Simultaneous measurement of refractive index and temperature for prism-based surface plasmon resonance sensors

Wei Luo; Rujing Wang; Hairong Li; Jieting Kou; Xinhua Zeng; He Huang; Xiaobo Hu; Wei Huang

doi:10.1364/OE.27.000576

1. Introduction

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 [1]. 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 [5].

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 [6]. 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 [11] 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 $ε_{1} = {ε^{'}}_{1} + {ε^{″}}_{1}$ ) is evaporated onto the prism (dielectric permittivity $ε_{0}$ ). 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 [14]. 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 [15], a comprehensive temperature Drude theoretical model can be established to study the temperature effect on the SPR sensor.

Fig. 1 Schematic of angular-interrogation mode surface plasmon resonance (SPR) sensor.

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Fig. 2 Schematic of wavelength-interrogation mode surface plasmon resonance (SPR) sensor.

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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 as

ε (ω) = {(n_{r} + j n_{i})}^{2} = 1 - \frac{ω_{p}^{2}}{ω (ω + i ω_{c})},

where

ε

, n_r, and n_i are the dielectric permittivity and real and imaginary parts of the RI of the metal, respectively; and

ω_{c}

,

ω_{p}

, and

ω

represent the collision frequency, plasma frequency, and angular frequency of the electromagnetic wave, respectively. As a result of the volumetric effects, the variation in plasma frequency can be calculated as

ω_{p} = \sqrt{\frac{4 π N e^{2}}{m^{*}}},

where N and m* represent the density and effective mass of an electron, respectively, and both are related to temperature. For the negligible temperature dependence of m*,

ω_{p}

can be expressed as [16]

ω_{p} = ω_{p 0} {[1 + γ_{e} (T - T {}_{0})]}^{- \frac{1}{2}},

where

ω_{p o}

is the plasma frequency at the reference temperature

T_{0}

, and

γ_{e}

is the expansion coefficient of the metal.

Temperature affects the collision frequency in two ways: phonon–electron scattering ( $ω_{c p}$ ) and electron–electron scattering ( $ω_{c e}$ ) [17,18]. The combined effect of these two aspects is

ω_{c} = ω_{c p} (T) + ω_{c e} (T)

The

ω_{c p}

can be calculated by the Hubbard–Holstein model as [19,20]

ω_{c p} (T) = ω_{0} [\frac{2}{5} + 4 {(\frac{T}{T_{D}})}^{5} \int_{0}^{T_{D} / T} \frac{z^{4} d z}{e^{z} - 1}],

where

T_{D}

is the Debye temperature of the metal and

ω_{0}

is a constant related to the electric conductivity. Owing to the direct current conductivity,

ω_{0}

can be calculated by the static limit of the above expressions [6], and

ω_{c e}

can be expressed by the Lawrence model as [21]

ω_{c e} (T) = \frac{1}{6} π^{4} \frac{Γ Δ}{ℏ E_{F}} [{(k_{B} T)}^{2} + {(\frac{ℏ ω}{4 π^{2}})}^{2}],

where

Γ

,

Δ

,

ℏ

,

E_{F}

, and

k_{B}

are all constants, representing the average over the Fermi surface of scattering probability, fractional umklapp scattering, Planck's constant, Fermi energy, and Boltzmann constant, respectively.

The temperature variation also has an influence on the metal layer thickness $d$ , 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 [22], the thickness can be expressed as

d (T) = d_{0} [1 + (T - T_{0}) γ \frac{1 + μ}{1 - μ}],

where

d_{0}

is the thickness at the reference temperature

T_{0}

; and

γ

is defined as the thermal linear expansion coefficient and

μ

is Poisson’s number of the film material. The values of the above-mentioned parameters used in our calculations were chosen from Table 1 [6,9].

Table 1. Parameter values used in the calculation of temperature effects on gold sensor film

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2.2 Temperature effect on the prism

The thermo-optic coefficient $d n / d T$ can be used to express the temperature dependence of RI in the prism $n (T)$ :

n (T) = n (T_{0}) + (T - T_{0}) \frac{d n}{d T}

Assuming that the most commonly used prisms in SPR sensors (e.g., BK9) have very low thermo-optic coefficients [18], 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].

R = \frac{λ^{2}}{(λ^{2} - λ_{i g}^{2})},

2 n (λ) \frac{d n}{d T} (λ) = G R + H R^{2},

n (λ) = {(1 + \frac{A_{1} λ^{2}}{λ^{2} - B_{1}^{2}} + \frac{A_{2} λ^{2}}{λ^{2} - B_{2}^{2}} + \frac{A_{3} λ^{2}}{λ^{2} - B_{3}^{2}})}^{1 / 2},

where A₁, A₂, A₃, B₁, B₂, and B₃ are the material-dependent values, and

λ

and

λ_{i g}

represent the incident wavelength and the band gap wavelength in units of µm. The prism parameters used for the numerical simulation are listed in Table 2 [23–26].

Table 2. Prism parameters used for the numerical simulation

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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

r = {| \frac{r_{01} + r_{12} \exp (2 i d_{1} k_{z 1})}{1 + r_{01} r_{12} \exp (2 i d_{1} k_{z 1})} |}^{2} .

where r_ij is the reflective coefficient of p-polarized light at the boundary between media i and j, and it can be calculated as

r_{i j} = \frac{k_{z i} / ε_{i} - k_{z j} / ε_{j}}{k_{z i} / ε_{i} + k_{z j} / ε_{j}},

k_{z i} = \frac{2 π}{λ} \sqrt{ε_{i} - ε_{0} {(\sin θ)}^{2}},

where

θ

is the angle of incidence and k_zi is the component of the wave vector in medium i along the z-direction. In our case (i.e., detecting the RI of an analyte), media 0, 1, and 2 represent the prism, gold metal film, and analyte, respectively.

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% [9]. 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.

Fig. 3 Angular-interrogation mode with an incident wavelength of 632.8 nm and a gold film thickness of 50 nm at a reference temperature of 300 K: (a) surface plasmon resonance (SPR) reflectance curves at different temperatures from 100 to 800 K with a refractive index (RI) of 1.330. (b) Fitting lines for resonance angle versus RI at different temperatures. (c) Variation of SPR resonance sensitivity with RI at different temperatures. The RI ranges from 1.330 to 1.346. (d) Variation of SPR resonance sensitivity with temperature at different RIs. The temperature variation ranges from 270 to 380 K, and R² is the coefficient of determination for the fitting lines.

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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 as

Δ A = m_{n} Δ n + m_{T} Δ T,

where

Δ A

,

Δ n

, and

Δ T

represent the resonance shift, RI variation, and temperature variation, respectively; and m_n and m_T are the sensitivities of the SPR resonance with RI and temperature. However, in an actual measurement of observed SPR resonance shift to detect the RI variation, another independent relationship similar to Eq. (15) is necessary. Assuming its existence and according to the principle of matrix theory, it can be expressed as

(\begin{matrix} Δ A_{i} \\ Δ A_{j} \end{matrix}) = (\begin{matrix} m_{n i} & m_{T i} \\ m_{n j} & m_{T j} \end{matrix}) (\begin{matrix} Δ n \\ Δ T \end{matrix}) = M (\begin{matrix} Δ n \\ Δ T \end{matrix}),

where i and j represent different detecting situations. M, a full rank matrix (the sensitivity matrix) that considers the cross-sensitivity between the RI variation and temperature variation. Therefore, the variation in RI and temperature can be derived by the inverse matrix as

Fig. 4 Surface plasmon resonance (SPR) resonance shift versus the simultaneous changes of refractive index (RI) and temperature for the angular-interrogation mode.

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(\begin{matrix} Δ n \\ Δ T \end{matrix}) = M^{- 1} (\begin{matrix} Δ A_{i} \\ Δ A_{j} \end{matrix})

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 [11]. 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.

Fig. 5 Angular-interrogation mode: (a) Variation in m_n with wavelength. The refractive index (RI) ranges from 1.330 to 1.346 with a temperature of 300 K. (b) Variation in m_T with incident wavelength. The temperature changes from 270 to 380 K with an RI of 1.330.

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The red curve indicates that m_n (the sensitivity to RI) decreases with the detecting wavelength; the change in temperature sensitivity (m_T) 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.

Fig. 6 Wavelength-interrogation mode with a fixed angle of incidence of 70°: (a) surface plasmon resonance (SPR) reflectance curves at different temperatures ranging from 100 to 800 K with a refractive index (RI) of 1.330. (b) Fitting lines for resonance wavelength versus RI at different temperatures. (c) Variation of SPR resonance sensitivity with RI at different temperatures. The RI ranges from 1.330 to 1.340. (d) Variation of SPR resonance sensitivity with temperature at different RIs. The temperature variation ranges from 270 to 380 K.

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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 (m_n) is shown in Fig. 6 (c). With the change in temperature from 270 to 380 K, m_n 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 (m_T) at different RIs is depicted in Fig. 6(d), in which m_T 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.

Fig. 7 Wavelength-interrogation mode: (a) Variation in m_n with fixed angle of incidence. The refractive index (RI) ranges from 1.330 to 1.340 at a temperature of 300 K. (b) Variation in m_T with angle of incidence. The temperature changes from 270 to 380 K with an RI of 1.330.

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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

(\begin{matrix} Δ A_{i} \\ Δ A_{j} \end{matrix}) = (\begin{matrix} f_{n i} (T) & f_{T i} (R I) \\ f_{n j} (T) & f_{T j} (R I) \end{matrix}) (\begin{matrix} Δ n \\ Δ T \end{matrix}) = M (\begin{matrix} Δ n \\ Δ T \end{matrix}) .

The f_n(T) and f_T(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

(\begin{matrix} Δ A_{i} \\ Δ A_{j} \end{matrix}) = (\begin{matrix} f_{n i} (T) & m_{T i} \\ f_{n j} (T) & m_{T j} \end{matrix}) (\begin{matrix} Δ n \\ Δ T \end{matrix}) = M (\begin{matrix} Δ n \\ Δ T \end{matrix}) .

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 [11]. 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.

Fig. 8 Experimental setup.

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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.

Fig. 9 Resonance wavelength shift versus refractive index (RI) and the corresponding linear fitting lines.

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Fig. 10 Resonance wavelength shift versus temperature and the corresponding linear fitting lines.

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With the linear sensitivity of NaCl solution [27] 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

(\begin{matrix} Δ n \\ Δ T \end{matrix}) = {(\begin{matrix} 8.7861 & - 1.1318 \\ 6.9026 & - 0.7905 \end{matrix})}^{- 1} (\begin{matrix} Δ A_{1} \\ Δ A_{2} \end{matrix}) .

where

Δ n

and

Δ T

are the RI variation and temperature change in units of 10⁻³ RIU and 10°C, respectively; and

Δ A_{1}

and

Δ A_{2}

are the resonance wavelength changes at the two fixed detection angles of 48° and 45°, respectively.

7. Conclusions

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.

Funding

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).

Disclosures

The authors declare that there are no conflicts of interest related to this article

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Symbol	Value	Unit	Symbol	Value	Unit
$γ_{e}$	1.42	10⁻⁵ K^–1	E _F	5.53	eV
T _D	185	K	$ℏ$	1.0546	10⁻³⁴ Js
$Γ$	0.55	/	$γ$	14.2	10⁻⁶
$Δ$	0.77	/	$μ$	0.42	/

Symbol	Value	Symbol	Value
G	−1.6548 × 10⁻⁶ K^–1	A ₃	0.8974794
H	31.7794 × 10⁻⁶ K^–1	B ₁	0.0684043
A ₁	0.6961663	B ₂	0.1162414
A ₂	0.4079426	B ₃	9.896161

Symbol	Value	Unit	Symbol	Value	Unit
$γ_{e}$	1.42	10⁻⁵ K^–1	E _F	5.53	eV
T _D	185	K	$ℏ$	1.0546	10⁻³⁴ Js
$Γ$	0.55	/	$γ$	14.2	10⁻⁶
$Δ$	0.77	/	$μ$	0.42	/

Symbol	Value	Symbol	Value
G	−1.6548 × 10⁻⁶ K^–1	A ₃	0.8974794
H	31.7794 × 10⁻⁶ K^–1	B ₁	0.0684043
A ₁	0.6961663	B ₂	0.1162414
A ₂	0.4079426	B ₃	9.896161

Simultaneous measurement of refractive index and temperature for prism-based surface plasmon resonance sensors

Abstract

1. Introduction

2. Theoretical analysis

2.1 Temperature effect on the gold film

2.2 Temperature effect on the prism

2.3 Reflectance of the SPR sensor

3. Sensitivity matrix method for angular-interrogation SPR

4. Sensitivity matrix method for wavelength-interrogation SPR

5. Correction of sensitivity matrix coefficients

6. Experimental results and discussion

7. Conclusions

Funding

Disclosures

References

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Figures (10)

Tables (2)

Equations (20)

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