ABSTRACT
Early in the development of photonic crystals, it became evident that the refractive index contrast played a vital role in opening up photonic gaps. A minimum value around ~2.5-3 was found to be the necessary threshold. It was also shown that not any periodic arrangement of dielectric scatterers yields a photonic gap. To date, all the crystal structures that have yielded a full three-dimensional (3D) gap belong to the A7 family of structures [1]. The A7 crystal structures consist of a rhombohedral
= ± β + + , where 1 2 3, ,anda a a
1 cos cos ( ,1,1), (1, ,1), and (1,1, ), with 1
cos a a a a a a
+ α − α = ε = ε = ε ε = −
α
where α is the angle between any two primitive lattice vectors. All full 3D gap structures can be produced from this group by proper selection of the parameters, α and β. For example, by choosing α = 60° and β = 1/8, the diamond structure results as in Figure 7.1. Setting α = 60° and β = 0, and joining the lattice points by cylinders, the Yablonovite structure results as in Figure 7.2. Similarly, the ISU layer-by-layer structure in Figure 7.3, the spiral rod structure in Figure 7.4, and even the simple cubic structures can be generated by the appropriate choice of parameters. To better understand the rules of thumb for yielding a full 3D band gap, it is imperative to understand how the photonic gap arises. In the next section, we shall follow the argument presented by John et al. [2].
