隐身衣、旋转衣和声子晶体

论文中英文摘要
作者姓名：陈焕阳
论文题目：各向异性材料中的波：隐身衣、旋转衣和声子晶体
作者简介：陈焕阳，男，1983年08月出生，2005年09月师从于上海交通大学马红孺教授，于2008年06月获博士学位。

2009年09月起任苏州大学物理科学与技术学院特聘教授。

中文摘要
转换介质理论是基于麦克斯韦方程组在伽利略变换下的坐标协变性得到的，这个理论也称为“转换光学”,它有点类似爱因斯坦的广义相对论，只是广义相对论把质量和空间进行等效，这里把电容率、磁导率张量和空间进行等效。

如果光在某一个想象的空间中按照我们所设计的路线传播，利用这种等效关系，我们就可以设计一种“转换介质”,使得光在它里面的传播路线和在假想空间里的传播一样，这就达到了对光传播的自由控制。

通过一种特殊的坐标变换，人们提出了隐身衣的概念。

本文讨论波在各向异性材料中的传播，着重研究转换介质的特性。

包括隐身衣的色散特性，新型的转换介质――旋转衣的性质。

附带研究了声子晶体的一些特性。

全文分为六章。

论文第一章介绍了研究背景和当前国内外的研究进展。

第二章介绍转换介质理论，并由此提出了双极隐身衣。

然后利用声波方程和直流电导方程的一一对应关系给出了声波的转换介质理论，并由此设计了三维隐声衣。

在第三章的第一部分里，我们根据转换介质的概念证明了隐身衣只能在单一频率下工作。

然后讨论了在适当牺牲隐身效果的前提下拓宽隐身衣频宽的可能性。

我们发现因果率的要求给转换介质的系统参数强加了许多严格的限制。

但是通过一定形式的“弱化”,可以设计出一个色散隐身衣，使得在一个有限频段内可以降低物体的散射。

另外，还给出了一个简单的不等式用于限制工作频率的频宽。

在第三章的第二部分，我们首先利用哈密尔顿几何光学的射线追踪法发现了三维隐身衣本身所具有的无限大时间延迟，然后结合群速度分布或能量传输速度分布，进一步证实了这种效应。

在第四章里，我们首次给出了一种特殊的转换介质，旋转衣。

这种旋转衣的有效电容率和磁导率可以通过引入旋转变换得到。

在旋转衣内部的观察者看来，外面的信息好像转了一个固定的角度入射进来。

我们发现旋转衣本身是一个在圆周切线方向的波的移动装置。

而这种波的移动装置可以作为基本单位组合成许多其它器件，如波的分束装置，波的组合装置，一维隐身衣等等。

这种基本单位可以通过由两种均匀介质交替所组成的斜的层状结构来实现。

基于同样的原理，就可以利用一些扇状层状结构来实现完美旋转衣。

在此基础上，我们提出了一种“简化的旋转衣”,这种简化的旋转衣本身具有一定的散射，但是仍然具有场的旋转效应。

在微波波段，我们在实验上实现了这样的旋转衣，并发现其旋转效应发生在很宽的频段里。

在第五章，基于简单晶格声子晶体的层多重散射理论，我们给出了复杂晶格声子晶体的
层多重散射理论，然后由此计算了NaCl 型声子晶体的声学特性。

最后，我们设计了一种手性声子晶体，并利用所推广的层多重散射理论计算了它们的特性。

投射率曲线和相应的能带结构表明，这种结构拥有较大的偏振禁带。

手性结构破坏了晶
体的对称性，从而使原先简并的横波模分裂成一对左右旋模式。

低频时的这种分裂在大填充
率系统较为明显。

如果这种手性结构带有强共振单元，将会产生弹性波的负群速度效应，共
振越强，负群速度的能带越宽。

关键词：转换介质，隐身衣，隐声衣，旋转衣，声子晶体，手性声子晶体，偏振禁带
Waves in anisotropic materials: Invisibility Cloaking, Rotation Cloaking
and Phononic Crystal
Chen Huanyang
ABSTRACT
The theory of t ransformation media, or “the transformation optics”, is proposed based on the covariant invariance of Maxwell's equations under the Galilean coordinate transformation. It is to some extent similar to Einstein's general theory of relativity. The general theory of relativity equates the mass and the space while the theory of transformation media puts the tensors of permittivity and permeability to be equivalent to the space. For the case that the light propagates along our designed route in an imaginary space, by the concept of the e quivalent principle, a kind of “transformation media” can be obtained inside which the light will propagate with the same route as the one in the imaginary space. Then we can substantiate the demand of freely controlling the propagation of light. For example, the concept of the invisibility cloaking would be suggested by one of the “transformation media”.
The thesis focuses on the propagation of wave in an anisotropic material, especially of those transformation media. The dispersive properties of cloaks area intensively studied and a new transformation media, the rotation cloak, is suggested, and some of its interesting properties are presented. In addition, the properties of the phononic crystals are studied.
There are six chapters in the thesis. The background of the thesis is introduced in Chapter one. In Chapter two, we mainly present the theory of the transformation media, based on which the
bipolar cloak is proposed. Then from the one to one mapping of the acoustic equation and the direct current conductivity equation, the theory of acoustic transformation media is built up, which leads to the concept of the three dimensional acoustic cloaking. In the first part of Chapter three, using the idea of a transformation medium, a designation of a cloak is shown that it makes a domain invisible to one target frequency. The possibility of extending the bandwidth of such a cloak is examined. The causality requirements impose severe constraints on the system parameters of the transformation medium, and a specific form of “reduction”can be helpful to create a cloak that offers a reduced cross section in a finite frequency range. A simple inequality is derived to limit the bandwidth of operation. In the second part of chapter three, the three dimensional ideal cloak is found to have infinite time delay from the ray tracing of the Hamiltonian geometry optics, and this effect is also examined by the calculated distributions of group velocity or energy transport velocity. In Chapter four, we propose a spectacular transformation media, the rotation cloak. The effective permittivity and permeability of the rotation cloak can be obtained by introducing a rotational mapping of coordinates. Inside the enclosed domain of rotation cloak, the information from the outside will appear as if it is coming from a different angle. Thus the rotation cloak can be regarded as a wave shifter in the polar coordinate. The wave shifter itself can be used as a universal element to build a myriad of interesting functional optical components such as the wave splitters, wave combiners, one dimensional cloak, etc. In addition, the element - wave shifter, can be realized by using an oblique layered system with two kinds of isotropic materials. Based on the same principle, we design the perfect rotation cloak by using a kind of fan-shaped layered system. Then the general form, “the reduced rotation cloak”, is proposed. This reduced version has the scattering itself, at the same time it inherits the field rotation effect. Such a rotation cloak is observed in a microwave experiment, and the rotation effect is found in a broad band of frequency range. In Chapter five, from the original theory for the phononic crystal with simple lattice, we propose a new kind of layer multiple-scattering theory which can handle the phononic crystal with complex lattice. Then the properties of NaCl-type phononic crystals are derived. Finally, we design the chiral phononic crystals, and their properties by the same extended theory are investigated. The transmittance curves and the corresponding band structures show that this kind of structure possesses significant polarization gaps. The chiral structures break the symmetry so that the degenerate transverse modes split into a pair of right-hand polarized mode and left-hand polarized mode. The polarization splitting in the low frequency range is enhanced in systems of large filling ratios. We also demonstrate that chiral structures containing strongly resonant units which can induce negative group velocity in elastic waves, and stronger resonance brings a wider band of negative group velocity. Chapter six concludes my work and outlines some future attempts.
Key words: transformation media, invisibility cloaking, acoustic cloaking, rotation cloaking, phononic crystal, chiral phononic crystal, polarization gaps。