J. Smajic, Computational Optics Group, IFH, ETH,
8092

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Fig. 1. Photonic crystal structure (original lattice) with two lattice vectors

In the description of the 2D
photonic crystal with triangular lattice, all details about reciprocal lattice,
the 1^{st} Brillouin zone construction and
periodic constants calculation are given. Following this background, the
results of these calculations are presented here.

The primitive lattice vectors for the square lattice are shown in the following figures.

Fig. 2. The original lattice |
Fig. 3. The reciprocal lattice |

Fig. 4. The 1^{st} Brollouin
zone construction and detail of the irreducible part

__Example:__

Let us consider a dielectric 2D photonic crystal with: a =
10^{-6} (m), r = 0.18a, ε_{r} =
11.56 (inside of the dielectric rods), square lattice (Fig. 1.). The wave
vector components and periodic constants are:

_{}, and _{},

The results of the periodic constant calculation for the
three corners M, K, and Г of the irreducible 1^{st} BZ are:

__Г: _{},
_{},__

_{}, _{},

__M: _{}, _{},__

_{}, _{},

__X: _{}, _{},__

_{}, _{}.

The results of the MaX-1 bandgap computations are given in the Fig. 8. and Fig. 9.

Fig. 7. Band structure of the 2D dielectric photonic crystal for the TE modes

Fig. 8. Band structure of the 2D dielectric photonic crystal for the TM modes

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