When the incident light at the surface of the medium is totally reflected, part of the energy will be coupled into the surface of the incident medium for transmission, when the evanescent wave caused by the incident light and the plasma wave at the surface of the metal resonate, surface plasmon resonance (SPR) occurs, and at this time, the energy in the incident medium leaks to the surface of the metal, and the loss peak appears. The data analyses and simulations in this paper were carried out using the finite element method (FEM) through COMSOL 6.0 software. The fine meshes and the number of degrees of freedom 137,646 are adopted in order to ensure the accuracy of the simulation. The cross - section and 3D view of the proposed sensor are shown in Fig. 1, and the internal air holes are arranged in hexagonal geometry with the diameter of d, e represents the distance between air holes, the polishing depth of D-shaped surface is denoted by f and the gold film thickness is denoted by t.
A possible manufacturing method of the proposed sensor in this paper is shown in Fig. 2. The air holes of the PCF structure can be constructed by stacking hollow rods and solid core rods of the same size, at which time the intermediate transition structure is obtained, and then the desired D-shaped PCF can be obtained by stretching and grinding, and finally the desired sensor can be obtained by coating the D-shaped surface with CVD method [23].
It can be seen in the electric field schematic of Fig. 3 that the energy of the fundamental mode of the proposed sensor is well confined in the dual-core at non-resonant wavelength. When the SPR occurs, the energy in the dual-core is able to leak to the surface of the gold film due to the equal effective refractive index (neff) of the fundamental mode and the SPP mode.
The dispersion curves of the odd and even modes for the two dielectric combinations at a RI of 1.40 are shown in Fig. 4. PCF confines energy to the fiber core through special arrangement of air holes. As the incident light wavelength changes to the resonant wavelength, the limitation of the energy in the fibre core is weakened, and the energy leakage of the fibre core reaches the maximum at the resonant wavelength of 1780 nm and 1840 nm for odd mode and even mode respectively.
Sellmier equation can be used to describe the RI of silica according to Eq. 1 [24]:
where the RI of silica is denoted by n, the wavelength of incident light is denoted by λ, and the other parameters are A1 = 0.6961663, A2 = 0.4079426, A3 = 0.8974794, B1 = 0.0684043 µm, B2 = 0.1162414 µm and B3 = 9.896161 µm.
Drude-Lorentz model can be used to describe the dielectric constant of the plasma material gold for the proposed sensor[25] according to Eq. 2.
where the frequency of the dielectric constant is expressed as ε∞, ωD is plasma frequency, Δε denotes the weight factor, the angular frequency of the incident light is ω. ωD and γD represent the plasma frequency and damping frequency, respectively, values of all these parameters have been taken from[25].
The confinement loss can be used to analyse the loss spectrum to obtain various sensing parameters of the sensor and can be expressed as Eq. 3 [26]:
$$\:{\alpha\:}=8.686\times\:{\text{k}}_{0}\times\:\text{I}\text{m}\left({\text{n}}_{\text{e}\text{f}\text{f}}\right)\times\:{10}^{7}(\text{d}\text{B}/\text{c}\text{m})$$
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where the limiting loss can be expressed in terms of \(\:{\alpha\:}\), the wave vector \(\:{\text{k}}_{0}=2{\pi\:}/{\lambda\:}\) and λ is the wavelength of incident light。The imaginary part of the neff can be expressed as \(\:\text{I}\text{m}\left({\text{n}}_{\text{e}\text{f}\text{f}}\right)\).
The WS of SPR sensors is expressed as the variation of the resonance wavelength with the RI. [27] Generally, the larger resonance wavelength shift indicates a higher WS, and can be calculated according to Eq. 4 [28]:
$$\:{\text{S}}_{{\lambda\:}}=\frac{{{\Delta\:}{\lambda\:}}_{\text{p}\text{e}\text{a}\text{k}}}{{\Delta\:}{\text{n}}_{\text{a}}}(\text{n}\text{m}/\text{R}\text{I}\text{U})$$
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where the resonance peak shift and the change in RI are expressed as \(\:{\varDelta\:{\lambda\:}}_{\text{p}\text{e}\text{a}\text{k}}\) and Δna, respectively.
AS is used to describe the intensity of the confinement loss variation with wavelength and RI, the larger the AS indicates the more significant resonance peak variation with wavelength and RI, and is calculated from Eq. 5 [29]:
$$\:{\text{S}}_{\text{A}}=-\frac{1}{{\alpha\:}({\lambda\:},{\text{n}}_{\text{a}})}\frac{\partial\:{\alpha\:}({\lambda\:},{\text{n}}_{\text{a}})}{\partial\:{\text{n}}_{\text{a}}}\left({\text{R}\text{I}\text{U}}^{-1}\right)$$
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where the \(\:{\alpha\:}({\lambda\:},{\text{n}}_{\text{a}})\)is the value of loss that varies with resonance wavelength for a given RI, \(\:\partial\:{\alpha\:}({\lambda\:},{\text{n}}_{\text{a}})\) is the loss difference of two consecutive RI and \(\:\partial\:{\text{n}}_{\text{a}}\) indicates RI variation.
Figure-of-merit (FOM) of the SPR sensor, which affects the accuracy of detection and the FOM can be calculated according to Eq. 6[29]:
$$\:\text{F}\text{O}\text{M}=\frac{{\text{S}}_{{\lambda\:}}}{\text{F}\text{W}\text{H}\text{M}}\left({\text{R}\text{I}\text{U}}^{-1}\right)$$
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The resolution of the sensor can be calculated in both wavelength and amplitude measuring methods defined by Eq. 7 and Eq. 8 [30]:
$$\:{\text{R}}_{\text{w}}=\frac{{\Delta\:}{\lambda\:}}{{\text{S}}_{{\lambda\:}}}\left(\text{R}\text{I}\text{U}\right)$$
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$$\:{\text{R}}_{\text{A}}=\frac{{{\Delta\:}\text{n}}_{\text{a}}}{{\text{S}}_{\text{A}}}\left(\text{R}\text{I}\text{U}\right)$$
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where the resolution of the spectrometer is represented by \(\:{\Delta\:}{\lambda\:}\) and assumed to be 0.1 nm in this paper, and the \(\:{{\Delta\:}\text{n}}_{\text{a}}\) is the minimum change in RI which is 0.01 in this paper.