Designing non-scattering graded-index media

Abstract

With recent advances in metamaterials research, scientists are increasingly trying to understand how interesting wave phenomena emerge from new materials. A key branch of this subject is transformation optics: using coordinate transformations to design materials which don't scatter light, used, for example, in the design of cloaking devices. This thesis is a study of various mathematical techniques, such as transformation optics, for designing non-scattering materials. The materials studied in this thesis are characterised by their macroscopic electromagnetic material properties; their permittivity and permeability. These quantities will be assumed isotropic, and to vary smoothly in space (such media being described as `graded-index'). These inhomogeneous media can be understood as the limit of a stack of different infinitesimally thin homogeneous blocks. However, the theory of graded-index media is analytically more tractable than the theory of piecewise media than the piecewise homogeneous permittivity and permeability profiles that arise from placing blocks of different materials side by side.

The first part of the thesis introduces the necessary background for the rest of the thesis. Chapter 1 introduces the background electromagnetic theory used throughout the rest of the work and chapter 2 is a review of some of the existing literature on designing non-scattering media including an introduction to metamaterials, which are one route to realising graded-index media.

The second part of the thesis concerns the use of phase-integral methods for designing planar, reflectionless media, inhomogeneous in one dimension. Chapter 3 introduces the phase-integral method and describes how it can be used to understand reflection in the complex position plane. Chapter 4 uses the phase-integral method to derive a large family of index profiles reflectionless from one side for all frequencies of light and angles of incidence. Chapter 5 uses the phase-integral method to calculate exact reflection coefficients for some example index profiles. Chapter 6 is concerned with designing media which, in addition to being reflectionless, transmit all incident light. It is found that this perfect transmission property is exhibited by even very highly disordered media. Chapter 7 looks further at the reflectionless media designed in chapter 4, deriving a subfamily which, in addition to being non-reflecting, are also perfectly absorbing.

The third and final part of the thesis deals with the design of planar, non-scattering media which are inhomogeneous in two dimensions. One way to handle this problem is to write the electromagnetic field in terms of its amplitude and phase, and use local conservation of energy in a lossless medium to understand how the amplitude and phase are related to each other. Chapter 8 uses this idea to find explicit expressions for non-scattering index profiles, including a generalisation of an existing version of transformation optics, and the design of a `beam-shifter'. Chapter 9 uses the characteristic method to solve the relationship between amplitude and phase and how it can be used to design non-scattering profiles, such as periodic media which don't diffract. Finally, chapter 10 discusses nodes of the electromagnetic field and how they can arise from sending a wave onto a medium, in particular looking into the possibility of diffraction of a plane wave from a periodic structure into a pair of complementary modes in transmission; the corresponding transmitted field of which contains nodes.