Enhanced optical Kerr effect method for a detailed characterization of the third-order nonlinearity of two-dimensional materials applied to graphene

Using an enhanced optically heterodyned optical Kerr effect method and a theoretical description of the interactions between an optical beam, a single layer of graphene, and its substrate, we provide experimental answers to questions raised by theoretical models of graphene third-order nonlinear optical response. In particular, we measure separately the time response of the two main tensor components of the nonlinear susceptibility, we validate the assumption that the out-of-plane tensor components are small, and we quantify the optical impact of the substrate on the measured coefficients. Our method can be applied to other two-dimensional materials, as it relies mainly on the small ratio between the thickness and the wavelength.

Figure 1

(a) Interaction of the pump-and-probe beams with the sample. Half- and quarter-wave plates are denoted, respectively, by $\lambda /2$ and $\lambda /4$. Iris denotes an iris diaphragm. (b) The pump-and-probe beams define the horizontal plane that intersects the sample along the $y$ axis. Sample tilt around $\widehat{y}$ is denoted by ${\alpha }_i$.

Figure 2

Experimental results: (a) 2D-OHD-OKE signal of graphene at positive and negative heterodyne angle. (b) Normalized difference and sum of 2D-OHD-OKE signals from (a) compared with the pulse autocorrelation obtained by the OHD-OKE signal from the silicon reference sample.

Figure 3

Experimental results: (a) comparison between $|S_{\text{dif}}^{\theta }|$ and $|S_{\text{dif}}^{\eta }|$; (b) normalized $S_{\text{sum}}^{\theta }$ for parallel pump-and-probe polarizations, providing $\mathrm{Im}\left({\chi }_{xxyy}\right)$ compared with the autocorrelation. Smoothed data (Savitzky-Golay) are provided as a guide for the eye.

Figure 4

Experimental data and fitting of the odd part of the 2D-OHD-OKE signal for a vertical input polarization of the probe ($x$ axis). The linear pump polarization is continuously set from vertical ($-{90}^{\circ }$) to horizontal ($0^{\circ }$) and vertical (${90}^{\circ }$). $C_{xyxy}$ and $C_{xzxz}$ denote, respectively, the coefficients of the real parts of ${\chi }_{xyxy}+{\chi }_{xyyx}$ and ${\chi }_{xzxz}+{\chi }_{xzzx}$. Data are fitted to these curves: black, total fit; blue, $xyxy$ component; red, $xzxz$ component. Coefficients $C_{xyxy}$ and $C_{xzxz}$ are calculated using (a) the symmetric surface field, and (b) without taking the field in the substrate into account.