skip to main content
OSTI.GOV title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: Perfectly matched layers in photonics computations: 1D and 2D nonlinear coupled mode equations

Abstract

Extending the general approach for first-order hyperbolic systems developed in [D. Appeloe, T. Hagstrom, G. Kreiss, Perfectly matched layers for hyperbolic systems: general formulation, well-posedness and stability, SIAM J. Appl. Math., 2006, to appear], we construct PML equations for the mixed-type system governing propagation of optical wave packets in both 1D and 2D Bragg resonant photonic waveguides with a cubic nonlinearity, i.e. the coupled mode equations. We prove that in the linear case the layer equations are absorbing and perfectly matched. We also prove they are stable for constant parameters. A number of numerical experiments are performed to assess the layer's performance in both the linear and nonlinear regimes.

Authors:
 [1];  [2]
  1. Seminar for Applied Mathematics, ETH-Zentrum, CH-8092, Zuerich (Switzerland). E-mail: dohnal@math.ethz.ch
  2. Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM 87131 (United States). E-mail: hagstrom@math.unm.edu
Publication Date:
OSTI Identifier:
20991576
Resource Type:
Journal Article
Resource Relation:
Journal Name: Journal of Computational Physics; Journal Volume: 223; Journal Issue: 2; Other Information: DOI: 10.1016/j.jcp.2006.10.002; PII: S0021-9991(06)00469-4; Copyright (c) 2006 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; CALCULATION METHODS; EQUATIONS; NONLINEAR PROBLEMS; ONE-DIMENSIONAL CALCULATIONS; PERFORMANCE; STABILITY; TWO-DIMENSIONAL CALCULATIONS; WAVE PACKETS; WAVEGUIDES

Citation Formats

Dohnal, Tomas, and Hagstrom, Thomas. Perfectly matched layers in photonics computations: 1D and 2D nonlinear coupled mode equations. United States: N. p., 2007. Web. doi:10.1016/j.jcp.2006.10.002.
Dohnal, Tomas, & Hagstrom, Thomas. Perfectly matched layers in photonics computations: 1D and 2D nonlinear coupled mode equations. United States. doi:10.1016/j.jcp.2006.10.002.
Dohnal, Tomas, and Hagstrom, Thomas. Tue . "Perfectly matched layers in photonics computations: 1D and 2D nonlinear coupled mode equations". United States. doi:10.1016/j.jcp.2006.10.002.
@article{osti_20991576,
title = {Perfectly matched layers in photonics computations: 1D and 2D nonlinear coupled mode equations},
author = {Dohnal, Tomas and Hagstrom, Thomas},
abstractNote = {Extending the general approach for first-order hyperbolic systems developed in [D. Appeloe, T. Hagstrom, G. Kreiss, Perfectly matched layers for hyperbolic systems: general formulation, well-posedness and stability, SIAM J. Appl. Math., 2006, to appear], we construct PML equations for the mixed-type system governing propagation of optical wave packets in both 1D and 2D Bragg resonant photonic waveguides with a cubic nonlinearity, i.e. the coupled mode equations. We prove that in the linear case the layer equations are absorbing and perfectly matched. We also prove they are stable for constant parameters. A number of numerical experiments are performed to assess the layer's performance in both the linear and nonlinear regimes.},
doi = {10.1016/j.jcp.2006.10.002},
journal = {Journal of Computational Physics},
number = 2,
volume = 223,
place = {United States},
year = {Tue May 01 00:00:00 EDT 2007},
month = {Tue May 01 00:00:00 EDT 2007}
}
  • A new absorbing boundary technique for the paraxial wave equations is proposed and analyzed. Numerical results show the efficiency of the method. 15 refs., 8 figs., 4 tabs.
  • We consider the two-dimensional Maxwell's equations in domains external to perfectly conducting objects of complex shape. The equations are discretized using a node-centered finite-difference scheme on a Cartesian grid and the boundary condition are discretized to second order accuracy employing an embedded technique which does not suffer from a ''small-cell'' time-step restriction in the explicit time-integration method. The computational domain is truncated by a perfectly matched layer (PML). We derive estimates for both the error due to reflections at the outer boundary of the PML, and due to discretizing the continuous PML equations. Using these estimates, we show how themore » parameters of the PML can be chosen to make the discrete solution of the PML equations converge to the solution of Maxwell's equations on the unbounded domain, as the grid size goes to zero. Several numerical examples are given.« less
  • Abstract not provided.
  • For numerical simulations of highly relativistic and transversely accelerated charged particles including radiation fast algorithms are needed. While the radiation in particle accelerators has wavelengths in the order of 100 {mu}m the computational domain has dimensions roughly five orders of magnitude larger resulting in very large mesh sizes. The particles are confined to a small area of this domain only. To resolve the smallest scales close to the particles subgrids are envisioned. For reasons of stability the alternating direction implicit (ADI) scheme by Smithe et al. [D.N. Smithe, J.R. Cary, J.A. Carlsson, Divergence preservation in the ADI algorithms for electromagnetics,more » J. Comput. Phys. 228 (2009) 7289-7299] for Maxwell equations has been adopted. At the boundary of the domain absorbing boundary conditions have to be employed to prevent reflection of the radiation. In this paper we show how the divergence preserving ADI scheme has to be formulated in perfectly matched layers (PML) and compare the performance in several scenarios.« less