Dissertations, Theses, and Capstone Projects

Date of Degree

9-2026

Document Type

Doctoral Dissertation

Degree Name

Doctor of Philosophy

Program

Physics

Advisor

Azriel Z Genack

Committee Members

Mohammad-Ali Miri

Patrick Sebbah

Alexander Khanikaev

Alexander Lisyansky

Subject Categories

Condensed Matter Physics | Optics | Other Physics | Physical Sciences and Mathematics | Physics

Keywords

Transmission matrix, Flux matrix, Modal alignment, Anderson localization, Transmission zeros, Velocity zeros

Abstract

This thesis presents microwave measurements and numerical simulations of wave propagation and transmission through random mesoscopic waveguides. We analyze the scaling of mesoscopic microwave conductance, including departures from Ohm’s law near channel openings, which are frequencies at which new propagating modes enter the system, and the evolution of modal phase alignment within the medium, which leads to the observed diversity of transmission eigenvalues. We extend the transmission matrix formalism to all sample depths using the flux matrix, which relates the flux amplitude within the sample to the incident flux, and decompose the flux at each sample depth into forward- and backward-propagating components. From the determinant of the transmission matrix, we extract zeros and poles of transmission as singularities in the complex frequency plane. We study the impact and characteristics of transmission zeros, eigenchannel velocity zeros, disorder-induced modal alignment, the statistics of singularities in the complex frequency plane, and the relationship between the field correlation function and eigenchannel velocities inside a disordered system. We have also constructed an apparatus and begun experiments to search for Anderson localization in three-dimensional random mixtures of metallic spheres and dielectric support. Studies of the transmission matrix will enable the distinguishing of exponential decay due to absorption from the effect of Anderson localization in this strongly absorbing medium.

We present the first experimental observations of transmission zeros, made in analysis of transmission matrix measurements originally performed by Zhou Shi. The analysis shows that departures from Ohm’s law near channel openings are governed by the transmission eigenchannel carrying the least flux, whose transmission vanishes at topological transmission zeros. These departures are associated with three related phenomena explored in microwave measurements of the transmission matrix of a multichannel waveguide.

The first phenomenon explored in detail is the transmission zero. At frequencies near transmission zeros, transmission in the lowest-transmission eigenchannel is measured up to nine orders of magnitude below the transmission of the highest-transmission eigenchannel, which is seven orders of magnitude below the experimental noise level. The evolution of modal alignment throughout the medium will provide an explanation for how this measurement can be possible. Transmission zeros have a characteristic quadratic spectral shape arising from the meromorphic structure of the determinant of the transmission matrix. They are also identifiable by Lorentzian peaks and troughs in the transmission time of the lowest-transmission channel under controlled gain and loss. They correspond to singularities in the map of the phase of the determinant of the transmission matrix in the complex frequency plane. Tracking the positions of poles and zeros and in the complex plane as the medium changes reveals that while poles repel one another, zeros can coalesce on the real axis and undergo type conversion between single zeros on the real axis and conjugate pairs off the axis. Starting from a uniform sample and increasing the magnitude of its disorder reveals that zeros are brought into existence on the real axis at the channel opening frequency, with each zero or conjugate pair accompanied by a pole. The second phenomenon is the velocity zero: the longitudinal velocity of a channel entering the system vanishes precisely at the channel opening where it enters. The third phenomenon is the impact of both types of zeros on the transmittance, which is equivalent to the electronic conductance, as described by Landauer. Because transmission eigenvalues are mutually correlated through level repulsion, even though the lowest-transmission channel carries only a small fraction of the total flux, the vanishing of the lowest eigenvalue at a transmission zero is felt across the entire transmission matrix, producing dips in the transmittance. This correlation is short-ranged, however; as the number of propagating channels increases, the correlation between the lowest-transmission channel and the higher-transmission channels weakens. As sample width increases, correlation among transmission eigenchannels decreases, the width of the probability distribution function of transmission zeros increases, and the specific impact of TZs is lessened, bringing transmittance into the linear, Ohmic scaling regime of particle diffusion. This is an instance of the correspondence principle in mesoscopic wave transport: the classical result is recovered as the number of propagating channels becomes large. The dips in conductance are proportional to peaks in the density of states, with a proportionality constant that decreases with sample length and is independent of sample width for fixed length.

We further demonstrate the mechanism by which the broad range in the flux and energy density, from complete to vanishing, arises from the evolution of phase alignment among the contributions from incident waveguide modes to the internal modal components of each eigenchannel. Transmission zeros originate from near-perfect phase cancellation of individually measurable modal flux amplitudes. This disorder-induced modal phase alignment, together with the flux matrix, determines the depth-dependent profiles of flux, energy density, and eigenchannel velocity throughout the medium.

Finally, we describe an experimental apparatus designed to search for Anderson localization of microwave photons in a three-dimensional random mixture of aluminum spheres in a dielectric background. Three-dimensional photon localization has not been unambiguously confirmed experimentally; earlier observations in strongly absorbing samples were inconclusive, because the signature of localization could not be distinguished from the effect of absorption. Recent numerical simulations of waveguides randomly packed with non-absorbing metallic spheres confirmed that a three-dimensional localization transition can occur. The aim of the experiment is to measure the dimensionless conductance and the dwell time, from which the density of states can be extracted. The conductance will be determined using the variance of the total transmission and the spatial field correlation function, both of which remain valid localization indicators in the presence of absorption. The technical hurdles of measurement repeatability, temperature stabilization, and sample settling over time have been addressed with the engineering of an automated antenna movement system, PID temperature control, and choice of dielectric in which to hold the spheres.

Together, these results show that the conductance of diffusive random waveguides is shaped by three interconnected mechanisms: the coupling between incident and internal modes in the flux matrix, the modal weights in the incident eigenchannel, and modal phase alignment. These mechanisms are also visible in the behavior of transmission zeros, velocity zeros, and eigenvalue correlation. Understanding these phenomena establishes a framework for the study of three-dimensional photon localization, where the same transmission matrix formalism can be brought to bear. Moreover, the disorder-induced modal phase alignment identified here determines the energy density, flux, and eigenchannel velocity profiles throughout the scattering medium, with implications for ultrasensitive detection, wavefront control, and targeted energy delivery in complex media.

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