  We recently reported on the first realization of ultraviolet photonic crystal
laser based on zinc oxide [Appl. Phys. Lett. {\bf 85}, 3657 (2004)]. Here we
present the details of structural design and its optimization. We develop a
computational super-cell technique, that allows a straightforward calculation
of the photonic band structure of ZnO photonic crystal slab on sapphire
substrate. We find that despite of small index contrast between the substrate
and the photonic layer, the low order eigenmodes have predominantly
transverse-electric (TE) or transverse-magnetic (TM) polarization. Because
emission from ZnO thin film shows strong TE preference, we are able to limit
our consideration to TE bands, spectrum of which can possess a complete
photonic band gap with an appropriate choice of structure parameters. We
demonstrate that the geometry of the system may be optimized so that a sizable
band gap is achieved.
