  In this paper we study the effect of positional randomness on transmissional
properties of a two dimensional photonic crystal as a function of a randomness
parameter $\alpha$ ($\alpha=0$ completely ordered, $\alpha=1$ completely
disordered). We use finite-difference time-domain~(FDTD) method to solve the
Maxwell's equations in such a medium numerically. We consider two situations:
first a $90\degr$ bent photonic crystal wave-guide and second a centrally
pulsed photonic crystal micro-cavity. We plot various figures for each case
which characterize the effect of randomness quantitatively. More specifically,
in the wave-guide situation, we show that the general shape of the normalized
total output energy is a Gaussian function of randomness with
wavelength-dependent width. For centrally pulsed PC, the output energy curves
display extremum behavior both as a function of time as well as randomness. We
explain these effects in terms of two distinct but simultaneous effects which
emerge with increasing randomness, namely the creation of semi-localized modes
and the shrinking (and eventual destruction) of the photonic band-gaps.
Semi-localized (i.e. Anderson localized) modes are seen to arise as a
synchronization of internal modes within a cluster of randomly positioned
dielectric nano-particles. The general trend we observe shows a sharp change of
behavior in the intermediate randomness regime (i.e. $\alpha \approx 0.5$)
which we attribute to a similar behavior in the underlying overlap probability
of nano-particles
