Giant enhancement of the third harmonic in graphene integrated in a layered structure

Abstract

Graphene was shown to have strongly nonlinear electrodynamic properties. In particular, being irradiated by an electromagnetic wave with the frequency ω, it can efficiently generate higher frequency harmonics. Here we predict that in a specially designed structure "graphene -dielectricmetal" the third-harmonic (3ω) intensity can be increased by more than two orders of magnitude as compared to an isolated graphene layer.PACS numbers: 78.67. Wj, 42.65.Ky, 73.50.Fq It was theoretically predicted [1] that, due to the "ultra-relativistic", massless energy dispersion of graphene electrons,it should demonstrate a strongly nonlinear electrodynamic response; here v F ≈ 10 8 cm/s is the Fermi velocity of graphene and k is the electron wave-vector. Physically, this is due to the absence of inertia of graphene electrons: According to (1), electrons can move only with the velocity v F in any directions, therefore, being placed in the oscillating external electric field they have to "instantaneously" change their velocity from +v F to −v F in the return points. This leads to the emission of radiation at higher (multiple) frequencies as well as to other nonlinear phenomena. The efficiency of the nonlinear effects in graphene was predicted to be many orders of magnitude larger than in many other nonlinear materials [1].Experimentally, a strong nonlinearity of the graphene response has been confirmed at microwave [2] and optical [3] frequencies. Further experimental studies of different nonlinear electrodynamic effects in graphene can be found in Refs. [4][5][6][7][8][9][10][11]. A quasi-classical theory of the nonlinear electrodynamic response of graphene, which is valid at relatively low (microwave, terahertz) frequencies hω ≪ 2E F , has been developed in Refs. [1,[12][13][14][15][16][17][18]; here E F is the Fermi energy. This theory takes into account only the intra-band electronic transitions and ignores the inter-band ones. More general, quantum theories, which take into account both contributions, have been recently proposed in Refs. [19][20][21][22][23][24]. It was shown that, apart from a strong resonance at low (hω ≪ 2E F ) frequencies, the third-order nonlinear conductivity σ αβγδ (ω 1 , ω 2 , ω 3 ) demonstrates a number of resonances at the frequencies corresponding to the one-, two-and three-photon interband absorption.In Refs. [19][20][21][22][23][24] the third-order nonlinear response functions of graphene have been calculated for a single, freely hanging in vacuum (or in air) mono-atomic graphene layer. In reality graphene lies of a dielectric substrate (of thickness d). In this Letter we study the influence of the dielectric environment on the efficiency of the third harmonic generation and show that, depending on the ratio d/λ ω , as well as on physical properties of layers supporting graphene, the output third harmonic intensity can be both several orders of magnitude smaller and several orders of magnitude larger than in the isolated graphene layer. A proper choice of the geometrical parameters and physica...

Quantum theory of the third-order nonlinear electrodynamic effects of graphene

“…If graphene lies on a dielectric substrate metalized from the back side the intensity of the third harmonic emitted from such a structure can be more than two orders of magnitude larger, see Ref. 53 .…”

Optical Kerr effect in graphene: Theoretical analysis of the optical heterodyne detection technique

“…The influence of different types of the substrate resonances (Fabry-Pérot, optical phonon resonances) on the third harmonic generation effect was comprehensively studied in Refs. 51,52 ; the role of these resonances in the Kerr response can be understood similarly.…”

Nonlinear optics of graphene and other 2D materials in layered structures

“…It has been pointed out that for weakly coupled 2D layers the atomic spacing between the layers should guarantee coherent radiation from the layers [8,60]. Despite many investigations using standard software [22,61,62], and nonlinear boundary conditions [63][64][65][66], there is still no systematic treatment of the response of graphene, taking into account its 2D nature, which could later be generalized to treat multilayer samples.…”