Emergence of semi-localized Anderson modes in a disordered photonic crystal as a result of overlap probability

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