## 2 Dyadic Green's Function for the Electric Field
### Renormalized Polarizability of an Isotropic Quantum Emitter Near a Plasmonic Graphene Grating
#### 3.4.1 Optical Properties of a Plasmonic Graphene Grating  
For a metamaterial such as the graphene-based grating depicted in Figure 1b, the description of the interaction of the material with a quantum emitter can be quite complex. One possible method for  
Figure 4: Real (**a**) and imaginary (**b**) parts of the renormalized polarizability of a CdSe nanoparticle with \(R=50\,\mathrm{nm}\) located at a distance of \(z_{0}=151\,\mathrm{nm}\) from a graphene sheet with a Fermi energy of 1 eV and damping parameter of \(\hbar\gamma=4.1\,\mathrm{meV}\), supported by a dielectric of permittivity \(e_{2}=2\). The solid red line represents the \(xx\) component of the polarizability, and the black dotted line represents the \(zz\) component. The dashed blue line is the \(xx\) component of the polarizability in the absence of graphene. The parameters used in both panels for the Lorentz model for the dielectric function of CdSe are: \(\epsilon_{\infty}=6.2\), \(\omega_{\text{LO}}=211\,\mathrm{cm}^{-1}\), \(\omega_{\text{TO}}=169\,\mathrm{cm}^{-1}\) and \(\Gamma_{\text{TO}}=5\,\mathrm{cm}^{-1}\). overcoming such a difficulty is by computing the effective conductivity of the metamaterial, in this case the plasmonic graphene grating. The general method for accomplishing this was given in [60] and was later applied to the problem of tuning total absorption in graphene [61], but no details of its calculation were given. Instrumental to the calculation of the effective conductivity is the knowledge of the reflection and transmission Fresnel coefficients. These coefficients were computed in approximated analytical form in [62] and we give here only the final results:  
\[r_{p,m} =-\delta_{m,0}+t_{p,m}+\mu_{0}\chi(\omega)\frac{w}{4}J_{1}(m\pi w /L) \tag{72}\] \[t_{p,m} =\frac{\epsilon_{2}\beta_{1,m}}{\epsilon_{1}\beta_{2,m}+\epsilon _{2}\beta_{1,m}}\left(2\delta_{m,0}-\mu_{0}\chi(\omega)\frac{w}{4}J_{1}(m\pi w /L)\right) \tag{73}\]  
where \(r_{p,0}\) and \(t_{p,0}\) are the reflection and transmission coefficients, respectively, of the zero diffraction-order of the grating (the only propagating order for a sub-wavelength grating), \(w\) is the width of the graphene ribbons in the grating, \(L\) is the period of the grating, and the function \(\chi(\omega)\) reads  
\[\chi(\omega)=\frac{2\beta_{2,0}\beta_{1,0}}{\epsilon_{1}\beta_{2,0}+\epsilon _{2}\beta_{1,0}}\frac{\sigma_{L}(\omega)c^{2}}{\omega\Lambda(\omega)} \tag{74}\]  
which encodes the information about the plasmonic resonance in the grating. With \(\Lambda(\omega)\) given by  
\[\Lambda(\omega)=\frac{w}{4}\sum_{n=-\infty}^{\infty}\frac{1}{n}J_{1}(n\pi w/L) \left[1+\frac{\sigma_{L}(\omega)}{\omega\epsilon_{0}}\frac{\beta_{2,n}\beta_{ 1,n}}{\epsilon_{1}\beta_{2,n}+\epsilon_{2}\beta_{1,n}}\right] \tag{75}\]  
where \(\beta_{1,n}=\sqrt{k_{1}^{2}-k_{x}^{2}-q_{n}^{2}}\) and \(\beta_{2,n}=\sqrt{k_{2}^{2}-k_{x}^{2}-q_{n}^{2}}\), with \(q_{n}=k_{y}+n2\pi/L\), \(J_{1}(x)\) is the Bessel function of order \(1\). Here, the summation in \(\Lambda(\omega)\) is delicate due to the oscillatory nature of the Bessel function (see [62]). For simplicity of the calculation, we approximate \(\beta_{j,n\
eq 0}\) by \(\beta_{j,n\
eq 0}\approx\sqrt{k_{j}^{2}-p_{\parallel}^{2}-n^{2}4\pi^{2}/L^{2}}\). In addition to \(r_{p,0}\) and \(t_{p,0}\) there is an infinite number of other coefficients associated with higher diffraction order, but they are all evanescent in nature for the parameters chosen in the figures. Therefore, we approximate the optical properties of the grating considering only \(r_{p,0}\) and \(t_{p,0}\), and \(r_{p,1}\) and \(t_{p,1}\) (we have checked that introducing more evanescent terms does not change the results). This gives us an analytical description of its optical properties. As noted above, from the knowledge of \(r_{p,0}\) and \(t_{p,0}\), and \(r_{p,1}\) and \(t_{p,1}\) we can derive an effective conductivity for the graphene grating along the direction perpendicular to the axis of the graphene ribbon.