4 A Solid-State Theoretical Approach 59 0.8 206 0.2 (a) 0.0 N(o)(a.u.) Fig.4.1.Density of States (a)and photonic band structure (b)for TM-polarized radiation in a square lattice (lattice constant a)of cylindrical air pores of radius Rpore =0.475a in dielectric with e=12 (silicon).This PC exhibits a large fun- damental gap extending from w=0.238 x 2mc/a to w=0.291 x 2c/a.A higher order band gap extends from w=0.425 x 2rc/a to w=0.464 x 2rc/a. to PWM.Additional refinements such as a finite element discretization will further increase the efficiency of the MG-approach. In Fig.4.1(b),we show the bandstructure for TM-polarized radiation in a 2D PC consisting of a square lattice (lattice constant a)of cylindrical air pores (radius rpore =0.475a)in a silicon matrix (ep =12).This structure exhibits two 2D PBGs.The larger,fundamental bandgap(20%of the midgap frequency)extends between w=0.238 x 2mc/a to w=0.291 x 2mc/a and the smaller,higher order bandgap (8%of the midgap frequency)extends from w=0.425×2πc/atow=0.464×2πc/a.Furthermore,.in Fig..4.1(a)we depict the DOS for our model system,where the photonic band gaps are manifest as regions of vanishing DOS.Characteristic for 2D systems is the linear behavior for small frequencies as well as the logarithmic singularities, the so-called van Hove singularities,associated with vanishing group velocities for certain frequencies inside the bands (compare with Fig.4.1(a)) 4.3 Defect Structures in Photonic Crystals To date,the overwhelming majority of theoretical investigations of cavities and waveguiding in PCs has been carried out using Finite-Difference Time- Domain(FDTD)and/or Finite-Element (FE)techniques.However,applying general purpose methodologies such as FDTD or FE methods to defect struc- tures in PCs largely disregards information about the underlying PC struc- ture which is readily available from photonic bandstructure computation.As a result,only relatively small systems can be investigated and the physical insight remains limited.4 A Solid-State Theoretical Approach 59 N(ω) (a.u.) 0.0 0.2 0.4 0.6 0.8 Frequency ( ωa/2πc) Γ X M Γ -π/a 0 π/a Γ X M ky kx x y a (a) (b) Fig. 4.1. Density of States (a) and photonic band structure (b) for TM-polarized radiation in a square lattice (lattice constant a) of cylindrical air pores of radius Rpore = 0.475a in dielectric with ε = 12 (silicon). This PC exhibits a large fun￾damental gap extending from ω = 0.238 × 2πc/a to ω = 0.291 × 2πc/a. A higher order band gap extends from ω = 0.425 × 2πc/a to ω = 0.464 × 2πc/a. to PWM. Additional refinements such as a finite element discretization will further increase the efficiency of the MG-approach. In Fig. 4.1(b), we show the bandstructure for TM-polarized radiation in a 2D PC consisting of a square lattice (lattice constant a) of cylindrical air pores (radius rpore = 0.475a) in a silicon matrix (εp = 12). This structure exhibits two 2D PBGs. The larger, fundamental bandgap (20% of the midgap frequency) extends between ω = 0.238 × 2πc/a to ω = 0.291 × 2πc/a and the smaller, higher order bandgap (8% of the midgap frequency) extends from ω = 0.425 × 2πc/a to ω = 0.464 × 2πc/a. Furthermore, in Fig. 4.1(a) we depict the DOS for our model system, where the photonic band gaps are manifest as regions of vanishing DOS. Characteristic for 2D systems is the linear behavior for small frequencies as well as the logarithmic singularities, the so-called van Hove singularities, associated with vanishing group velocities for certain frequencies inside the bands (compare with Fig. 4.1(a)). 4.3 Defect Structures in Photonic Crystals To date, the overwhelming majority of theoretical investigations of cavities and waveguiding in PCs has been carried out using Finite-Difference Time￾Domain (FDTD) and/or Finite-Element (FE) techniques. However, applying general purpose methodologies such as FDTD or FE methods to defect struc￾tures in PCs largely disregards information about the underlying PC struc￾ture which is readily available from photonic bandstructure computation. As a result, only relatively small systems can be investigated and the physical insight remains limited