Mathematical diffraction of aperiodic structures

Kinematic diffraction is well suited for a mathematical approach via measures, which has substantially been developed since the discovery of quasicrystals. The need for further insight emerged from the question of which distributions of matter, beyond perfect crystals, lead to pure point diffraction, hence to sharp Bragg peaks only. More recently, it has become apparent that one also has to study continuous diffraction in more detail, with a careful analysis of the different types of diffuse scattering involved. In this review, we summarise some key results, with particular emphasis on non-periodic structures. We choose an exposition on the basis of characteristic examples, while we refer to the existing literature for proofs and further details

Breaking symmetries in ordered materials : spin polarized light transport in magnetized noncentrosymmetric 1D photonic crystals, and photonic gaps and fabrication of quasiperiodic structured materials from interference lithography

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Materials Science and Engineering, 2006."February 2006."Includes bibliographical references.Effects of breaking various symmetries on optical properties in ordered materials have been studied. Photonic crystals lacking space-inversion and time-reversal symmetries were shown to display nonreciprocal dispersion relations, and to exhibit a remarkable set of symmetry-related properties. Even in 1D, these materials are found to display indirect photonic band gaps, backward wave propagating modes (antiparallel phase and group velocities) which enable negative refraction at the air-crystal interface, ability to allow bending light with perpendicular magnetic fields, unidirectional superprism effects, etc. By calculating the complex photonic band structure, we show that the gap modes differ fundamentally from the commonly assumed evanescent modes with purely imaginary wave vectors - solely due to symmetry, we show that the gaps of nonreciprocal photonic crystals have complex wave vectors with both imaginary components and non-zero, frequency dependent real components. This basic finding is further studied in the context of tunneling dynamics, by considering the problem of tunneling time for nonreciprocal photonic band gap barriers (the tunneling wave packet has an energy in the middle of the gap).(cont.) It was found that the classical Hartman effect (independence of tunneling time on barrier length, beyond a certain length), previously implied as universal, is forbidden solely due to symmetry. Instead of a classical zero group delay, we find that tunneling wave packets with opposite spins display non-zero group delays, with opposite signs. Due to analogies based on symmetry, these results directly impact the problem of spin-polarized electronic tunneling in magnetized noncentrosymmetric semiconductors, such as GaMnAs or carbon nanotubes with applied axial magnetic fields. An interference lithography based fabrication process was developed to produce 2D and 3D quasiperiodically structured materials, which have long-range order but break translational symmetry. Multiple exposure interference lithography was used to fabricate 2D quasicrystals with feature sizes as small as 100nm. Replica molding was used to fabricate transparent and conformable 2D quasiperiodic phase masks, which subsequently allowed the fabrication of 3D structured materials with quasiperiodicity by coherent diffraction lithography. The effect of the higher point group symmetries of 2D quasicrystals on photonic band gap formation (TM polarized only) was studied by finite difference time domain calculations, and it was found that increasing the rotational symmetry does not always lead to wider gaps.by Ion Bita.Ph.D