Quantum Theory of the Third-Harmonic Generation in Graphene

TL;DR

This paper develops a quantum theoretical framework for third-harmonic generation in graphene, deriving an analytical nonlinear conductivity tensor and predicting resonant enhancement at specific frequencies.

Contribution

It provides the first analytical formula for the nonlinear conductivity tensor in graphene related to third-harmonic generation, including resonance conditions.

Findings

Resonant maxima occur at low frequencies and around twice the Fermi energy.

Predicted third-harmonic output power of about 50 W/cm² at typical laser intensities.

Analytical expression enables better understanding of nonlinear optical responses in graphene.

Abstract

A quantum theory of the third-harmonic generation in graphene is presented. An analytical formula for the nonlinear conductivity tensor $\sigma^{(3)}_{\alpha\beta\gamma\delta}(\omega,\omega,\omega)$ is derived. Resonant maxima of the third harmonic are shown to exist at low frequencies $\omega\ll E_F/\hbar$, as well as around the frequency $\omega=2E_F/\hbar$, where $E_F$ is the Fermi energy in graphene. At the input power of a CO$_2$ laser ($\lambda\approx 10$ $\mu$m) of about 1 MW/cm$^2$ the output power of the third-harmonic ($\lambda\approx 3.3$ $\mu$m) is expected to be $\simeq 50$ W/cm$^2$.