Research On Light Wave Propagation Properties In Honeycomb Lattice And Gradient Index Potentials

The research on the transmission dynamics of light in different media has always been a hot spot in optics.In a specific carrier and medium,different external conditions can be imposed or changed to flexibly adjust and control the transmitted beam to achieve the desired modulation purpose,which has important application value in optical communication,optical interconnection and other fields.Honeycomb lattice,as an ideal carrier for the study of light propagation,its special symmetrical structure leads to the existence of special energy band structure of Dirac point,which will produce the phenomenon of conical diffraction of light.The modulation of conical diffraction is an important research topic.At the same time,when nonlinearity is added to the fractional Schr?dinger equation without potential,there are few studies on the dynamics of light propagation in it.In addition,the application of the refractive index gradient potential provides another effective way to control the beam propagation characteristics.This paper is based on the Schr?dinger-like equation or the fractional Schr?dinger equation used to describe the propagation of light.By changing the potential energy conditions in the Schr?dinger-like equation and the fractional Schr?dinger equation and controlling the introduction of different types of nonlinearities or different forms of incident modes,further relevant modulation and research on the transmitted light are carried out.The specific content can be summarized into the following three parts:The generation and destruction of asymmetric conical diffraction at Dirac point in honeycomb lattice are studied.By changing the rotation angle of the three potential vectors,the Dirac eigenmodes in the honeycomb lattice are excited,symmetric conical diffraction are obtained.With the change of the rotation angle of the position vector,a dark notch appears in the conical diffraction pattern and will move clockwise or anticlockwise along the outer edge of the bright ring.In addition,it is also demonstrated that both Kerr and saturated nonlinearities can cause the triangular deformation of the conical diffraction pattern.Under the same nonlinearity,two identical wave packets centered on different Dirac points can evolve to triangular rings with opposite directions.For the self-focusing and self-defocusing nonlinear cases,a dark notch at a point of triangular appears and moves in a clockwise or anti-clockwise way.The nonlinear propagation dynamics of Gaussian beams in fractional Schr?dinger equation are studied.When nonlinearity is introduced into fractional Schr?dinger equation without an external potential,the evolution behaviors of incident Gaussian beams are modulated regularly and some novel phenomena arise.It is found that the type or intensity of nonlinearity introduced in fractional Schr?dinger equation can produce significant modulation effect on the evolution behavior of incident Gaussian beams.In the weak nonlinear region,the splitting angle of one-dimensional incident beam will be flexibly modulated to a larger or smaller value.In addition,when the selffocusing intensity is moderate,the beam energy is highly concentrated,forming breathing soliton structure.In the two-dimensional case,Kerr or saturated nonlinearity modulates the energy to the middle or edge in a certain nonlinear region,corresponding to the decrease or increase of the conical diffraction radius.And there are two evolution periods under the saturated self-focusing nonlinearity.The effective manipulation of Talbot image using dynamic gradient index potential is studied theoretically and numerically.The exact recurrent solutions for two kinds of incident periodic modes are obtained.In each case,by designing a dynamic gradient refractive index potential,the periodic initial light intensity distribution can propagate along a predetermined trajectory and form the same repeated self-imaging as the incident field.In addition to the lateral displacement caused by the predefined trajectory,the integer or fractional Talbot image self-occurrence phenomenon is clearly observed at a specific Talbot distance.Finally,the results are extended to the twodimensional case,and the numerical simulation is completely consistent with the theoretical expectation.