Surface plasmons in graphene may provide an attractive alternative to noble-metal plasmons due to their tighter confinement, peculiar dispersion, and longer propagation distance. We present theoretical studies of the nonlinear difference frequency generation of terahertz surface plasmon modes supported by two-dimensional layers of massless Dirac electrons, which includes graphene and surface states in topological insulators. Our results demonstrate strong enhancement of the DFG efficiency near the plasmon resonance and the feasibility of phase-matched nonlinear generation of plasmons over a broad range of frequencies.PACS numbers: 81.05.ue, 42.65.-k Graphene exhibits many interesting electronic properties because of its chiral symmetry and gapless linear spectrum of free carriers near the Dirac point. In recent years, graphene has also been recognized as a promising broadband optoelectronic material in the infrared (IR) and terahertz (THz) region, especially when utilizing a surface plasmon resonance [1][2][3]. Surface plasmon is a collective mode of coupled charge-density and field oscillations at an interface between a free-carrier system and a dielectric or vacuum. Surface plasmons guided by graphene are expected to have low losses and be highly tunable by gating and doping, making graphene an attractive alternative to metal plasmonics. Surface states in certain topological insulators (TIs) have a massless Dirac-cone electron dispersion at low energies with a slope similar to that in graphene. They provide a potentially even more interesting host medium for surface plasmons due to lower scattering rates of two-dimensional (2D) surface electrons that are topologically protected from scattering on non-magnetic impurities [4]. In particular, Bi 2 Se 3 has a large bulk band gap of about 0.3 eV, suitable for THz and mid-infrared plasmonics, and a tunable Fermi level which can be put within the bulk gap [5]. The combination of highly efficient light-matter interaction, relatively long propagation distances, and tight confinement of surface plasmons in graphene and TIs promises interesting applications including compact room-temperature THz sources for imaging, spectroscopy and telecommunications; integrated photonic circuits; THz modulation of telecom signals, and compact THz sensors. Furthermore, optical methods [6] may provide a better access to characterization and manipulation of massless fermion states than transport measurements that are affected by conductivity in the bulk.Nonlinear optics of massless Dirac fermions has received little attention so far, especially in the THz range where many basic devices and components are lacking. Here we show that the difference frequency generation (DFG) in 2D layers of massless Dirac electrons, e.g. graphene and TIs, is an efficient and controllable way of generating surface plasmons over a broad range of frequencies. Second-order nonlinear processes such as DFG are usually assumed to be forbidden in an isotropic medium [14] such as the plane of a graphene layer. How...

We present quantum-mechanical density-matrix formalism for calculating the nonlinear optical response of magnetized graphene, valid for arbitrarily strong magnetic and optical fields. We show that magnetized graphene possesses by far the highest third-order optical nonlinearity among all known materials. The giant nonlinearity originates from unique electronic properties and selection rules near the Dirac point. As a result, even one monolayer of graphene gives rise to appreciable nonlinear frequency conversion efficiency for incident infrared radiation.PACS numbers: 81.05.ue, 42.65.-k Graphene, a two-dimensional monolayer of carbon atoms arranged in a hexagonal lattice, holds many records as far as its mechanical, thermal, electrical, and optical properties are concerned; see. e.g. [1] for the review. With this Letter we would like to add yet another distinction to this list of superlatives: we show that graphene in a strong magnetic field has the highest infrared optical nonlinearity of all materials we know.Strong optical nonlinearity of graphene, like most of its unique electrical and optical properties, stems from linear dispersion of carriers near the K,K' points of the Brillouin zone. As a result, the electron velocity induced by an incident electromagnetic wave is a nonlinear function of induced electron momentum. Nonlinear electromagnetic response of classical charges with linear energy dispersion has been studied theoretically in [2]. Recently, fourwave mixing in mechanically exfoliated graphene flakes without magnetic field has been observed at near-infrared wavelengths [3]. Effective bulk third-order susceptibility was estimated to have a very large value, χ (3) ∼ 10esu. This is comparable in magnitude to the resonant intersubband χ (3) nonlinearity observed in the mid-infrared range for low-doped quantum cascade laser structures [4], which are essentially asymmetric coupled quantum well heterostructures.Nonlinear cyclotron resonance in graphene was considered theoretically in [5], again in the classical limit, by solving the equation of motion F = dp/dt for a massless charge. Classical approximation can be applied to electrons in low magnetic field that are occupying highly excited Landau levels n 1, when energy and momentum quantization are neglected. Here we present rigorous quantum mechanical description of the nonlinear optical response of magnetized graphene, which is valid for arbitrary magnetic fields and electron distributions over Landau levels (LLs). After finding matrix elements of the optical transitions between LLs, we calculate the thirdorder nonlinear susceptibility using the density-matrix formalism and then evaluate the efficiency of the fourwave mixing process. The magnitude of χ (3) turns out * Electronic address: belyanin@tamu.edu to be astonishingly large, of the order of 10 −1 esu at mid/far-infrared wavelengths in the field of several Tesla. This leads to a surprisingly high four-wave mixing efficiency of the order of 10 −4 W/W 3 per monolayer.Linear (one-photon) absorption...

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A theoretical and experimental study of multimode operation regimes in quantum cascade lasers (QCLs) is presented. It is shown that the fast gain recovery of QCLs promotes two multimode regimes: One is spatial hole burning (SHB) and the other one is related to the Risken-Nummedal-Graham-Haken instability predicted in the 1960s. A model that can account for coherent phenomena, a saturable absorber, and SHB is developed and studied in detail both analytically and numerically. A wide variety of experimental data on multimode regimes is presented. Lasers with a narrow active region and/or with metal coating on the sides tend to develop a splitting in the spectrum, approximately equal to twice the Rabi frequency. It is proposed that this behavior stems from the presence of a saturable absorber, which can result from a Kerr lensing effect in the cavity. Lasers with a wide active region, which have a weaker saturable absorber, do not exhibit a Rabi splitting and their multimode regime is governed by SHB. This experimental phenomenology is well-explained by our theoretical model. The temperature dependence of the multimode regime is also presented

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