# American Institute of Mathematical Sciences

October  2012, 5(5): 989-1020. doi: 10.3934/dcdss.2012.5.989

## A lattice model for resonance in open periodic waveguides

 1 Department of Mathematics, Embry-Riddle Aeronautical University, Daytona Beach, FL 32114, United States 2 Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, United States

Received  April 2011 Revised  September 2011 Published  January 2012

We present a discrete model of resonant scattering of waves by an open periodic waveguide. The model elucidates a phenomenon common in electromagnetics, in which the interaction of plane waves with embedded guided modes of the waveguide causes sharp transmission anomalies and field amplification. The ambient space is modeled by a planar lattice and the waveguide by a linear periodic lattice coupled to the planar one along a line. We show the existence of standing and traveling guided modes and analyze a tangent bifurcation, in which resonance is initiated at a critical coupling strength where a guided mode appears, beginning with a single standing wave and splitting into a pair of waves traveling in opposing directions. Complex perturbation analysis of the scattering problem in the complex frequency and wavenumber domain reveals the complex structure of the transmission coefficient at resonance.
Citation: Natalia Ptitsyna, Stephen P. Shipman. A lattice model for resonance in open periodic waveguides. Discrete & Continuous Dynamical Systems - S, 2012, 5 (5) : 989-1020. doi: 10.3934/dcdss.2012.5.989
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