small (|dn_p/dλ| < 10⁻⁴ RIU/nm), such as fused silica and SF11, the second term in the denominator of Equation (2) is much smaller than the first one. We neglect the second term in the denominator and Equation (2) is simplified into

$$S_{\lambda }={\displaystyle \frac{d\lambda }{d{n}_a}}={\displaystyle \frac{-2{\epsilon }_m^2}{n_a^3\cdot \frac{d{\epsilon }_m}{d\lambda }}}$$

(3)

and Equation (3) agrees well with Equation (2) when the RI of the analyte is less than 1.40 RIU, as shown in Figure 2, so Equation (3) is used in the following analysis.

Put Equation (1) into Equation (3); Equation (3) is rewritten as

$$S_{\lambda }=-{\displaystyle \frac{2n_a\cdot {\left(n_p\cdot \sin \theta \right)}^4}{{\left(n_a^2-{\left(n_p\cdot \sin \theta \right)}^2\right)}^2\cdot \frac{d{\epsilon }_m}{d\lambda }}}$$

(4)

From Equation (4), the spectral sensitivity is the function of the RIs of the analyte and the prism, the incident angle of the light, and the derivation of the dielectric permittivity of SPR active material to the wavelength. But, it is well known that there are no total reflections if the RI of the analyte is larger than the product of the RI of the prism and the sine of the incident angle of the light (i.e., the effective RI of the incident light, which is defined as $n_{eff}=k_{in}/k_0$, where k_in is the component of the incident light wave vector in the prism along the metal film.), so there is no ATR. The quadratic component of the denominator in Equation (4) covers up the situation of no total reflections, so the root of the spectral sensitivity is adopted as shown in Equation (5).

$$S_{\lambda root}={\displaystyle \frac{\sqrt{2n_a}\cdot {\left(n_p\cdot \sin \theta \right)}^2}{\left({\left(n_p\cdot \sin \theta \right)}^2-n_a^2\right)\cdot \sqrt{-\frac{d{\epsilon }_m}{d\lambda }}}}$$

(5)

The root of the spectral sensitivities as a function of the RI of the analyte and the derivation of the dielectric permittivity of SPR active material to the wavelength are demonstrated in Figure 3 with three effective RIs of the incident light, 1.1426, 1.3347, and 1.4356, respectively.

As shown in Figure 3, the root of spectral sensitivity is rapidly promoted when the RI of the analyte is close to the effective RI of the incident light resulting from the denominator of Equation (5) approaching zero. Further, the root of spectral sensitivity is proportional to the (−dε_m/dλ)^−1/2, but this effect to the sensitivity is not obvious compared with the RI of the analyte since the variation in the derivation of the dielectric permittivity of the SPR active material to the wavelength is usually small. dε_m/dλ is set from −0.4 to −0.02/nm in the analysis.

3. Sensitivity Enhancement and Dynamic Range

The maximum of the refractive index of the analyte is supposed to be known, and then the spectral sensitivity could be enhanced by arranging the effective RI of the incident light to be a little larger than the maximum of the RI of the analyte, leading to the denominator approximating zero and/or adopting prisms with large RI values, resulting in a larger numerator, according to Equation (5). The large dielectric permittivity of the SPR active materials is one of the key points regarding high spectral sensitivity. Only a large dielectric permittivity could satisfy the coupling condition of SPR according to Equation (1) when the effective RI of the incident light is posited a little larger than the maximum of the RI of the analyte to increase the spectral sensitivity, as plotted in Figure 4, where the high spectral sensitivity is accompanied with the large dielectric permittivity of the SPR active materials. The effective RI of the incident light and the derivation of the dielectric permittivity of the SPR active material to the wavelength are set at 1.3347 (the dotted line) and −0.2 /nm, respectively, in Figure 4. The result is consistent with the previous reports that prism-based SPR sensors have large spectral sensitivity at a long wavelength [3,4,6,10,12] since metals have large dielectric permittivity at a long wavelength.

The dynamic range is constrained by the sensitivity’s decline as the analyte’s RI diminishes. However, the dynamic range is 0.032 RIU (from 1.097 to 1.129 RIU, the purple case in Figure 3b), which is sufficient in most cases since the RI variation is small, usually 10⁻⁴~10⁻² RIU as the SPR sensor used for RI or bio-reaction measurements in the case of the effective RI of the incident light and the derivation of the dielectric permittivity of SPR active material to the wavelength are set at 1.1426 and −0.4 /nm, respectively, and the root of spectral sensitivity is from 30 to 100; i.e., the sensitivity is from 900 to 10,000 nm/RIU, which is about one magnitude larger than the reported results [7,24]. The dynamic range is broadened when an analyte with a large RI is measured with an appropriate effective RI of the incident light. For example, the dynamic range is 0.040 RIU (from 1.278 to 1.318 RIU, the cyan case in Figure 3b), expanded 1.25 times compared with the last one in the same sensitivity range when the effective RI of the incident light is set at 1.3347. Moreover, the spectral sensitivity would reach a very high value, such as 10⁶~10⁸ nm/RIU, when the effective RI of the incident light is managed more closely to the RI of the analyte, but the dynamic range is small for this high sensitivity. A trade-off is required between the sensitivity and the dynamic range in the measurements. It should be noted that the large real dielectric permittivity of the SPR active material would increase the wavelength of the incident light from visible to infrared and might need to be greater than 50 to achieve a sensitivity larger than 10⁵ nm/RIU. A material with such a large dielectric permittivity may be unknown to us, but it might be possible for some metamaterials [25]. In addition, the imaginary part of the dielectric permittivity of the SPR active material ignored in our study also becomes large as the real part increases, which would deteriorate the resolution of the SPR sensor.

4. Conclusions

The spectral sensitivity is numerically investigated from the perspective of the RI of the analyte with respect to prism-based SPR sensors and might attain more than 10⁵ nm/RIU at a large RI when gold and fused silica are employed as the SPR active material and the prism, respectively. The correlations between the RI of the analyte, the effective RI of the incident light, the derivation of the dielectric permittivity of the SPR active material to the wavelength, and the spectral sensitivity are illustrated, and it is found that the spectral sensitivity is greatly increased when the RI of the analyte is close to the effective RI of the incident light. It is suggested to set the effective RI of the incident light 0.01~0.02 RIU larger than the maximal RI of the analyte to enhance the spectral sensitivity to 10⁴ nm/RIU, and in general adjusting the incident angle is more convenient than modifying the RI of the prism to achieve a desirable effective RI of the incident light. The dynamic range is 0.040 RIU when the effective RI of the incident light is set at 1.3347, which, although confined, is sufficient for the majority of applications and can be enlarged when measuring an analyte with a large RI. The high sensitivity is fundamentally due to the large dielectric permittivity of the SPR active material, prompting a reassessment of the parameter design and the measurement accuracy of the SPR sensor based on wavelength interrogations.