Over the past decade, graphene and other two-dimensional (2D) materials have been widely recognized as promising media for establishing nonlinear light-matter interactions. The scope of this Special Topic is to present new insights in the nonlinear-optical characteristics of these 2D crystals, both from a theoretical and an experimental point of view, and to address their potential for practical applications. Particular attention is paid to the role of intraband and interband transitions, the dynamics of the photocarriers excited in the materials, the analogies and differences between graphene and its 3D and 1D counterparts, and the combination of 2D materials with waveguiding and/or resonance structures. This editorial concludes with a general outlook for future research in the field.

The emergence of 2D materials, pioneered by the first isolation of graphene in 2004, has opened up a wealth of new research topics in the fields of physics, chemistry, electronics, mechanics, and also photonics. Whereas it all started with graphene, the family of 2D crystals has been expanding rapidly and nowadays also comprises, amongst others, transition-metal-doped chalcogenides (TMDs), hexagonal boron nitride, and black phosphorus. Each of these materials features extraordinary optical properties, and several of them are particularly suited for establishing nonlinear light-matter interactions such as saturable absorption,1–3 nonlinear wave mixing,4–8 and self-phase modulation and (de)focusing effects.9–13 When furthermore combined with, for example, optical fiber technology or on-chip waveguiding structures,5,6,12,13 they could give rise to various novel nonlinear-optical devices with unprecedented characteristics and performances.

Although there has been important research progress over the past several years in the field of 2D-material-based nonlinear optics, there is still much to be discovered and explored. The goal of this Special Topic is to provide new insights in the nonlinear-optical properties of 2D crystals, both from a theoretical and an experimental point of view, and to touch upon their potential for practical applications.

The physics that underpins the unusual optical properties of 2D crystals mostly resides in the material band structure. The Invited Perspective Article of J. L. Cheng et al. included in this Special Topic focuses on the intraband contributions to the third-order nonlinear optical response of 2D materials, and shows by means of analytic expressions that these contributions can diverge at optical frequencies and lead to large nonlinear conductivities.14 In the Invited Article of L. Mennel et al., interband second-harmonic generation is experimentally investigated in several TMD monolayers subjected to strain. The authors measure a very high strain sensitivity of the TMDs’ second-harmonic generation response and can connect it with the strain dependence of the A-exciton energies for the different materials.15 

To fully understand the nonlinear-optical response of 2D materials, it is also of crucial importance to study its dynamics. R. J. Suess et al. report on optoelectronic mixing experiments in black phosphorus using the nonlinear dependence of the photocarrier lifetime on the optical power. A model based on radiative recombination of the photoexcited carriers allows explaining the mixing origin as well as the upper bandwidth observed in their experiments.16 

Drawing upon the long-existing knowledge of bulk materials, researchers often employ three-dimensional macroscopic quantities also when discussing 2D materials. S. A. Mikhailov shows that special care must be taken when addressing the macroscopic nonlinear-optical properties of 2D crystals and introduces two-dimensional nonlinear susceptibilities obtained from averaging Maxwell’s equations along a new approach.17 At the same time, there has been an increasing interest in so-called three-dimensional Dirac semimetals (3D DSMs), hosting gapless Dirac nodal points in the bulk–in other words, the “graphene analogue” in 3D bulk material. The Invited Article of K. J. A. Ooi et al. presents a theoretical study of the nonlinear plasmonic response of 3D DSMs. Their calculations show a lower yet considerable nonlinear performance for these materials as compared to that of graphene, while the structural benefits of bulk metals are maintained in the 3D DSMs.18 This comparison of 3D DSMs with graphene is complemented by the Invited Tutorial Article of S. Yamashita comparing graphene with its 1D counterpart, namely, carbon nanotubes. Besides discussing the basics of nonlinear optics in low-dimensional materials, this Tutorial also addresses some key applications and devices demonstrated in recent years.19 

A natural strategy towards device development relies on the combination of the nonlinear-optical 2D materials with waveguiding and/or resonance structures. In the Invited Article of P. Navaeipour and M. M. Dignam, the authors consider a metallic parallel-plate waveguide design with graphene embedded in the middle for third harmonic generation of terahertz radiation. Their calculations show how the waveguide dimensions should be tuned to improve phase matching and hence increase the power efficiency of the third harmonic generation process.20 Another study on 2D-material nonlinearities combined with waveguides or resonators is presented in the Invited Article of M. Tokman et al. Here the authors introduce a formalism describing the Purcell enhancement of parametric down-conversion in 2D crystals embedded in subwavelength cavities or waveguides. This enhancement effect is found to enable a strong reduction in the parametric instability threshold for realistic material and cavity parameters.21 Finally, the Invited Article of Y. Yang et al. reports on the experimental demonstration of enhanced four-wave mixing in a graphene-oxide-cladded silica waveguide. The authors also discuss in what way the waveguide cross-section could be redesigned to improve the modal overlap with the graphene oxide layer and as such further boost the four-wave mixing efficiency.22 

In conclusion, this Special Topic showcases different examples of recent advances in the challenging yet fascinating research field of nonlinear optics in graphene and its structural analogues. Although this research domain still is in its early years, efforts are already being made to exploit the extraordinary physical phenomena in discrete or integrated devices useful for practical applications. At the same time, the transition of 2D materials out of the lab and into the marketplace will be most effective if it is supported by in-depth fundamental research results with a thorough understanding of the physics taking place. In view of the complex nature of nonlinear optics especially in low-dimensional material systems, further theoretical and experimental investigations will have to be performed by researchers with different areas of expertise. Employing an interdisciplinary approach and keeping an open mind that thinks beyond previously made assumptions are both key requirements for obtaining a complete picture of the nonlinear light-matter interactions at work; these advances in fundamental science can then pave the way towards the full exploitation of nonlinear-optical 2D materials in real-life applications.

We hope this Special Topic will be relevant and interesting for researchers both in and outside the field. Finally we would like to acknowledge all authors who contributed to this Special Topic, as well as Associate Editors Christelle Monat and Baohua Jia, Journal Manager Erinn Brigham, and Editor-in-Chief Benjamin Eggleton.

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