Photon Transmission Control Based On Kramers-Kronig Potential In Discrete Lattices

Light waves traveling through an optical interface or transmitting in an inhomogeneous medium are usually partially reflected.Wave reflection is a basic phenomenon,generally arises from a sharp change of the refractive index on a spatial scale of the order of the optical wavelength.In many applications,the reflection are bad phenomena,people have developed several ways to avoid it.For example,applying an anti-reflection coating can eliminate the reflection of the sharp interface.There are some well-known examples of non-reflecting profile,such as the hyperbolic secant profile.Generalized reflectionless potential was put forward by Kay and Moses,in 1956.Although the medium can be designed without reflection(no matter what is the angel of incidence),it does not usually are invisible.Because the homogeneity introduces some shape distortion and the change of the propagation of beam.Invisible scattering medium is locally heterogeneous,if there does not have any objects.In recent years,the wave reflection in engineered photonic structures has received a renewed interest,and new kinds of reflectionless potentials have been introduced,such as those based on parity-time symmetry,supersymmetry and spatial Kramers-Kronig relations.In such structures,the refractive index is allowed to be complex,i.e.spatial regions with optical loss or gain are introduced.Compared with Kay and Moses potential,non-Hermitian potential can be designed as unidirectionally or bidirectionally invisible rather than simply reflectionless.In all previous articles,the wave propagation was usually designed in the Helmholtz equations or stationary Schr?dinger equation,the framework of Helmholtz equations or stationary Schr?dinger equations is used to describe the scattering phenomenon of waves in continuous system.However,in several physical contexts,such as in the quantum or classical transport on a lattice or in quantum mechanical models of discretized space,wave transport can be described by the discrete version of the Schr?dinger equations better.Researchers have found that in continuous Schr?dinger equations,when the potential is static,the incident wave would be transmitted with no reflection.And when the potential is moving,its scattering properties will not change.This is mainly because the continuous Schr?dinger equation is relativistic wave equation,so under the Galilean transformation it is a constant.When considering discrete Schr?dinger equation,the phenomenon will be different.In this dissertation,we first briefly introduce some background knowledge of the reflectionless phenomenon,and introduce the derivation process of the Kramers-Kronig relations,in order to choose a right Kramers-Kronig potential in the third chapter.In the second chapter,we have a summary of predecessors' work and introduce the unidirectional reflectionless potential and bidirectional invisible potential under some conditions.In the third chapter,we take a little more complicated Kramers-Kronig potential,and give a change of the theoretical model.Through the comparison with previous work,and the summary of the reflection and transmission phenomena under different conditions and situations,we perfect the content of this part.