## 4 Near Field Coupling and Ruler Equation  
The variation in near-field coupling could be well labeled by a simple dipolar model. According to such a model, the near-field strength strongly depends on the polarizability of the dimer and the decay length constant. Both polarizability and decay constant has an inverse relation with particle volume \(D^{3}\) and inter-particle distance (\(1/\text{g}^{3}\)), respectively [23]. This means that near-field coupling is highly  
Figure 5: Far-field properties of the SCS dimer. (**a**) shows the Major bonding dipole at \(f=405\) THz while (**b,c**) are HEMDM at \(f=597\) THz and \(684\) THz respectively.  
sensitive to inter-particle distance normalized by the size of the particle \((r/D)^{-3}\). Hence, the first-order exponential decay Ruler equation has been used to fit the calculated near-field coupling data [17; 24]:  
\[\frac{\Delta\lambda}{\lambda_{o}}=Ae^{-\frac{g}{T}}. \tag{4}\]  
The fit parameters \(\tau\) and free exponential factor \(A\) represent the decay length and near-field coupled strength, respectively. Equation (4) correlates with the fractional plasmonic shift and normalized inter-particle distance \((g/D)\).  
The peak position or the fractional plasmonic shift (\(\Delta\lambda/\lambda_{o}\)) of the major plasmonic mode was monitored as a function of inter-particle distance \(g\) normalized by particle size \(D\), as depicted in Figure 6a; \(\Delta\lambda\) and \(\lambda_{o}\) = 504 nm represent the peak shift and reference resonance wavelength of a single SCS nanoparticle, respectively. In this figure, only the shift of the major boning dipole peaks is addressed because the other HEMDM peaks are absent at larger \(g\) values. It is clear from Figure 6a that exponential decay Equation (4), with decay constant \(\tau\) = 0.2864 \(\pm\) 0.091 and amplitude \(A\) = 0.075 \(\pm\) 0.05, describes the calculated data completely with a few exceptions. It should be noted that expected exponential decay does not agree with calculated data when \(g/D\leq\) 0.05 nm (\(g\leq\) 2 nm). This deviation is due to the breeding of two other HEMDM peaks when nanoparticles are very close to each other [5; 16]. These modes have already been explained in the aforementioned sections of the article. However, this section includes the deviation of near-field coupling from the exponential Ruler equation due to an abrupt decrease in the fractional plasmonic shift (\(\Delta\lambda/\lambda_{o}\)) and in the intensity of the major dipolar mode. As described earlier, two additional states are generated between the particles of the SCS dimer antenna when \(g\leq\) 2 nm. To understand this, Figure 2 and Table 1 should be consulted. It is clear that two peaks, ii and iii, corresponding to the HEMDM, appear when \(g\leq\) 12 nm and \(g\leq\) 2 nm, respectively. In this case, the energy of the incident planewave is divided over all three dipolar modes. This causes the resultant energy of the major bonding dipolar mode to decrease, and as a result the fractional plasmonic phase shift (\(\Delta\lambda/\lambda_{o}\)) becomes negative when the gap \(g\) is decreases beyond 2 nm. Obviously, this will affect far-field strength (major dipolar mode), as shown in Figure 6b. Figure 6b shows the maximum scattering cross section of the SCS dimer in dB as a function of the linear inter-particle gap \(g\). The maximum scattering cross section (Peak maxima) increases when the \(g\) value decreases. However, a sudden fall in the scattering cross section when \(g\leq\) 2 nm can be seen. The QCT is not included here, because we believe that the QCT may only affect the strength of dipolar modes present in the SCS dimer antenna, but not their origin of generation.  
Figure 6: \(Cont.\)