Conformal Transformation Optics

The principle of transformation optics is the equivalent relationship between electromagnetic properties of materials and metric properties of space.This equivalency is similar to that between mass-energy tensors and metric properties of space in general relativity.Based on this,we can consider the curved behavior of light in inhomogeneous media as the curvature of space.On the other side,we can find a coordinate transformation to design a certain kind of materials,which builds up the relationship between the original space and the transformed space.Therefore,transformation optics provides us with a geometric perspective to understand electromagnetic properties of materials.Meanwhile,with the development of metamaterials,transformation optics also gives a new approach to design novel optical devices.However,transformed medium based on general transformation optics is quite unnatural and complicated,hence undesirable for achieving good performances even in a reduced cloak.Here,we introduce an easier method to control propagations of electromagnetic waves in two dimensions based on inhomogeneous refractive index profiles,namely conformal transformation optics.In the beginning,the range of refractive index profiles designed by conformal transformation optics is usually hard to achieve,hindering the fabrication of conformal transformation devices in practice.Besides,there is discontinuity along the branch cut in traditional conformal transformation optics,which compromises the functionalities of the original optical designs.With the spirit of conformal transformation optics,we develop a new method,known as geodesic conformal transformation optics,to obtain media with continuous refractive index profiles.There are five chapters in this thesis.In Chapter One,we summarize the relationship between optics and geometry,introduce the origin of transformation optics and transformed medium,discuss the properties of metamaterials and their effective medium theory,and sum up the history and current situation of invisibility cloaks.We give the background of this thesis in this chapter.In Chapter Two,we start from the differential geometry and transformation optics to discuss the equivalent relationship between electromagnetic properties of media and geometry.Then,by introducing Fabry-Perot resonances to transformation optics,we fabricate an electromagnetic concentrator and design other optical devices.Moreover,we prove that there are a series of two dimensional concentrators with perfect performance.In this chapter,we mainly focus on two dimensional transformation optics and extreme anisotropic medium.In Chapter Three,we derive conformal transformation optics from transformation optics,and explain the relationship between media with inhomogeneous refractive index profile and two dimensional spaces.Then we introduce some conformal transformation devices to exhibit its application.We finally elaborate a conformal lens based on a two dimensional gradient refractive index micro-structured waveguide.We observe self-focused light rays and Talbot effect in such a lens.In this chapter,we mainly focus on two dimensional conformal transformation optics and isotropic medium.In Chapter four,we systematically introduce the design and optimization of conformal invisibility cloaks based on dual logarithm conformal mapping.Then,we explain the relationship between geodesic lenses and absolute instruments.We analyze the properties of geodesic lenses both in geometry optics and in wave optics.Furthermore,we fabricate two geodesic lenses in optical spectrum,which demonstrates closed trajectories of geodesic lenses.In the end,based on the properties of geodesic lenses,we use two dimensional geodesic conformal mappings to design continuous refractive index profiles to manipulate the propagations of light.Here,we particularly dicuss its application to conformal transparency and invisibility.In this chapter,we mainly focus on geodesic conformal transformation optics and isotropic medium with continuous refractive index profile.