Photonic Crystals: Computational Studies


Three-dimensional photonic crystals / tetragonal lattice

R. Hillebrand and W.Hergert


In this page photonic crystal structures (PCs) of tetragonal lattice type are introduced and studied. They feature complete three-dimensional (3D) photonic band gaps.



Fig. 1 (left) shows the model of the "Hallite" - a family of tetragonal photonic crystals (PCs) having a large complete band gap. The design is based on two systems of ordered parallel pores  being perpendicular to each other. For increasing radii, the pore systems interpenetrate and a complete band gap arises. The band gap position and the gap size can be tuned varying the lattice parameters Lx,z / Ly and the radii r/a of the pores (r_drilled, R_cyl). The photonic crystal may be considered as an inverted woodpile structure. The parameters of the PCs have been systematically varied according to the scheme given in Fig 1 (right) (see also: [1], [2]).


The photonic band structures presented in Fig. 2 have been performed with the mit-package ( The eigenvalue problems are solved applying a frequency-domain based numerical procedure. The fully-vectorial  algorithm allows to calculate the frequency eigenstates and the electromagnetic field modes. Iterative eigensolvers are applied to find an adjustable number of the lowest eigenstates of Maxwell’s equations. Varying the aspect ratio Lx,z [0.70 … 0.58] / Ly of the crystal structure (see Fig. 1a) allows to shift the lattice structure from cubic-fcc to hexagonal. Left: fcc cell, maximum gap for r/a: 0.24, f_c: 0.565 ; right: hexagonal cell, maximum gap for r/a: 0.22, f_c: 0.687.


Fig. 3 shows the absolute sizes of the complete photonic band gaps indicating the influence of the tetragonal lattice aspect ratio Lx,z / Ly on the gap attainable. It is interesting to note that the absolute maximum is between the fcc and the hexagonal geometries. It has been proven with other ε-values that the Lx,z value of the lattice that provides the maximum gap is material dependent.


In Fig. 4 we display the gap maps indicating the influence of the tetragonal lattice aspect ratio Lx,z / Ly on the energetic position of the 3D band gaps. The area within the circumscribed sectors always represents a complete photonic band gap, i.e. NO states, NO propagation. The maximum gap/midgap-ratio attainable is 26.9%. As obvious from Fig. 2, the midgap frequencies strongly vary with the lattice modifications. The experimental fabrication of the "Hallite" structure is described in [3]. The theoretical optical properties could be confirmed.


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