Publications

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We will discuss general mathematical ideas arising in the problems of Laser beam shaping and splitting. We will be particularly concerned with questions concerning the scaling and symmetry of such systems.

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We present an analysis of gas chromatographic columns where the stationary phase is not assumed to be a thin uniform coating along the walls of the cross section. We also give an asymptotic analysis assuming that the parameter {beta} = KD{sup II}{rho}{sup II}/D{sup I}{rho}{sup I} is small. Here K is the partition coefficient, and D{sup i} and {rho}{sup i}, i = I, II are the diffusivity and density in the mobile (i = I) and stationary (i = II) regions.

This paper demonstrates that the conditions for the existence of a dissipation-induced heteroclinic orbit between the inverted and noninverted states of a tippe top are determined by a complex version of the equations for a simple harmonic oscillator: the modified Maxwell-Bloch equations. A standard linear analysis reveals that the modified Maxwell-Bloch equations describe the spectral instability of the noninverted state and Lyapunov stability of the inverted state. Standard nonlinear analysis based on the energy momentum method gives necessary and sufficient conditions for the existence of a dissipation-induced connecting orbit between these relative equilibria.

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In gas chromatography, a chemical sample separates into its constituent components as it travels along a long thin column. As the component chemicals exit the column they are detected and identified, allowing the chemical makeup of the sample to be determined. For correct identification of the component chemicals, the distribution of the concentration of each chemical along the length of the column must be nearly symmetric. The prediction and control of asymmetries in gas chromatography has been an active research area since the advent of the technique. In this paper, we develop from first principles a general model for isothermal linear chromatography. We use this model to develop closed-form expressions for terms related to the first, second, and third moments of the distribution of the concentration, which determines the velocity, diffusion rate, and asymmetry of the distribution. We show that for all practical experimental situations, only fronting peaks are predicted by this model, suggesting that a nonlinear chromatography model is required to predict tailing peaks. For situations where asymmetries arise, we analyze the rate at which the concentration distribution returns to a normal distribution. Numerical examples are also provided.

In this paper we present an analysis of a new configuration for achieving spin stabilized magnetic levitation. In the classical configuration, the rotor spins about a vertical axis; and the spin stabilizes the lateral instability of the top in the magnetic field. In this new configuration the rotor spins about a horizontal axis; and the spin stabilizes the axial instability of the top in the magnetic field.

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Analytic solutions are useful for code verification. Structural vibration codes approximate solutions to the eigenvalue problem for the linear elasticity equations (Navier's equations). Unfortunately the verification method of 'manufactured solutions' does not apply to vibration problems. Verification books (for example [2]) tabulate a few of the lowest modes, but are not useful for computations of large numbers of modes. A closed form solution is presented here for all the eigenvalues and eigenfunctions for a cuboid solid with isotropic material properties. The boundary conditions correspond physically to a greased wall.

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Synthetic Aperture Radar systems are being driven to provide images with ever-finer resolutions. This, of course, requires ever-wider bandwidths to support these resolutions in a number of frequency bands across the microwave (and lower) spectrum. The problem is that the spectrum is already quite crowded with a multitude of users, and a multitude of uses. For a radar system, this manifests itself as a number of 'stay-out' zones in the spectrum mandated by regulatory agencies; frequencies where the radar is not allowed to transmit. Even frequencies where the radar is allowed to transmit might be corrupted by interference from other legitimate (and/or illegitimate) users, rendering these frequencies useless to the radar system. In a SAR image, these spectral holes (by whatever source) degrade images, most notably by increasing objectionable sidelobe levels, most evident in the neighborhood of bright point-like objects. For contiguous spectrums, sidelobes in SAR images are controlled by employing window functions. However, those windows that work well for contiguous spectrums don't seem to work well for spectrums with significant gaps or holes. In this paper we address the question "Can some sorts of window functions be developed and employed to advantage when the spectrum is not contiguous, but contains significant holes or gaps?" A window function that minimizes sidelobe energy can be constructed based on prolate spheroidal wave functions. This approach is extended to accommodate spectral notches or holes, although the guaranteed minimum sidelobe energy can be quite high in this case.

We present a technique for determining non-linear resistances, capacitances, and inductances from ring down data in a non-linear RLC circuit. Although the governing differential equations are non-linear, we are able to solve this problem using linear least squares without doing any sort of non-linear iteration.

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Imaging systems such as Synthetic Aperture Radar collect band-limited data from which an image of a target scene is rendered. The band-limited nature of the data generates sidelobes, or ''spilled energy'' most evident in the neighborhood of bright point-like objects. It is generally considered desirable to minimize these sidelobes, even at the expense of some generally small increase in system bandwidth. This is accomplished by shaping the spectrum with window functions prior to inversion or transformation into an image. A window function that minimizes sidelobe energy can be constructed based on prolate spheroidal wave functions. A parametric design procedure allows doing so even with constraints on allowable increases in system bandwidth. This approach is extended to accommodate spectral notches or holes, although the guaranteed minimum sidelobe energy can be quite high in this case. Interestingly, for a fixed bandwidth, the minimum-mean-squared-error image rendering of a target scene is achieved with no windowing at all (rectangular or boxcar window).

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LOCA, the Library of Continuation Algorithms, is a software library for performing stability analysis of large-scale applications. LOCA enables the tracking of solution branches as a function of a system parameter, the direct tracking of bifurcation points, and, when linked with the ARPACK library, a linear stability analysis capability. It is designed to be easy to implement around codes that already use Newton's method to converge to steady-state solutions. The algorithms are chosen to work for large problems, such as those that arise from discretizations of partial differential equations, and to run on distributed memory parallel machines. This manual presents LOCA's continuation and bifurcation analysis algorithms, and instructions on how to implement LOCA with an application code. The LOCA code is being made publicly available at www.cs.sandia.gov/loca.

Superresolution concepts offer the potential of resolution beyond the classical limit. This great promise has not generally been realized. In this study we investigate the potential application of superresolution concepts to synthetic aperture radar. The analytical basis for superresolution theory is discussed. The application of the concept to synthetic aperture radar is investigated as an operator inversion problem. Generally, the operator inversion problem is ill posed. A criterion for judging superresolution processing of an image is presented.

The three most important factors effecting the difficulty of a beam shaping problems were discussed. These factors were scaling, smoothness, and coherence. Algorithms were developed to counteract these factors encountered in the design of any beam shaping system.

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