Sensitivity of Optical Fiber Sensor Based on Surface Plasmon Resonance: Modeling and Experiments

Abstract

In this paper, surface plasmon resonance curves of an optical fiber-based sensor were investigated. From an experimental and theoretical perspective, the response curves were analyzed and discussed. Precisely, such curves were calculated by modeling the analyte/metallic layer interface using a multilayer system, including the effects of roughness. Then, the experimental response curves observed in solutions with different refractive indices were compared to the simulated curves. Good agreement was obtained with respect to the resonance peak location and the shape of the curves. Consequently, these results enabled us to predict the ideal functioning conditions of the sensor, i.e., the working parameters corresponding to the best sensitivities of detection.

Appendix

Transfer matrix formalism for a multilayer system

In this appendix, we briefly explain the concept employed to realize the numerical modeling. The reflectance R of the light on our multilayer system was computed using the transfer matrix formalism. This formalism, which is based on the calculation of light propagation through a multilayer medium consisting of (N − 1) isotropic and homogeneous layers, has already been extensively described [28]. A computer simulation was performed on three- and four-layer systems (optical fiber–metal–analyte), as depicted in Fig. 3.

This electromagnetic analysis of light reflection on a multilayer system was solved by the Maxwell’s equations subjected to boundary conditions. The Maxwell’s equation states that the relationships between fundamental electromagnetic quantities, as the electric field vector E and the magnetic field vector H. A schematic illustration of the system is presented in Fig. 11. The amplitude of E and H vectors are related to the formula within the framework of transfer matrix formalism, Eq. 11:

$$\left[ {\begin{array}{*{20}c} {E_0 } \\ {H_0 } \\ \end{array} } \right] = \left[ M \right] \times \left[ {\begin{array}{*{20}c} {E_N } \\ {H_N } \\ \end{array} } \right]$$

(11)

where [M] is the characteristic matrix of the layered system defined by the following:

$$\left[ M \right] = \left[ {\begin{array}{*{20}c} {M_{11} } {M_{12} } \\ {M_{21} } {M_{22} } \\ \end{array} } \right] = \prod\limits_{k = 1}^{N - 1} {\left( {\left[ {\begin{array}{*{20}c} {\cos \delta _k } {\frac{{ - i\,\sin \delta _k }}{{\eta _k }}} \\ { - i\eta _k \sin \delta _k } {\cos \delta _k } \\ \end{array} } \right]} \right)} $$

δ _k , the phase factor of the kth layer is a function of the refractive index $n_k = \left( {\varepsilon _k \mu _k } \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}$, the thickness $d_k = \left( {z_k - z_{k - 1} } \right)$ of this kth layer distributed along the z-axis, the incident angle θ ₀, and wavelength λ. This phase factor is defined as:

$$\delta _k = \frac{{2\pi }}{\lambda }n_k \cos \theta _k \times \left( {z_k - z_{k - 1} } \right) = \frac{{2\pi d_k }}{\lambda }\left( {\varepsilon _k - n_0^2 {\kern 1pt} \sin ^2 \theta _0 } \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} $$

$\eta _{_k }^{}$, the optical admittance is defined as a function of the polarization states as:

• $\eta _k^s = \left( {\frac{{\varepsilon _k }}{{\mu _k }}} \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} \,\cos \theta _k = \left( {\varepsilon _k - n_0^2 {\kern 1pt} \sin ^2 \theta _0 } \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}$ for s-wave (TE).

• $\eta _k^p = \left( {\frac{{\varepsilon _k }}{{\mu _k }}} \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} \times \frac{1}{{\cos \theta _k }} = \frac{{\varepsilon _k }}{{\eta _k^s }}$ for p-wave (TM).

Fig. 11

Electric and magnetic field vectors of p wave at the inner surface of the multilayer system

Finally, the reflectance R of the whole multilayer structure is provided in terms of Fresnel reflection coefficients (r _s and r _p ) and then in terms of M elements as (Eqs. 12 and 13):

$$R_s = \left| {r_s } \right|^2 = \left| {\frac{{\left( {M_{11}^s + M_{12}^s \cdot \eta _N^s } \right)\eta _0^s - \left( {M_{21}^s + M_{22}^s \cdot \eta _N^s } \right)}}{{\left( {M_{11}^s + M_{12}^s \cdot \eta _N^{s,p} } \right)\eta _0^s + \left( {M_{21}^s + M_{22}^s \cdot \eta _N^s } \right)}}} \right|^2 $$

(12)

$$R_p = \left| {r_p } \right|^2 = \left| {\frac{{\left( {M_{11}^p + M_{12}^p \cdot \eta _N^p } \right)\eta _0^p - \left( {M_{21}^p + M_{22}^p \cdot \eta _N^p } \right)}}{{\left( {M_{11}^p + M_{12}^p \cdot \eta _N^p } \right)\eta _0^p + \left( {M_{21}^p + M_{22}^p \cdot \eta _N^p } \right)}}} \right|^2 $$

(13)

Power distribution launched in the optical fiber

The power distribution P _in (θ _in ) included into a solid angle dθ _in arriving at the fiber-end face of an optical fiber is generally expressed as $P_{in} \left( {\theta _{in} } \right)d\theta _{in} \propto \left( {{{\tan \theta _{in} } \mathord{\left/ {\vphantom {{\tan \theta _{in} } {\cos ^2 }}} \right. \kern-\nulldelimiterspace} {\cos ^2 }}\theta _{in} } \right)d\theta _{in}$, where θ _in is the launched angle of the rays (Fig. 6). θ _in can be expressed as a function of θ ₀ from the Snell’s law: $\theta _{in} = \arcsin \left[ {n_0 \cos \theta _0 } \right]$.

Finally, the power distribution P(θ ₀ ) can be written as [17]:

$$P\left( {\theta _0 } \right)d\theta _0 \propto \frac{{\varepsilon _0 \sin \theta _0 \cos \theta _0 }}{{\left( {1 - \varepsilon _0 \cos ^2 \theta _0 } \right)^2 }}d\theta _0 $$