Symmetries and Wavefunctions of Photons Confined in 3D Photonic Band Gap Superlattices

TL;DR

This work computationally characterizes 3D light confinement in a diamond-like inverse woodpile photonic crystal with a defect pore superlattice. Using a scaling analysis and plane-wave expansion, it classifies confinement dimensionalities and generates confinement maps across the regular and defect pore radii $R$ and $R'$, revealing threshold deviations and trends that larger $R$ promote more and stronger confinement. The authors show that photonic bands exhibit symmetry patterns and degeneracies rooted in the lattice Bloch states rather than atomic-like orbitals, offering richer photonic orbital geometries. For cQED relevance, donor-like defects ($R'<R$) yield superior LDOS enhancements in air, providing a practical path toward 3D networks of strongly coupled cavities with embedded quantum dots.

Abstract

We perform a computational study of confined photonic states that appear in a three-dimensional (3D) superlattice of coupled cavities, resulting from a superstructure of intentional defects. The states are isolated from the vacuum by a 3D photonic band gap, using a diamond-like inverse woodpile crystal structure, and exhibit 'Cartesian' hopping of photons in high-symmetry directions. We investigate the confinement dimensionality to verify which states are fully 3D confined, using a recently developed scaling theory to analyze the influence of the structural parameters of the 3D crystal. We create confinement maps that trace the frequencies of 3D confined bands for select combinations of key structural parameters, namely the pore radii of the underlying regular crystal and of the defect pores. We find that a certain minimum difference between the regular and defect pore radii is necessary for 3D confined bands to appear, and that an increasing difference between the defect pore radii from the regular radii supports more 3D confined bands. In our analysis we find that their symmetries and spatial distributions are more varied than electronic orbitals known from solid state physics. We also discover pairs of degenerate 3D confined bands with p-like orbital shapes and mirror symmetries matching the symmetry of the superlattice. Finally, we investigate the enhancement of the local density of optical states (LDOS) for cavity quantum electrodynamics (cQED) applications. We find that donor-like superlattices, i.e., where the defect pores are smaller than the regular pores, provide greater enhancement in the air region than acceptor-like structures with larger defect pores, and thus offer better prospects for doping with quantum dots and ultimately for 3D networks of single photons steered across strongly-coupled cavities.

Figures (14)

• Figure 1: 3D superlattice of defect pores embedded in a 3D inverse woodpile photonic band gap crystal. For certain states, confined light is conceived to hop between neighboring cavities in Cartesian directions, giving rise to 'Cartesian' light.
• Figure 2: (a) Structure of the perfect inverse woodpile photonic crystal. We use a tetragonal unit cell with lattice constants $b$ and $a$, and the pore radius is denoted by $R$. The figure shows a $2\times2\times2$ supercell. (b) Design of a cavity. The radius of two proximal defect pores (shown in green) is altered, resulting in a region with excess of either silicon or air, which behaves as a cavity confining the light (orange glow).
• Figure 3: Band gap frequencies as a function of pore radius $R$ in an inverse woodpile photonic crystal with $\varepsilon = 12.1$ typical for a silicon backbone. The 3D photonic band gap exists for pore radii $0.15\le R/a\le 0.29$, with the maximum width at $R/a = 0.245$.
• Figure 4: Band structure of an inverse woodpile crystal with regular pore radius $R=0.24a$, and the defect pore radius $R^\prime=0.5R$. Bands that are identified as confined in $c = 3$ dimensions are colored, with red designating individual bands and blue pairs of degenerate bands. The bands are also labeled by their band number $N_\mathrm{b}$.
• Figure 5: 3D isosurface plot of the energy density in confined band $N_\mathrm{b} = 111$ in an inverse woodpile crystal with regular pore radius $R = 0.24a$, and defect pore radius $R^\prime=0.5R$. The energy profile exhibits specific symmetries inherited from the parent defect superlattice. Besides the mirror symmetries along the $z/b\approx1.5$ and $x/b\approx1.5$ planes, it is also symmetric with respect to mirroring with according to the $y/a\approx 1.15$ plane combined with 90$^{\circ}$ rotation about the $(x/b,z/b)\approx(1.5,1.5)$ axis. a) Birds-eye view; b) view of the $y$-$z$ plane.