The Effect Of Long-Range Correlations On One-Dimensional Disordered Systems And Optics Properties Of Disordered Chains Of Coupled Microspherical Resonators

To study the effect of long-range correlations (LRC) of the on-site energies or of the hopping integrals on the localization properties of one-dimensional (1D) systems, we firstly generated a sequence of random numbers with LRC by using the Fourier filtering method. This kind of sequence has a power-law spectral density of the form S(k)?k-?and describes the trace of a fractional Brownian motion. It can be used to simulate stochastic processes in nature such as the correlations in the nucleotide sequence of DNA molecules and height-height correlations in the interface roughness appearing during growth. We then introduced this kind of LRC into the on-site energies and compared the localization properties of different localized regimes in a 1D system with long-range correlated diagonal disorder. We also introduced this kind of LRC into the hopping integrals and studied the phase transitions between the localized states and extended states in a ID system with long-range correlated off-diagonal disorder.As a 1D system, which has practical applications, the disordered chain of coupled microspherical resonators also demonstrates Anderson localization. Since its component—microspherical resonator—is characterized by an ultra-high quality factor and extremely small mode volume, it has been proposed for applications in slow light, all-optical buffer, optical storage and intensification of light-matter interaction. In practice, however, the structural fabrication imperfections such as the random variations in the size of the microspherical resonators and in the distance between the nearest-neighbor microspherical resonators, and the surface roughness, collectively termed as "disorder", significantly influence the transport efficiency and dramatically enhance the radiative losses when light is propagating in such a chain. To study the effect of size disorder on the transport properties and to investigate statistical properties of radiative losses in a chain of coupled microspherical resonators, we defined the transmission coefficient, reflection coefficient and radiative losses by using Mie scattering theory, transfer matrix method and the nearest-neighbor approximation. We also numerically studied transport properties of two types of structures:asymptotically long chain with system size is much larger than the localization length and relatively short chain with system size is shorter larger than the localization length.In 1D system with long-range correlated diagonal disorder, we found that, under the combined effect of LRC and disorder, the localization properties of localized regime with weak disorder and weak correlations (regime I) are significantly different from that of localized regime with strong disorder and strong correlations (regime II) and the localization properties of these two regimes are different from that of 1D standard Anderson model. In regime I, the wave functions are exponentially localized, the localization length?has a power-law relation with the system size N,??N?with?increases with the increasing correlation exponent?; the distribution of Lyapunov exponent?is normal and its variance var(?) scales as var(?)?N-v with v also increases with the increasing?. In regime II, the localized mode exhibits a butterflylike form, which totally differs from exponentially localized modes excited in regime I; the distribution of Lyapunov exponent is not normal anymore and both of the localization length and var(?)are constants, which don't change with the changing system size. Moreover, we found that the single parameter scaling theory is not valid anymore in both of these two regimes and LRC plays dual role in the system:it may either suppress or enhance localization.In 1D system with long-range correlated off-diagonal disorder, we found that, even though such a system has the property of chiral symmetry, like the introducing of LRC in on-site energies induces localization-delocalization transition (LDT), the introducing of LRC in hopping integrals also induces LDT and the critical correlation strength is also?c= 2.0. On the other hand, under the condition of fixed average hopping integral t0 and fixed eigenenergy E, when?>?c and disorder strength W is small, the increasing of W will induce delocalization-localization transition (DLT) and the critical disorder strength is Wc= 2t0-|E|. In addition, under the condition of fixed t0 and fixed W, when?>?and W is small, the changing of E will also induce DLT and the critical eigenenergy is Ec=±|2t0-W|.After studied the optical transport properties of the disordered chain of microspherical resonators, we found that, for a long chain, even though the chain behaves in some aspects as a typical strong Anderson localized system, its radiative loss statistics is strongly different from that observed for other lossy optical systems. The distribution function of loss A in this chain is found as f(A)?A-2 exp[-(a-A)2/(b2A2)], the mean value and variance of A, which are related to parameters a and b, are the functions of a single scaling parameter:ratio between the localization length and the loss length in ordered chains. For a short chain, on another hand, we found that the band edges are much more sensitive to disorder than the band center showing that the frequencies near the band edges are more suitable to be used to study the effect of slow light.