To begin, Eq. (1) is simplified using the curl of curl identity: Display Formula
$\u2207[\u2207\xb7E(r)]\u2212\u22072E(r)=(\omega c)2\u03f5r(r)E(r).$(2)
We assume that there are no free charges, which demands that Display Formula$\u2207\xb7E(r)=0.$
Equation (2) now simplifies to Display Formula$\u2212\u22072E(r)=(\omega c)2\u03f5r(r)E(r).$(3)
Expressing Eq. (3) in terms of second order central finite differences gives the following: Display Formula$\u2212(Ei+1,jx\u22122Ei,jx+Ei\u22121,jxhx2+Ei,j+1y\u22122Ei,jy+Ei,j\u22121xhy2,0)=(\omega c)2\u03f5ri,jEi,j(Ei,jx,Ei,jy,0),$(4)
where $Ex$ and $Ey$ are the electric field vectors along $x$ and $y$, and $hx$ and $hy$ are the physical distances between the mesh elements along the $x$ and $y$ axis, respectively. Equation (4) is fully decoupled; that is, changes in $Ex$ happen only along $i$ ($x$-axis), and changes in $Ey$ happen only along $j$ ($y$-axis). This allows it to be rewritten as Display Formula$\u2212(Ei+1,j\u22122Ei,j+Ei\u22121,jhx2+Ei,j+1\u22122Ei,j+Ei,j\u22121hy2)=(\omega c)2\u03f5ri,jEi,j.$(5)
To solve this eigenvalue problem, a mesh is generated along a unit cell of the PC, with $i,j$ values corresponding to mesh elements. Figure 5 shows an example of the $n\xd7n$ mesh in which the electric field at each position is labeled $En$. In the finite differences equation, the field at a given position, $Ei,j$, corresponds to one position, $En$. The $Ei\xb11$ and $Ej\xb11$ terms correspond to adjacent mesh cells in the lattice. Elements within the unit cell are shown as bounded by solid lines and those within adjacent cells are represented by dashed lines. The ^{1} further discusses the mesh elements of the unit cell, as well as the required mesh size at which calculated bands converge to a common value.