LatticeCopies¶

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Many geometries like photonic crystals are constructed by a replication of a unit cell geometry at certain positions of a lattice. The following figure shows such a geometry with several circles copied to the lattice positions of a square lattice while omitting some lattice points (see the example in the GeoTutorial Lattice Copies):

A lattice is defined by two lattice vectors $\pvec{a}_1$ and $\pvec{a}_2$ , the structures is replicated at positions displaced by

$\begin{eqnarray*} \pvec{d}_{l_1, l_2} & = & \pvec{a}_1\cdot l_1 + \pvec{a}_2 \cdot l_2 \end{eqnarray*}$

with integers $l_1$ and $l_2$ . The lattice vectors are defined by their lengths $[a_1, a_2]$ (LatticeVectorLengths) and the angle $\alpha$ between them (LatticeAngle), that is

$\begin{eqnarray*} \pvec{a}_1 & = & a_1\cdot[1; 0] \\ \pvec{a}_2 & = & a_2\cdot [\cos(\alpha); \sin(\alpha)]. \end{eqnarray*}$

The lattice positions where the replicated geometry is actually placed can be defined in sub-sections Positions.