Formation of optimal-order necklace modes in one-dimensional random photonic superlattices

Abstract

We study the appearance of resonantly coupled optical modes, optical necklaces, in Anderson localized one-dimensional random superlattices through numerical calculations of the accumulated phase. The evolution of the optimal necklace order m * shows a gradual shift towards higher orders with increasing the sample size. We derive an empirical formula that predicts m * and discuss the situation when in a sample length L the number of degenerate in energy resonances exceeds the optimal one. We show how the extra resonances are pushed out to the miniband edges of the necklace, thus reducing the order of the latter by multiples of two.Wave interference phenomena play a crucial role in transport properties of various physical systems from periodic [1] to disordered [2,3]. Among these, originally studied for electronic systems, Anderson localization seems the more intriguing one [4]. It predicts a phase transition form metallic-like conductivity to an insulating regime, when the transport can come to halt with increasing the randomness. On the other hand, in a onedimensional (1D) disordered system, such a transition happens when the sample size L extends beyond the socalled localization length ξ. In such conditions, the wavefunctions become localized within an extension ξ and decay exponentially with distance l. Being essentially an interference phenomenon, Anderson localization has been studied as for electromagnetic and acoustic waves [5], as well as for degenerate atomic gases [6]. Very recently, Anderson localization of optical waves in the microwave regime has been demonstrated in experiments on 1D random multilayer dielectric stacks [7,8].Initially, it has been widely accepted that the conductivity (transmittivity) of a disordered chain is mainly supported by states which are closer situated to the sample center [9]. Later, this was questioned, since for long enough specimen, the states in the center possess significantly reduced probability to support a two-step hopping transport through these states. In late 80's, Pendry [10] and Tartakovskii et al. [11], independently, argued that the conductivity in such systems should be dominated by so-called necklaces -few homogeneously distributed through the sample states, degenerate in energy, and coupled resonantly to delocalize and extend through the chain. The number m of resonant states forming a necklace, was calculated to scale as L 1/2 , while the probability of their occurrence was shown to drop as exp(−L 1/2 ), thus predicting them to be increasingly improbable events in long samples [10]. Therefore, for a certain sample length, a trade-off between the expected number m and their occurrence probability would determine the optimal order of the necklace.In this Letter we study the appearance of optical necklaces in finite 1D random superlattices through numerical calculations of the accumulated phase and follow the evolution of the optimal necklace order when increasing the sample size. We suggest an empirical formula which predicts the optimal necklace order m *...

Characterization of short necklace states in the logarithmic transmission spectra of localized systems

“…Such a periodic on average model is an optical counterpart of the Anderson model of electronic systems and is widely used for localization studies 12,14,15,19,25 . The transmission coefficients of optical waves are calculated by standard transfer matrix method 25 .…”

“…Even though NSs can contribute very large transmission, their formation has rigorous conditions, i.e., the nearly-degenerated localized states are required to be evenly distributed inside the system 7,14,16,17 . In general the localized states with similar frequency are not such well distributed in a specific random configuration.…”

Periodically oscillating Anderson localization in random photonic superlattices with resonant units

“…[11]). As mentioned in the beginning, we observe an even more increased α/β ratio, which is 3.4 now.…”

“…Such a reduction then should result in the formation of optimal-order necklaces [7]. The question of how optimal-order necklaces form in 1D random dielectric stack of resonant layers was addressed very recently [11]. Numerical studies showed that the optimal necklace order m tends to shift gradually towards higher orders with increasing the sample size.…”

Periodic Oscillations in Transmission Decay of Anderson Localized One-Dimensional Dielectric Systems

“…This reduction then should result in the formation of optimal-order necklaces [7]. Very recently, the question of how optimal-order necklaces form in 1D random dielectric stacks of resonant layers was addressed [11]. It was shown through numerical studies that the optimal necklace order m shifts gradually towards higher orders with increasing the sample size.…”