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Department of Mathematics

Karlsruhe Institute of Technology
D-76128 Karlsruhe
Germany
Tel.: +49 721 608-43800

Mathematical topics on photonic crystals (Winter Semester 2007/08)

Lecturer: Tomas Dohnal, Christian Engström
Classes: Lecture (1100)
Weekly hours: 2
Audience: Matheamtics (from 7. semester)

FIRST HALF


lecturer:

Christian Engström

topics (tentative):

1- A short introduction to homogenization in periodic and random media (asymptotic expansions, two-scale convergence, H-convergence)
2- Bounds on effective material parameters (Herglotz functions, Pade approximants)
3- Inverse homogenization
4- Constitutive relations and dispersion
5- Applications to plasmonic photonic crystals and metamaterials. Limitations of the theory (Homogenization)
6- A short introduction to the Floquet-Bloch theory
7- Possible corrections to the effective material parameters

literature:

  • Bensoussan A, Lions J L and Papanicolaou G. Asymptotic analysis for periodic structures, 1978
  • Milton G, The theory of composites, 2002
  • Jikov V. V, Kozlov S. M. and Oleinik. Homogenization of Differential Operators and Integral Functionals, 1994

SECOND HALF


lecturer:

Tomas Dohnal

topics (tentative):

1- wavePackets, group velocity, group velocity dispersion
2- nonlinearity in photonic crystals
3- gap solitons in Kerr nonlinear photonic crystals
4- modeling of weakly nonlinear gap solitons in 1 and 2 spatial dimensions via coupled mode equations
5- quasi gap solitons in low contrast 2D photonic crystals
6- linear (in)stability of 1D gap solitons
7- interactions of 1D gap solitons and 2D quasi gap solitons with localized defetcs in the photonic crystal

literature:

  • G. P. Agrawal,, Nonlinear Fiber Optics, Academic Press,. New York, 1995.
  • R.E. Slusher and B.J. Eggleton, Nonlinear Photonic Crystals, Springer Verlag, Berlin (2003).
  • I.V. Barashenkov, D.E. Pelinovsky, and E.V. Zemlyanaya, Vibrations and Oscillatory Instabilities of Gap Solitons, Phys. Rev. Lett. 80, 5117-5120 (1998).
  • R. H. Goodman, R. E. Slusher, and M. I. Weinstein, "Stopping light on a defect ," J. Opt. Soc. Am. B 19, 1635-1652 (2002).
  • T. Dohnal and A.B. Aceves, ``Optical soliton bullets in (2+1)D nonlinear Bragg resonant periodic geometries, J. Yang, editor, Nonlinear Wave Phenomena in Periodic Photonic Structures, Studies in Applied Math. 115:209-232 (2005).
  • T. Dohnal, D. Pelinovsky and G. Schneider, ``Coupled-mode equations and gap solitons in a two-dimensional nonlinear elliptic problem with a separable periodic potential, submitted to J. Nonlin. Sci., 2007.

Schedule
Lecture: Wednesday 9:45-11:15 Seminarraum 34 Begin: 24.10.2007, End: 13.2.2008