Fig. 1. Photonic crystal structure (original lattice) with two lattice vectors
For the dielectric constant is possible to write:
, where is one of the original lattice vectors.
According to the Bloch’s theorem, it is possible to write:
, and is also the periodic function.
On the base of these equations, it is easy to prove the following equation:
Non-trivial solutions of this equation are only found when:
, i.e. , where N is an integer.
This allows us to define the so-called reciprocal lattice space, spanned by the reciprocal lattice vectors. We first define the original lattice vectors as follows:
, where , , are three independent lattice vectors and , , are integer numbers. Note that is missing in 2D crystals.
Similarly we write for the primitive reciprocal lattice vectors:
If we want to make construction of the reciprocal lattice, one possible solution is given by the following equations:
These equations are derived from the definition of the reciprocal lattice vector space. For the 2D situation, the vector should be omitted and the vector will become the unit vector.
The primitive lattice vectors for the triangular lattice are shown in the following figures.
Fig. 2. The original lattice
Fig. 3. The reciprocal lattice
On the base of the equations above, it is possible to make an important conclusion. The discrete translational symmetry of a photonic crystal leads us to the fact that the modes with the wave vector and with the wave vector are identical, i.e. we have periodicity also in the reciprocal space. The basic cell for this periodicity is called the first Brillouin zone. It can be defined as a zone around any lattice point in the reciprocal space with points that are closer to that lattice point than to any other lattice point.
The Brillouin zone construction (using Bragg’s planes – blue lines) for a triangular lattice is shown in the Figure 4. Because of the symmetry, we need to analyze only a part of the 1st BZ. This part is called irreducible 1st BZ, as illustrated in Figure 4.
For the photonic bandgap computation using the MMP method contained in MaX-1, it is useful to give some details about the periodic constants (Cx and Cy) for the triangular lattice case.
For the triangular lattice, we first define the components: Xx, Yx and Yy, of the vectors spanning the original cell. These values are shown in Figure 5. When we set these values, MaX-1 works with the non-orthogonal coordinate system X’Y’ shown in Figure 5.
In the point “P” we can write the field on the base of the field value in the point “R” as follows:
On the other hand, if we write this equation in the usual Cartesian coordinate system, we have:
For the constants Xx, Yx and Yy we can write:
,, , () and obtain:
We therefore have:
, and (for ).
This allows us to calculate the periodic constants in the case of the triangular lattice.
Fig. 4. The 1st Brollouin zone construction and detail of the irreducible part
Fig. 5. The coordinate systems with respect to the symmetry
For the bandgap calculation, the Cx and Cy values are calculated on the base of the 1st irreducible BZ in the reciprocal space. After this step, it is necessary to calculate the Cx’ and Cy’ values and with these values we can go in the eigenvalue search process using MaX-1, like in the figure 6.
Fig. 6. Detail of the periodic constant (Cx’, Cy’) calculation for the point M
Let us consider a dielectric 2D photonic crystal with: a = 10-6 (m), εr = 8.41 (inside the dielectric rods), r = 0.15ּa, triangular lattice (Fig. 1.). The wave vector components and periodic constants with:
, and ,
we obtain for the three corners in the M, K and Г of the irreducible 1st BZ.
Г: , ,
M: , ,
K: , ,
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