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The quantum conductance and its classical wave analogue, the transmittance, are given by the sum of the eigenvalues of the transmission matrix. The lowest transmission eigenvalue in diffusive media might be expected to play a negligible role in the conductance, and, in any case, to be too small to be observed. Here, we observe the lowest transmission eigenchannel in microwave waveguides, though it is orders of magnitude below the nominal noise level, and show that the transmittance is pulled down by global correlation among transmission eigenvalues and among zeros and poles of the transmission matrix. Transmission vanishes either when the energy density on the sample output vanishes at topological transmission zeros or when the longitudinal velocity vanishes precisely at the crossover to a new channel. This lowers the conductance by an amount proportional to the modulation of the density of states. In accord with the correspondence principle, the conductance approaches Ohms law as the number of channels increases with sample width. The exploration of the transmission matrix opens the door to a new understanding of mesoscopic transport and ultrasensitive detection techniques.

The diffusion model is used to calculate 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 acknowledged 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 remain distinct 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 scatter9ng parameters, including the scattering mean free path, oscillate with width of the sample as the number and shape of the propagating channels in the medium change.

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., Comment: 30 pages, 5 figures

The eigenvalues of the transmission matrix provide the basis for a full description of the statistics of steady-state transmission and conductance. At the same time, the ability to excite the sample with the waveform of specific transmission eigenchannels allows for control over transmission. However, the nature of pulsed transmission of transmission eigenchannels and their spectral correlation, which would permit control of propagation in the time domain, has not been discussed. Here we report the dramatic variation of the dynamic properties of transmission with incident waveform. Computer simulations show that lower-transmission eigenchannels respond more promptly to an incident pulse and are correlated over a wide frequency range. We explain these results together with the puzzlingly large dynamic range of transmission eigenvalues in terms of the way quasi-normal modes of the medium combine to form specific transmission eigenchannels. Key factors are the closeness of the illuminating waves to resonance with the modes comprising an eigenchannel, their spectral range, and the interference between the modes. We demonstrate in microwave experiments that the modal characteristics of eigenchannels provide the optimum way efficiently excite specific modes of the medium., Comment: This paper is an expansion of a previous paper http://arxiv.org/abs/1406.3673 and treats many new issues including pulsed transmission of transmission eigenchannels, correlation between modes and transmission eigenchannels, and the efficient and selective excitation of modes. The previous article is no longer under active

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., Comment: 28 Pages, 18 Figures (Review)

Chapter 2 in the book "Light localisation and lasing Random and Pseudo-random Photonic Structures" eds by Mher Ghulinyan and Lorenzo Pavesi, Comment: arXiv admin note: text overlap with arXiv:1210.0279 by other authors

Because 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 for diffusive waves in terms of auxiliary localization lengths determined by the corresponding transmission eigenvalues. The spatial profile of each eigenchannel is shown to be a solution of a generalized diffusion equation. These results reveal the rich structure of transmission eigenchannels and enable the control of wave propagation and the energy distribution inside random media.

Transmission eigenchannels and quasi-normal modes are powerful bases for describing wave transport and controlling transmission and energy storage in disordered media. Here we elucidate the connection between these approaches by expressing the transmission matrix (TM) at a particular frequency as a sum of TMs for individual modes drawn from a broad spectral range. The wide range of transmission eigenvalues and correlation frequencies of eigenchannels of transmission is explained by the increasingly off-resonant excitation of modes contributing to eigenchannels with decreasing transmission and by the phasing between these contributions.

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., Comment: Supplementary Information included at the end of the document

This thesis describes the measurement and analysis of the transmission matrix (TM) for microwave radiation propagating through multichannel random waveguides in the crossover to Anderson localization. Eigenvalues of the transmission matrix and the associated eigenchannels are obtained via a singular value decomposition of the TM. The sum of the transmission eigenvalues yields the transmittance $\it T$, which is the classical analog of the dimensionless conductance $\textsl g$. The dimensionless conductance $\textsl g$ is the electronic conductance in units of the quantum conductance, $G/(e^2/h)$.

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 in contrast observed reflects the distribution of decay rates of the quasi-normal modes within the sample., Comment: 4 pages, 3 figures

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., Comment: 9 pages, 4 figures

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, $L\gg \xi$, 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 wavefunction 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 analogue of the dimensionless conductance. We show that for $L\sim4\xi$ a single eigenvalue of the transmission matrix (TM) dominates transmission and the distribution of the $\ln T$ is Gaussian with a variance equal to the average of $-\ln T$, as conjectured by SPS. For samples in the crossover to localization, $L\sim\xi$, we find a one-sided distribution for $\ln T$. This anomalous distribution is explained in terms of a charge model for the eigenvalues of the transmission matrix $\tau$ in which the Coulomb interaction between charges mimics the repulsion between the eigenvalues of transmission matrix. 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., Comment: 5 pages, 6 figures. PNAS (2014)

Using random matrix calculations, we show that, the contrast between maximally focused intensity through random media and the background of the transmitted speckle pattern for diffusive waves is, \mu_N =1 +N_{eff}, where N eff is the eigenchannel participation number for the transmission matrix. For diffusive waves, N_{eff} is the inverse of the degree of intensity correlation, \kappa. The profile of the focused beam relative to the ensemble average intensity is expressed in terms of the square of the normalized spatial field correlation function, F(\Delta r), and \kappa. These results are demonstrated in microwaves experiments and provide the parameters for optimal focusing and the limits of imaging.

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 $\tau_n$ of the transmission matrix $tt^\dagger$ whose sum equals g. In diffusive samples, the highest eigenvalue, $\tau_1$, is close to unity corresponding to a transmission of nearly 100%, while for localized waves, the average of $\tau_1$, is nearly equal to g. We find that the spacing between average values of $\ln\tau_n$ is constant and demonstrate that when surface interactions are taken into account it is equal to the inverse of the bare conductance., Comment: 5 pages, 5 figures

We report the realization of an active one-dimensional Fibonacci photonic quasi-crystal via spin coating. Manipulation of the luminescence properties of an organic dye embedded in the quasi-crystal is presented and compared to theoretical simulations. The luminescence occurs via the pseudo-bandedge mode and follows the dispersion properties of the Fibonacci crystal. Time resolved luminescence measurement of the active structure shows faster spontaneous emission rate, indicating the effect of the large photon densities available at the bandedge due to the presence of critically localized states. The experimental results are in excellent agreement with the theoretical calculations., Comment: PDF file, 14 pages 4 figures