1. Photonic band gaps in 12-fold symmetric quasicrystals B. P. Hiett *, D. H. Beckett *, S. J. Cox †, J. M. Generowicz *, M. Molinari *, K. S. Thomas * * Department of Electronics and Computer Science, † School of Engineering Sciences; University of Southampton, Southampton, SO17 1BJ, UK ABSTRACT The 12-fold symmetric quasicrystal shows great potential as a novel photonic band gap (PBG) structure exhibiting a band gap for relatively low filling fractions and dielectric contrasts. The band gaps are highly homogeneous with respect to the angle of incidence of the incoming light due to the crystals high degree of rotational symmetry. These crystals have been analysed using a finite element method (FEM) developed specifically for modelling PBG structures. We present and discuss quasicrystal structures and their optical properties.

2. High Performance Computing Photonic Crystal Overview Photonic band gap structures exhibit a photonic band gap analogous to the electronic band gap present in semi- conductors. ‘Photonic Band Gap Device’ ‘Photonic Band Gap Device with Defect Waveguide’ Electronic band structure (Gallium Arsenide) Photonic band structure (Triangular Lattice) A band gap arises due to destructive interference from Bragg like diffraction of electromagnetic waves through the crystal. Photons in the frequency range of the band gap are completely excluded so that atoms within such materials are unable to spontaneously absorb and re- emit light in this region. The construction of integrated optical circuits would allow the revolutionary shift from electronic to photonic technology to take place. This step is widely viewed as the ‘holy grail’ in communication technology. Energy Band Gap ELECTRONICPHOTONIC Photonic Crystals offer enormous potential in the development of: F Sharp angle wave guides, F Highly efficient single mode lasers F Integrated optical circuits F High-speed optical communication networks

3. High Performance Computing The Finite Element Method Domain Discretisation Selection of Interpolation Function Derivation of the Elemental Equations Matrix Assembly Solution of the Eigensystem Visualisation Bridge waveguide structure courtesy of Martin Charlton, Southampton Microelectronics Research Group. Unit-Cell Real-Thing pitch=300nm Local to Global Node Mapping Generalised Eigenvalue Problem LinearQuadratic Approximate & discretise the governing equations Band diagrams & Field Plots

4. High Performance Computing Quasicrystal Structures The crystal structure is based on the tiling of squares and equilateral triangles in a dodecagonal configuration. The red lines indicate the symmetries Tiling of these dodecagons allows the lattice vectors to be constructed (seen here in red). These vectors form the boundary of the unit cell which, when periodic boundary conditions are applied, allows us to model a crystal of infinite extent using the mesh shown. 1. ‘Triangle – Triangle’ Configuration and mesh. 2. ‘Square – Square’ Configuration and mesh.

5. High Performance Computing Results The mirror symmetry shown between dispersion relations as one moves along K-  and  -X in reciprocal lattice space proves that a crystal rotation of 30  produces the same band structure. This supports the claim that quasicrystals are highly homogeneous with respect to the angle of incidence of incoming light. On the left there are dispersion relations for the triangle- triangle and square-square configurations. Rod radius to pitch length ratio (r/a) at 0.5 for air rods in silicon nitride (SiN,  = 4.1) and gallium arsenide (GaAs,  = 11.4). On the right there are ‘gap-maps’ plotting normalised frequency range of TE and TM mode band-gaps against the filling fraction. For meshes with an r/a of 0.1 to 0.5 at 0.01 intervals for both configurations and for each substrate material. KK  X KK  X KK  X KK  X Triangle SiN Square Triangle GaAs Square Triangle Air rods in SiN Triangle Air rods in GaAs Square Air rods in GaAs Triangle GaAs rods in air

6. High Performance Computing Conclusion The 12-fold symmetric quasicrystal does produce complete band gaps but not for especially low dielectric contrasts, e.g. air rods in SiN or glass. Complete band gaps appear as the dielectric contrast increases, and as the rod radius to pitch length increases. Swapping rod/substrate materials such that the PC consists of high dielectric rods in a low dielectric substrate also increases the frequency range of the band gaps.The high order of symmetry makes the structure less sensitive to the angle of incidence of the incoming light resulting in a more homogeneous band-gap. Comparison of results produced with the finite element code show excellent agreement with those produced using other numerical methods, including finite difference time domain and also with experimental data. The accuracy of the accuracy of the finite element code coupled with its efficiency, both in terms of computation and memory requirements make it a very attractive approach to photonic crystal modelling. Finite Element Method Results F ‘Triangle-triangle’ configuration F Air-rods in Silicon Nitride F Filling fraction = 30%. Finite Difference Time Domain Results Experimental Results

7. High Performance Computing Further Research 2D 3D Developing a finite element code to allow the freedom to model fully 3D periodic structures. This involves the development of vector rather than scalar based interpolation functions. This is necessary to avoid correctly model the continuity requirements of the magnetic field at element boundaries and to address the problem of spurious (non-physical) modes. DPP17 DPP21 3D Vector Finite Element Analysis PBG Design Optimisation The infinite arrangement of rods in a unit cell can be classified into a finite set of groups of similar structure. These groups can be explored using minimisation of the objective function to discover a parameter space that produces an optimum band gap. Several examples of rod configuration groups Mesh representation of those groups