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

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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 1^{st} BZ. This part is
called *irreducible 1 ^{st} 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 1^{st} 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 1^{st} 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

__Example:__

Let us consider a dielectric 2D photonic crystal with: a =
10^{-6} (m), ε_{r} = 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 1^{st} 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

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