Research Of Nonlinear Properties Of Waveguide Arrays With Negative Index Channels

The metamaterials(MM)is a kind of artificial synthetic materials,which have enabled the realization of unconventional electromagnetic properties not found in nature.Negaitve-index metamaterials are characterized by simultaneously negative dielectric permittivity and magnetic permeability.Such media are usually termed left-handed materials,since negative sign of the index of refraction leads to the left-hand orientation of the fundamental tippet E,H,k.This artificial electromagnetic media that are structured on the subwavelength scale,were initially suggested for the negative-index “superlens”.Later metamaterials became a paradigm for engineering electromagnetic space and controlling propagation of waves: the field of transformation optics was born.For some years,the unique electromagnetic properties of metamaterials provoke us to rethink the established rules of optics in both the linear and nonlinear regimes.Recently,the research agenda is now shifting towards achieving tunable,switchable,nonlinear and sensing functionalities.Therefore,it is timely to discuss the emerging field of metadevices,where we define the devices as having unique and useful functionalities that are realized by structuring of functional matter on the subwavelength scale.An array of waveguides is a basic system for a majority of nonlinear phenomena in discrete optic,which provide a convenient setup for experimental investi gation of periodic nonlinear systems.Their ability to modify diffraction and refraction laws has also been regarded as a powerful tool in designing novel micro-structured photonic devices.A wide variety of new phenomena such as optical Bloch oscillations,modulation instability,discrete solitons,Zener tunneling,diffraction management,dynamic localization,Rabi oscillations,and so on,has been studied and observed in waveguide arrays.Moreover,diffractive properties of light in modulated arrays of evanescently-coupled waveguides are strongly affected by the complex spatial dispersive properties of the medium,with the relevant possibility of controlling the magnitude and sign of diffraction.Lattice engineering enables us to mold,in a rather flexible way,the flow of discretized light,hence providing altogether new opportunities for applications.In recently,researchers use negative index metameterials to design optical waveguide arrays.In further,nonlinear optical waveguide arrays(NOWA)with positive and negative index will have many useful applications.However,only few nonlinear optical phenomemen of such NOWA have been studied.In this paper,we mainly studied the nonlinear optical phenomenon of waveguide arrays with positive and negative index,and the research results are listed below:Firstly,the modulation instability(MI)in second-harmonic generation in metamaterials(MMs)with quadratic nonlinearity is studied.Within a in a certain frequency range,we supposed that FF wave locates at the negative-index regime and SH wave locates at the positive-index regime.The analytical expressions for MI gain spectra are obtained,from which the generation condition of MI and the respective roles of dispersion,group velocity mismatch and phase mismatch in MI are identified.It is found that the region of phase mismatch where the MI can take place in MMs is in contrast with its counterpart in conventional materials due to the negative wave vector direction of FF wave.The MI gains mainly rely on the phase mismatch.Since magnetic permeability is dominant for negative refraction,so the refractive index difference of the FF and SH wave can fluctuate in a large range in MMs.While the frequency is near the critical point,the refractive index difference of the FF and SH wave is small,so the phase mismatch is small and the MI gain is low.However,while the frequency is far away from the critical point,the refractive index difference of the FF and SH wave becomes large,this leads to a large phase mismatch,and the giant gain with more sidelobes can be obtained for the large phase mismatch occurring in MMs with quadratic nonlinearity.Secondly,inspired by the propagation properties of the waves in the metamaterials,the dispersion relation and modulation instability(MI)of array of alternating positive-index material(PIM)and negative-index material(NIM)waveguides are investigated.Due to the negative index,the transverse wave vector of the stability wave solution has important effect on the dispersion relation and MI regime.The dispersion relation is linear for staggered case,yet it is nonlinear for unstaggered case.The MI region for staggered case is larger than the MI region for unstaggered case,and only for unstaggered case,the self-focusing or self-defocusing nonlinear property of NIM channel affects the MI gain structure.Except the nonlinear property of the NIM channels,the nonlinear interaction between the PIM-NIM channels also has great influence on MI.These findings indicate that we may manipulate MI in arrays with positive and negative index waveguides by controlling the NIM channel.Thirdly,in order to investigate the influence of the negative index metamaterials on the wave propagation in waveguide arrays.The modulation instability(MI)in zigzag waveguide arrays,which are composed of alternating one negative-index materials(NIM)waveguide and two different positive-index materials(PIM)waveguides,has been studied.It is demonstrated that the effects of nonlinearity of NIM channel and power of light on MI are similar,only affect partially the sidelobes of the MI gain spectra,and can be modulated flexibly through the variation of the nonlinear strength or focusing property of PIM channels due to the interactions between PIM-NIM channels.If the the wave vector jumps from unstaggered to staggered case,the MI region shrinks.However,while the coupling coefficient between NIM-NIM channels increases,the MI region will become wide.The study of MI in such waveguide arrays offers useful instructions for further study of solitons.Finaly,the modulation instability(MI)has been studied in one-dimensional bi-periodic optical lattice of negative index metamaterials waveguides,of which the propagation constant of adjacent waveguide is out-of-phase modulated.Compared with the dispersion curve of the convetional optical lattice with same structure,it is found that negative index cancels out the effect of the longitudinal modulation depth on the dispersion curve,the diffraction platform of the dispersion curve disappear.Within the longitudinal modulation depth,the incident power has important effect on MI.Low incident power corresponds to low gain,and the variation of the lattice vector basically does not affect MI gain;high incident power corresponds to high gain,but the variation of the lattice vector may affects MI gain dramatically.In a low gain region,it may easy to generate discrete soliton due to the coupling effect and nonlinear effect of the lattice on MI gain is opposite;but in a high gain region,the modulation range of coupling coefficient and nonlinearity coefficient that may generate discrete soltion will reduce.