Nonlinear Surface Waves And Guided Modes Near The Dirac Point In Negative-zero-positive Index Metamaterials

In graphene, the conduction and valence bands touch each other at DP with a double-cone structure. Near DP, the dispersion is linear with two branches. The valence electron dynamics in graphene is governed by a massless Dirac equation. Due to the novel dynamics of valance electronics, Graphene exhibits many unique electronic properties , including half integer and unconventional quantum Hall effect, observation of minimum conductivity, and Klein tunneling.The optical-like behaviors of electron waves in graphene have also drawn considerable attention recently, such as focusing, collimation , Bragg reflection, and Goos-Hanchen effect. On the other hand, the DP with a double-cone structure in photonic crystals for the Bloch states is found from the similarity of the photonic bands of the two-dimensional photonic crystals with the electronic bands of solids. Several novel optical transport properties near the DP have been shown, such as conical diffraction, a'pseudodiffusive scaling'.Recently, a novel metamaterial with simultaneous negative permittivity and negative permeability was demonstrated in 2000. The unique properties of this metamaterial, such as negative refraction and negative phase velocity, enriched the field of research on waveguide optics. Motivated by the realization of the Dirac point (DP) with a double-cone structure for optical field in the negative-zero-positive index metamaterial (NZPIM),many novel properties, such as pseudodiffusive property, Zitterbewegung effect, GH effect and nonlinear surface polarization have been investigated near DP in NZPIM. In this thesis, we will study the nonlinear surface waves near the DP in a NZPIM and the guided modes in nonlinear NZPIM waveguide.The main contents include the following three aspects:1. The nonlinear surface waves localized at the single interface separating a nonlinear conventional dielectric medium and a NZPIM are studied. It is found that, near the DP, there is a forbidden band for the nonlinear surface waves and the nonlinear surface waves also have a frequency threshold. In contrast to the negative permeability of NZPIM in some frequency regions, we show that the nonlinear surface waves are always forward with positive power fluxes. The frequency property, the group velocity and the power flux of the surface waves are also discussed in detail.2. We make theoretical investigations of the guided modes in the NZPIM waveguide near the DP by using the graphical method. Due to the linear Dirac dispersion, the fundamental mode is absent when the angular frequency is smaller than the DP, while the behaviors of NZPIM waveguide are similar to the conventional dielectric waveguide when the angular frequency is larger than the DP. The unique properties of the guided modes are analogous to the propagation of electron waves in grapheme waveguide, corresponding to the classical motion and the Klein tunneling.3. The nonlinear guided modes near the DP are theoretically investigated in non-linear NZPIM waveguide. When the nonlinearity is self-focusing, there exists an asymmetric forbidden band near DP which can be modulated by the strength of the nonlinearity. However, the self-defocusing nonlinearity can completely eliminate the asymmetric band gap. We also study the nonlinear surface waves in such nonlinear NZPIM waveguide.