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Wave simulations in infinite spacetime
Chad McKell, Mohammad Sina Nabizadeh, Stephanie Wang, Albert Chern
Solving the wave equation on an infinite domain has been an ongoing challenge
in scientific computing. Conventional approaches to this problem only generate numerical solutions on a
small subset of the infinite domain. In this paper, we present a method for solving the wave equation on
the entire infinite domain using only finite computation time and memory. Our method is based on the
conformal invariance of the scalar wave equation under the Kelvin transformation in Minkowski spacetime. As
a result of the conformal invariance, any wave problem with compact initial data contained in a causality
cone is equivalent to a wave problem on a bounded set in Minkowski spacetime. We use this fact to perform
wave simulations in infinite spacetime using a finite discretization of the bounded spacetime with no
additional loss of accuracy introduced by the Kelvin transformation.
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Geometric Boundary Modeling for Wave Simulation
Chad McKell
This work presents novel geometric methods for handling open exterior boundaries in wave field simulations.
First, I discuss extensions to the reflectionless discrete perfectly matched layer for the wave and Helmholtz
equations. Then, I demonstrate that the conformal invariance of the wave equation under a Kelvin transform in
Minkowski spacetime allows one to convert an infinite domain problem into a bounded domain problem that can be
solved using standard numerical methods with no additional loss of accuracy introduced by the transform.
I conclude by discussing parallel computing strategies for the geometric boundary models, applications in
architectural acoustics and binaural audio, and future work in obstacle boundary flattening. Advisors: Albert
Chern and Miller Puckette. (UCSD Seal: By Source, Fair use, wikipedia.org).
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Optical corral using a standing-wave Bessel beam
Chad McKell, Keith Bonin
Here we create a series of optical corrals and calculate their potential energy profile.
A standing-wave Bessel beam is used to form traps in 1D (along the optical axis) and corrals in 2D, in planes
perpendicular to the optical axis at the antinodal regions of the standing waves. These optical corrals are formed
by an axicon-generated Bessel beam that is retro-reflected back onto itself. We report on Mie calculations of the
2D optical corrals and then compare the resulting probability distributions to those observed for latex particles
of diameters 100, 200, and 300 nm. The experimental radial probability density function of tracked particles closely
mimics the theoretical optical structure of a Bessel standing-wave corral. The Bessel standing-wave corrals we have
characterized are being developed to measure rotational diffusion and torques on micro- and nanorods to help
understand microfluidic behavior. The maximum forces on our small beads in the diffraction-free central zone of the
Bessel beam standing wave are 𝐹||=0.5 pN and 𝐹⊥=0.1 pN.
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Sonification of optically-ordered Brownian motion
Chad McKell
In this paper, a method is outlined for the sonification of experimentally-observed Brownian motion organized into optical
structures. Sounds were modeled after the tracked, three-dimensional motion of Brownian microspheres confined in the
potential wells of a standing-wave laser trap. Stochastic compositions based on freely-diffusing Brownian particles are
limited by the indeterminacy of the data range and by constraints on the data size and dimensions. In this study, these
limitations are overcome by using an optical trap to restrict the random motion to an ordered stack of two-dimensional
regions of interest. It is argued that the confinement of the particles in the optical lattice provides an artistically
appealing geometric landscape for constructing digital audio effects and musical compositions based on experimental
Brownian motion. A discussion of future work on data mapping and computational modeling is included. The present study
finds relevance in the fields of stochastic music and sound design.
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Confinement and Tracking of Brownian Particles in a Bessel Beam Standing Wave
Chad McKell
Optical trapping is a useful tool for manipulating microscopic particles and probing the physical
interactions of matter. However, previous optical trapping techniques introduced complications for analyzing Brownian particle
diffusion in viscous media because they either restricted the particles' motion or trapped the particles too close to a surface.
To our knowledge, this thesis presents the first realization of two-dimensional, transverse tracking of Brownian microparticles
in multiple, surface-isolated traps. To accomplish this, we used an axicon-generated, zeroth-order Bessel beam standing wave whose
parameters were adjusted to allow tight axial confinement and loose transverse confinement of microscopic-sized particles in the
central maximum of the Bessel beam. We chose a Bessel beam because its unique non-diffracting and self-healing properties provided
distinct advantanges over a Gaussian beam. In particular, a Bessel beam standing wave was shown to produce optical potential wells
that are more abundant, uniform, and stable than those of a Gaussian standing wave. Advisor: Keith Bonin. (WFU Seal: By Source, Fair use, wikipedia.org).
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