Localization of electromagnetic waves in two-dimensional disordered systems

Localization of plasmon modes in a 2D photonic nanostructure with a controlled disorder

In this paper, we focus on the optical properties of disordered hole arrays etched in a gold thin film. The disorder is induced and controlled using hole displacements following a Gaussian distribution and starting from a periodic array. The nanostructures present a transition from ordered arrays to short-range ordered arrays and random arrays by increasing the disorder amount. The associated optical properties are characterized in far and near fields by complementary approaches (absorption spectroscopy, classical scanning near field optical microscopy (SNOM) and Finite Difference Time Domain (FDTD) simulations). By increasing the disorder, a broadened absorption up to 30% in the far-field is achieved. Experiments in agreement with FDTD simulations point out the energy localization induced by the disorder and the dependence on the amount of disorder and on the excitation wavelength. By using a controlled disorder, we also show that the effect of these two parameters is also closely linked.

Level spacing statistics for light in two-dimensional disordered photonic crystals

We study the distribution of eigenfrequency spacings (the so-called level spacing statistics) for light in a two-dimensional (2D) disordered photonic crystal composed of circular dielectric (silicon) rods in air. Disorder introduces localized transverse-magnetic (TM) modes into the band gap of the ideal crystal. The level spacing statistics is found to approach the Poisson distribution for these modes. In contrast, for TM modes outside the band gap and for transverse-electric (TE) modes at all frequencies, the level spacing statistics follows the Wigner-Dyson distribution.

Two-dimensional disorder for broadband, omnidirectional and polarization-insensitive absorption

The surface of thin-film solar cells can be tailored with photonic nanostructures to allow light trapping in the absorbing medium. This in turn increases the optical thickness of the film and thus enhances their absorption. Such a coherent light trapping is generally accomplished with deterministic photonic architectures. Here, we experimentally explore the use of a different nanostructure, a disordered one, for this purpose. We show that the disorder-induced modes in the film allow improvements in the absorption over a broad range of frequencies and impinging angles.

Photon management in two-dimensional disordered media

Elaborating reliable and versatile strategies for efficient light coupling between free space and thin films is of crucial importance for new technologies in energy efficiency. Nanostructured materials have opened unprecedented opportunities for light management, notably in thin-film solar cells. Efficient coherent light trapping has been accomplished through the careful design of plasmonic nanoparticles and gratings, resonant dielectric particles and photonic crystals. Alternative approaches have used randomly textured surfaces as strong light diffusers to benefit from their broadband and wide-angle properties. Here, we propose a new strategy for photon management in thin films that combines both advantages of an efficient trapping due to coherent optical effects and broadband/wide-angle properties due to disorder. Our approach consists of the excitation of electromagnetic modes formed by multiple light scattering and wave interference in two-dimensional random media. We show, by numerical calculations, that the spectral and angular responses of thin films containing disordered photonic patterns are intimately related to the in-plane light transport process and can be tuned through structural correlations. Our findings, which are applicable to all waves, are particularly suited for improving the absorption efficiency of thin-film solar cells and can provide a new approach for high-extraction-efficiency light-emitting diodes.

Transmission of electromagnetic waves through a two-layer plasma structure with spatially nonuniform electron density

Transmission of a p-polarized electromagnetic wave through a two-layer plasma structure with spatially nonuniform distributions of electron density in the layers is studied. The case, when the electromagnetic wave is obliquely incident on the structure and is evanescent in both plasma layers, is considered. The conditions for total transparency of the two-layer structure are found for the thin slab case and when the plasma inhomogeneity is weak. It is shown that the transmission coefficient of the p-polarized wave can be about unity, even if the plasma inhomogeneity is large. The effects of plasma inhomogeneity on transparency of the structure are more important if the slabs are thick, comparing with the case of thin layers.

Resonant transparency of a two-layer plasma structure in a magnetic field

Transparency of a two-layer plasma structure in an external steady-state magnetic field, perpendicular to the wave incidence plane, is studied. The case of the p-polarized electromagnetic wave is considered. The electromagnetic wave is obliquely incident on the two-layer structure and is evanescent in both layers. The conditions for total transparency of the two-layer structure are found. The parametric dependencies of the transparency coefficient on the plasma slab widths, the magnitude of the wave number component, as well on the magnetic field magnitude are obtained.

The effects of periodic and quasi-periodic orders on the photonic bandgap structures of microring coupled-resonator optical waveguides

We present a coupling matrix formalism to investigate the effects of periodic and quasi-periodic orders on the photonic bandgap (PBG) structures of coupled-resonator optical waveguides (CROWs) based on microring resonators. For the periodic order case, size-tuned defects are introduced at periodic locations among the regular rings, which are size-untuned, to form a periodic ordered CROW system. The periodic coupled defects result in multiple localization states that lead to the formation of mini-defect bands and mini-PBGs within the PBG of a defect-free CROW. The position and number of such mini-defect bands depend on the size tuning of the defects. For the quasi-periodic order case, the arrangement of the defects and the regular rings in the ring cascade is an intermediate between periodic order and randomness, thus forming a quasi-periodic ordered CROW system. The effects of quasi-periodicity on the PBG structures are illustrated using the Fibonacci sequences, which result in a single high-Q localized state to appear that gradually transits to a mini-band within a wide photonic stop band as the number of lattice cells increases.

Stochastic differential equation approach for waves in a random medium

We present a mathematical approach that simplifies the theoretical treatment of electromagnetic localization in random media and leads to closed-form analytical solutions. Starting with the assumption that the dielectric permittivity of the medium has delta-correlated spatial fluctuations, and using Ito's lemma, we derive a linear stochastic differential equation for a one-dimensional random medium. The equation leads to localized wave solutions. The localized wave solutions have a localization length that scales as L approximately omega(-2) for low frequencies whereas in the high-frequency regime this length behaves as L approximately omega(-2/3) .

Approximate equivalence between guided modes in a low-contrast photonic bandgap fiber and Maxwell TM modes of a high-contrast two-dimensional photonic structure

We present a formal analogy between the eigenvalue problem for guided scalar modes in a low-contrast photonic bandgap fiber and quasi-stationary TM modes of a two-dimensional (2D) photonic structure. Using this analogy, we numerically study the confinement losses of disordered microstructured fibers through the leakage rate of an open 2D system with high refractive index inclusions. Our results show that for large values of the disorder, the confinement losses increase. However, they also suggest that losses might be improved in strongly disordered fibers by exploring ranges of physical parameters where Anderson localization sets in.

Localized modes in a finite-size open disordered microwave cavity

We present measurements of the spatial intensity distribution of localized modes in a two-dimensional open microwave cavity randomly filled with cylindrical dielectric scatterers. We show that each of these modes displays a range of localization lengths, and we successfully relate the largest value to the measured leakage rate at the boundary. These results constitute unambiguous signatures of the existence of strongly localized electromagnetic modes in two-dimensional open random media.

Localization of phonon polaritons in disordered polar media

The localization of the hybrid modes of phonons and photons in polar matter is investigated in the presence of random scatterers theoretically. We employ the self-consistent generalized Born-Huang approach to derive effective equations describing the phonon-polariton fields. Based on these equations, the density of states and various localization properties are exploited in two-dimensional systems both analytically and numerically within the framework of the Anderson model with a non-Hermitian effective Hamiltonian. Consequently, it is shown that the disorder effect brings some intriguing features which include the appearance of the localized states in the polariton bottleneck in the energy spectrum and the collapse of the energy gap. In addition, an analysis is given of the polariton level-spacing distribution.

High-transmission waveguide with a small radius of curvature at a bend fabricated by use of a circular photonic crystal

A bent photonic crystal waveguide was fabricated by use of a lattice pattern of a circular photonic crystal that allowed high transmission for a broad band of wavelengths with a small radius of curvature at a bend. The waveguide was fabricated by use of alumina rods with a diameter of 3 mm. Windows of high transmission as a result of waveguiding were observed near 9 and 15 GHz. By measurement of the relative wave intensity [E]2 along the line defects, the propagation losses in the straight and the bent sections were estimated at 9.3 GHz to be 0.04 and 0.03 dB/mm, respectively.

Localized modes in random arrays of cylinders

Anderson localization of classical waves in random arrays of dielectric cylinders is numerically investigated as a function of the distribution of their diameters. We show that using polydispersed resonant scatterers increases the localization length, while using identical resonant scatterers fosters Anderson localization. We discuss this collective process and link it to the effect of proximity resonances that has been studied in the case of a small number of resonant scatterers.

Localization of classical waves in weakly scattering two-dimensional media with anisotropic disorder

We study the localization of classical waves in weakly scattering two-dimensional systems with anisotropic disorder. The analysis is based on a perturbative path-integral technique combined with a spectral filtering that accounts for the first-order Bragg scattering only. It is shown that in the long-wavelength limit the radiation is always localized, and the localization length is independent of the direction of propagation, the latter in contrast to the predictions based on an anisotropic tight-binding model. For shorter wavelengths that are comparable to the correlation scales of the disorder, the transport properties of disordered media are essentially different in the directions along and across the correlation ellipse. There exists a frequency-dependent critical value of the anisotropy parameter, below which waves are localized at all angles of propagation. Above this critical value, the radiation is localized only within some angular sectors centered at the short axis of the correlation ellipse and is extended in other directions.

Linear and nonlinear optical studies in photonic crystal alloys

We report on linear transmittance and reflectance as well as on third-harmonic generation in photonic crystal alloys formed by various compositions of polystyrene and poly (methyl methacrylate) colloidal spheres of the same size. These photonic crystal alloys are structurally ordered but contain refractive-index disorder and thus provide a random variation of scattering potential. The stopgap shows a monotonic shift in wavelength as a function of composition that can be fitted by assuming an effective dielectric constant for the colloidal spheres. In each alloy a dramatic enhancement of third-harmonic generation is observed, always on the short-wavelength side of the stopgap.

Propagation inhibition and localization of electromagnetic waves in two-dimensional random dielectric systems

We rigorously calculate the propagation and scattering of electromagnetic waves by rectangular and random arrays of dielectric cylinders in a uniform medium. For regular arrays, the band structures are computed and complete bandgaps are discovered. For random arrays, the phenomenon of wave transmission and scattering is investigated and compared in two scenarios: (1) Wave propagating through the array of cylinders; this is the scenario which has been commonly considered in the literature, and (2) wave transmitted from a source located inside the ensemble. We show that within complete band gaps, results from the two scenarios are similar. Outside the gaps, however, there could be a distinct difference, that is, wave transmission can be inhibited by disorders in the first scenario, but such an inhibition may not prevail in the second scenario.

Enhanced second-harmonic generation from planar photonic crystals

Strongly enhanced second-harmonic generation is observed from a two-dimensional square lattice GaAs/AlGaAs photonic crystal waveguide when the fundamental beam, the second-harmonic beam, or both beams resonantly couple to a leaky eigenmode. P-polarized second-harmonic spectra are obtained for s-polarized, 150-fs pump pulses that are tuned from 5000 to 5600 cm(-1) and directed along the gamma-chi direction of the crystal for various angles of incidence. Compared with off-resonant conditions, enhancements of >1200x in the second-harmonic conversion are observed for resonant coupling of both the fundamental and the second-harmonic fields to leaky eigenmnodes. The angular and spectral positions of the peaks are in good agreement with simulations.

Simulation study of localization of electromagnetic waves in two-dimensional random dipolar systems

We study the propagation and scattering of electromagnetic waves by random arrays of dipolar cylinders in a uniform medium. A set of self-consistent equations, incorporating all orders of multiple scattering of the electromagnetic waves, is derived from first principles and then solved numerically for electromagnetic fields. For certain ranges of frequencies, spatially localized electromagnetic waves appear in such a simple but realistic disordered system. Dependence of localization on the frequency, radiation damping, and filling factor is shown. The spatial behavior of the total, coherent, and diffusive waves is explored in detail, and found to comply with a physical intuitive picture. A phase diagram characterizing localization is presented, in agreement with previous investigations on other systems.

Diffusive and localization behavior of electromagnetic waves in a two-dimensional random medium

In this paper, we discuss the transport phenomena of electromagnetic waves in a two-dimensional random system which is composed of arrays of electrical dipoles, following the model presented earlier by Erdogan et al. [J. Opt. Soc. Am. B 10, 391 (1993)]. A set of self-consistent equations is presented, accounting for the multiple scattering in the system, and is then solved numerically. A strong localization regime is discovered in the frequency domain. The transport properties within, near the edge of, and nearly outside the localization regime are investigated for different parameters such as filling factor and system size. The results show that within the localization regime, waves are trapped near the transmitting source. Meanwhile, the diffusive waves follow an intuitive but expected picture. That is, they increase with traveling path as more and more random scattering incurs, followed by a saturation, then start to decay exponentially when the travelling path is large enough, signifying the localization effect. For the cases where the frequencies are near the boundary of or outside the localization regime, the results of diffusive waves are compared with the diffusion approximation, showing less encouraging agreement as in other systems [Asatryan et al., Phys. Rev. E 67, 036605 (2003)].

Localization of classical waves in two-dimensional random media: a comparison between the analytic theory and exact numerical simulation

The localization length of classical waves in two-dimensional random media is calculated exactly, and is compared with the theoretical prediction from the current analytic theory. Significant discrepancies are observed. It is also shown that as the frequency varies, critical changes in the localization behavior can occur. However, by a rescaling of parameters the two results tend to match each other for weak scattering. Possible reasons for the discrepancies are discussed.

Localization of electromagnetic waves in a two-dimensional random medium

Motivated by previous investigations on the radiative effects of the electric dipoles embedded in structured cavities, localization of electromagnetic waves in two dimensions is studied ab initio for a system consisting of many randomly distributed two-dimensional dipoles. A set of self-consistent equations, incorporating all orders of multiple scattering of the electromagnetic waves, is derived from first principles and then solved numerically for the total electromagnetic field. The results show that spatially localized electromagnetic waves are possible in such a simple but realistic disordered system. When localization occurs, a coherent behavior appears and is revealed as a unique property differentiating localization from either the residual absorption or the attenuation effects.

Effects of geometric and refractive index disorder on wave propagation in two-dimensional photonic crystals

The effects of disorder in the geometry and refractive index on the transmittance of two-dimensional photonic crystals composed of dielectric circular cylinders are considered, including randomness of radii, positions of the cylinder centers, and thickness of each layer of the photonic crystal. The effects of combinations of different types of strong disorder are also considered. The localization and homogenization properties of disordered photonic crystals are investigated. Analytical expressions for the two-dimensional localization length in the form of integrals are presented for both polarizations. It is shown numerically that the slope of the exponential divergence of the localization length in two dimensions is proportional to the inverse of the square of randomness for strong disorder and proportional to the inverse of the randomness for weak disorder. The effective dielectric constants for both polarizations in the case of strong disorder are also found. The transition from localization to homogenization is discussed and the terms responsible for this transition are identified and investigated.

Calculation of electromagnetic properties of regular and random arrays of metallic and dielectric cylinders

A method is developed to calculate electromagnetic properties of arrays of metallic and dielectric cylinders. It incorporates and exploits cylindrical boundary conditions and Rayleigh identities for efficient, high-accuracy calculation of scattering off individual layers that are stacked into arrays using scattering matrices. The method enables absorption, dispersion, and randomness to be incorporated efficiently, and reproduces known results with vastly improved speed and accuracy. It is used to demonstrate existence of states introduced into photonic band gaps of a dielectric array by disorder, and anomalous absorption behavior in arrays of aluminum cylinders.

Effects of disorder on wave propagation in two-dimensional photonic crystals

The electromagnetic transmittance of disordered two-dimensional photonic crystals composed of circular cylinders is investigated as a function of wavelength and polarization. At short wavelengths, the transmittance shows a band structure similar to that found in the optical absorption spectrum of amorphous semiconductors, with impurity states increasingly appearing on the long wavelength side of the band gaps as the degree of disorder is increased. In the long-wavelength limit, Anderson localization of waves is found, provided that the wavelength is not so large that the random photonic crystal can be viewed as homogeneous. The localization properties in this regime are studied and an analytic expression for the dependence of the localization length on wavelength is derived. In the limit of extremely long wavelengths, the system homogenizes and can be replaced by an equivalent one with uniform effective refractive index, whose form is derived for both polarizations. Analysis of the crossover between localization and homogenization is also presented.