**2. Principle of SPR phenomena**

Metals are composed of positively charged nuclei with a lot of free electrons in their conduction band (surface of the metal). If an external electric field is applied close to the metal surface, free electrons are dislocated, resulting in an electric dipole [61]. A longitudinal oscillation has resulted from such electron transportation in a metal surface known as surface plasmons (SPs) [49]. To support the generated SPs a metal and dielectric interface is needed [46] whereas excitation of these SPs leads to an enhanced electromagnetic field resulting in a collective oscillation of free electrons or electron plasma [46, 61, 62]. The basic principle of the construction of SPR based sensors lies in the generation and propagation of electromagnetic waves called surface plasmon wave (SPW) due to the interaction of irradiating electric fields and the generated fields for dislocation of the electrons between the metal-dielectric interface [4]. The SPWs can only be produced by the incidence of a transverse magnetic (TM-) or plane (p-) polarized field as Maxwell's equations supports no solution for transverse electric (TE-) polarized case [46]. Furthermore, the fact that electron oscillation means resistive losses. Thus, when an optical field appears at the metal-dielectric boundary, the SPW produces due to

optical absorption of exponentially decaying evanescent waves. Mathematically, when the wave vector of the SPW is equal to the propagation constant of the irradiating lightwave, maximum absorption of evanescent field is observed leading to a strong SPW generation [63, 64]. This condition is called resonance condition. The propagating evanescent wave can be characterized by propagation constant *β*ev as follows [10, 65]:

$$
\beta\_{ev} = \frac{2\pi}{\lambda} n \times \sin \theta \tag{1}
$$

Where *λ*, n, *θ* indicate the incident light wavelength, refractive index of the medium, and angle of incident of light at the metal surface, respectively. The equation as follows characterizes the SPW [66]:

$$\mathcal{J}\_{SPW} = \mathcal{J}\_f \sqrt{\frac{\mathcal{E}\_m \mathcal{E}\_d}{\mathcal{E}\_m + \mathcal{E}\_d}} \tag{2}$$

Where <sup>2</sup> *f* = π β λ, is the wave vector of light at free space. Also, *ε*m and *ε*d indicate

the dielectric constants of plasmonic material and dielectric medium, respectively. Eq. (2) can also be rewritten in terms of the refractive index as follows [31]:

$$\beta\_{SPW} = \frac{2\,\pi}{\mathcal{A}} \sqrt{\frac{n\_m^2 n\_d^2}{n\_m^2 + n\_d^2}} \tag{3}$$

Where the RI of plasmonic material ( *nm* ) and sensing medium ( *nd* ) are related to the dielectric constants as <sup>2</sup> *nm m* =∈ and <sup>2</sup> *nd d* =∈ , respectively. The resonance condition is located in SPR based sensor, where the propagation constant of incident light (*β*ev) is matched with the SPW's wave vector *β*SPW [67]. From Eqs. (1) and (3), it can be related that by controlling the incident angle for a particular frequency of light, the resonance condition can be achieved. This method is called the angular interrogation method. Similarly, the light wavelength can be modified to achieve the SPR condition for a particular angle of incident light for the wavelength interrogation method [68]. At this condition, the electrons start to resonate triggering the generation of SPW on the metal-dielectric interface where a sharp loss peak called the SPR point appears. The SPR point is extremely responsive to the refractive index (RI) of the surrounding medium where a minor change in RI of the dielectric (sensing) medium shifts the SPR point to a new state [69].
