Figure 1 depicts the three layers of an SPR sensor arrangement, the first of which is a BK7 glass prism with a refractive index of n1. The second layer is silver, which is then followed by a sensing medium. An incoming light beam travelling through the prism generates an evanescent wave due to complete internal reflection at the prism-silver layer contact. This produced evanescent wave passes through the silver layer and propagates in the x-direction (Fig. 1). In the x-direction, the amplitude of the propagating wave vector may be written as[13]:

$${k_x}=\left( {2\pi {n_1}/\lambda } \right)\sin \theta$$

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The first layer in these devices is the BK7 prism layer, which has a refractive index of Eq. (3) for wavelengths ranging from 0.3 to 2.5 um[13].

$${n_p}^{2} - 1=\frac{{1.03961212{\lambda ^2}}}{{{\lambda ^2} - 0.0060006986}}+\frac{{0.23179344{\lambda ^2}}}{{{\lambda ^2} - 0.0200179144}}+\frac{{1.01046945{\lambda ^2}}}{{{\lambda ^2} - 103.560653}}$$

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Because Ag is an excellent absorbing material, it creates a large number of surface plasmons (SPs) at the metal-sensing medium interface, resulting in a greater value of resonance angle and a short peak of the reflectance curve with a considerable shift in the curve[14]. The metal's refractive index (nmetal) is computed using the Drude model, which is expressed as[15]:

$${n_m}={\left[ {1 - \left( {\frac{{{\lambda ^2}{\lambda _c}}}{{({\lambda _c}+i\lambda )\lambda _{p}^{2}}}} \right)} \right]^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}}$$

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In the following equation, λc = 1.7914×10− 5 m is the collision wavelength and λp = 1.4541× 10− 7 m is the plasma wavelength of silver (Ag).

Using Eq. (4) for the incident wavelength (λ), we determined the complex refractive index of graphene to be[16]:

$${n_G}=3+i\frac{{c\lambda }}{3}$$

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Where c equal to 5.446 µm− 1. Each graphene sheet layer has a thickness of (0.34* *d**G* ) nm, and *d**G* is the layer number[17].

The refractive index of the sensing medium (blood) in the visible range can be described

by[4]:

$${n_s}={n_{Hb}}=n+i \times k$$

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n is the real part of refractive of hemoglobin, it's change us function of wavelength and concentration is given by Friebel in the form[18]:

$${n_{Hb}}=n(\lambda ,{C_{Hb}})={n_{{H_2}O}}(\lambda ) \times \left( {\beta (\lambda ) \times {C_{Hb}}+1} \right)$$

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Where nH2O is the refractive index of water us function of wavelength, β(λ) is specific refractive increment dependent on the wavelength and CHb is the concentration of hemoglobin.

The imaginary component (k) of the refractive index that varies with hemoglobin content may be determined at any wavelength using Prahl's molar extinction coefficient data[4][19].

$$k=(2.303 \times e \times {C_{Hb}} \times \lambda )/(4\pi \times {M_{Hb}})$$

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Where e is the molar extinction coefficient and MHb is the molar mass of hemoglobin.

The transfer matrix approach and the Fresnel equation was used to calculate the outcomes of an n-layer (polymeric layer) design. MATLAB software was used to compute the analogy of SPR modification. For a multilayer structure, the input of each layer at amplitudes equivalent to a distance z in the layer, the field amplitudes at the first limit are related to those at the last limit by the total characteristic matrix[20]:

$$\left[ {\begin{array}{*{20}{c}} {H_{{yk}}^{0}} \\ { - E_{{xk}}^{0}} \end{array}} \right]={M_k}\left[ {\begin{array}{*{20}{c}} {{H_{yk}}(z)} \\ { - {E_{xk}}(z)} \end{array}} \right]$$

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where \(H_{{yk}}^{0}\)and \(E_{{xk}}^{0}\), are respectively the tangential components of the electric and magnetic fields at the boundary of the first layer. \({H_{yk}}(z)\), \({E_{xk}}(z)\) are the fields corresponding to the Nth layer boundary. Here, Mk is known as the characteristic transfer matrix of the combined structure and is given by[13]:

$$M=\prod\limits_{{k=1}}^{{N - 1}} {{M_k}=\left[ {\begin{array}{*{20}{c}} {{M_{11}}}&{{M_{12}}} \\ {{M_{21}}}&{{M_{22}}} \end{array}} \right]}$$

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Or, M11, M12, M21, M22 are the components of the transfer matrix, with[21]:

$${M_k}=\left[ {\begin{array}{*{20}{c}} {\cos {\beta _k}}&{ - i\sin {\beta _k}/{q_k}} \\ { - i{q_k}\sin {\beta _k}}&{\cos {\beta _k}} \end{array}} \right]$$

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Mk named the characteristic matrix for the k-layer and related to the optical properties and the thickness of each layer (dk) where[21]:

$${q_k}={\left( {\frac{{{\mu _k}}}{{{\varepsilon _k}}}} \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}}\cos {\theta _k}=\frac{{{{({\varepsilon _k} - n_{0}^{2}{{\sin }^2}\theta )}^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}}}}{{{\varepsilon _k}}}$$

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$${\beta _k}=\frac{{2\pi }}{\lambda }{n_k}\cos {\theta _k}({z_k} - {z_{k - 1}})=\frac{{2\pi }}{\lambda }{d_k}{({\varepsilon _k} - n_{0}^{2}{\sin ^2}\theta )^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}}$$

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Here n0 mean np in the figure (the refractive index of prism) the incident level of the system.

The reflectivity (R) for incident light is represented as[13], [21]:

$${r_p}{\text{=}}\left[ {\frac{{\left( {{{\text{M}}_{{\text{11}}}}{\text{+}}{{\text{M}}_{{\text{12}}}}{{\text{q}}_{\text{k}}}} \right){{\text{q}}_{\text{1}}}{\text{-}}\left( {{{\text{M}}_{{\text{21}}}}{\text{+}}{{\text{r}}_{{\text{22}}}}{{\text{q}}_{\text{k}}}} \right)}}{{\left( {{{\text{M}}_{{\text{11}}}}{\text{+}}{{\text{M}}_{{\text{12}}}}{{\text{q}}_{\text{k}}}} \right){{\text{q}}_{\text{1}}}{\text{+}}\left( {{{\text{M}}_{{\text{21}}}}{\text{+}}{{\text{M}}_{{\text{22}}}}{{\text{q}}_{\text{k}}}} \right)}}} \right]$$

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The reflection coefficient (rs) of an incoming light is determined using the reflectivity (R) parameters and is provided by the relation[13], [21]:

$$R={\left| {{r_P}} \right|^2}$$

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By changing Eq. (14) to polar form, the phase sensitivity, defined as the ratio between the phase shift and the refractive index fluctuation of the Nth layer (sensing medium ns), may be calculated as follows[22]:

$${r_p}=\left| {{r_p}} \right|\exp (\iota \varphi )$$

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$$\varphi =\arg ({r_p})$$

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