## 4.1 Single Particle

When AuNPs are bounded to an Au-surface, the system symmetry is destroyed. In addition, the plasmon coupling between the AuNPs and Au-surface occurs, generating an induced charge in the Au-surface, which interacts with AuNPs. However, the polarity of these charges is opposite, and consequently, the local electric field of the free charges between AuNP and Au-surface increases under near-field coupling [29].

An exciting feature of the WF-SPRM is that it can detect any individual particle bound to the surface. Therefore, the time dependence of relative intensities of a binding event of AuNPs (40, 60, and 80 nm) to the gold-sensor substrate represents three-time stages, i.e., relatively low flat intensity before AuNPs binding, a sudden increase in local intensity in a moment of AuNPs binding, and high flat intensity after AuNPs binding (Fig. 2a). It can be seen that the magnitude of the intensity step is proportional to the AuNPs size [30]. The signal intensity increases linearly from 208 a.u. for 40 nm AuNPs to 583 a.u.. This clearly shows that the signal-building mechanism is essentially different according to Rayleigh's theory [31]. Additionally, the signal noise of AuNP increases as increasing AuNP size from 40 to 80 nm, which can be attributed to increasing surface plasmon polaritons produced by the coupling effect between the AuNP and Au-surface.

When the surface plasmon polaritons produced by Au-surface are coupled to the AuNPs, the surface charges between them are redistributed, and the local electric field increases. Figure 2b illustrates the plasmon coupling between the AuNPs and the Au-surface, generating the local electromagnetic field. The red regions represent the coupling enhancement regions between AuNPs and Au-surface. 80 nm AuNPs have higher enhancement and confinement than 40 nm AuNPs, indicating that the electric-field intensity between the AuNPs and Au-layer increases with increasing AuNPs size. A 40 nm diameter confined incident light between AuNPs and Au-layer within a closer vicinity of the particle surface than 80 nm AuNPs, indicating to lower efficiency of confined field between AuNPs and Au-layer, such an efficient excitation of the gap-mode plasmon by 80 nm AuNPs. This finding is in good agreement with the experimental data (Fig. 2b). In addition, the presence of AuNPs near the sensor surface generates a disturbance in the electric field in the analyte region [32].

Based on Eq. 12, we can calculate the intensity distribution of the bounding event between the AuNP and Au-surface. Figure 2c-e illustrates the intensity profile distribution for experiment AuNP bounded to the Au-surface compared with the calculated intensity profile. It can be seen the calculated intensity profile is accepted with the experimental data. This means the success of our derived model.

## 4.2 Two-Particle Interference

When two gold nanoparticles (2-AuNPs) are bound to the Au-layer surface at a distance smaller than the surface plasmon propagation length, the surface plasmon waves around this pair are enhanced from the constructive and destructive interferences of the surface plasmon waves of each particle (Fig. 3). Figures 3a and 3b illustrate the COMSOL simulation electric field intensity distribution and a WF-SPRM image for 2-AuNPs of 80 nm size. The surface plasmon waves around the two particles are generated from the electric field superposition between these waves. Superposing plasmon intensities of two AuNPs with an initial phase difference (\(\phi\)) as \({\left|a+a\right|}^{2}\) (or \({\left|2a\right|}^{2}\)) gives us a quantum superposition of \({\left|2a\right|}^{2}={\left|a\right|}^{2}+{\left|a\right|}^{2}+2{\left|a\right|}^{2}\text{cos}\phi\), and consequently yields complex and cumulative superposing plasmon intensities [33]. This can be deduced from Eq. 11, which can be rewritten for two particles:

$$I\left(r\right)={I}_{1}+{I}_{2}+c {\epsilon }_{0} n\left|{E}_{s1}\right|\left|{E}_{s2}\right|\text{cos}\left({\phi }_{1}-{\phi }_{2}\right)$$

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with,

$${E}_{sj}={E}_{sp}^{0}\left({r}_{j}\right){e}^{-\kappa \left|r-{r}_{i}^{{\prime }}\right|}{e}^{-ik\left|r-{r}_{i}^{{\prime }}\right|}$$

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$${I}_{j}\left(r,{r}^{{\prime }}\right)=\frac{1}{2}c {\epsilon }_{0} n\left[\left({E}_{r}^{2}+{\left({E}_{sp}^{0}\left(r\right)+\alpha {E}_{sj}\right)}^{2}+2{E}_{r}\left({E}_{sp}^{0}\left(r\right)+\alpha {E}_{sj}\right)\text{cos}\phi \right)-\left({E}_{r}^{2}+{E}_{sp}^{2}+2{E}_{r}{E}_{sp}\text{cos}\phi \right)\right]$$

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Therefore, we can calculate the intensity distribution of the bounding event between the 2-AuNPs and Au-surface. Figure 3c illustrates the intensity profile distribution for experiment 2-AuNPs bounded to the Au-surface compared with the calculated intensity profile. It can be seen the calculated intensity profile is accepted with the experimental data. This means the success of our derived model. The intensity enhancement of the two AuNPs compared to that of the single AuNP is due to increasing the local surface plasmon coupling between the AuNPs with Au-surface and each other [34].

## 4.3 Multiple-Particle Interference

The multiple particles near the Au-layer can be represented by effective media, described by the effective dielectric constant. The effective dielectric constant of the AuNPs in the analyte based on Maxwell-Garnet equations is given by [35]:

$${\epsilon }_{eff}\left(\omega \right)={\epsilon }_{eff}^{{\prime }}+i{\epsilon }_{eff}^{{\prime }{\prime }}$$

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with \({\epsilon }_{eff}^{{\prime }}\) and \({\epsilon }_{eff}^{{\prime }{\prime }}\) are the effective real and imaginary parts of effective dielectric constants, respectively, which are given by:

$${\epsilon }_{eff}^{{\prime }}={\epsilon }_{w}+\frac{f\left({\epsilon }_{Au}^{{\prime }}-{\epsilon }_{w}\right)\times \left\{{\epsilon }_{w}+\beta \left({\epsilon }_{Au}^{{\prime }}-{\epsilon }_{w}\right)-f\left(\gamma {\epsilon }_{Au}^{{\prime }}-{\epsilon }_{w}\right)\right\}-f{\epsilon }_{Au}^{{\prime }{\prime }}\times \left(\beta {\epsilon }_{Au}^{{\prime }{\prime }}-f\gamma {\epsilon }_{Au}^{{\prime }{\prime }}\right)}{{\left\{{\epsilon }_{w}+\beta \left({\epsilon }_{Au}^{{\prime }}-{\epsilon }_{w}\right)-f\left(\gamma {\epsilon }_{Au}^{{\prime }}-{\epsilon }_{w}\right)\right\}}^{2}+{\left(\beta {\epsilon }_{Au}^{{\prime }{\prime }}-f\gamma {\epsilon }_{Au}^{{\prime }{\prime }}\right)}^{2}}$$

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$${\epsilon }_{eff}^{{\prime }{\prime }}=\frac{f{\epsilon }_{Au}^{{\prime }{\prime }}\times \left\{{\epsilon }_{w}+\beta \left({\epsilon }_{Au}^{{\prime }}-{\epsilon }_{w}\right)-f\left(\gamma {\epsilon }_{Au}^{{\prime }}-{\epsilon }_{w}\right)\right\}-f\left({\epsilon }_{Au}^{{\prime }}-{\epsilon }_{w}\right)\times \left(\beta {\epsilon }_{Au}^{{\prime }{\prime }}-f\gamma {\epsilon }_{Au}^{{\prime }{\prime }}\right)}{{\left\{{\epsilon }_{w}+\beta \left({\epsilon }_{Au}^{{\prime }}-{\epsilon }_{w}\right)-f\left(\gamma {\epsilon }_{Au}^{{\prime }}-{\epsilon }_{w}\right)\right\}}^{2}+{\left(\beta {\epsilon }_{Au}^{{\prime }{\prime }}-f\gamma {\epsilon }_{Au}^{{\prime }{\prime }}\right)}^{2}}$$

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where: \({\epsilon }_{Au}^{{\prime }}\) and \({\epsilon }_{Au}^{{\prime }{\prime }}\) are components of the complex dielectric constant of AuNPs, \({\epsilon }_{w}\) is the dielectric constant of water, \(f\) is the filling factor, defined as the total volume of particles divided by the total volume of the composite analyte, \(\beta\) is the shape factor of the particle (\(\beta =1/3\) for sphere), and \(\gamma\) is a factor, which is given by:

$$\gamma =\frac{1}{3{\epsilon }_{w}}+\frac{K}{4\pi {\epsilon }_{w}}$$

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where: \(K\) is the interaction parameter between the electric field generated by adjacent particles; since the AuNPs in the analyte have low concentrations, the AuNPs are far enough. Consequently, the dipolar interaction must be negligible, and \(K=0\) [36]. The calculated \({\epsilon }_{eff}^{{\prime }}\) and \({\epsilon }_{eff}^{{\prime }{\prime }}\) of AuNPs in the analyte as a function of filling factor \(f \left(0<f<1\right)\) are illustrated in Fig. 4a and 4b. \({\epsilon }_{eff}^{{\prime }{\prime }}\) values increases with increasing filling factor up to about \(f=0.7\), and the transition between the positive and negative part of \({\epsilon }_{eff}^{{\prime }}\) occurs at \(f=0.71\). Therefore, the effective dielectric constant of AuNPs in the analyte exhibits a nonlinear response as a function of the filling factor, which is in agreement with K. Tamada et al. study [35].

Based on the above discussion, the theoretical reflectivity curves of the Au-layer with different sizes of AuNPs (40, 60, and 80 nm) and a constant filling factor of 0.02 are calculated using WINSPALL software (Fig. 4c) [37]. For all AuNPs sizes, increasing of filling factor increases the incident photon angle at minimum reflectivity. At a constant filling factor, the incident photon angle at minimum reflectivity increases with increasing the AuNPs size (Fig. 4c). The SPR sensitivity is calculated by \(S=\varDelta \theta /\varDelta n\), where: \(\varDelta \theta\) represents the SPR angle shift [degree], and \(\varDelta n\) is refractive index change [RIU]. The sensitivity of 40 nm AuNPs is 50.6 deg./RIU, which increases linearly to 93.5 deg./RIU for 80 nm AuNPs detection (Fig. 4d). Therefore, AuNPs of a larger size provided a better detection sensitivity. This finding is in good agreement with the experimental and simulation data in Fig. 4e-g.