We investigate the nonlinear optical properties of graphene flakes using four-wave mixing. The corresponding third-order optical susceptibility is found to be remarkably large and only weakly dependent on the wavelength in the near-infrared frequency range. The magnitude of the response is in good agreement with our calculations based on the nonlinear quantum response theory. DOI: 10.1103/PhysRevLett.105.097401 PACS numbers: 78.67.Wj, 42.65.Ky, 78.47.nj Graphene, a single sheet of carbon atoms in a hexagonal lattice, is the basic building block for all graphitic materials. Although it has been known as a theoretical concept for some time [1], a layer of graphene has only recently been isolated from bulk graphite and deposited on a dielectric substrate [2]. The great interest in studying graphene is driven by its linear, massless band structureand many unusual electrical, thermal, mechanical, and optical properties [3,4] [here the upper (lower) sign corresponds to the electron (hole) band, p is the quasimomentum, and V % 10 6 m=s is the Fermi velocity]. For example, the optical absorption of graphene has been shown to be wavelength independent ('2:3% per layer) in a broad range of optical frequencies [5][6][7]. Recently, it has been predicted that the linear dispersion described by Eq. (1) should lead to strongly nonlinear optical behavior at microwave and terahertz frequencies [8]. At higher, optical frequencies one can also expect an enhanced optical nonlinearity as, due to graphene's band structure, interband optical transitions occur at all photon energies. Here we report on the first observation of the coherent nonlinear optical response of graphene at visible and nearinfrared frequencies. We show that graphene has an exceptionally high nonlinear response, described by the effective nonlinear susceptibility j ð3Þ j $ 10 À7 esu (electrostatic units). This nonlinearity is shown to be essentially dispersionless over the wavelength range in our experiments (emission at e ' 760-840 nm). These results are in good agreement with predictions derived from nonlinear quantum response theory. The large optical nonlinearity of graphene can be used for exceptionally high-contrast imaging of single and multilayered graphene flakes.Single-and few-layer graphene samples are fabricated using the method of mechanical exfoliation [2] and deposited onto a 100 m thick glass cover slip. Prior to investigation in the nonlinear microscope, the layer thickness is estimated via contrast measurements under an optical microscope, using a method similar to Ref. [9]. To investigate the nonlinear response of graphene flakes, we employ the four-wave mixing technique [10]. This involves the generation of mixed optical frequency harmonics 2! 1 À ! 2 under irradiation by two monochromatic waves with the frequencies ! 1 and ! 2 .Figure 1(a) illustrates the principle of the method: two incident pump laser beams with wavelengths 1 (tunable from 670 nm to 980 nm) and 2 (1130 nm to 1450 nm) are focused collinearly onto a sample and mix together...

A new, weakly damped, transverse electromagnetic mode is predicted in graphene. The mode frequency ω lies in the window 1.667

It is shown that the massless energy spectrum of electrons and holes in graphene leads to the strongly non-linear electromagnetic response of this system. We predict that the graphene layer, irradiated by electromagnetic waves, emits radiation at higher frequency harmonics and can work as a frequency multiplier. The operating frequency of the graphene frequency multiplier can lie in a broad range from microwaves to the infrared.In the past two years a great deal of attention has been attracted by a recently discovered, new two-dimensional (2D) electronic system -graphene, built out of a single monolayer of carbon atoms with a honeycomb 2D crystal structure [1,2]. The band structure of the charge carriers in this system consists of six Dirac cones at the corners of the hexagonshaped Brillouin zone [3,4], with the massless, linear electron/hole dispersion. The massless electron spectrum leads to unusual transport and electrodynamic properties, which have been intensively studied in the literature, see e.g. [5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31] and for review [32,33].Electrodynamic properties of graphene have been theoretically studied in Refs. [16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31]. The frequency dependent conductivity [16,17,20,21,22], as well as plasmon [23,25,27,29,30], plasmon-polariton [24], and transverse electromagnetic wave spectra [31] have been investigated. In all these papers electrodynamic response of the system has been studied within the linear response theory (for instance, using the Kubo formalism, or the random phase approximation, or the self-consistent-field approach). In this Letter we show that, apart from all the fascinating and non-trivial properties of graphene predicted and observed so far, this material should also demonstrate strongly non-linear electrodynamic behavior. In particular, irradiation of the graphene sheet by a harmonic electromagnetic wave with the frequency Ω should lead to the emission of the higher harmonics with the frequencies mΩ, m = 3, 5, . . ., from the system. The operating frequency of such a frequency multiplier can vary from microwaves up to infrared, and the required ac electric field is rather low, especially at low carrier densities and low temperatures. The predicted non-linear electrodynamic properties of graphene may open up new exciting opportunities for building electronic and optoelectronic devices based on this material.To qualitatively demonstrate the non-linear behavior of graphene electrons consider a classical 2D particle with the charge −e and the energy spectrum ǫ p = V p = V p 2 x + p 2 y in the external electric field E x (t) = E 0 cos Ωt. Here V is the velocity of 2D electrons in the energy band (in graphene V ≈ 10 8 cm/s [1, 2]). According to the classical equations of motion dp x /dt = −eE x (t) the momentum p x will then be equal to p x (t) = −(eE 0 /Ω) sin Ωt, and the velocity v x = ∂ǫ p /∂p x is then v x (t) = −V sgn(sin Ωt). If there are n s particles per unit area, the corresponding ac ...

Graphene is a recently discovered carbon based material with unique physical properties. This is a monolayer of graphite, and the two-dimensional electrons and holes in it are described by the effective Dirac equation with a vanishing effective mass. As a consequence, electromagnetic response of graphene is predicted to be strongly non-linear. We develop a quasi-classical kinetic theory of the non-linear electromagnetic response of graphene, taking into account the self-consistent-field effects. Response of the system to both harmonic and pulse excitation is considered. The frequency multiplication effect, resulting from the non-linearity of the electromagnetic response, is studied under realistic experimental conditions. The frequency up-conversion efficiency is analysed as a function of the applied electric field and parameters of the samples. Possible applications of graphene in terahertz electronics are discussed.

The linear energy dispersion of graphene electrons leads to a strongly nonlinear electromagnetic response of this material. We develop a general quantum theory of the third-order nonlinear local dynamic conductivity of graphene σ αβγδ (ω1, ω2, ω3), which describes its nonlinear response to a uniform electromagnetic field. The derived analytical formulas describe a large number of different nonlinear phenomena such as the third harmonic generation, the four wave mixing, the saturable absorption, the second harmonic generation stimulated by a dc electric current, etc., which may be used in different terahertz and optoelectronic devices.

An analytical theory of the nonlinear electromagnetic response of a two-dimensional (2D) electron system in the second order in the electric field amplitude is developed. The second-order polarizability and the intensity of the second harmonic signal are calculated within the self-consistent-field approach both for semiconductor 2D electron systems and for graphene. The second harmonic generation in graphene is shown to be about two orders of magnitude stronger than in GaAs quantum wells at typical experimental parameters. Under the conditions of the 2D plasmon resonance the second harmonic radiation intensity is further increased by several orders of magnitude.

Metamaterials and plasmonics are powerful tools for unconventional manipulation and harnessing of light. Metamaterials can be engineered to possess intriguing properties lacking in natural materials, such as negative refractive index. Plasmonics offers capabilities of confining light in subwavelength dimensions and enhancing light–matter interactions. Recently, the technological potential of graphene-based plasmonics has been recognized as the latter features large tunability, higher field-confinement and lower loss compared with metal-based plasmonics. Here, we introduce hybrid structures comprising graphene plasmonic resonators coupled to conventional split-ring resonators, thus demonstrating a type of highly tunable metamaterial, where the interaction between the two resonances reaches the strong-coupling regime. Such hybrid metamaterials are employed as high-speed THz modulators, exhibiting ∼60% transmission modulation and operating speed in excess of 40 MHz. This device concept also provides a platform for exploring cavity-enhanced light–matter interactions and optical processes in graphene plasmonic structures for applications including sensing, photo-detection and nonlinear frequency generation.

A general electrodynamic theory of a grating coupled two dimensional electron system (2DES) is developed. The 2DES is treated quantum mechanically, the grating is considered as a periodic system of thin metal strips or as an array of quantum wires, and the interaction of collective (plasma) excitations in the system with electromagnetic field is treated within the classical electrodynamics. It is assumed that a dc current flows in the 2DES. We consider a propagation of an electromagnetic wave through the structure, and obtain analytic dependencies of the transmission, reflection, absorption and emission coefficients on the frequency of light, drift velocity of 2D electrons, and other physical and geometrical parameters of the system. If the drift velocity of 2D electrons exceeds a threshold value, a current-driven plasma instability is developed in the system, and an incident far infrared radiation is amplified. We show that in the structure with a quantum wire grating the threshold velocity of the amplification can be essentially reduced, as compared to the commonly employed metal grating, down to experimentally achievable values. Physically this is due to a considerable enhancement of the grating coupler efficiency because of the resonant interaction of plasma modes in the 2DES and in the grating. We show that tunable far infrared emitters, amplifiers and generators can thus be created at realistic parameters of modern semiconductor heterostructures.