Geometric Properties of Optimal Photonic Crystals

Photonic crystals can be designed to control and confine light. Since the introduction of the concept by Yablonovitch and John two decades ago, there has been a quest for the optimal structure, i.e., the periodic arrangement of dielectric and air that maximizes the photonic band gap. Based on numerical optimization studies, we have discovered some surprisingly simple geometric properties of optimal planar band gap structures. We conjecture that optimal structures for gaps between bands $n$ and $n+1$ correspond to $n$ elliptic rods with centers defined by the generators of an optimal centroidal Voronoi tessellation (transverse magnetic polarization) and to the walls of this tessellation (transverse electric polarization).

Figure 1

Band diagram for a periodic arrangement of circular dielectric cylinders with ${ϵ}_r=11.56$ and radius $r/a=0.25$ in air. The gray rectangles indicate band gaps where TM-polarized waves cannot propagate. Insets show the reciprocal lattice with the irreducible Brillouin zone indicated by the gray triangle (left) and the periodic lattice (right) with lattice constant $a$.

Figure 2

Geometrical procedure for generating (near) optimal planar band gap structures demonstrated on a square cell for band $n=10$. (a) Initial point positions satisfying symmetry and periodicity for Lloyd’s algorithm with free geometric parameters $b_1$ and $b_2$. (b) Converged point positions and associated Voronoi tessellation. (c) (Near) optimal TM structure (blue), TE structure (red), in air (gray).

Figure 3

Composite picture showing optimal point positions (black dots) obtained from Lloyd’s algorithm and corresponding Voronoi tessellations (dashed lines) for band numbers increasing from $n=1$ (upper left) to $n=15$ (lower right). Top: square unit cell; bottom: rhombic unit cell. Colors indicate topology optimized material distributions. Blue is the optimal distribution of dielectric for TM polarization, red is the optimal distribution for TE polarization, and gray indicates air.

Figure 4

Two-step optimization process. (a) Coarse grid with symmetry lines for (a) square unit cell with $N=10$ and 15 free design variables, and (b) rhombic unit cell with $N=11$ and 16 design variables. (c) Result of optimization process for coarse rhombic grid for band number $n=10$ (left) and fine grid ($N=89$) solution (right).

Figure 5

Relative band gaps and geometric energies vs band number for square cell (top) and rhombic cell (bottom). Band gaps for TM polarization (blue lines and empty circles), TE polarization (red lines and filled circles), and inverse normalized energy (black dashed lines and stars).