Terahertz shifted optical sideband generation in graphene

Exploration of optical nonlinear response of graphene predominantly relies on ultrashort time domain measurements. Here we propose an alternate technique that uses frequency shifted continuous wavefront optical fields, thereby probing graphene's steady state nonlinear response. We predict frequency sideband generation in the reflected field that originates from coherent electron dynamics of the photoexcited carriers. The corresponding threshold in input intensity for optimal sideband generation provides a direct measure of the third-order optical nonlinearity in graphene. Our formulation yields analytic forms for the generated sideband intensity, is applicable to generic two-band systems and suggests a range of applications that include switching of frequency sidebands using nonlinear phase shifts and generation of frequency combs.

Figure 1

Schematic of graphene illuminated by a CW pump beam (red) of frequency ${\omega }_p$ and a frequency shifted probe beam (green) of frequency ${\omega }_p+{\omega }_s$. The third harmonic related nonlinear interband polarization combines with optical Bloch oscillations to generate a new side beam (purple) at frequencies ${\omega }_p-{\omega }_s$. The color plot in the center shows the momentum resolved photoexcited population inversion corresponding to the pump beam: $n_{\mathit{k}}^{\left(0\right)}={\rho }_{cc}^{\left(0\right)}-{\rho }_{vv}^{\left(0\right)}$. The new sideband can be detected in the backward propagating beam via the optical intensity and polarization angle dependence of the reflectivity and polarization angle of the sideband.

Figure 2

(a) Real part of the momentum resolved conductivity kernel corresponding to the pump (${\sigma }_{xx\mathit{k}}^{\left(0\right)}$, in red), probe (${\sigma }_{xx\mathit{k}}^{\left(1\right)}$, in green), and the newly generated sideband (${\sigma }_{xx\mathit{k}}^{(-1)}$, in blue) as a function of ${log}_{10}\zeta$ for ${\mathrm{\Theta }}_p=0$ (solid curve) and ${\mathrm{\Theta }}_p=\pi /4$ (dotted curve). Here ${\omega }_{\mathit{k}}=0.8{\omega }_p$ and $\widehat{\mathit{k}}=(0,1)$. (b) The corresponding integrated optical conductivities (in units of ${\sigma }_0=e^2/4\hslash )$ as a function of ${log}_{10}\zeta$. As expected, both ${\sigma }_{xx}^{\left(0\right)}$ and ${\sigma }_{xx}^{\left(1\right)}$ display linear response behavior ($\to {\sigma }_0$) as $\zeta \to 0$. However, the newly generated ${\sigma }_{xx}^{(-1)}$ is finite only after the onset of the nonlinear response regime ($\zeta \approx 1$). (c) The polarization angle dependence of longitudinal conductivities for $\zeta =1$. Other parameters are ${\omega }_p=5\times {10}^{14}{\text{s}}^{-1}$, ${\gamma }_1=1\times {10}^{12}{\text{s}}^{-1}$, ${\gamma }_2=5\times {10}^{13}{\text{s}}^{-1}$, and $\mu =0$.

Figure 3

Variation of (a) reflection probability (b) and its phase as a function of the pump field strength (${log}_{10}\zeta$), for the probe and ${\omega }_{-1}$ frequencies. The nonlinear ${\omega }_{-1}$ sideband can also be probed via interference experiments sensitive to the phase of the reflection coefficient. We have chosen ${\mathrm{\Theta }}_p=\pi /4$ for both panels, and other parameters are identical to those of Fig. 2.

Figure 4

Dependence of (a) the probability and (b) the phase of the reflection coefficient for the probe and ${\omega }_{-1}$ sideband as a function of ${\mathrm{\Theta }}_p$. The polarization rotation of the reflected beam (Kerr angle) for $s$- and $p$-polarized beams as a function of ${\mathrm{\Theta }}_p$ are shown in (c) for the pump and probe frequencies and in (d) for the ${\omega }_{-1}$ frequency. Here $\zeta =1$ and other parameters are identical to those of Fig. 2.

Figure 5

Real part of population inversion corresponding to probe frequency ($\mathrm{Re}\left[n_{\mathbf{k}}^{\left(1\right)}\right]$) for (a) ${\widehat{\mathbf{e}}}_p=\widehat{\mathit{x}}$, ${\widehat{\mathbf{e}}}_s=\widehat{\mathit{x}}$, (b) ${\widehat{\mathbf{e}}}_p=(\widehat{\mathit{x}}+\widehat{\mathit{y}})/\sqrt{2}$, ${\widehat{\mathbf{e}}}_s=\widehat{\mathit{x}}$, and (c) ${\widehat{\mathbf{e}}}_p=\widehat{\mathit{y}}$, ${\widehat{\mathbf{e}}}_s=\widehat{\mathit{x}}$. Other parameters are ${\omega }_p=5\times {10}^{14}{\text{s}}^{-1}$, ${\gamma }_1=1\times {10}^{12}{\text{s}}^{-1}$, ${\gamma }_2=5\times {10}^{13}{\text{s}}^{-1}$, and the chemical potential $\mu =0$.

Figure 6

Longitudinal optical conductivities [scaled with ${\sigma }_0=e^2/\left(4\hslash \right)]$ as a function of electric field dependent dimensionless parameter $\zeta$ and polarization. (a) Real part (shown as a solid curve) and imaginary part (shown as dashed) corresponding to the pump (${\sigma }_{xx}^{\left(0\right)}$), probe (${\sigma }_{xx}^{\left(1\right)}$), and the newly generated sideband (${\sigma }_{xx}^{(-1)}$) as a function of ${log}_{10}\zeta$ for ${\mathrm{\Theta }}_p=0$. (b) and (c) demonstrate the response of ${\sigma }_{xx}^{\left(1\right)}$ and ${\sigma }_{xx}^{(-1)}$ for three different values of the sideband frequencies, ${10}^{11}{\mathrm{s}}^{-1}$, $5\times {10}^{12}{\mathrm{s}}^{-1}$, and $5\times {10}^{13}{\mathrm{s}}^{-1}$, showing that the former increases while the latter decreases with increase in ${\omega }_s$. In (d), (e), and (f) we show the dependence of ${\sigma }_{xx}^{\left(0\right)}$, ${\sigma }_{xx}^{\left(1\right)}$, and ${\sigma }_{xx}^{(-1)}$ with respect to ${\mathrm{\Theta }}_p$ for $\zeta =1$. Please note that we show $\mathrm{Re}\left[{\sigma }_{xx}^{(-1)}\right]$ as dashed in Fig. 6, because of the negative sign, as can also be seen from Fig. 6. Other parameters are the same as those of Fig. 5.

Figure 7

Real and imaginary parts of transverse optical conductivities [scaled with ${\sigma }_0=e^2/\left(4\hslash \right)]$ as a function of electric field dependent dimensionless parameter $\zeta$ and polarization. We show (a) Re $\left[{\sigma }_{xy}^{\left(0\right)}\right]$ and $\mathrm{Im}\left[{\sigma }_{xy}^{\left(0\right)}\right]$, (b) $\mathrm{Re}\left[{\sigma }_{xy}^{\left(1\right)}\right]$ and $\mathrm{Im}\left[{\sigma }_{xy}^{\left(1\right)}\right]$, and (c) $\mathrm{Re}\left[{\sigma }_{xy}^{(-1)}\right]$ and $\mathrm{Im}\left[{\sigma }_{xy}^{(-1)}\right]$ as a function of ${log}_{10}\zeta$ for ${\mathrm{\Theta }}_p=\pi /4,$ whereas in (d), (e), and (f) we show the same set of variables with respect to ${\mathrm{\Theta }}_p$ for $\zeta =1$. Please note that the dashed curves represent the negative values of the variables. Other parameters are the same as those of Fig. 5.

Figure 8

Kerr angle for $s$ and $p$ components of pump, probe, and the new sideband, as a function of ${log}_{10}\zeta$. Other parameters are the same as those of Fig. 5.

Figure 9

Ellipticity for $s$ and $p$ components as a function of ${log}_{10}\zeta$ for (a) the pump and probe frequencies and (b) the new sideband. Dependence of ellipticity for $s$ and $p$ component as a function of ${\mathrm{\Theta }}_p$ for (c) the pump and probe frequencies and (d) the new sideband. Here ${\mathrm{\Theta }}_p=\pi /4$ for (a) and (b) and $\zeta =1$ for (c) and (d). Other parameters are the same as those of Fig. 5.