Electromagnetic wave propagation and scattering in a randomly-inhomogeneous dielectric sphere

Electromagnetic wave propagation and scattering in a sphere composed of an inhomogeneous medium having random variations in its permittivity are studied by utilizing the Born approximation in solving the vector wave equation. The variations in the permittivity are taken to be isotropic and homogeneous, and are spatially characterized by a Gaussian correlation function. Temporal variations in the medium are not considered.

Two particular problems are considered: i) finding the far-zone electric field when an electric or magnetic dipole is situated at the center of the sphere, and ii) finding the electric field at the sphere's center when a linearly polarized plane wave is incident upon it. Expressions are obtained for the mean-square magnitudes of the scattered field components; it is found that the mean of the product of any two transverse components vanishes. The cases where the wavelength is much shorter than correlation distance of the medium and where it is much longer than it are both considered.

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

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.

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.

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.

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.

Phase-space analysis of wavefront coding imaging systems

We explore the use of the Radon-Wigner transform, which is associated with the fractional Fourier transform of the pupil function, for determining the point spread function (PSF) of an incoherant defocused optical system. Then we introduce these phase-space tools to analyse the wavefront coding imaging system. It is shown that the shape of the PSF for such a system is highly invarient to the defocous-related aberrations except for a lateral shift. The optical transfer function of this system is also investigated briefly from a new understanding of ambiguity function.

Equilibration of a liquid droplet-vapor-gas mixture downstream of a shock wave

The equations of motion for the flow of a mixture of liquid droplets, their vapor, and an inert gas through a normal shock wave are derived. A set of equations is obtained which is solved numerically for the equilibrium conditions far downstream of the shock. The equations describing the process of reaching equilibrium are also obtained. This is a set of first-order nonlinear differential equations and must also be solved numerically. The detailed equilibration process is obtained for several cases and the results are discussed.

Optimum linear coding for additive noise systems using information feedback

The feedback coding problem for Gaussian systems in which the noise is neither white nor statistically independent between channels is formulated in terms of arbitrary linear codes at the transmitter and at the receiver. This new formulation is used to determine a number of feedback communication systems. In particular, the optimum linear code that satisfies an average power constraint on the transmitted signals is derived for a system with noiseless feedback and forward noise of arbitrary covariance. The noisy feedback problem is considered and signal sets for the forward and feedback channels are obtained with an average power constraint on each. The general formulation and results are valid for non-Gaussian systems in which the second order statistics are known, the results being applicable to the determination of error bounds via the Chebychev inequality.

An Optically recombined laser interferometer for gravitational wave detection

Kilometer scale interferometers for the detection of gravitational waves are currently under construction by the LIGO (Laser Interferometer Gravitational-wave Observatory) and VIRGO projects. These interferometers will consist of two Fabry-Perot cavities illuminated by a laser beam which is split in half by a beam splitter. A recycling mirror between the laser and the beam splitter will reflect the light returning from the beam splitter towards the laser back into the interferometer. The positions of the optical components in these interferometers must be controlled to a small fraction of a wavelength of the laser light. Schemes to extract signals necessary to control these optical components have been developed and demonstrated on the tabletop. In the large scale gravitational wave detectors the optical components must be suspended from vibration isolation platforms to achieve the necessary isolation from seismic motion. These suspended components present a new class of problems in controlling the interferometer, but also provide more exacting test of interferometer signal and noise models.

This thesis discusses the first operation of a suspended-mass Fabry-Perot-Michelson interferometer, in which signals carried by the optically recombined beams are used to detect and control all important mirror displacements. This interferometer uses an optical configuration and signal extraction scheme that is planned for the full scale LIGO interferometers with the simplification of the removal of the recycling mirror. A theoretical analysis of the performance that is expected from such an interferometer is presented and the experimental results are shown to be in generally good agreement.

Development of roll waves in open channels

This study is concerned with some of the properties of roll waves that develop naturally from a turbulent uniform flow in a wide rectangular channel on a constant steep slope . The wave properties considered were depth at the wave crest, depth at the wave trough, wave period, and wave velocity . The primary focus was on the mean values and standard deviations of the crest depths and wave periods at a given station and how these quantities varied with distance along the channel.

The wave properties were measured in a laboratory channel in which roll waves developed naturally from a uniform flow . The Froude number F (F = u_n/√gh_n, u_n = normal velocity , h_n = normal depth, g =acceleration of gravity) ranged from 3. 4 to 6. 0 for channel slopes S_o of . 05 and . 12 respectively . In the initial phase of their development the roll waves appeared as small amplitude waves with a continuous water surface profile . These small amplitude waves subsequently developed into large amplitude shock waves. Shock waves were found to overtake and combine with other shock waves with the result that the crest depth of the combined wave was larger than the crest depths before the overtake. Once roll waves began to develop, the mean value of the crest depths h_nmax increased with distance . Once the shock waves began to overtake, the mean wave period T_av increased approximately linearly with distance.

For a given Froude number and channel slope the observed quantities h^-_max/h_n , T' (T' = S_o T_av √g/h_n), and the standard deviations of h^-_max/h_n and T', could be expressed as unique functions of l/h_n (l = distance from beginning of channel) for the two-fold change in h_n occurring in the observed flows . A given value of h^-_max/h_n occurred at smaller values of l/h_n as the Froude number was increased. For a given value of h /hh^-_max/h_n the growth rate of δh^-_max/h^-_maxδl of the shock waves increased as the Froude number was increased.

A laboratory channel was also used to measure the wave properties of periodic permanent roll waves. For a given Froude number and channel slope the h^-_max/h_n vs. T' relation did not agree with a theory in which the weight of the shock front was neglected. After the theory was modified to include this weight, the observed values of h^-_max/h_n were within an average of 6.5 percent of the predicted values, and the maximum discrepancy was 13.5 percent.

For h^-_max/h_n sufficiently large (h^-_max/h_n > approximately 1.5) it was found that the h^-_max/h_n vs. T' relation for natural roll waves was practically identical to the h^-_max/h_n vs. T' relation for periodic permanent roll waves at the same Froude number and slope. As a result of this correspondence between periodic and natural roll waves, the growth rate δh^-_max/h^-_maxδl of shock waves was predicted to depend on the channel slope, and this slope dependence was observed in the experiments.

Wave induced oscillations in harbors of arbitrary shape

Theoretical and experimental studies were conducted to investigate the wave induced oscillations in an arbitrary shaped harbor with constant depth which is connected to the open-sea.

A theory termed the “arbitrary shaped harbor” theory is developed. The solution of the Helmholtz equation, ∇²f + k²f = 0, is formulated as an integral equation; an approximate method is employed to solve the integral equation by converting it to a matrix equation. The final solution is obtained by equating, at the harbor entrance, the wave amplitude and its normal derivative obtained from the solutions for the regions outside and inside the harbor.

Two special theories called the circular harbor theory and the rectangular harbor theory are also developed. The coordinates inside a circular and a rectangular harbor are separable; therefore, the solution for the region inside these harbors is obtained by the method of separation of variables. For the solution in the open-sea region, the same method is used as that employed for the arbitrary shaped harbor theory. The final solution is also obtained by a matching procedure similar to that used for the arbitrary shaped harbor theory. These two special theories provide a useful analytical check on the arbitrary shaped harbor theory.

Experiments were conducted to verify the theories in a wave basin 15 ft wide by 31 ft long with an effective system of wave energy dissipators mounted along the boundary to simulate the open-sea condition.

Four harbors were investigated theoretically and experimentally: circular harbors with a 10° opening and a 60° opening, a rectangular harbor, and a model of the East and West Basins of Long Beach Harbor located in Long Beach, California.

Theoretical solutions for these four harbors using the arbitrary shaped harbor theory were obtained. In addition, the theoretical solutions for the circular harbors and the rectangular harbor using the two special theories were also obtained. In each case, the theories have proven to agree well with the experimental data.

It is found that: (1) the resonant frequencies for a specific harbor are predicted correctly by the theory, although the amplification factors at resonance are somewhat larger than those found experimentally,(2) for the circular harbors, as the width of the harbor entrance increases, the amplification at resonance decreases, but the wave number bandwidth at resonance increases, (3) each peak in the curve of entrance velocity vs incident wave period corresponds to a distinct mode of resonant oscillation inside the harbor, thus the velocity at the harbor entrance appears to be a good indicator for resonance in harbors of complicated shape, (4) the results show that the present theory can be applied with confidence to prototype harbors with relatively uniform depth and reflective interior boundaries.

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

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.

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.

Three-dimensional coupled wave study for finite-sized anisotropic volume holographic gratings under ultrashort pulsed beam readout

The three-dimensional coupled wave theory is extended to systematically investigate the diffraction properties of finite-sized anisotropic volume holographic gratings (VHGs) under ultrashort pulsed beam (UPB) readout. The effects of the grating geometrical size and the polarizations of the recording and readout beams on the diffraction properties are presented, in particular under the influence of grating material dispersion. The wavelength selectivity of the finite-sized VHG is analyzed. The wavelength selectivity determines the intensity distributions of the transmitted and diffracted pulsed beams along the output face of the VHG. The distortion and widening of the diffracted pulsed beams are different for different points on the output face, as is numerically shown for a VHG recorded in a LiNbO3 crystal. The beam quality is analyzed, and the variations of the total diffraction efficiency are shown in relation to the geometrical size of the grating and the temporal width of the readout UPB. In addition, the diffraction properties of the finite-sized and one-dimensional VHG for pulsed and continuous-wave readout are compared. The study shows the potential application of VHGs in controlling spatial and temporal features of UPBs simultaneously. (C) 2007 Optical Society of America