Graphene is an allotrope of carbon in the form of a two-dimensional, atomic-scale, honey-comb lattice in which one atom forms each vertex. It is largely responsible for the recent revolution of two-dimensional materials, owing to a combination of unique properties: one-atom thickness, remarkable mechanical strength, and high thermal and electrical conductivity. Naturally, this placed graphene in the forefront of several research fields in the optical spectral region and down to the far-infrared (THz). In fact, graphene can support **surface-plasmon polariton (SPP) waves in the THz band**, similar to metal films in the optical band, but at significantly reduced dimensions as compared to the operating wavelength.

One of the most interesting features for photonic and optoelectronic applications is the **tunable conductivity** of graphene, achieved by controlling its Fermi level (also called chemical potential) by electrical gating or by strong magnetic fields. This tunable conductivity can be used to design absorption and phase modulators by placings graphene in a capacitor-like structure and applying a voltages of a few Volts.

Apart from its tunable linear-regime properties, graphene also exhibits high second- and **third-order nonlinearity**. This can make graphene the platform for phenomena like second- and third-harmonic generation (SHG, THG) or four-wave mixnig (FWM) and all its variations, namely the Kerr effect (self-phase modulation, self-focusing), cross-phase modulation etc. In order to maximize the magnitude of these nonlinear effects in optical waveguides, graphene must be placed in the vicinity of tightly confining nano-photonic or hybrid-plasmonic structures. The same principles also stand for nanophotonic resonant structures, where the inherited intensity build up lowers the power requirement for the nonlinearity to manifest, while more sophisticated phenomena such as **optical bistability **can be observed.

Along with the phase-influencing nonlinear effect, graphene also exhibits important resistive loss saturation under moderate input power, a phenomenon commonly referred to as **saturable absorption (SA)**, as well as power-dependent nonlinear losses due to two-photon absorption (TPA). The saturable absorption process in graphene is widely exploited in mode-locking and Q-switching fiber laser. Likewise, incorporating graphene in nanophotonic waveguides and/or resonant structures allows for the implementation of high-performance, all-optical switching elements.

Graphene can be modeled either as a **"true" 2D material** (zero thickness) via a complex surface conductivity or as an "equivalent" bulk material layer (of very fine thickness) via an appropriate anisotropic susceptibility tensor. In terms of the finite-element method (FEM), the 2D/sheet representation of graphene is preferable as it reduces the computational burden associated with meshing ultra-fine layers, apart from retaining its physical properties.

## References

- Chatzidimitriou D. and Kriezis Em. E., "Light propagation in nanophotonic waveguides considering graphene's saturable absorption",
*Physical Review A*,**102**(5), 053512, (2020). [pdf] - Christopoulos T., Ataloglou V. G., and Kriezis Em. E., "All-optical nanophotonic resonant element for switching and routing applications exploiting graphene saturable absorption",
*Journal of Applied Physics*,**127**(22), 223102, (2020). [pdf] - Christopoulos T., Tsilipakos O., Sinatkas G., and Kriezis Em. E., "Degenerate four-wave mixing in nonlinear resonators comprising two-dimensional materials: A coupled-mode theory approach",
*Physical Review B*,**98**(23), 235421, (2018). [pdf] - Ataloglou V. G., Christopoulos T., and Kriezis Em. E, "Nonlinear coupled-mode-theory framework for graphene-induced saturable absorption in nanophotonic resonant structures",
*Physical Review A*,**97**(6), 063836, (2018). [pdf] - Christopoulos T., Tsilipakos O., and Kriezis Em. E, "Low-power bistability in graphene-comprising 3D photonic resonant circuits",
*Journal of Applied Physics*,**122**, 233101, (2017). [pdf] - Christopoulos T., Tsilipakos O., Grivas N., and Kriezis Em. E., "Coupled-mode-theory framework for nonlinear resonators comprising graphene,"
*Physical Review E*,**94**(6), 062219, (2016). [pdf] - Chatzidimitriou D., Pitilakis A., Kriezis Em. E., "Rigorous calculation of nonlinear parameters in graphene-comprising waveguides,"
*Journal of Applied Physics*,**118**, 023105, (2015). [pdf]

**Fig. 1:** Silicon wire overlaid with a graphene monolayer. Bottom right inset: Sample finite-element meshing of the waveguide cross-section, where graphene is modeled as a "true" sheet/2D material.

**Fig. 2:** Tuning graphene (a) surface conductivity and (b) equivalent permittivity at λ=1550 nm via its chemical potential (*μ _{c}*). Note the transitional point at

*μ*

_{c}~0.5hf.