Photonic Crystal Properties of Self-Assembled Kagome Lattices
Binary colloidal mixtures are at the frontier for the creation of new photonic crystals (PhCs) because they spontaneously undergo complex ordering at the mesoscale and have the potential to vastly increase the range of structures that can be self-assembled. It is now known that binary mixtures of sub-micrometer shapes form colloidal alloys, i.e., analogous to metalic alloys, determined by the size ratio of the particle populations, the relative concentrations, and the total packing fraction. What is currently unknown is how the multicomponent basis and particle composition in binary colloidal crystals effect photonic dispersion relations. This gap in our understanding of structure-optocal property relationships in colloidal alloys is an important problem because this knowledge will aid in establishing a new paradigm in the colloid-based photonics field and will guide experimentalists to high payoff targets. Inspired by recent studies showing that binary mixtures of (shape-anisotropic) patchy particles are predicted to self-assemble into a range of Archimedian tilings (ATs), we investigate the photonic properties of the Kagome lattice, i.e., Schläfli symbol (3 ∙ 6)2, using electromagnetic calculations (MIT photonic bands package, MPB) and simulations (MIT Electromagnetic Equation Propagation package, MEEP). The Schläfli symbol represents the cyclic order of triangular- and hexagonal cross-section rods surrounding the vertices, with repeated elements collected as the exponent. We expected that this strucure would support strong light-matter interactions because it meets the structure factor criteria for large photonic bandgaps, i.e., the static structure factor approaches zero as the wavevector magnitude approaches zero. In this thesis, we find large bandgaps up to 30% (TM, gap-to-midgap ratio) in the direct (3 ∙ 6)2 structure. The dielectric constants for non-close-packed hexagonal- and triangular rods are varied independently, between 2 and 16, consistent with binary compositions. Mode field distributions indicate that the bandgaps originate from Lorenz-Mie scattering. For inverse structures, bandgaps arise due to dielectric band-air band transitions. Equifrequency contour analysis and finite difference time domain (FDTD) simulations show that negative refraction occurs over all angles of incidence (AANR) for normalized frequencies of 0.27-0.28 (ε(hexagon) = 16, ε(triangle) = 2), 0.32-0.34 (ε(triangle) = 16, ε(hexagon) = 2), and 0.32-0.34 (ε(matrix) = 12). The effective refractive indices reach negative one within these ranges. Sub-wavelength imaging and self-collimation are demonstrated for flatlenses having the (3 ∙ 6)2 structure. These dielectic PhCs provide an alternative to lossy metallic materials for realizing negative refraction in the optical and near-IR region. The photonic properties predicted here are important for applications in waveguides, solid state lighting, nonlinear optics, and superlenses, i.e., imaging beyond the diffraction limit.
Kagome lattice; Negative Refraction; Photonic Crystals; Self-assembly
Materials Science and Engineering
M.S., Materials Science and Engineering
Master of Science
dissertation or thesis