## Dissertations, Theses, and Capstone Projects

## Date of Degree

6-2017

## Document Type

Dissertation

## Degree Name

Ph.D.

## Program

Physics

## Advisor

Azriel Z. Genack

## Committee Members

Alexander Lisyansky

Alexander Khanikaev

Chushun Tian

Victor Gopar

## Subject Categories

Condensed Matter Physics | Optics

## Keywords

Random media, Wave scattering, Statistical optics

## Abstract

This thesis presents results of studies of wave scattering within and transmission through random and periodic systems. The main focus is on energy profiles inside quasi-1D and 1D random media.

The connection between transport and the states of the medium is manifested in the equivalence of the dimensionless conductance, g, and the Thouless number which is the ratio of the average linewidth and spacing of energy levels. This equivalence and theories regarding the energy profiles inside random media are based on the assumption that LDOS is uniform throughout the samples. We have conducted microwave measurements of the longitudinal energy profiles within disordered samples contained in a copper tube supporting multiple waveguide channels with an antenna moving along a slit on the tube. These measurements allow us to determine the local density of states (LDOS) at a location which is the sum of energy from all incoming channels on both sides. For diffusive samples, the LDOS is uniform and the energy profile decays linearly as expected. However, for localized samples, we find that the LDOS drops sharply towards the middle of the sample and the energy profile does not follow the result of the local diffusion theory where the LDOS is assumed to be uniform. We analyze the field spectra into quasi-normal modes and found that the mode linewidth and the number of modes saturates as the sample length increases. Thus the Thouless number saturates while the dimensionless conductance g continues to fall with increasing length, indicating that the modes are localized near the boundaries. This is in contrast to the general believing that g and Thouless number follow the same scaling behavior. Previous measurements show that single parameter scaling (SPS) still holds in the same sample where the LDOS is suppressed. We explore the extension of SPS to the interior of the sample by analyzing statistics of the logrithm of the energy density lnW(x) and found that =-x/l where l is the transport mean free path. The result does not depend on the sample length, which is counterintuitive yet remarkably simple. More supprisingly, the linear fall-off of energy profile holds for totally disordered random 1D layered samples in simulations where the LDOS is uniform as well as for single mode random waveguide experiments and 1D nearly periodic samples where the LDOS is suppressed in the middle of the sample.

The generalization of the transmission matrix to the interior of quasi-1D random samples, which is defined as the field matrix, and its eigenvalues statistics are also discussed. The maximum energy deposition at a location is not the intensity of the first transmission eigenchannel but the eigenvalue of the first energy density eigenchannels at that cross section, which can be much greater than the average value. The contrast, which is the ratio of the intensity at the focused point to the background intensity, in optimal focusing is determined by the participation number of the energy density eigenvalues and its inverse gives the variance of the energy density at that cross section in a single configuration. We have also studied topological states in photonic structures. We have demonstrated robust propagation of electromagnetic waves along reconfigurable pathways within a topological photonic metacrystal. Since the wave is confined within the domain wall, which is the boundary between two distinct topological insulating systems, we can freely steer the wave by reconstructing the photonic structure. Other topics, such as speckle pattern evolutions and the effects of boundary conditions on the statistics of transmission eigenvalues and energy profiles are also discussed.

## Recommended Citation

Cheng, Xiaojun, "Wave Propagation Inside Random Media" (2017). *CUNY Academic Works.*

https://academicworks.cuny.edu/gc_etds/2067