# Analysis of All-Optical Generation of Graphene Surface Plasmons by a Frequency-Difference Process
## 2 Materials and Methods
### Reflection Coefficients of the Optical Beams  
The reflection coefficients of the optical beams are affected by the SP generation and this is how the latter was detected in Ref. [6]. Here we shall calculate the \"perturbed\" reflection and transmission coefficients using the results of the previous section. We consider a situation similar to the experiments [6], with graphene cladded by two dielectrics with dielectric constants \(\varepsilon_{1}=1\) and \(\varepsilon_{2}=2.4\) and two \(p-\)polarised optical waves impinging on it. We define the reflection and transmission coefficients as \(r_{j}=E_{j}^{R}/E_{j}^{I}\) and \(t_{j}=E_{j}^{T}/E_{j}^{I}\) (\(j=a,b\)). The calculation is carried out in a standard way and follows Ref. [5], where the graphene conductivity (entering boundary conditions for the magnetic field) now includes an NL term. If only the second-order processes that correspond to the DF generation are considered, the transmission coefficients are (see SI for details):  
\[t_{a}=t_{a}^{(0)}\left(1-2t_{a}^{(0)}|t_{b}^{(0)}|^{2}\pi^{2}i_{xxx}^{c(2)}( \mathbf{\Omega}_{a},\mathbf{\Omega}_{b},-\mathbf{\Omega})\sigma^{(2)*}_{xxx}(\mathbf{\Omega}, \mathbf{\Omega}_{b},-\mathbf{\Omega}_{a})|E_{b}^{I}|^{2}\cos^{2}\theta_{b}^{T}\over \sqrt{\varepsilon_{1}}c\omega D^{*}(\omega,q)}\cos\theta_{a}^{T}\right); \tag{13}\]  
\[t_{b}=t_{b}^{(0)}\left(1+2t_{b}^{(0)}|t_{a}^{(0)}|^{2}\pi^{2}i_{xxx}^{c(2)}( \mathbf{\Omega}_{b},\mathbf{\Omega}_{a},\mathbf{\Omega})\sigma^{(2)}_{xxx}(\mathbf{\Omega}, \mathbf{\Omega}_{b},-\mathbf{\Omega}_{a})|E_{a}^{I}|^{2}\cos^{2}\theta_{a}^{T}\over \sqrt{\varepsilon_{1}}c\omega D(\omega,q)}\cos\theta_{b}^{T}\right) \tag{14}\]  
Figure 3: Real and imaginary parts of \(\sigma^{xxx}(\mathbf{\Omega},\mathbf{\Omega}_{b},-\mathbf{\Omega}_{a})\) plotted _vs_\(\omega\) (**left** panel) and \(q\) (**right** panel). The solid and dashed lines were calculated using expression (8) with Fermi levels \(E_{F}=300\) and \(500\) meV, respectively. The triangles and squares are numerical calculations of expression (6). In the right panel, \(\theta_{b}\) was fixed at \(40^{\circ}\), while in the left plot, \(\omega\) was fixed at \(94\) ps\({}^{-1}\). In both cases \(\omega_{a}=3065\) ps\({}^{-1}\), \(\gamma=10.6\) meV, and \(\theta_{a}=20^{\circ}\).  
where \(t_{j}^{(0)}\) denotes the transmission coefficient in the absence of the second pulse,  
\[t_{j}^{(0)}=\frac{2(\varepsilon_{1}/k_{j_{2}}^{I})\sec\theta_{j}^{T}\cos\theta_{ j}^{I}}{(\varepsilon_{1}/k_{j_{2}}^{I})+(\varepsilon_{2}/k_{j_{2}}^{T})+\frac{4\pi i }{\omega_{j}}\sigma^{(1)}(\omega_{j})}\,, \tag{15}\]  
\(k_{j_{2}}^{I(T)}\) denotes the \(z-\)component of the incident (transmitted) wave,  
\[D(\omega,q)=\frac{\varepsilon_{2}}{\kappa_{2}}+\frac{\varepsilon_{1}}{\kappa_ {1}}+\frac{4\pi i}{\omega}\sigma^{(1)}(\omega,q)\,, \tag{16}\]  
\(\sigma^{(1)}\) is the linear conductivity of graphene (taken according to the Drude model for SPs and accounting for interband transitions [5] for the optical beams) and \(\kappa_{m}=\sqrt{q^{2}-\varepsilon_{m}\omega^{2}/c^{2}}\) (\(m=1,2\)). The corresponding reflection coefficients are given by:  
\[r_{j}=1-\sec\theta_{j}^{I}\cos\theta_{j}^{T}t_{j}\qquad(j=a,b) \tag{17}\]  
with \(\theta_{j}^{I}\) and \(\theta_{j}^{T}\) being the incidence and transmission angles.  
Notice that the condition  
\[\text{Re}D(\omega,q)=0 \tag{18}\]  
determines the SP dispersion relation [27]. We can see from Equations (13) and (14) that there is a transfer of energy between the two optical fields, which is mediated by the SP field formed in the vicinity of graphene and, therefore, is resonant under the condition (18). This process, in spite of being due to the second-order response of graphene, is, in fact, of the third-order in optical fields.