负折射率介质的特性研究
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摘要
负折射机制分为两种，局域共振机制和非局域布拉格散射机制。电磁波在多数左手材料中的负折射属于局域共振机制，在光子晶体中，布拉格散射机制起主要作用。局域共振机制不要求周期结构的晶格常数与电磁波的波长相比拟，周期结构的单元可以远小于波长，因此左手材料可以被视为均匀介质。而布拉格机制要求晶格常数与电磁波的波长相比拟，对于THz以及光波段，晶格常数在纳米量级以下，对工艺要求非常高，因此绝大多数光子晶体为各向异性材料，不能用有效介质理论来处理。如果将左手材料定义为介电常数和磁导率同时为负的材料，则光子晶体不属于左手材料，通过对光子晶体的结构及光子能带进行合理设计，也可以实现所需频段的负折射。本论文从研究左手材料的负折射特性着手，延伸到光子晶体中，对两类介质中的负折射机理进行深入剖析，为左手材料和负折射光子晶体在集成光学中的应用提供理论和仿真实验依据。论文的主要工作和创新点包括以下几个方面：
（1）利用有效介质理论对左手材料中的负折射现象进行了深入的研究，提出了负折射是负相速度的结果，与平面波的空间调制有关，与时间调制无关。代表时间调制特性的单频率脉冲波前和多频率干涉波前发生正折射维持了因果关系。推导了有损耗介质中的后向波条件，即同时满足(?)。在各向同性介质中，只存在两种可能情况，即群速度vg平行于波矢k，或vg与k反平行。
（2）证明了两个半无限大非磁性均匀介质界面上表面波存在的条件，即ε1ε2<0且ε1+ε2<0。若同时满足ε2<0和μ2<0，表面波可以既为TM模式又为TE模式。证明了改变右手介质或左手介质的结构参数，可以调制表面波透明窗口的大小，还可以改变TE和TM模式表面波的方向。建立了表面波群速度的理论模型，证明了表面波的波矢与坡印廷矢量可以反平行。仿真验证了通过设计材料参数可以控制表面波的传输方向。对均匀介质平板中的TM模式和TE模式场分布进行计算，得到了TM和TE模式表面波的色散关系，得到了平板中对称模式与非对称模式的场分布，分析了平板参数对这两种模式的影响，计算了介质平板的传输系数与反射系数，证明了左手介质平板为倏逝波提供了一个放大因子exp(k2ω2c2xd)，给出了平板系统中倏逝波的场强分布。对于介质平板厚度d→+∞的极限情况下倏逝波的衰减系数进行修正。
（3）将光子晶体中的负折射分为三类，即有效负折射率负折射、全角负折射，和由于特殊的等频线形状引起的负折射。利用正三角形晶格光子晶体的第二能带实现了有效负折射率负折射。提出了理论计算有效参数的方法，得到了有效折射率、有效介电常数与有效磁导率与频率的关系曲线。分析了有效负折射率光子晶体平板的成像特性，验证了入射角度对优先取向波导效应的影响。利用正方形晶格光子晶体的第一能带实现了全角负折射，证明了全角负折射光子晶体比有效负折射率光子晶体更利于实现超透镜现象。对菱形晶格光子晶体的第一、二频带进行分析，仿真验证了不同频率的光分别入射到沿Г-T和Г-M的界面方向均可以发生负折射现象，实现了两个界面方向的超棱镜效应。
（4）通过对半无限大光子晶体能带结构进行设计，仿真实现了对表面波方向的控制。对光子晶体的近场放大机理进行了理论分析，证明了倏逝波的放大依赖于表面波的谐振耦合机制，提出了光子晶体超透镜与均匀左手介质超透镜最大的不同之处在于光子晶体的周期对可放大的倏逝波横向波矢量加载了一个截止上限。证明了能带曲线中表面模式色散曲线的平坦度体现了倏逝波的放大倍数，光子晶体超透镜的分辨率仅与表面周期大小有关，设计了边缘调制的正方形晶格光子晶体超透镜，光斑的半峰值宽度为0.35λ，突破了衍射极限。
Negative refraction mechanism can be divided into two types, namely localresonance and non-local Bragg scattering. The former doesn’t impose a limit on thesizes of the lattice constants, the period of the structure could be much smaller than theworking wavelength. Left-handed material (LHM) with simultaneously ε<0and μ<0,which belongs to this mechanism, is approximated to be a isotropic homogeneous media.Photonic crystals (PhCs) are inhomogeneous materials whose lattice constants are inorder of the wavelength of light and Bragg scattering strength of each scatter is strong.The propagation of light waves inside such lattices will be modified by the photonicbandgap, and negative refraction will be realized in particular frequency ranges by finedesign the band structure. Currently negative refraction media promise a lot of interestsby the scientific community and are engineered materials functionality for applications.In this work, we study in detail the issue of negative refraction mechanism in bothisotropy and anisotropy metamaterials, typically in LHM and PhCs, the main works areas follows:
(1) By using the effective media theory, negative refraction phenomena in LHM isinvestigated in detail. It is emphasised that in this kind of isotropic media backwardwaves will appear because of the Poynting vector direction is antiparallel to the phasevelocity direction and negative refraction is a consequence of a negative phase velocity.It has been investigated that the group velocity is always positive without the need ofconsideration of the refractive index. It is point out that the spatial aspect of an incidentwave refracts negatively, and the temporal nature of the wave refracts positively whichpreserves the Causality. It has been confirmed that ε′<0and μ′<0is only anecessary condition for the existence of backward wave, when losses are present,(?)is also required.
(2) The richness properties of the surface waves generated at the surfaces ofnegative refraction index media is minutely elucidated. By analysising the dispersioncurves of TE and TM surface modes in the single interface configuration, it is fund thatwhen conditionsε1ε2<0andε1+ε2<0are satisfied, there exists surface wave on the interface between the air and semi-infinite non-magnetic media. Media withsimultaneouslyε2<0andμ2<0can support surface wave with both TE and TMpolarization. An interesting idea of controling the group velocity of the surface waves ispresented. Dispersion relationships of symmetric and anti-symmetric surface modesdistribution in the LHM slab interfaces configuration are discussed in detail, whichprovided an amplification factor ofexp(k2ω2c2xd)for the evanescent wave incidentfrom air, and the field distribution of the evanescent wave in the slab system is alsopresented. The attenuation factor of the evanescent wave is modified in the condition ofd→∞.
(3) Negative refraction effects of light propagating in two-dimensional square,hexagonal and rhombus lattice silicon PhCs are demonstrated. Theoretical analyses andnumerical simulations are presented. Three different kinds of negative refractions,namely, negative refraction with effective negative index, all-angle negative refraction(AANR) without a negative index, and negative refraction as a result of the complexequal-frequency contour (EFC) shapes have been verified and compared. It is fund thatlight focusing effect occurs in both effective negative index PhC slab and AANR PhCslab. The negative refraction in former system has high angular dependence. The AANRPhC slab is more suitable for high-resolution superlensing application. Thepreferential-direction waveguide effect in hexagonal lattice PhCs and superprism effectof both Г-T and Г-M interface direction in rhombus lattice PhCs are also investigated.
(4) The principles of subwavelength imaging in PhCs slab have been explained indetail. Propagation wave can focal in AANR PhCs without an effective negative index.For evanescent wave with large transverse vector component, PhCs permit existence ofspactial bound surface states which can used to amplify the evanescent wave thatexponentially decay in air. It is imposed an upper cutoff for superlensing using PhCs.The resolution limit of a PhC superlens is derived, and a high resolution PhCs superlensis designed, a full width at half maximum (FWHM) of0.35λ is obtained on the imageplane.
（1）利用有效介质理论对左手材料中的负折射现象进行了深入的研究，提出了负折射是负相速度的结果，与平面波的空间调制有关，与时间调制无关。代表时间调制特性的单频率脉冲波前和多频率干涉波前发生正折射维持了因果关系。推导了有损耗介质中的后向波条件，即同时满足(?)。在各向同性介质中，只存在两种可能情况，即群速度vg平行于波矢k，或vg与k反平行。
（2）证明了两个半无限大非磁性均匀介质界面上表面波存在的条件，即ε1ε2<0且ε1+ε2<0。若同时满足ε2<0和μ2<0，表面波可以既为TM模式又为TE模式。证明了改变右手介质或左手介质的结构参数，可以调制表面波透明窗口的大小，还可以改变TE和TM模式表面波的方向。建立了表面波群速度的理论模型，证明了表面波的波矢与坡印廷矢量可以反平行。仿真验证了通过设计材料参数可以控制表面波的传输方向。对均匀介质平板中的TM模式和TE模式场分布进行计算，得到了TM和TE模式表面波的色散关系，得到了平板中对称模式与非对称模式的场分布，分析了平板参数对这两种模式的影响，计算了介质平板的传输系数与反射系数，证明了左手介质平板为倏逝波提供了一个放大因子exp(k2ω2c2xd)，给出了平板系统中倏逝波的场强分布。对于介质平板厚度d→+∞的极限情况下倏逝波的衰减系数进行修正。
（3）将光子晶体中的负折射分为三类，即有效负折射率负折射、全角负折射，和由于特殊的等频线形状引起的负折射。利用正三角形晶格光子晶体的第二能带实现了有效负折射率负折射。提出了理论计算有效参数的方法，得到了有效折射率、有效介电常数与有效磁导率与频率的关系曲线。分析了有效负折射率光子晶体平板的成像特性，验证了入射角度对优先取向波导效应的影响。利用正方形晶格光子晶体的第一能带实现了全角负折射，证明了全角负折射光子晶体比有效负折射率光子晶体更利于实现超透镜现象。对菱形晶格光子晶体的第一、二频带进行分析，仿真验证了不同频率的光分别入射到沿Г-T和Г-M的界面方向均可以发生负折射现象，实现了两个界面方向的超棱镜效应。
（4）通过对半无限大光子晶体能带结构进行设计，仿真实现了对表面波方向的控制。对光子晶体的近场放大机理进行了理论分析，证明了倏逝波的放大依赖于表面波的谐振耦合机制，提出了光子晶体超透镜与均匀左手介质超透镜最大的不同之处在于光子晶体的周期对可放大的倏逝波横向波矢量加载了一个截止上限。证明了能带曲线中表面模式色散曲线的平坦度体现了倏逝波的放大倍数，光子晶体超透镜的分辨率仅与表面周期大小有关，设计了边缘调制的正方形晶格光子晶体超透镜，光斑的半峰值宽度为0.35λ，突破了衍射极限。
Negative refraction mechanism can be divided into two types, namely localresonance and non-local Bragg scattering. The former doesn’t impose a limit on thesizes of the lattice constants, the period of the structure could be much smaller than theworking wavelength. Left-handed material (LHM) with simultaneously ε<0and μ<0,which belongs to this mechanism, is approximated to be a isotropic homogeneous media.Photonic crystals (PhCs) are inhomogeneous materials whose lattice constants are inorder of the wavelength of light and Bragg scattering strength of each scatter is strong.The propagation of light waves inside such lattices will be modified by the photonicbandgap, and negative refraction will be realized in particular frequency ranges by finedesign the band structure. Currently negative refraction media promise a lot of interestsby the scientific community and are engineered materials functionality for applications.In this work, we study in detail the issue of negative refraction mechanism in bothisotropy and anisotropy metamaterials, typically in LHM and PhCs, the main works areas follows:
(1) By using the effective media theory, negative refraction phenomena in LHM isinvestigated in detail. It is emphasised that in this kind of isotropic media backwardwaves will appear because of the Poynting vector direction is antiparallel to the phasevelocity direction and negative refraction is a consequence of a negative phase velocity.It has been investigated that the group velocity is always positive without the need ofconsideration of the refractive index. It is point out that the spatial aspect of an incidentwave refracts negatively, and the temporal nature of the wave refracts positively whichpreserves the Causality. It has been confirmed that ε′<0and μ′<0is only anecessary condition for the existence of backward wave, when losses are present,(?)is also required.
(2) The richness properties of the surface waves generated at the surfaces ofnegative refraction index media is minutely elucidated. By analysising the dispersioncurves of TE and TM surface modes in the single interface configuration, it is fund thatwhen conditionsε1ε2<0andε1+ε2<0are satisfied, there exists surface wave on the interface between the air and semi-infinite non-magnetic media. Media withsimultaneouslyε2<0andμ2<0can support surface wave with both TE and TMpolarization. An interesting idea of controling the group velocity of the surface waves ispresented. Dispersion relationships of symmetric and anti-symmetric surface modesdistribution in the LHM slab interfaces configuration are discussed in detail, whichprovided an amplification factor ofexp(k2ω2c2xd)for the evanescent wave incidentfrom air, and the field distribution of the evanescent wave in the slab system is alsopresented. The attenuation factor of the evanescent wave is modified in the condition ofd→∞.
(3) Negative refraction effects of light propagating in two-dimensional square,hexagonal and rhombus lattice silicon PhCs are demonstrated. Theoretical analyses andnumerical simulations are presented. Three different kinds of negative refractions,namely, negative refraction with effective negative index, all-angle negative refraction(AANR) without a negative index, and negative refraction as a result of the complexequal-frequency contour (EFC) shapes have been verified and compared. It is fund thatlight focusing effect occurs in both effective negative index PhC slab and AANR PhCslab. The negative refraction in former system has high angular dependence. The AANRPhC slab is more suitable for high-resolution superlensing application. Thepreferential-direction waveguide effect in hexagonal lattice PhCs and superprism effectof both Г-T and Г-M interface direction in rhombus lattice PhCs are also investigated.
(4) The principles of subwavelength imaging in PhCs slab have been explained indetail. Propagation wave can focal in AANR PhCs without an effective negative index.For evanescent wave with large transverse vector component, PhCs permit existence ofspactial bound surface states which can used to amplify the evanescent wave thatexponentially decay in air. It is imposed an upper cutoff for superlensing using PhCs.The resolution limit of a PhC superlens is derived, and a high resolution PhCs superlensis designed, a full width at half maximum (FWHM) of0.35λ is obtained on the imageplane.
引文
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