Ohm's law lost and regained: observation and impact of transmission and velocity zeros.

The quantum conductance and its classical wave analogue, the transmittance, are given by the sum of the eigenvalues of the transmission matrix. However, neither measurements nor theoretical analysis of the transmission eigenchannels have been carried out to explain the dips in conductance found in simulations as new channels are introduced. Here, we measure the microwave transmission matrices of random waveguides and find the spectra of all transmission eigenvalues, even at dips in the lowest transmission eigenchannel that are orders of magnitude below the noise in the transmission matrix.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

Universal structure of transmission eigenchannels inside opaque media.

As the desire to explore opaque materials is ordinarily frustrated by multiple scattering of waves, attention has focused on the transmission matrix of the wave field. This matrix gives the fullest account of transmission and conductance and enables the control of the transmitted flux; however, it cannot address the fundamental issue of the spatial profile of eigenchannels of the transmission matrix inside the sample. Here we obtain a universal expression for the average disposition of energy of transmission eigenchannels within random diffusive systems in terms of auxiliary localization lengths determined by the corresponding transmission eigenvalues.

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.

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.

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.

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.

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.

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.

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.

Mesoscopic speckle.

We have measured first and second order statistics of the velocity of phase singularities v in evolving speckle patterns of microwave radiation transmitted through random quasi-1D samples as the frequency is swept and relate these to global statistics of speckle evolution. When v is normalized by the standard deviation of the fractional intensity change, the probability distribution and correlation function of v approach those for random Gaussian fields even for localized waves. Analogous results are found for transmitted intensity normalized by the total transmission.

Intensity statistics and photon localization beyond one dimension.

We find a transformation in the probability density of intensity for light transmitted through stacks of glass slides of increasing thickness from one-dimensional (1D) to quasi-1D statistics. We argue that this represent a crossover to universal first order intensity statistics beyond 1D associated with the appearance of phase singularities at intensity nulls in the speckle pattern.

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.

Photon delocalization transition in dimensional crossover in layered media.

We report a crossover in optical propagation in random layered media from localization towards diffusion as the interaction of the wave with the sample is transformed from one to three dimensional due to nonuniformity in the layer thickness. The crossover occurs at the point that the lateral spread of the wave equals the transverse coherence length in the transmitted speckle pattern.