Predicting Graphene's Nonlinear‐Optical Refractive Response for Propagating Pulses

Abstract

Nonlinear‐optical refraction is typically described by means of perturbation theory near the material's equilibrium state. Graphene, however, can easily move far away from its equilibrium state upon optical pumping, yielding strong nonlinear responses that cannot be modeled as mere perturbations. So far, one is still lacking the required theoretical expressions to make predictions for these complex nonlinear effects and to account for their evolution in time and space. Here, this long‐standing issue is solved by the derivation of population‐recipe‐based expressions for graphene's nonperturbative nonlinearities. The presented framework successfully predicts and explains the various nonlinearity magnitudes and signs observed for graphene over the past decade, while also being compatible with the nonlinear pulse propagation formalism commonly used for waveguides.

A generic model for the study of supercontinuum generation in graphene-covered nanowires

“…In particular, these references put forth an explanation for graphene's nonlinearity by means of saturable photoexcited-carrier refraction (SPCR). Not only SPCR accounts well for the values of n 2 reported in the literature [45], but it also allows to adequately model the nonlinear propagation of pulses in graphene-decorated waveguides [44].…”

“…Observe that N G t is a linear density due to the 2D nature of graphene, in contrast to N W t which is a volume density. The SPCR coefficient can be calculated as [45]…”

“…where α ≈ 1/137 is the fine-structure constant, ∆σ (1) = σ (1) (N sat /2) − σ (1) (0), σ (1) is the conductivity of graphene, σ 0 = e 2 /4ℏ is the universal conductivity, and n eff is the effective mode index. Castelló-Lurbe et al [45] presents details of the calculation of σ (1) as a function of the carrier density, as well as a derivation of equation (10), modeling the evolution of photoexcited carriers, on the basis of the theoretical model in [49,50]. The two-dimensional graphene layers are assumed to be on a line segment G x .…”

“…We choose different dimensions in order to obtain similar values of the SPCR coefficient σ SP (see figure 4), and we vary the number of graphene layers in order to study its influence. We consider modest doping of graphene with a hole density of p 0 = 6.5 × 10 12 cm −2 as in [44,45]. Figure 2 shows the cross-section of the graphene-covered SiN (top) and silicon (bottom) waveguides and the transverse-electric mode distribution, computed by means of a finite-difference time-domain (FDTD) algorithm [52].…”

“…There is a wide range of values for the nonlinear refractive index of graphene reported in the literature. In [44,45] it is argued that this wide range stems from measurement conditions, such as the wavelength, duration and intensity of the propagated pulses. In particular, these references put forth an explanation for graphene's nonlinearity by means of saturable photoexcited-carrier refraction (SPCR).…”