An in-depth examination of two-dimensional Laplace inversion and application to three-dimensional holography

Date of Award

2014

Degree Name

M.S. in Electrical Engineering

Department

Department of Electrical and Computer Engineering

Advisor/Chair

Advisor: Monish Ranjan Chatterjee

Abstract

An analytic examination of 3-D holography under a recording geometry was carried out earlier in which 2-D spatial Laplace transforms were introduced in order to develop transfer functions for the scattered outputs under readout [1]. Thereby, the resulting reconstructed output was obtained in the 2-D Laplace domain whence the spatial information would be found only by performing a 2-D Laplace inversion. Laplace inversion in 2-D was attempted by testing a prototype function for which the analytic result was known using two known inversion algorithms via the Brancik and the Abate [2]. The results indicated notable differences in the 3-D plots between the algorithms and the analytic result, and hence were somewhat inconclusive. In this research, we take a close look at the Brancik algorithm in order to understand better the implications of the choices of key parameters such as the real and imaginary parts of the Bromwich contour and the grids sizes of the summation operations [3]. To assess the inversion findings, three prototype test cases are considered for which the analytic solutions are known. For specific choices of the algorithm parameters, optimal values are determined that would minimize errors in general. It is found that even though errors accumulate near the edges of the grid, overall reasonably accurate inversions are possible to obtain with optimal parameter choices that are verifiable via cross-sectional views. For a holographic problem, a 90-deg geometry recording model is established to derive two important coupled equations [4]. The optimum parameters are used to find the output field profiles under readout for a uniform plane wave, a point source wave and a Gaussian profile input. To understand the results better, a convolutional approach and a holistic approach are compared. Further work may include recording and reconstructing a dynamic object wave whose wave representations are more complicated. Also, the observed right shift" phenomenon associated with an inversion needs further examination and is currently inconclusive."

Keywords

Holography Mathematical models, Three-dimensional imaging Mathematical models, Laplace transformation, Optics, Holography, 2-D Laplace Inversion, Brancik Algorithm, Hadamard Product, Lozenge Diagram, Fast Fourier Transform, Object Wave, Reference Wave, First Order Beam, Zeroth Order Beam

Rights Statement

Copyright © 2014, author

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