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Saturable absorption (SA) is an extreme nonlinear phenomenon that consists of the quenching of optical absorption under high-intensity illumination. This effect, which is an inherent property of photonic materials, constitutes a key element for passive mode-locking (PML) in laser cavities, where continuous waves are broken into a train of ultrashort optical pulses. Most materials undergo SA at very high optical intensities, in close proximity to their optical damage threshold. Currently, state-of-the-art semiconductor-based SA mirrors are routinely employed for PML lasers. However, these mirrors operate in a narrow spectral range, are poorly tunable, and require advanced fabrication techniques. Recently, carbon nanomaterials have emerged as an attractive, viable, and cost-effective alternative for the development of next-generation PML lasers. For example, carbon nanotubes undergo SA at rather modest light intensities, while their operation wavelength (determined by the energy band gap) can be manipulated by tuning their diameter. Broadband operation has been demonstrated by using an ensemble of CNTs with a wide distribution in diameter, at the expense of higher linear loss from off-resonance tubes. Graphene overcomes this limitation thanks to its peculiar conical band structure, which gives rise to broadband resonant SA at remarkably low light intensity that can further be tuned by means of an externally applied gate voltage. Graphene-based SA components have been used to achieve PML ultrafast laser operation, broadband tunability, and quality-factor switching. Graphene multilayers have also been employed to generate large energy pulses and to achieve PML in fiber lasers with normal dispersion. In addition, recent theoretical investigations predict single-mode operation of random lasers by embedding graphene flakes in a gain medium.
Here we calculate intraband and interband contributions to SA of extended graphene by nonperturbatively and semianalytically solving the single-particle Dirac equation for massless Dirac fermions (MDFs) in the presence of an external electromagnetic field retaining only one-photon processes. We further investigate the dependence of the intensity-saturated grapheme conductivity on doping, temperature, and optical frequency. Interestingly, we find a remarkably low intensity threshold for SA, which is consistent with available experimental reports. Our calculations indicate a strong quenching of absorption depth produced by electrical doping (which can be controlled through gating), as well as a weak dependence on electron temperature. Additionally, through time-domain simulations based on an atomistic tight-binding/single-particle density-matrix formalism, we study SA in graphene nanoribbons, including finite-size effects and electron-electron interactions that play a significant role in the optical response of nanostructured graphene. Surprisingly, we find that while the linear absorption predicted in atomistic simulations is reduced compared to that of extended graphene, its nonlinear saturation intensity threshold is in good quantitative agreement with predictions based on the MDF model. Deviations from the semianalytical treatment occur only at high doping, where SA is quenched and multiphoton processes lead to an intensity-dependent increase of absorption. We anticipate that the present findings will impact the future development of graphene-based PML fibre lasers and single-mode random lasers.
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