  A three-dimensional (3D) photonic band gap crystal is an ideal tool to
completely inhibit the local density of optical states (LDOS) at every position
in the crystal throughout the band gap. This notion, however, pertains to ideal
infinite crystals, whereas any real crystal device is necessarily finite. This
raises the question as to how the LDOS in the gap depends on the position and
orientation inside a finite-size crystal. Therefore, we employ rigorous
numerical calculations using finite-difference time-domain (FDTD) simulations
of 3D silicon inverse woodpile crystals filled with air or with toluene, as
previously studied in experiments. We find that the LDOS versus position
decreases exponentially into the bulk of the crystal. From the dependence on
dipole orientation, we infer that the characteristic LDOS decay length
$\ell_{\rho}$ is mostly related to far-field dipolar radiation effects, whereas
the prefactor is mostly related to near-field dipolar effects. The LDOS decay
length has a remarkably similar magnitude as the Bragg length for directional
transport, which suggests that the LDOS in the crystal is dominated by vacuum
states that tunnel from the closest interface towards the position of interest.
Our work leads to design rules for applications of 3D photonic band gaps in
emission control and lighting, quantum information processing, and in
photovoltaics.
