Reflectivity calculated for a three-dimensional silicon photonic band gap crystal with finite support

We study numerically the reflectivity of three-dimensional (3D) photonic crystals with a complete 3D photonic band gap. We employ the finite element method to study crystals with the cubic diamondlike inverse woodpile structure. The high-index backbone has a dielectric function similar to silicon. We study crystals with a range of thicknesses up to ten unit cells (L <= 10c). The crystals are surrounded by vacuum, and have a finite support as in experiments. The polarization-resolved reflectivity spectra reveal Fabry-Perot fringes related to standing waves in the finite crystal, as well as broad stop bands with nearly 100% reflectivity, even for thin crystals. The frequency ranges of the stop bands change little with angle of incidence, which is plausible since the stop bands are part of the 3D band gap. Moreover, this result supports the previous assertion that intense reflection peaks measured with a large numerical aperture provide a faithful signature of the 3D photonic band gap. For p-polarized waves, we observe an intriguing hybridization between the Fabry-Perot resonances and the Brewster angle that remains to be observed in experiments. From the strong reflectivity peaks, it is inferred that the maximum reflectivity observed in experiments is not limited by finite size. The frequency ranges of the stop bands agree very well with stop gaps in the photonic band structure that pertain to infinite and perfect crystals. The angle-dependent reflectivity spectra provide an improved interpretation of the reflectivity measurements performed with a certain numerical aperture and a new insight in the crystal structure, namely unequal pore radii in X and Z directions. The Bragg attenuation lengths L-B are found to be smaller by a factor 6 to 9 than earlier estimates that are based on the width of the stop band. Hence, crystals with a thickness of 12 unit cells studied in experiments are in the thick crystal limit (L >> L-B). Our reflectivity calculations suggest that the 3D silicon photonic band gap crystals are interesting candidates for back reflectors in a solar cell in order to enhance the photovoltaic efficiency.