Graphene-nonlinearity unleashing at lasing threshold in graphene-assisted cavities

We investigate the nonlinear optical features of a graphene sheet embedded in an active cavity and we show that, when tuned near its lasing threshold, the cavity is able to isolate the spatially localized graphene nonlinearity thus producing a very strong nonlinear device response with multivalued features. As opposed to standard situations where the small thickness of the graphene sheet hampers its remarkable nonlinear optical properties to be exploited, in our scheme the strong nonlinear optical regime is mainly triggered by the very intrinsic planar localization of graphene nonlinearity. The proposed strategy for exploiting graphene nonlinearity through its unleashing could open novel routes for conceiving ultraefficient nonlinear photonic devices.

Figure 1

Geometry of the graphene layer embedded within a gain medium $G$ enclosed by two metallic (Ag) mirrors. The cavity is illuminated by an inclined monochromatic wave $i$ that produces reflected $r$ and transmitted waves $t$. The cavity is also illuminated by an inverted pump beam.

Figure 2

(a) Modified cavity transmissivity $\tilde{T}$ versus the complex parameter $W$ characterizing the cavity status and excitation, showing the occurrence of a strong nonlinear system behavior. Due to normalization, such a surface characterizes any possible graphene-assisted cavity response. (b) Region $M$ of the complex plane $W$ where the transmissivity is multivalued. As explained in the text, this region physically corresponds to a cavity excited close to its lasing threshold.

Figure 3

(a) Logarithmic plot of the function $\mathrm{\Psi }$ whose zeros correspond to the cavity lasing thresholds. In this example all the cavity parameters are held fixed except for the incidence angle $\theta$ and the concentration of the dye molecules $N_T$. The sharp peak corresponds to the cavity threshold. (b) The oblique solid line represents the cavity states with fixed incident optical intensity $I_i=1.5\text{W}/{\mathrm{cm}}^2$ and varying chemical potential ${\mu }_c$. The shadowed region $M$ is the same as in Fig. 2 and it contains the states where the cavity response is multivalued. The intersection of the solid line with $M$ proves that, for the considered optical intensity, the cavity actually shows a highly nonlinear behavior with multivalued response features.

Figure 4

Cavity (a) transmissivity $T$ and (b) reflectivity $R$ plotted against the incident intensity $I_i$ and the chemical potential ${\mu }_c$.

Figure 5

(a) Real and imaginary parts of the electric field $E_y$ within the cavity corresponding to the higher value of the transmissivity pertaining to the excitation state $I_i=1.5\text{W}/{\mathrm{cm}}^2$ and ${\mu }_c=385\text{meV}$. The field enhancement produced by the cavity is particularly evident since for the considered intensity the incident electric field amplitude is $|E_i|=3.36\times {10}^3\text{V}/\mathrm{m}$. Note that, since the electric field is practically left invariant by the reflection $z\to -z$, the field almost replicates itself after a complete round-trip inside the cavity, the fundamental physical ingredient allowing the considered strong nonlinear regime to occur. (b) Optical intensity $I$ within the cavity corresponding to the situation of (a). Note that such optical intensity is much smaller than the considered saturation intensity $I_{\mathrm{se}}^{\prime }\simeq 1500\text{MW}/{\mathrm{cm}}^2$, so the model for the gain medium we have used is self-consistently correct.