Sideways adiabaticity: Beyond ray optics for slowly varying metasurfaces

Optical metasurfaces (subwavelength-patterned surfaces typically described by variable effective surface impedances) are typically modeled by an approximation akin to ray optics: the reflection or transmission of an incident wave at each point of the surface is computed as if the surface were "locally uniform", and then the total field is obtained by summing all of these local scattered fields via a Huygens principle. (Similar approximations are found in scalar diffraction theory and in ray optics for curved surfaces.) In this paper, we develop a precise theory of such approximations for variable-impedance surfaces. Not only do we obtain a type of adiabatic theorem showing that the "zeroth-order" locally uniform approximation converges in the limit as the surface varies more and more slowly, including a way to quantify the rate of convergence, but we also obtain an infinite series of higher-order corrections. These corrections, which can be computed to any desired order by performing integral operations on the surface fields, allow rapidly varying surfaces to be modeled with arbitrary accuracy, and also allow one to validate designs based on the zeroth-order approximation (which is often surprisingly accurate) without resorting to expensive brute-force Maxwell solvers. We show that our formulation works arbitrarily close to the surface, and can even compute coupling to guided modes, whereas in the far-field limit our zeroth-order result simplifies to an expression similar to what has been used by other authors

2D Gravity and Random Matrices

We review recent progress in 2D gravity coupled to $d<1$ conformal matter, based on a representation of discrete gravity in terms of random matrices. We discuss the saddle point approximation for these models, including a class of related $O(n)$ matrix models. For $d<1$ matter, the matrix problem can be completely solved in many cases by the introduction of suitable orthogonal polynomials. Alternatively, in the continuum limit the orthogonal polynomial method can be shown to be equivalent to the construction of representations of the canonical commutation relations in terms of differential operators. In the case of pure gravity or discrete Ising--like matter, the sum over topologies is reduced to the solution of non-linear differential equations (the Painlev\'e equation in the pure gravity case) which can be shown to follow from an action principle. In the case of pure gravity and more generally all unitary models, the perturbation theory is not Borel summable and therefore alone does not define a unique solution. In the non-Borel summable case, the matrix model does not define the sum over topologies beyond perturbation theory. We also review the computation of correlation functions directly in the continuum formulation of matter coupled to 2D gravity, and compare with the matrix model results. Finally, we review the relation between matrix models and topological gravity, and as well the relation to intersection theory of the moduli space of punctured Riemann surfaces.Comment: 190 pages (harvmac l mode), 400kb (don't even dream of requesting hardcopy

Three Stratified Fluid Models: Benjamin-Ono, Tidal Resonance, and Quasi-Geostrophy.

In this dissertation, we study three models arising in stratified fluid dynamics: the Benjamin-Ono equation, a two-layer tidal model, and the quasi-geostrophic equations. First, we compute the scattering data of the Benjamin-Ono equation for arbitrary rational initial conditions with simple poles, under mild restrictions. For this class of initial conditions, we are able to obtain explicit formulas for the Jost solutions and eigenfunctions of the associated spectral problem, yielding an Evans function for the eigenvalues and formulas for the phase constants and reflection coefficient. We proceed to use these exact formulas to deduce precise asymptotics for the reflection coefficient, the location of the eigenvalues and their density, and the asymptotic dependence of the phase constant on the eigenvalue. Second, we present a simplified analytical tidal model that demonstrates the influence of stratification on both large- and small-scale surface tidal elevations in a qualitatively similar manner as in previous global realistic-domain numerical simulations. Our analytical results demonstrate the potential for the presence of stratification and changes to stratification to alter the large-scale (barotropic) tide. We find that changes in stratification may change large-scale tidal elevations at the same order as small-scale tidal elevations if there is significant bottom topography and the damping acts primarily on the bottom layer. Third, we investigate the influence of bottom friction on quasi-geostrophic turbulence dynamics in a multi-level forced-dissipated model that includes buoyancy advection at the boundaries. Motivated by earlier studies of two-layer quasi-geostrophic turbulence, we study the influence of bottom friction on the horizontal scales and vertical structure of eddy kinetic energy with the aid of a ‘surface-aware’ modal decomposition.PhDApplied and Interdisciplinary MathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/116662/1/wreagan_1.pd

Biomolecular electrostatics with continuum models: a boundary integral implementation and applications to biosensors

The implicit-solvent model uses continuum electrostatic theory to represent the salt solution around dissolved biomolecules, leading to a coupled system of the Poisson-Boltzmann and Poisson equations. This thesis uses the implicit-solvent model to study solvation, binding and adsorption of proteins. We developed an implicit-solvent model solver that uses the boundary element method (BEM), called PyGBe. BEM numerically solves integral equations along the biomolecule-solvent interface only, therefore, it does not need to discretize the entire domain. PyGBe accelerates the BEM with a treecode algorithm and runs on graphic processing units. We performed extensive verification and validation of the code, comparing it with experimental observations, analytical solutions, and other numerical tools. Our results suggest that a BEM approach is more appropriate than volumetric based methods, like finite-difference or finite-element, for high accuracy calculations. We also discussed the effect of features like solvent-filled cavities and Stern layers in the implicit-solvent model, and realized that they become relevant in binding energy calculations. The application that drove this work was nano-scale biosensors-- devices designed to detect biomolecules. Biosensors are built with a functionalized layer of ligand molecules, to which the target molecule binds when it is detected. With our code, we performed a study of the orientation of proteins near charged surfaces, and investigated the ideal conditions for ligand molecule adsorption. Using immunoglobulin G as a test case, we found out that low salt concentration in the solvent and high positive surface charge density leads to favorable orientations of the ligand molecule for biosensing applications. We also studied the plasmonic response of localized surface plasmon resonance (LSPR) biosensors. LSPR biosensors monitor the plasmon resonance frequency of metallic nanoparticles, which shifts when a target molecule binds to a ligand molecule. Electrostatics is a valid approximation to the LSPR biosensor optical phenomenon in the long-wavelength limit, and BEM was able to reproduce the shift in the plasmon resonance frequency as proteins approach the nanoparticle

Semi-spectral Chebyshev method in Quantum Mechanics

Traditionally, finite differences and finite element methods have been by many regarded as the basic tools for obtaining numerical solutions in a variety of quantum mechanical problems emerging in atomic, nuclear and particle physics, astrophysics, quantum chemistry, etc. In recent years, however, an alternative technique based on the semi-spectral methods has focused considerable attention. The purpose of this work is first to provide the necessary tools and subsequently examine the efficiency of this method in quantum mechanical applications. Restricting our interest to time independent two-body problems, we obtained the continuous and discrete spectrum solutions of the underlying Schroedinger or Lippmann-Schwinger equations in both, the coordinate and momentum space. In all of the numerically studied examples we had no difficulty in achieving the machine accuracy and the semi-spectral method showed exponential convergence combined with excellent numerical stability.Comment: RevTeX, 12 EPS figure

Feedback control of sector-bound nonlinear systems with applications to aeroengine control

This dissertation is divided into two parts. In the first part we consider the problem of feedback stabilization of nonlinear systems described by state-space models. This approach is inherited from the methodology of sector bounded or passive nonlinearities, and influenced by the concept of absolute and quadratic stability. It aims not only to regionally stabilize the nonlinear dynamics asymptotically but also to maximize the estimated region of quadratic attraction and to ensure nominal performance at each equilibrium. In close connection to gain scheduling and switching control, a path of equilibria is programmed based on the assumption of centered-epsilon-cover which leads to a sequence of linear controllers that regionally stabilize the desired equilibrium asymptotically. In the second part we tackle the problem of control for fluid flows described by the incompressible Navier-Stokes equation. We are particularly interested in film cooling for gas turbine engines which we model with the jet in cross-flow problem setup. In order to obtain a model amenable to the controller design presented in the first part, the well-known Proper Orthogonal Decomposition (POD)/Galerkin projection is employed to obtain a nonlinear state-space system called the reduced order model (ROM). We are able to stabilize the ROM to an equilibrium point via our design method and we also present direct numerical simulation (DNS) results for the system under state feedback control