# Analysis of All-Optical Generation of Graphene Surface Plasmons by a Frequency-Difference Process
## 3 Results and Discussion
### How to Detect the Optically Generated SPs? As pointed out in the previous section, the differential reflectance method [6,21] relies on a process that is of the third order with respect to the optical fields. Considering the optical beams as plane waves, it seems that the variation of the reflection coefficients of these beams is small and consequently hard to detect unless very high intensities of the pump beam are used. Then other (non-electronic) types of nonlinearity can arise, for instance, due to photothermal effects [22], which is less interesting from the point of view of applications. Thus, it might be preferable to try and detect the optically generated surface plasmons using another route. Time-resolved THz spectroscopy is a powerful technique in the spectral range relevant to graphene SPs [28] and it could help detect the flux of plasmons escaping from the optical spots where they were generated. Below we shall evaluate this flux. Let us assume for simplicity that the optical beams are focused by cylindrical lenses so that the system is uniform along the \(y\)-direction. As a first approximation, we neglect the uncertainty of the in-plane components of the wave vectors \(\mathbf{k}_{a}\) and \(\mathbf{k}_{b}\) (\(q_{a}\) and \(q_{b}\)) as if they were plane waves. In this situation, the energy flux in the \(x-\)direction associated with the generated SPs, per unit length along \(y\), is (see SI for details):  
\[J=q\frac{\omega}{16\pi}\sum_{m=1,2}\frac{\varepsilon_{m}}{\kappa_{m}^{3}}|E_{m,x}(\omega,q)|^{2}\propto|D(\omega,q)|^{-2}\,, \tag{19}\]  
with \(D(\omega,q)\) defined by Equation (16). At the plasmon resonance, the evanescent field is strongly enhanced and the energy flux increases. Since SPs are dissipated because of the Ohmic losses, we can neglect the inverse process of optical photon generation with the propagation of SPs. Thus, the flux (19) is responsible for the removal of energy from the optical beams. If the incident energy fluxes are \(J_{a}^{(I)}\), \(J_{b}^{(I)}\) (per unit length along \(y\)), and if both reflected and transmitted beams \(a\) and \(b\) are measured, it should be possible to detect a variation of the total energy of both optical beams since, at the plasmon resonance, it should be diminished by \(J\) because of the energy conservation,  
\[J_{a}^{(I)}+J_{b}^{(I)}=\left(J_{a}^{(R)}+J_{a}^{(T)}+J_{b}^{(R)}+J_{b}^{(T)}+ \mathcal{L}_{a}+\mathcal{L}_{b}\right)+J\,, \tag{20}\]  
where \(\mathcal{L}\) stands for losses associated with the transmission and reflection of the optical beams at the graphene-covered surface, which are related to \(\text{Re}\sigma^{(1)}(\omega_{a,b})\) and, therefore, can be evaluated. Detection of a positive difference between the left-hand side and the parenthesis in the right-hand side of (20) will indicate the SP generation, \(J>0\). Another possibility is to try to detect the SP flux by converting it into propagating THz light, as was  
Figure 4: Differential reflectance (**left**, solid lines), transmittance (**left**, dashed lines), and absorbance (**right**) of the pump (blue curves) and probe (green curves) beams as functions of \(\omega\). Here the Fermi energy \(E_{F}=500\) meV, \(\theta_{a}^{I}=20^{\circ}\), and \(\theta_{b}^{I}=40^{\circ}\), while other parameters are the same as in Figure 3. The plasmon resonance is observed at approximately \(\omega_{\text{SP}}=80\) ps\({}^{-1}\). We assumed the same intensities for both optical beams, \(I_{a}=I_{b}=10^{16}\) erg cm\({}^{-2}\)s\({}^{-1}\), in order to facilitate the comparison. done in the first demonstrations of graphene surface plasmons [2,3]. Yet another possibility to detect the generated plasmons is to measure the electric current arising due to the heating up of charge carriers in graphene, following SP decay [29].