基于界面反射本征态的外尔超材料偏振特性研究

Polarization characteristics of Weyl metamaterial based on interfacial reflection eigenmodes

近年来,因为体现出诸多异常拓扑现象,外尔超材料一直备受关注。此前的研究表明外尔超材料的反射场具有偏振转换特性。然而,关于反射本征态的讨论却不够深入,原因在于入射电磁场与反射电磁场处于不同的态空间,给分析计算带来了诸多不便。文中从电场为极矢量、磁场为轴矢量的基本特性出发,通过引入镜像算符将入射态空间与反射态空间联系起来,最后对全反射界面的本征态进行了明确的定义。依据上述定义,笔者对外尔超材料的反射本征态进行了计算,结果显示:在入射电磁场为线性偏振的前提下,外尔超材料的反射本征态均为线性偏振态。反射本征态为线性偏振态,这一性质是由系统能量守恒与洛伦兹互易性共同保证的。经过进一步探讨发现,当动量空间的路径绕外尔点一圈时,本征态的电磁场方向旋转角度为90°。这也意味着如果电磁场要回到初始状态,动量空间的路径需绕外尔点4圈。这一现象的拓扑结构可以类比为四阶莫比乌斯环。此外,依据反射本征态为互相垂直的线性偏振态这一特性,笔者定义了旋转角度θ用以描述本征态朝向的变化规律。值得一提的是,这种计算方法不仅仅适用于外尔超材料,满足条件的全反射界面的反射本征态均可以依照上述定义与方法进行分析求解。

Objective As a massless relativistic fermion,Weyl fermions play a crucial role in quantum theory and the standard model.To mimic the physical properties of Weyl fermions,constructing Weyl points in momentum space needs breaking the inversion or time-reversal symmetry,and those Weyl points are topologically protected.Weyl points possess unique characteristics,including positive and negative chiralities corresponding to the sources and sinks of the Berry curvature,respectively.Consequently,Weyl points are regarded as magnetic monopoles in momentum space.Weyl points have also attracted significant attention due to their scattering and transport properties.For example,Weyl semimetal exhibits chiral zero modes and corresponding chiral magnetic effects in condensed matter physics.It is worth mentioning that Weyl points are widely acknowledged as singularities in the reflection phase,but there has been relatively little study on the reflection eigenmodes of the interface between a Weyl metamaterial and air.Methods This article utilizes the ideal Weyl metamaterial as a research platform,which offers a relatively large frequency range for exploring the fundamental properties of Weyl points.By applying the effective media theory,the constitutive relation of saddle-shaped metallic structures(Fig.1)can be described concisely.Additionally,the band structure of Weyl metamaterial can be calculated using simulation software.Results and Discussions For a totally reflected interface,the wave vectors of the incident and reflected state space are different.Before solving for the reflection eigenmodes,it is necessary to define the basis separately for the two different state spaces of the incident and reflected electromagnetic fields.Since the electric field is a polar vector and the magnetic field is an axial vector,the mirror operation introduces different responses in the directions perpendicular and parallel to the interface.By applying the mirror operation,we can connect the incident and reflected state spaces,allowing us to solve for the eigenmodes using conventional methods and determine their matrix representation.Given this definition,energy conservation ensures that the reflection coefficient matrix must be unitary,while Lorentz reciprocity guarantees that the reflection coefficient matrix must be symmetric.Such a unique reflection coefficient matrix must have real eigenstates,resulting in both reflection eigenmodes being linearly polarized.Using this method to analyze the reflection eigenmodes of Weyl metamaterials,it is found that all reflection eigenstates are linearly polarized.The two eigen electromagnetic fields are perpendicular to each other,forming a cross shape.As the scanning path changes continuously in the Brillouin zone,the orientation of the cross shape also varies.When the scanning path surrounds the Weyl points in momentum space,the eigen field(cross shape)undergoes an additional phase shift ofπ/2.A quadratic Möbius strip can describe this feature.Conclusions In the case of total reflection at an interface,the incident state space and the reflected state space are bridged via a mirror operator.Combining the interface reflection operator with the mirror operator allow us to define the eigenstates of the total reflection interface.When the incident basis is chosen as linear polarized,this unique definition results in the reflection eigenstates of the total reflection interface being linear polarization modes protected by energy conservation and Lorentz reciprocity.Taking the Weyl metamaterial as an example,even if the interface reflection between the Weyl metamaterial and air exhibits polarization conversion characteristics,given this unique definition,the eigenmodes of the interface reflection can be obtained as linear polarized.Since the two linear polarization eigenstates are perpendicular to each other,a rotation angle could be defined to characterize the change in the rotation angle of the eigenmode.In addition,when scanning a loop path around the Weyl points in momentum space,the eigen field acquires an additional phase shift ofπ/2,which can be described using a quadratic Möbius strip.