# Analysis of All-Optical Generation of Graphene Surface Plasmons by a Frequency-Difference Process
## 1 Introduction  
Graphene, a monolayer-thick transparent conductor, is known to support evanescent waves called surface plasmons (the correct name for these excitations that involve charge density oscillations and the associated electromagnetic field is \"surface plasmon-polariton\", we abbreviate it for convenience) (SPs) [1]. Since the optical conductivity of graphene can be tuned electrically by adjusting the Fermi energy, making the SPs' electrical control feasible, it gave rise to a broad field of intense research named graphene plasmonics [1; 2; 3; 4; 5]. Graphene SPs, occurring in the THz-to-IR spectral range, are characterized by large in-plane wavevectors, beyond the light cone, and cannot be excited directly by propagating electromagnetic (EM) waves because the conditions of energy and momentum matching cannot be fulfilled for an SP and a propagating photon. Even though several methods of SP excitation by light do exist, such as using the attenuated total internal reflection (ATR) configuration, different kinds of grating, or just a symmetry-breaking object placed in the vicinity of the graphene sheet [5], all-optical generation of SPs in homogeneous graphene would be highly desirable [6; 7]. Several years ago, it was proposed theoretically [8] to use two optical beams with slightly different frequencies to generate SPs via a frequency-difference (DF) nonlinear (NL) process. A similar idea exploiting a four-wave mixing process was presented in Ref. [7]. Graphene had been predicted and demonstrated to have a strong third-order nonlinearity (the Kerr effect) in the THz-to-IR spectral range [9; 10; 11; 12; 13; 14], with a number of potentially interesting effects following from this, such as the third [11] and also higher-order [15] harmonics generation, optical bistability [16], saturable absorption [17], etc. The reason is the Dirac-type electronic energy spectrum of graphene [9]. Although it is a centrosymmetric materialwhere the second-order conductivity (or susceptibility) should vanish by symmetry [18], this restriction can be bypassed if one uses oblique optical beams with non-zero and unequal in-plane components of the wavevector, thus breaking the symmetry (see Figure 1). Indeed, second-harmonic generation signals were observed for suspended graphene, caused by curvature fluctuations over the graphene sheet [19]. Another possibility to leverage the second-order nonlinearity is, for instance, to use the surface asymmetry of a dual-layer graphene heterostructure, which has been shown to enable to the generation of a difference-frequency signal at THz frequencies [20]. The idea illustrated in Figure 1 was implemented experimentally in Ref. [6] and investigated further in subsequent articles [21; 22]. These experiments used graphene on top of SiO\({}_{2}\) and two femtosecond laser pulses, a pump, and a probe, to excite the SPs. The probe wavelength was fixed at \(\lambda_{a}=615\) nm while the pump wavelength, \(\lambda_{b}\), was varied between 615 and 545 nm. By recording the differential reflectivity of the probe, at the condition of energy and in-plane momentum matching for plasmons (see Supplementary Information (SI)), a resonance associated with the SP generation was observed [6; 21], as had been predicted theoretically [8]. By recording the differential reflectivity of the probe, at the condition of energy and in-plane momentum matching for plasmons (see Supplementary Information (SI)), a resonance associated with the SP generation was observed [6; 21]. However, contrary to what was predicted theoretically [8], this resonance appeared as a narrow signal, symmetric with respect to the pulse peak, on top of a broader asymmetric band which was also observed outside of the plasmon resonance. This asymmetric temporal variation of the differential reflectance was tentatively attributed to hot-electron effects [21] and it is not well understood so far.