Mathematical topics on photonic crystals (Wintersemester 2007/08)
- Dozent*in: 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 | |||
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| Vorlesung: | Mittwoch 9:45-11:15 | Seminarraum 34 | Beginn: 24.10.2007, Ende: 13.2.2008 |