A detailed theoretical model is provided to analyze the effects of temperature on prism-based surface plasmon resonance (SPR) sensors, including temperature dependence of the metal and prism. A complete sensitivity matrix simultaneously measures variations in refractive index (RI) and temperatures using measurements at two wavelengths for the angular-interrogation mode, or at two angles of incidence for the wavelength-interrogation mode. Correction of matrix coefficients improves accuracy of the two modes. Validation is performed using a self-designed wavelength SPR system with an adjustable incident angle perform. This method provides a new way to detect the RI and may lead to the better design and fabrication of prism-based SPR sensors.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Surface plasmon resonance (SPR) is an extremely sensitive technique for detecting very small changes in the refractive index (RI) of an analyte in contact with a sensor metal film . With a bio-coating sensor layer on the metal film, high sensitivity, and real-time, label-free detection, the SPR technique has been applied in several fields, including adsorption process monitoring of molecule and concentration detections [2–4], and as an indispensable method in the study of molecular biology .
To understand the various applications of the SPR technique, studies have focused on the detection of tiny changes in the RI or other quantities converted into an equivalent RI . Among the several equivalent quantities, temperature has multiple effects on SPR sensor detection, leading to the inability to discriminate between RI-induced and temperature-induced SPR changes. Currently, temperature interference is suppressed by constructing a sensor probe with a pair of sensing channels, where one of them is the reference, or more directly, to passively control the temperature. However, this kind of constructed probe is not suitable for compact sensor fabrication. Furthermore, temperature control may not be practical in applications where the sample temperature changes during the measurements or the RI is detected at different temperature levels [7,8].
Lin et al. [9,10] analyzed the temperature dependence of the resonance position and the sensitivity for prism-based SPR sensors, including the angular-interrogation and wavelength-interrogation modes, respectively. A proof-of-phenomenon experiment was carried out without providing a way to account for the temperature interference. A dual-wavelength method was proposed by Xiao  to simultaneously measure the RI and temperature, but only for the angular-interrogation mode; the wavelength-interrogation mode was note considered.
In this study, we developed a complete sensitivity matrix method, both for the angular-interrogation mode and wavelength-interrogation mode, to distinguish RI-induced SPR shifts without temperature control. With the most commonly used Kretschmann SPR configuration, and using the measurement of two wavelengths for the angular-interrogation mode or two angles of incidence for the wavelength-interrogation mode, both the RI and temperature variation can be obtained simultaneously.
2. Theoretical analysis
In a typical case of measuring the RI of an analyte, the prism-based Kretschmann SPR configuration can be abstracted, as shown in Figs. 1 and 2. For the angular-interrogation mode (Fig. 1), a metal film (thickness d1, dielectric permittivity ) is evaporated onto the prism (dielectric permittivity ). For simplicity, but without loss of generality, the thickness of the detected medium is assumed to be sufficiently large and thus treated as infinite. P-polarized light illuminates the prism. Under the condition of total internal reflection, evanescent waves penetrate through the metal film and are then absorbed by the surface plasmons, leading to absorption in the reflected light intensity [12,13]. For the wavelength-interrogation mode, as shown in Fig. 2, collimated light strikes the prism at a fixed angle, resulting in the measurement of the domain of reflected waves . Therefore, the multiple effects of temperature on the SPR sensor can be transformed into the physical parameters of these constituent units, including the prism and metal layer. Then, according to the thermo-optic and the dispersion effects in the prism, as well as the effects of phonon–electron scattering and electron–electron scattering in the metal layer , a comprehensive temperature Drude theoretical model can be established to study the temperature effect on the SPR sensor.
2.1 Temperature effect on the gold film
Thermal changes significantly affect metal film parameters, and according to the Drude model, the frequency-dependent dielectric function can be appropriately represented as16]
Temperature affects the collision frequency in two ways: phonon–electron scattering () and electron–electron scattering () [17,18]. The combined effect of these two aspects is19,20]6], and can be expressed by the Lawrence model as 
The temperature variation also has an influence on the metal layer thickness , which is an important parameter of SPR sensors. As the thickness variation only corresponds to the expansion in the normal direction, by means of a corrected thermal-expansion coefficient , the thickness can be expressed asTable 1 [6,9].
2.2 Temperature effect on the prism
The thermo-optic coefficient can be used to express the temperature dependence of RI in the prism :
Assuming that the most commonly used prisms in SPR sensors (e.g., BK9) have very low thermo-optic coefficients , typically on the order of dn/dT ~10−6 K–1, it can be observed from the wavelength-interrogation mode in Eqs. (9)–(11) that both the RI of the prism and the thermal-optic coefficient are functions of wavelength, and the dispersion of glass must be seriously considered [23,24].Table 2 [23–26].
2.3 Reflectance of the SPR sensor
For the SPR geometry, shown in Figs. 1 and 2 and according to the Fresnel equations, the reflectance r can be expressed as
3. Sensitivity matrix method for angular-interrogation SPR
For an analyte RI of 1.330, the SPR resonance curves at different temperatures are presented in Fig. 3(a), in which the resonance angle decreases with temperature. Figure 3(b) shows the fitting lines between the resonance angle and RI at different temperatures, and the variation of their slopes with temperature is shown in Fig. 3(c). From these curves, it is clear that the resonance angle changes linearly with RI. With the variation in temperature from 270 to 380 K, the line-fitting sensitivity shifts slightly, from 130.875°/RIU to 130.125°/RIU, which is a negligible change of 0.57% . This implies that the linear sensitivity of the resonance with RI has little temperature dependence. Further calculation reveals that the resonance angle also changes linearly with temperature, and the line-fitting sensitivity with temperature at different RIs is presented in Fig. 3(d). With the RI changing from 1.330 to 1.346 RIU, the sensitivity shifts from 0.986 to 1.095 x10−3 °/k, resulting in a shift of 11.05%. Furthermore, from the relative variation in the linear sensitivities with RI and temperature, it is clear that the SPR angle shift is considerably more sensitive to RI variation than to temperature variation.
Detailed scenarios for the dependence of the SPR resonance shift on the simultaneous variation of RI and temperature are presented in Fig. 4. It can be seen that the line surface of the relationships governing the resonance angle change versus the RI and temperature changes is almost an inclined plane, which reveals that by ignoring the tiny changes in the line-fitting sensitivity, the dependences of the SPR resonance shift on the RI and temperature are linear and independent of each other. Therefore, the SPR resonance angle shift can be determined asEq. (15) is necessary. Assuming its existence and according to the principle of matrix theory, it can be expressed as
Then, we confront the detecting situation to obtain another relationship similar to Eq. (15). For the angular-interrogation mode, when measuring the variation in reflected light intensity with different angles of incidence, the relationships governing the resonance angle change versus the RI change or temperature change are dependent on the light wavelength used in the measurement . Further calculation reveals that, with different detecting light wavelengths, the resonance angle also changes linearly with RI or temperature, independently. The variations in sensitivity coefficients at different wavelengths are presented in Fig. 5.
The red curve indicates that mn (the sensitivity to RI) decreases with the detecting wavelength; the change in temperature sensitivity (mT) is represented as a blue curve. Therefore, the other independent relationship similar to the form of Eq. (15) can be observed at another detecting wavelength. Thus, by detecting with two particular wavelengths, a full rank sensitivity matrix M can be obtained, and the variations in RI and temperature can be derived as Eq. (17). With arbitrarily chosen RI and temperature references, both can be measured.
4. Sensitivity matrix method for wavelength-interrogation SPR
The effects of temperature are also analyzed for the wavelength-interrogation mode, as shown in Fig. 6. The angle of incidence of parallel light is fixed at 70°, and the reflectivity is a function of wavelength. The temperature dependences of the physical parameters of the gold sensor film and prism are identical to the angular-interrogation mode.
Figure 6(a) presents the variation of resonance curves at different temperatures, which is similar to the case of the angular-interrogation mode, as shown in Fig. 3(a). However, a detailed analysis exposes differences. Fitting lines between the resonance wavelength and RI at different temperatures are presented in Fig. 6(b). It can be seen that the resonance wavelength changes linearly with RI; the variation in the line-fitting sensitivity with RI (mn) is shown in Fig. 6 (c). With the change in temperature from 270 to 380 K, mn changes between 9453.312 and 9131.859 nm/RIU, corresponding to a relative non-negligible change of 3.40%. Further calculation reveals that the resonance wavelength also changes linearly with temperature; the change in the line-fitting sensitivity (mT) at different RIs is depicted in Fig. 6(d), in which mT changes from −0.111 to −0.161 nm/K for a change in RI between 1.330 and 1.346 RIU, resulting in a non-negligible difference of 45.05%. However, according to the observation that the resonance mainly depends on the RI, and for the practical detection range of RI and temperature, it is reasonable to ignore the slight variation in the sensitivity matrix coefficients. Then, the dependence of the resonance shift on RI and temperature can be expressed in the form of Eq. (15).
A more specific analysis, calculated with different fixed angles of incidence, reveals that the linear relationships governing the resonance wavelength change versus the RI change or temperature change are dependent on the angles of incidence used in the measurements. The variations in line-fitting coefficients at different angles of incidence are presented in Fig. 7. Therefore, relationships in the form of Eq. (16) can be obtained in the detecting cases at two particular angles of incidence. Then, with the inverse matrix, both the variation in RI and temperature can be derived for the wavelength-interrogation mode.
5. Correction of sensitivity matrix coefficients
To meet the requirements for higher-precision detection, the coefficients of the sensitivity matrix must be corrected. From Figs. 3(c) and 3(d) or Figs. 6(c) and 6(d), it can be seen that the two sensitivity coefficients change linearly with temperature or RI. Therefore, the coefficients of the matrix can be corrected as
The fn(T) and fT(RI) are linear functions of temperature and RI for the detecting cases i and j, respectively. In a practical measurement, the temperature can be detected. However, the RI is unknown and must be obtained. Moreover, the SPR shift is much more sensitive to the RI variation than temperature variation, and the resonance mainly depends on the RI rather than the temperature. Therefore, the correction of sensitivity between the SPR resonance and RI with temperature can be established as
With the linear correction of the sensitivity coefficients at detecting fragments, the accuracy of the matrix method can be improved. We present a complete sensitivity matrix method for prism-based SPR sensors to simultaneously detect the RI and temperature. This approach is very accurate when the dependences of SPR resonance on both the temperature and RI are linear and independent. Any nonlinear factor will reduce the accuracy. In this case, a more complex model would be needed to address the problem.
6. Experimental results and discussion
For angular-interrogation mode SPR, the technique has been proved experimentally by Xiao . Here, a proof-of-concept experiment was carried out to demonstrate the ability of the proposed technique for wavelength-interrogation mode SPR sensors. As shown in Fig. 8(a), the optical setup of a prism-based spectral SPR is constructed. A broadband optical light source, with an emission spectrum span from 400 to 1200 nm, is used to excite the surface plasmon wave. The emitted light beam is collimated through sets of compounds lenses and an adjustable iris, where it then impinges onto the bottom of the coupling prism coated with gold film that is 47 nm thick. The reflected light beam is coupled into the transfer fiber, measuring the spectrum with a spectrometer. For the wavelength-interrogation mode SPR sensor, two different fixed incident angles are needed to carry out the validation experiment. With the experiment setup, as shown in Fig. 8(b), the angle of incidence parallel light can be adjusted conveniently.
The RI of the analyte changes with different concentrations of aqueous NaCl solution, as confirmed using an Abel refractive index measuring instrument. Temperature was changed using a hot plate. In order to eliminate common-mode noise components arising from the light source, we recorded the reflectance characteristics of the incident light before the experiment, and obtained SPR curves by calculating the differences of spectra intensity between the real-time detected and the record reflectance light.
Figure 9 depicts the resonance shift versus the RI at two fixed detection angles with the temperature fixed at 20°C. The fitting straight-lines show that the resonance has a significant linear response to RI variation. With RI of the analyte changes from 1.3335 to 1.3407 RIU, the resonance wavelength increases from 787.9 to 851.2 nm at a detection angle of 48°, and shifts from 706.7 to 756.4 nm at 45°, corresponding to linear slopes of 8786.1 and 6902.6 nm/RIU, respectively. Figure 10 shows that, with an analyte RI of 1.3384 RIU at a temperature of 20°C, the resonance also changes linearly with the increase of temperature, resulting in measured slopes of −1.1440 and −0.8743 nm/°C, respectively.
With the linear sensitivity of NaCl solution  to temperature (−1.1521 × 10−4 RIU/°C) detected during calibration, and the linear coefficients obtained from Figs. 9 and 10, the linear dependence of SPR resonance shift on the temperature can be calculated. Therefore, with the proposed matrix detecting technique, the variation of the analyte’s RI and temperature can be expressed as
In this study, the effects of temperature on a prism-based SPR sensor, including the dependences of the thermal coefficient of the metal and the dispersion of the prism, are theoretically investigated in the Kretschmann configuration. A complete sensitivity matrix is presented and analyzed for the angular-interrogation mode and wavelength-interrogation mode. For the wavelength-interrogation mode SPR sensor, a proof-of-concept validation experiment was performed. With the proposed method, both the RI and temperature variation can be obtained simultaneously without temperature control or calibration processes. This approach paves the way toward an SPR sensor that can detect the RI at different temperature levels and may expand the application domain of SPR sensors.
General Programs of the National Natural Science Foundation of China (31671586); National Key Research and Development Program of Ningxia Autonomous Region (Y723A215B2); National Science and Technology Major Project (2018YFC0831102); National Natural Science Foundation of China (61475163).
The authors declare that there are no conflicts of interest related to this article
3. C. L. Wong, G. C. K. Chen, B. K. Ng, S. Agarwal, Z. Lin, P. Chen, and H. P. Ho, “Multiplex spectral surface plasmon resonance imaging (SPRI) sensor based on the polarization control scheme,” Opt. Express 19(20), 18965–18978 (2011). [CrossRef] [PubMed]
4. Y. Zeng, L. Wang, S. Y. Wu, J. He, J. Qu, X. Li, H. P. Ho, D. Gu, B. Z. Gao, and Y. Shao, “High-throughput imaging surface plasmon resonance biosensing based on an adaptive spectral-dip tracking scheme,” Opt. Express 24(25), 28303–28311 (2016). [CrossRef] [PubMed]
5. N. Blow, “Proteins and proteomics: life on the surface,” Nat. Methods 6(5), 389–393 (2009). [CrossRef]
6. S. K. Ozdemir and G. Turhan-Sayan, “Temperature effects on surface plasmon resonance: design considerations for an optical temperature sensor,” J. Lightwave Technol. 21(3), 805–814 (2003). [CrossRef]
7. J. B. Fiche, A. Buhot, R. Calemczuk, and T. Livache, “Temperature effects on DNA chip experiments from surface plasmon resonance imaging: isotherms and melting curves,” Biophys. J. 92(3), 935–946 (2007). [CrossRef] [PubMed]
8. W. Luo, S. Chen, L. Chen, H. Li, P. Miao, H. Gao, Z. Hu, and M. Li, “Dual-angle technique for simultaneous measurement of refractive index and temperature based on a surface plasmon resonance sensor,” Opt. Express 25(11), 12733–12742 (2017). [CrossRef] [PubMed]
9. L. Kai-Qun, W. Lai-Ming, Z. Dou-Gou, Z. Rong-Sheng, W. Pei, L. Yong-Hua, and M. Hair, “Temperature effects on prism-based surface plasmon resonance sensor,” Chin. Phys. Lett. 24(11), 3081–3084 (2007). [CrossRef]
10. K. Lin, Y. Lu, Z. Luo, R. Zheng, P. Wang, and H. Ming, “Numerical and experimental investigation of temperature effects on the surface plasmon resonance sensor,” Chin. Phys. Lett. 7, 428–431 (2009).
11. F. Xiao, D. Michel, G. Li, A. Xu, and K. Alameh, “Simultaneous measurement of refractive index and temperature based on surface plasmon resonance sensors,” J. Lightwave Technol. 32(21), 4169–4173 (2014). [CrossRef]
12. A. Otto, “Excitation of nonradiative surface plasma waves in silver by the method of frustrated total reflection,” Zeitschrift Für Physik A Hadrons and Nuclei 216, 398–410 (1968).
13. M. S. Tame, K. R. McEnery, Ş. K. Özdemir, J. Lee, S. A. Maier, and M. S. Kim, “Quantum plasmonics,” Nat. Phys. 9(6), 329–340 (2013). [CrossRef]
14. I. Watad and I. Abdulhalim, “Comparative study between polarimetric and intensity-based surface plasmon resonance sensors in the spectral mode,” Appl. Opt. 56(27), 7549–7558 (2017). [CrossRef] [PubMed]
15. H.-P. Chiang, Y.-C. Wang, and P. T. Leung, “Effect of temperature on the incident angle-dependence of the sensitivity for surface plasmon resonance spectroscopy,” Thin Solid Films 425(1-2), 135–138 (2003). [CrossRef]
16. H.-P. Chiang, P. T. Leung, and W. S. Tse, “Remarks on the substrate-temperature dependence of surface-enhanced raman scattering,” J. Phys. Chem. B 104(10), 2348–2350 (2000). [CrossRef]
17. A. K. Sharma and B. D. Gupta, “Influence of temperature on the sensitivity and signal-to-noise ratio of a fiber-optic surface-plasmon resonance sensor,” Appl. Opt. 45(1), 151–161 (2006). [CrossRef] [PubMed]
18. G. Ghosh, Handbook of Thermo-Optic Coefficients of Optical Materials with Applications (Academic Press, 1998).
19. R. T. Beach and R. W. Christy, “Electron-electron scattering in the intraband optical conductivity of Cu, Ag, and Au,” Phys. Rev. B Condens. Matter Mater. Phys. 16(12), 5277–5284 (1977). [CrossRef]
20. T. Holstein, “Optical and infrared volume absorptivity of metals,” Phys. Rev. 96(2), 535–536 (1954). [CrossRef]
21. W. E. Lawrence, “Electron-electron scattering in the low-temperature resistivity of the noble metals,” Phys. Rev. B Condens. Matter Mater. Phys. 13(12), 5316–5319 (1976). [CrossRef]
22. S. Herminghaus and P. Leiderer, “Surface plasmon enhanced transient thermoreflectance,” Appl. Phys., A Mater. Sci. Process. 51(4), 350–353 (1990). [CrossRef]
23. G. Ghosh, “Temperature dispersion of refractive indexes in some silicate fiber glasses,” IEEE Photonics Technol. Lett. 6(3), 431–433 (1994). [CrossRef]
24. I. H. Malitson, “Interspecimen comparison of the refractive index of fused silica,” J. Opt. Soc. Am. 55(10), 1205–1209 (1965). [CrossRef]
25. G. Ghosh, “Temperature dispersion of refractive indices in crystalline and amorphous silicon,” Appl. Phys. Lett. 66(26), 3570–3572 (1995). [CrossRef]
27. W. M. Yunus, “Temperature dependence of refractive index and absorption of NaCl, MgCl2, and Na2SO4 solutions as major components in natural seawater,” Appl. Opt. 31(16), 2963–2964 (1992). [CrossRef] [PubMed]