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 |