Investigations in the theory of electromagnetic scattering from expanding dielectric obstacles

<p>An equation for the reflection which results when an expanding dielectric slab scatters normally incident plane electromagnetic waves is derived using the invariant imbedding concept. The equation is solved approximately and the character of the solution is investigated. Also, an equation for the radiation transmitted through such a slab is similarly obtained. An alternative formulation of the slab problem is presented which is applicable to the analogous problem in spherical geometry. The form of an equation for the modal reflections from a nonrelativistically expanding sphere is obtained and some salient features of the solution are described. In all cases the material is assumed to be a nondispersive, nonmagnetic dielectric whose rest frame properties are slowly varying. </p>

Aberration measurement of projection optics in lithographic tools based on two-beam interference theory

The degradation of image quality caused by aberrations of projection optics in lithographic tools is a serious problem in optical lithography. We propose what we believe to be a novel technique for measuring aberrations of projection optics based on two-beam interference theory. By utilizing the partial coherent imaging theory, a novel model that accurately characterizes the relative image displacement of a fine grating pattern to a large pattern induced by aberrations is derived. Both even and odd aberrations are extracted independently from the relative image displacements of the printed patterns by two-beam interference imaging of the zeroth and positive first orders. The simulation results show that by using this technique we can measure the aberrations present in the lithographic tool with higher accuracy. (c) 2006 Optical Society of America.

Algebraic Techniques in Coding Theory: Entropy Vectors, Frames, and Constrained Coding

<p>The study of codes, classically motivated by the need to communicate information reliably in the presence of error, has found new life in fields as diverse as network communication, distributed storage of data, and even has connections to the design of linear measurements used in compressive sensing. But in all contexts, a code typically involves exploiting the algebraic or geometric structure underlying an application. In this thesis, we examine several problems in coding theory, and try to gain some insight into the algebraic structure behind them.</p> <p>The first is the study of the entropy region - the space of all possible vectors of joint entropies which can arise from a set of discrete random variables. Understanding this region is essentially the key to optimizing network codes for a given network. To this end, we employ a group-theoretic method of constructing random variables producing so-called "group-characterizable" entropy vectors, which are capable of approximating any point in the entropy region. We show how small groups can be used to produce entropy vectors which violate the Ingleton inequality, a fundamental bound on entropy vectors arising from the random variables involved in linear network codes. We discuss the suitability of these groups to design codes for networks which could potentially outperform linear coding.</p> <p>The second topic we discuss is the design of frames with low coherence, closely related to finding spherical codes in which the codewords are unit vectors spaced out around the unit sphere so as to minimize the magnitudes of their mutual inner products. We show how to build frames by selecting a cleverly chosen set of representations of a finite group to produce a "group code" as described by Slepian decades ago. We go on to reinterpret our method as selecting a subset of rows of a group Fourier matrix, allowing us to study and bound our frames' coherences using character theory. We discuss the usefulness of our frames in sparse signal recovery using linear measurements.</p> <p>The final problem we investigate is that of coding with constraints, most recently motivated by the demand for ways to encode large amounts of data using error-correcting codes so that any small loss can be recovered from a small set of surviving data. Most often, this involves using a systematic linear error-correcting code in which each parity symbol is constrained to be a function of some subset of the message symbols. We derive bounds on the minimum distance of such a code based on its constraints, and characterize when these bounds can be achieved using subcodes of Reed-Solomon codes. </p>

The vacuum Regge Trajectory in conventional field theory

<p>The effect on the scattering amplitude of the existence of a pole in the angular momentum plane near J = 1 in the channel with the quantum numbers of the vacuum is calculated. This is then compared with a fourth order calculation of the scattering of neutral vector mesons from a fermion pair field in the limit of large momentum transfer. The presence of the third double spectral function in the perturbation amplitude complicates the identification of pole trajectory parameters, and the limitations of previous methods of treating this are discussed. A gauge invariant scheme for extracting the contribution of the vacuum trajectory is presented which gives agreement with unitarity predictions, but further calculations must be done to determine the position and slope of the trajectory at s = 0. The residual portion of the amplitude is compared with the Gribov singularity. </p>

A theory of strong and weak scintillations with applications to astrophysics

<p>The propagation of waves in an extended, irregular medium is studied under the "quasi-optics" and the "Markov random process" approximations. Under these assumptions, a Fokker-Planck equation satisfied by the characteristic functional of the random wave field is derived. A complete set of the moment equations with different transverse coordinates and different wavenumbers is then obtained from the characteristic functional. The derivation does not require Gaussian statistics of the random medium and the result can be applied to the time-dependent problem. We then solve the moment equations for the phase correlation function, angular broadening, temporal pulse smearing, intensity correlation function, and the probability distribution of the random waves. The necessary and sufficient conditions for strong scintillation are also given.</p> <p>We also consider the problem of diffraction of waves by a random, phase-changing screen. The intensity correlation function is solved in the whole Fresnel diffraction region and the temporal pulse broadening function is derived rigorously from the wave equation.</p> <p>The method of smooth perturbations is applied to interplanetary scintillations. We formulate and calculate the effects of the solar-wind velocity fluctuations on the observed intensity power spectrum and on the ratio of the observed "pattern" velocity and the true velocity of the solar wind in the three-dimensional spherical model. The r.m.s. solar-wind velocity fluctuations are found to be ~200 km/sec in the region about 20 solar radii from the Sun.</p> <p>We then interpret the observed interstellar scintillation data using the theories derived under the Markov approximation, which are also valid for the strong scintillation. We find that the Kolmogorov power-law spectrum with an outer scale of 10 to 100 pc fits the scintillation data and that the ambient averaged electron density in the interstellar medium is about 0.025 cm^-3. It is also found that there exists a region of strong electron density fluctuation with thickness ~10 pc and mean electron density ~7 cm^-3 between the PSR 0833-45 pulsar and the earth.</p>

The resolution of the thermodynamic paradox and the theory of guided wave propagation in anisotropic media

<p>The resolution of the so-called thermodynamic paradox is presented in this paper. It is shown, in direct contradiction to the results of several previously published papers, that the cutoff modes (evanescent modes having complex propagation constants) can carry power in a waveguide containing ferrite. The errors in all previous ���proofs��� which purport to show that the cutoff modes cannot carry power are uncovered. The boundary value problem underlying the paradox is studied in detail; it is shown that, although the solution is somewhat complicated, there is nothing paradoxical about it.</p> <p>The general problem of electromagnetic wave propagation through rectangular guides filled inhomogeneously in cross-section with transversely magnetized ferrite is also studied. Application of the standard waveguide techniques reduces the TM part to the well-known self-adjoint Sturm Liouville eigenvalue equation. The TE part, however, leads in general to a non-self-adjoint eigenvalue equation. This equation and the associated expansion problem are studied in detail. Expansion coefficients and actual fields are determined for a particular problem. </p>

I. The effect of droplet solidification upon two-phase flow in a rocket nozzle. II. A kinetic theory investigation of some condensation-evaporation phenomena by a moment method

<p>Part I:</p> <p>The perturbation technique developed by Rannie and Marble is used to study the effect of droplet solidification upon two-phase flow in a rocket nozzle. It is shown that under certain conditions an equilibrium flow exists, where the gas and particle phases have the same velocity and temperature at each section of the nozzle. The flow is divided into three regions: the first region, where the particles are all in the form of liquid droplets; a second region, over which the droplets solidify at constant freezing temperature; and a third region, where the particles are all solid. By a perturbation about the equilibrium flow, a solution is obtained for small particle slip velocities using the Stokes drag law and the corresponding approximation for heat transfer between the particle and gas phases. Singular perturbation procedure is required to handle the problem at points where solidification first starts and where it is complete. The effects of solidification are noticeable.</p> <p>Part II:</p> <p>When a liquid surface, in contact with only its pure vapor, is not in the thermodynamic equilibrium with it, a net condensation or evaporation of fluid occurs. This phenomenon is studied from a kinetic theory viewpoint by means of moment method developed by Lees. The evaporation-condensation rate is calculated for a spherical droplet and for a liquid sheet, when the temperatures and pressures are not too far removed from their equilibrium values. The solutions are valid for the whole range of Knudsen numbers from the free molecule to the continuum limit. In the continuum limit, the mass flux rate is proportional to the pressure difference alone. </p>

Essays in revealed preference theory and behavioral economics

Time, risk, and attention are all integral to economic decision making. The aim of this work is to understand those key components of decision making using a variety of approaches: providing axiomatic characterizations to investigate time discounting, generating measures of visual attention to infer consumers' intentions, and examining data from unique field settings. <br/><br/> Chapter 2, co-authored with Federico Echenique and Kota Saito, presents the first revealed-preference characterizations of exponentially-discounted utility model and its generalizations. My characterizations provide non-parametric revealed-preference tests. I apply the tests to data from a recent experiment, and find that the axiomatization delivers new insights on a dataset that had been analyzed by traditional parametric methods. <br/><br/> Chapter 3, co-authored with Min Jeong Kang and Colin Camerer, investigates whether "pre-choice" measures of visual attention improve in prediction of consumers' purchase intentions. We measure participants' visual attention using eyetracking or mousetracking while they make hypothetical as well as real purchase decisions. I find that different patterns of visual attention are associated with hypothetical and real decisions. I then demonstrate that including information on visual attention improves prediction of purchase decisions when attention is measured with mousetracking. <br/><br/> Chapter 4 investigates individuals' attitudes towards risk in a high-stakes environment using data from a TV game show, Jeopardy!. I first quantify players' subjective beliefs about answering questions correctly. Using those beliefs in estimation, I find that the representative player is risk averse. I then find that trailing players tend to wager more than "folk" strategies that are known among the community of contestants and fans, and this tendency is related to their confidence. I also find gender differences: male players take more risk than female players, and even more so when they are competing against two other male players. <br/><br/> Chapter 5, co-authored with Colin Camerer, investigates the dynamics of the favorite-longshot bias (FLB) using data on horse race betting from an online exchange that allows bettors to trade "in-play." I find that probabilistic forecasts implied by market prices before start of the races are well-calibrated, but the degree of FLB increases significantly as the events approach toward the end.

A kinetic theory description for external spherical flows with arbitrary Knudsen number by a moment method

<p>The Maxwell integral equations of transfer are applied to a series of problems involving flows of arbitrary density gases about spheres. As suggested by Lees a two sided Maxwellian-like weighting function containing a number of free parameters is utilized and a sufficient number of partial differential moment equations is used to determine these parameters. Maxwell's inverse fifth-power force law is used to simplify the evaluation of the collision integrals appearing in the moment equations. All flow quantities are then determined by integration of the weighting function which results from the solution of the differential moment system. Three problems are treated: the heat-flux from a slightly heated sphere at rest in an infinite gas; the velocity field and drag of a slowly moving sphere in an unbounded space; the velocity field and drag torque on a slowly rotating sphere. Solutions to the third problem are found to both first and second-order in surface Mach number with the secondary centrifugal fan motion being of particular interest. Singular aspects of the moment method are encountered in the last two problems and an asymptotic study of these difficulties leads to a formal criterion for a "well posed" moment system. The previously unanswered question of just how many moments must be used in a specific problem is now clarified to a great extent. </p>