Title page for ETD etd-06032009-172258


Type of Document Dissertation
Author Ptitsyna, Natalia Grigoryevna
Author's Email Address ptitsyna@math.lsu.edu
URN etd-06032009-172258
Title A Discrete Model of Guided Modes and Anomalous Scattering in Periodic Structures
Degree Doctor of Philosophy (Ph.D.)
Department Mathematics
Advisory Committee
Advisor Name Title
Stephen Shipman Committee Chair
Jimmie Lawson Committee Member
Peter Wolenski Committee Member
Robert Lipton Committee Member
William Adkins Committee Member
Keith Gonthier Dean's Representative
Keywords
  • resonant enhancement
  • bifurcation
  • asymptotic analysis
  • variational form
  • Dirichlet-to-Neumann map
  • spatial Fourier harmonics
  • spectral theory
  • lattice
  • scattering problem
Date of Defense 2009-04-30
Availability unrestricted
Abstract
We study a discrete prototype of anomalous scattering associated with the interaction of guided modes of a periodic scatterer and plane waves incident upon the scatterer. The transmission anomalies arise because of the non-robustness of a guided mode, a mode that exists only at a specific frequency and wave number pair. The simplicity of the discrete prototype allows one to make certain explicit calculations and proofs, and to examine details of important resonant phenomena of the open wave guides. The main results are (1) a formula for transmission anomalies near a non-robust guided mode with rigorous error estimates that extends the formula of Shipman and Venakides [28] to non-zero Bloch wave number and (2) rigorous analysis of a bifurcation that connects the anomaly for non-zero wave number to that of zero wave number.
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