Generation of Wannier functions for photonic crystals

We present an approach for the efficient generation of Wannier functions for Photonic Crystal computations that is based on a combination of group-theoretical analysis and efficient minimization strategies. In particular, we describe the symmetry properties that allow for exponential localization of Wannier functions and how they are related to the underlying Bloch mode symmetries of the photonic band structure and we show that no exponentially localized Wannier functions can be created from the physical modes of a three-dimensional crystal. Moreover, we comment on the use of conjugate gradient and randomized minimization algorithms that—together with the group theoretical considerations—facilitate the efficient numerical determination of maximally localized Wannier functions for many bands. This is a requirement for the accurate computation of Photonic Crystal functional elements and devices.

Figure 1

Sketch to illustrate the interdependence of different symmetry groups and their representations within the problem of GWF generation. Here, irreps is an abbreviation for irreducible representations, HS is shorthand for high symmetry, and IBZ stands for irreducible Brillouin zone.

Figure 2

Symmetric hybridizations for the Wyckoff positions of a square lattice of cylindrical scatterers. Each dashed box encloses a sketch of $N^{\left(\text{G}\right)}$ GWFs at $\mathbf{R}=\mathbf{0}$ corresponding to one irreducible representation $D^{\left(\mathbf{q}\alpha \right)}$ in Table . A sketch of the hybridization pattern for Wyckoff position “g” omitted in order to save space. It consists of 8 copies of an asymmetric function located at a general point.

Figure 3

Symmetric hybridizations for the Wyckoff positions of a triangular lattice of cylindrical scatterers. Each dashed box encloses a sketch of $N^{\left(\text{G}\right)}$ GWFs at $\mathbf{R}=\mathbf{0}$ corresponding to one irreducible representation $D^{\left(\mathbf{q}\alpha \right)}$ in Table . A sketch of the hybridization pattern for Wyckoff position “f” omitted in order to save space. It consists of 12 copies of an asymmetric function located at a general point.

Figure 4

This figure illustrates how properly hybridized trial functions (left part of the equation) are constructed from truncated $\Gamma$-point Bloch modes (on the right-hand side). The Bloch modes were H-polarized bands 2–5 at the $\Gamma$ point for a square lattice of holes (relative radius $0.45$) in silicon ($\epsilon =12$).

Figure 5

Trial functions (top row), corresponding initial (nonminimized) and final (spread minimized) GWFs (middle and bottom row, respectively) with individual spreads $\Omega$ for H-polarized bands 2–7 of the square lattice of air holes ($r/a=0.45$) in silicon ($\epsilon =12$). Four of the trials have been hybridized as shown in Fig. 4 according to the site symmetry representations anticipated by Table . Note the striking similarity of the $\Gamma$-point trials and the GWFs.

Figure 6

Sketch of the reduced objective $\Psi$ (dashed line) for the simulated annealing. For a certain configuration (blue or red dot) it is defined as the final value of $\Omega$ (solid line) in a complete CG minimization starting from that very point (indicated by arrows). $\Psi$ is discontinuous because neighboring points (blue and red dot) in the configuration space can only have different values if the CG minimizer approaches different local minima of $\Omega$. For each local minimum of $\Omega$, $\Psi$ is constant throughout the whole catchment area.

Figure 7

Sketch of the first Brillouin zones of the square and the triangular lattices including the points of high symmetry that are relevant for our symmetry analysis. The outlines of the irreducible parts of either Brillouin zone are depicted in black.

Figure 8

Wyckoff positions for a square lattice (left panel) and triangular lattice (right panel) arrangement of circular scatterers. Black solid lines indicate the outline of the Wigner-Seitz cell; the position of a circular scatterer was included in light gray. Points that belong to the same Wyckoff set are depicted by the same symbol in the same color. Points that emerge via shifting by a lattice vector were omitted. Please note that the positions “d”–“f” for the square lattice and the positions “d” and “e” for the triangular lattice are lines rather than individual points. The interior of irreducible wedge belongs to the classes “g” and “f” for the square or triangular lattice, respectively. Finally, we shall mention that the line connecting positions “b” and “c” in the triangular case is an extension of the line “a”–“b” of a neighboring Wigner-Seitz cell. Thus, these points belong to the class “d”, too.