Waves In Anisotropic Materials: Invisibility Cloaking, Rotation Cloaking And Phononic Crystal

The theory of transformation 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 equivalent 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.