Invariance Principle for Wave Propagation inside Inhomogeneously Disordered Materials.

Disorder is more the rule than the exception in natural and synthetic materials. Nonetheless, wave propagation within inhomogeneously disordered materials has received scant attention. We combine microwave experiments and theory to find the spatial variation of generic wave propagation quantities in inhomogeneously disordered materials. We demonstrate that wave statistics within samples of any dimension are independent of the detailed structure of a material and depend only on the net strengths of distributed scattering and reflection between the observation point and each of the boundaries.

Robust reconfigurable electromagnetic pathways within a photonic topological insulator.

The discovery of topological photonic states has revolutionized our understanding of electromagnetic propagation and scattering. Endowed with topological robustness, photonic edge modes are not reflected from structural imperfections and disordered regions. Here we demonstrate robust propagation along reconfigurable pathways defined by synthetic gauge fields within a topological photonic metacrystal. The flow of microwave radiation in helical edge modes following arbitrary contours of the synthetic gauge field between bianisotropic metacrystal domains is unimpeded. This is demonstrated in measurements of the spectrum of transmission and time delay along the topological domain walls. These results provide a framework for freely steering electromagnetic radiation within photonic structures.

Transport through modes in random media.

Excitations in complex media are superpositions of eigenstates that are referred to as 'levels' for quantum systems and 'modes' for classical waves. Although the Hamiltonian of a complex system may not be known or solvable, Wigner conjectured that the statistics of energy level spacings would be the same as for the eigenvalues of large random matrices. This has explained key characteristics of neutron scattering spectra. Subsequently, Thouless and co-workers argued that the metal-insulator transition in disordered systems could be described by a single parameter, the ratio of the average width and spacing of electronic energy levels: when this dimensionless ratio falls below unity, conductivity is suppressed by Anderson localization of the electronic wavefunction. However, because of spectral congestion due to the overlap of modes, even for localized waves, a comprehensive modal description of wave propagation has not been realized. Here we show that the field speckle pattern of transmitted radiation--in this case, a microwave field transmitted through randomly packed alumina spheres--can be decomposed into a sum of the patterns of the individual modes of the medium and the central frequency and linewidth of each mode can be found. We find strong correlation between modal field speckle patterns, which leads to destructive interference between modes. This allows us to explain complexities of steady state and pulsed transmission of localized waves and to harmonize wave and particle descriptions of diffusion.

Microwave conductance in random waveguides in the cross-over to Anderson localization and single-parameter scaling.

The nature of transport of electrons and classical waves in disordered systems depends upon the proximity to the Anderson localization transition between freely diffusing and localized waves. The suppression of average transport and the enhancement of relative fluctuations in conductance in one-dimensional samples with lengths greatly exceeding the localization length, L>>ξ, are related in the single-parameter scaling (SPS) theory of localization. However, the difficulty of producing an ensemble of statistically equivalent samples in which the electron wave function is temporally coherent has so-far precluded the experimental demonstration of SPS. Here we demonstrate SPS in random multichannel systems for the transmittance T of microwave radiation, which is the analog of the dimensionless conductance. We show that for L∼4ξ, a single eigenvalue of the transmission matrix (TM) dominates transmission, and the distribution of the T is Gaussian with a variance equal to the average of −ln T, as conjectured by SPS. For samples in the cross-over to localization, L∼ξ, we find a one-sided distribution for T. This anomalous distribution is explained in terms of a charge model for the eigenvalues of the TM τ in which the Coulomb interaction between charges mimics the repulsion between the eigenvalues of TM. We show in the localization limit that the joint distribution of T and the effective number of transmission eigenvalues determines the probability distributions of intensity and total transmission for a single-incident channel.

Statistics and control of waves in disordered media.

Fundamental concepts in the quasi-one-dimensional geometry of disordered wires and random waveguides in which ideas of scaling and the transmission matrix were first introduced are reviewed. We discuss the use of the transmission matrix to describe the scaling, fluctuations, delay time, density of states, and control of waves propagating through and within disordered systems. Microwave measurements, random matrix theory calculations, and computer simulations are employed to study the statistics of transmission and focusing in single samples and the scaling of the probability distribution of transmission and transmittance in random ensembles. Finally, we explore the disposition of the energy density of transmission eigenchannels inside random media.

Motion of intensity maxima in averaged speckle patterns of transmitted radiation.

We show intensity maxima in speckle patterns averaged over a frequency interval diffuse as the frequency is scanned with a diffusion coefficient that decreases linearly with the width of the frequency interval for moderate intervals. This makes it possible to find the diffusion coefficient even with data averaged over a frequency window. These results apply as well to speckle patterns averaged over time in systems with internal motion and so provide a means for characterizing dynamic systems.

Transmission eigenchannels and the densities of states of random media.

We show in microwave measurements and computer simulations that the contribution of each eigenchannel of the transmission matrix to the density of states (DOS) is the derivative with angular frequency of a composite phase shift. The accuracy of the measurement of the DOS determined from transmission eigenchannels is confirmed by the agreement with the DOS found from the decomposition of the field into modes. The distribution of the DOS, which underlies the Thouless number, is substantially broadened in the Anderson localization transition. We find a crossover from constant to exponential scaling of fluctuations of the DOS normalized by its average value. These results illuminate the relationships between scattering, stored energy, and dynamics in complex media.

Focusing and energy deposition inside random media.

The degree of control over waves transmitted through random media is determined by characteristics of the singular values of the transmission matrix. This Letter explores focusing and energy deposition in the interior of disordered samples and shows that these are determined by the singular values of the matrix relating the field channels inside a medium to the incident channels. Through calculations and simulations, we discovered that the variation with depth of the maximal energy density and the contrast in optimal focusing are determined by the participation number M(z) of the energy density eigenvalues, while its inverse gives the variance of the energy density at z in a single configuration.

Phase singularity diffusion.

We follow the trajectories of phase singularities at nulls of intensity in the speckle pattern of waves transmitted through random media as the frequency of the incident radiation is scanned in microwave experiments and numerical simulations. Phase singularities are observed to diffuse with a linear increase of the square displacement 〈R2〉 with frequency shift. The product of the diffusion coefficient of phase singularities in the transmitted speckle pattern and the photon diffusion coefficient through the random medium is proportional to the square of the effective sample length. This provides the photon diffusion coefficient and a method for characterizing the motion of dynamic material systems.

Velocities of transmission eigenchannels and diffusion.

The diffusion model is used to calculate both the time-averaged flow of particles in stochastic media and the propagation of waves averaged over ensembles of disordered static configurations. For classical waves exciting static disordered samples, such as a layer of paint or a tissue sample, the flux transmitted through the sample may be dramatically enhanced or suppressed relative to predictions of diffusion theory when the sample is excited by a waveform corresponding to a transmission eigenchannel. Even so, it is widely assumed that the velocity of waves is irretrievably randomized in scattering media. Here we demonstrate in microwave measurements and numerical simulations that the statistics of velocity of different transmission eigenchannels are distinct and remains so on all length scales and are identical on the incident and output surfaces. The interplay between eigenchannel velocities and transmission eigenvalues determines the energy density within the medium, the diffusion coefficient, and the dynamics of propagation. The diffusion coefficient and all scattering parameters, including the scattering mean free path, oscillate with the width of the sample as the number and shape of the propagating channels in the medium change.

Transmission statistics and focusing in single disordered samples.

We show in microwave experiments and random matrix calculations that in samples with a large number of channels the statistics of transmission for different incident channels relative to the average transmission is determined by a single parameter, the participation number of the eigenvalues of the transmission matrix, M. Its inverse, M(-1), is equal to the variance of relative total transmission of the sample, while the contrast in maximal focusing is equal to M. The distribution of relative total transmission changes from Gaussian to negative exponential over the range in which M(-1) changes from 0 to 1. This provides a framework for transmission and imaging in single samples.

Focusing through random media in space and time: a transmission matrix approach.

We exploit the evolution in time of the transmission matrix following pulse excitation of a random medium to focus radiation at a selected time delay t' and position r. The temporal profile of a focused microwave pulse is the same as the incident Gaussian pulse. The contrast in space at time t' of the focused wave is determined by the participation number of transmission eigenvalues M' and the size N' of the measured transmission matrix. The initial rise and subsequent decay of the contrast observed reflect the distribution of decay rates of the quasi-normal modes within the sample.

Transmission eigenvalues and the bare conductance in the crossover to Anderson localization.

We measure the field transmission matrix t for microwave radiation propagating through random waveguides in the crossover to Anderson localization. From these measurements, we determine the dimensionless conductance g and the individual eigenvalues τ(n) of the transmission matrix tt(†) whose sum equals g. In diffusive samples, the highest eigenvalue, τ(1), is close to unity corresponding to a transmission of nearly 100%, while for localized waves, the average of τ(1), is nearly equal to g. We find that the spacing between average values of lnτ(n) is constant and demonstrate that when surface interactions are taken into account it is equal to the inverse of the bare conductance.

Chiral diffraction gratings in twisted microstructured fibers.

We observed dips in transmission spectra of uniformly twisted pure-silica microstructured fibers. The spectral positions of the dips and their insensitivity to the surrounding medium are consistent with Bragg diffraction from the helical structure. The reproducibility of the variation of the dip wavelength with temperature up to 1000 degrees C makes the chiral diffraction grating suitable for high-temperature sensing.

Selectively exciting quasi-normal modes in open disordered systems.

Transmission through disordered samples can be controlled by illuminating a sample with waveforms corresponding to the eigenchannels of the transmission matrix (TM). But can the TM be exploited to selectively excite quasi-normal modes and so control the spatial profile and dwell time inside the medium? We show in microwave and numerical studies that spectra of the TM can be analyzed into modal transmission matrices of rank unity. This makes it possible to enhance the energy within a sample by a factor equal to the number of channels. Limits to modal selectivity arise, however, from correlation in the speckle patterns of neighboring modes. In accord with an effective Hamiltonian model, the degree of modal speckle correlation grows with increasing modal spectral overlap and non-orthogonality of the modes of non-Hermitian systems. This is observed when the coupling of a sample to its surroundings increases, as in the crossover from localized to diffusive waves.

Diffusion in translucent media.

Diffusion is the result of repeated random scattering. It governs a wide range of phenomena from Brownian motion, to heat flow through window panes, neutron flux in fuel rods, dispersion of light in human tissue, and electronic conduction. It is universally acknowledged that the diffusion approach to describing wave transport fails in translucent samples thinner than the distance between scattering events such as are encountered in meteorology, astronomy, biomedicine, and communications. Here we show in optical measurements and numerical simulations that the scaling of transmission and the intensity profiles of transmission eigenchannels have the same form in translucent as in opaque media. Paradoxically, the similarities in transport across translucent and opaque samples explain the puzzling observations of suppressed optical and ultrasonic delay times relative to predictions of diffusion theory well into the diffusive regime.

Pseudo-spin-valley coupled edge states in a photonic topological insulator.

Pseudo-spin and valley degrees of freedom engineered in photonic analogues of topological insulators provide potential approaches to optical encoding and robust signal transport. Here we observe a ballistic edge state whose spin-valley indices are locked to the direction of propagation along the interface between a valley photonic crystal and a metacrystal emulating the quantum spin-Hall effect. We demonstrate the inhibition of inter-valley scattering at a Y-junction formed at the interfaces between photonic topological insulators carrying different spin-valley Chern numbers. These results open up the possibility of using the valley degree of freedom to control the flow of optical signals in 2D structures.

Author Correction: Interplay between evanescence and disorder in deep subwavelength photonic structures.

This corrects the article DOI: 10.1038/ncomms12927.

Observation of Anderson localization in disordered nanophotonic structures.

Anderson localization is an interference effect crucial to the understanding of waves in disordered media. However, localization is expected to become negligible when the features of the disordered structure are much smaller than the wavelength. Here we experimentally demonstrate the localization of light in a disordered dielectric multilayer with an average layer thickness of 15 nanometers, deep into the subwavelength regime. We observe strong disorder-induced reflections that show that the interplay of localization and evanescence can lead to a substantial decrease in transmission, or the opposite feature of enhanced transmission. This deep-subwavelength Anderson localization exhibits extreme sensitivity: Varying the thickness of a single layer by 2 nanometers changes the reflection appreciably. This sensitivity, approaching the atomic scale, holds the promise of extreme subwavelength sensing.

Interplay between evanescence and disorder in deep subwavelength photonic structures.

Deep subwavelength features are expected to have minimal impact on wave transport. Here we show that in contrast to this common understanding, disorder can have a dramatic effect in a one-dimensional disordered optical system with spatial features a thousand times smaller than the wavelength. We examine a unique regime of Anderson localization where the localization length is shown to scale linearly with the wavelength instead of diverging, because of the role of evanescent waves. In addition, we demonstrate an unusual order of magnitude enhancement of transmission induced due to localization. These results are described for electromagnetic waves, but are directly relevant to other wave systems such as electrons in multi-quantum-well structures.