## 2 Dyadic Green's Function for the Electric Field
### Renormalized Polarizability of an Isotropic Quantum Emitter Near a Plasmonic Graphene Grating
#### 3.4.2 Renormalization of the Polarizability of a Quantum Emitter  
In this section we study the renormalization of the polarizability of a quantum emitter near a plasmonic graphene-based grating. As explained above, we use the reflection coefficients \(r_{p,0}\) and \(r_{p,1}\) in the reflected \(p\)-Green's function and an effective Fermi energy, given by \(E_{F}^{\rm eff}=E_{F}w/L\) in the \(r_{s}\) coefficient in Equation (55), and use this in the reflected \(s\)-Green's function. We consider only the case of a metallic nanoparticle, as the results are qualitatively the same for a semiconductor one. In Figure 6 we depict the real and imaginary parts of the renormalized polarizability of a gold nanoparticle in the proximity of a graphene-based grating. A strong renormalization of the real part of the polarizability can be seen at the same frequency where the grating supports the excitation of surface plasmon polaritons (see Figure 5). The same happens in the imaginary part. However, the relative change of the imaginary part is much larger than for the real part. The results for the imaginary part of the polarizability in the case of grating should be compared to those given in Figure 3 for the same quantity. For the continuous sheet the enhancement of the imaginary part of \(\alpha\) is about that found in the present case. This is attributed to the approximate description of the reflection coefficients of the grating. Indeed, we would expect the renormalization to be larger in the case of the grating as the latter supports excitation of plasmons by far-field radiation, whereas in the case of the continuous graphene sheet the excitation of plasmons is due to near-field excitation only. We also note that the resonance peak in the imaginary part of the polarizability is not of broad band when compared to the same quantity in the continuous case.  
On other hand, the frequency where the maximum of the resonance is located is larger in the present case. This happens since we can tune the position of the resonance in the grating by varying both the Fermi energy and the geometric parameters of the grating. Therefore, the grating system has a versatility that cannot be found in the continuous sheet case. Indeed, using gratings with smaller periods, the resonance can be tuned across the electromagnetic spectrum, from the THz spectra to the infrared. We also note that the renormalization of \(\alpha_{zz}\) component (black dotted line) is substantially larger than the \(\alpha_{xx}\) component (red solid line) and the \(\alpha_{yy}\) one (brown dashed line). This happens because the \(zz-\)component of the Green's function is about twice as large compared to the \(xx-\)component.  
As noted above, tuning the grating parameters allows for an additional degree of freedom to control the impact of graphene on the polarizability of a nanoparticle. The tuning can be twofold: (1) changing the period of the grating and keeping the aspect ratio to 1/2 (constant filling factor); and (2) changing the aspect ratio, keeping the period fixed. In the first case, the procedure allows  
Figure 5: Real (blue dashed line) and imaginary (orange line) of the function \(\mu_{0}\chi(\omega)\). The parameters of the grating are \(L=0.5\,\upmu\)m and \(w=L/2\). The Fermi energy of graphene is \(E_{F}=1\) eV. The real part has a pronounced resonance due to the excitation of a surface plasmon polariton of that frequency (\(\sim\)87 THz).  
for large changes in the spectral position of the resonance; this is because the momentum of the SPP is essentially \(q=\pi/L\). Since the SPP dispersion is proportional to \(\sqrt{q}\), \(L\) fixes the position of the resonance. In the second case, a fine tuning of the position of the resonance is achieved. However, the effect of the renormalization of the polarizability is greater for the half-filled case. This is because in this regime the plasmonic resonance has the maximum intensity [16; 62]  
Finally, we have verified that when \(w\to L\), we recover the results of a continuous graphene sheet.