  We theoretically investigate the design of cavities in a three-dimensional
(3D) inverse woodpile photonic crystal. This class of cubic diamond-like
crystals has a very broad photonic band gap and consists of two perpendicular
arrays of pores with a rectangular structure. The point defect that acts as a
cavity is centred on the intersection of two intersecting perpendicular pores
with a radius that differs from the ones in the bulk of the crystal. We have
performed supercell bandstructure calculations with up to $5 \times 5 \times 5$
unit cells. We find that up to five isolated and dispersionless bands appear
within the 3D photonic band gap. For each isolated band, the electric-field
energy is localized in a volume centred on the point defect, hence the point
defect acts as a 3D photonic band gap cavity. The mode volume of the cavities
resonances is as small as 0.8 $\lambda^{3}$ (resonance wavelength cubed),
indicating a strong confinement of the light. By varying the radius of the
defect pores we found that only donor-like resonances appear for smaller defect
radius, whereas no acceptor-like resonances appear for greater defect radius.
From a 3D plot of the distribution of the electric-field energy density we
conclude that peaks of energy found in sharp edges situated at the point
defect, similar to how electrons collect at such features. This is different
from what is observed for cavities in non-inverted woodpile structures. Since
inverse woodpile crystals can be fabricated from silicon by CMOS-compatible
means, we project that single cavities and even cavity arrays can be realized,
for wavelength ranges compatible with telecommunication windows in the near
infrared.
