Anderson localization of classical waves in weakly scattering one-dimensional Levy lattices

© 2018 American Physical Society. Anderson localization of classical waves in weakly scattering one-dimensional Levy lattices is studied analytically and numerically. The disordered medium is composed of layers with alternating refractive indices and with thickness disorder distributed according to the Pareto distribution ∼1/x(α+1). In Levy lattices the variance (or both variance and mean) of a random parameter does not exist, which leads to a different functional form for the localization length. In this study an equation for the localization length is obtained, and it is found to be in excellent agreement with the numerical calculations throughout the spectrum. The explicit asymptotic equations for the localization lengths for both short and long wavelengths have been deduced. It is shown that the localization length tends to a constant at short wavelengths and it is determined by the layer interface Fresnel coefficient. At the long wavelengths the localization length is proportional to the power of the wavelength ∼λα for 12, where the variance of the random distribution exists, the localization length attains its classical long-wavelength asymptotic form ∼λ2

Localisation and disorder in the design of 2D photonic crystal devices

Photonic crystals are meta-materials that can inhibit the propagation of light in all directions for specific wavelength ranges. Material or structural defects can be introduced into the crystal to cause localised modes, providing the ability to mould the flow of light on the wavelength scale and allowing the development of miniaturised, integrated photonic devices. For this reason, photonic crystals will likely be key building blocks for future micro-optical and communication technology. In this paper, we examine the Bloch mode modelling of 2D photonic crystal structures with application to the analysis of photonic crystal waveguides and their susceptibility to disorder, which provides a framework for studying fabrication tolerances in realistic devices

Modeling waveguides in photonic woodpiles using the fictitious source superposition method

We extend the fictitious source superposition method in order to model linear defects in photonic woodpiles, and we use the method to model a waveguide that is created by changing either the radius or refractive index of a single rod of an infinite woodpile composed of chalcogenide glass cylinders. In one instance, a nearly constant dispersion was observed over a sizable kx interval, where kx is the Bloch vector in the waveguiding direction, making this a compelling geometry for slow-light waveguides. The principal advantage of the method is that it does not rely on a supercell, thus avoiding what is possibly the greatest source of inefficiency present in most of the other methods that are used for modeling these structures. Instead, the method proceeds by placing an artificial source inside each rod of the defect layer and then subsequently taking an appropriate field superposition to remove all but one of these sources. The remaining source can then be used to mimic the fields that would be produced by a defect rod. © 2011 Optical Society of America

Gap-edge asymptotics of defect modes in two-dimensional photonic crystals

We consider defect modes created in complete gaps of 2D photonic crystals by perturbing the dielectric constant in some region. We study their evolution from a band edge with increasing perturbation using an asymptotic method that approximates the Green function by its dominant component which is associated with the bulk mode at the band edge. From this, we derive a simple exponential law which links the frequency difference between the defect mode and the band edge to the relative change in the electric energy. We present numerical results which demonstrate the accuracy of the exponential law, for TE and TM polarizations, hexagonal and square arrays, and in each of the first and second band gaps. © 2007 Optical Society of America

Conductance of photons and Anderson localization of light

Conductance properties of photons in disordered two-dimensional photonic crystals is calculated using exact multipole expansions technique. The Landauers two-terminal formula is used to calculate average of the conductance, its variance and the probability density distribution

Shallow defect states in two-dimensional photonic crystals

We investigate localized defect states near the edge of a band gap in a two-dimensional photonic crystal. An asymptotic approach based on Green's functions leads to analytical results both for the frequency and for the spatial behavior of the defect states. In particular, we find a simple exponential law which relates the change in frequency of the defect states to the relative change in electrical energy of the Bloch modes on the band edge, and to the density of states in the photonic crystal. We find that the symmetries of the Bloch modes at band extrema play an important role in the manifestation and evolution of defect states. We confirm the analysis with numerical simulations based on the fictitious source superposition method. © 2008 The American Physical Society

Two-dimensional local density of states in two-dimensional photonic crystals

We calculate the two-dimensional local density of states (LDOS) for two-dimensional photonic crystals composed of a finite cluster of circular cylinders of infinite length. The LDOS determines the dynamics of radiation sources embedded in a photonic crystal. We show that the LDOS decreases exponentially inside the crystal for frequencies within a photonic band gap of the associated infinite array and demonstrate that there exist "hot" and "cold" spots inside the cluster even for wavelengths inside a gap, and also for wavelengths corresponding to pass bands. For long wavelengths the LDOS exhibits oscillatory behavior in which the local density of states can be more than 30 times higher than the vacuum level. © 2001 Optical Society of America

Exact modelling of generalised defect modes in photonic crystal structures

An exact theory for modelling modes of generalised defects in 2D photonic crystals (PCs) with a genuinely infinite cladding is presented. The approach builds on our fictitious source superposition method for simple defects and permits an elegant extension allowing the modelling of arbitrary defects. Numerical results that demonstrate the accuracy and efficiency of the extended method are presented. We also use the method to study the evolution of the mode generated by varying the refractive index of a single defect cylinder and find significant differences between the behaviour of defects in rod-type and hole-type PCs. © 2007 Elsevier B.V. All rights reserved

Diffusion and anomalous diffusion of light in two-dimensional photonic crystals

The transport properties of electromagnetic waves in disordered, finite, two-dimensional photonic crystals composed of circular cylinders are considered. Transport parameters such as the transport and scattering mean free paths and the transport velocity are calculated, for the case where the electromagnetic radiation has its electric field along the cylinder axes. The range of the parameters in which the diffusion process can take place is specified. It is shown that the transport velocity [Formula presented] can be as much as [Formula presented] times less than its free space value, while just outside the cluster [Formula presented] can be 0.3c. The effects of weak and strong disorders on the transport velocity are investigated. Different regimes of the wave transport—ordered propagation, diffusion, and anomalous diffusion—are demonstrated, and it is inferred that Anderson localization is incipient in the latter regime. Exact numerical calculations from the Helmholtz equation are shown to be in good agreement with the diffusion approximation. © 2003 The American Physical Society