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Fakultät für Mathematik

Karlsruher Institut für Technologie
D-76128 Karlsruhe
Tel.: +49 721 608-43800

Mathematical topics on photonic crystals (Wintersemester 2007/08)

Dozent: Dr. Tomas Dohnal, Dr. Christian Engström
Veranstaltungen: Vorlesung (1100)
Semesterwochenstunden: 2
Hörerkreis: Mathematics (ab 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- phase velocity, group velocity, group velocity dispersion, wavepackets
2- nonlinearity in dielectrics
3- multiple scales expansions
4- modeling of gap solitons in 1 and 2 spatial dimensions via coupled mode equations
5- Lyapunov-Schmidt reductions
6- asymptotic reductions of coupled mode equations
7- gap solitons and quasi gap solitons in Kerr nonlinear photonic crystals (1D and 2D)
8- linear (in)stability of 1D gap solitons

literature:

  • G. B. Whitham: Linear and Nonlinear Waves, Wiley, 1974.
  • 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).
  • 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.
  • M. Golubitsky and D.G. Schaeffer, "Singularities and groups in bifurcation theory," Springer-Verlag, 1985.

Termine
Vorlesung: Mittwoch 9:45-11:15 Seminarraum 34 Beginn: 24.10.2007, Ende: 13.2.2008