Nonlinear eigenvalue problems in optics

Wednesday, December 14, 2016 - 1:30pm - 2:30pm
Keller 3-180
Christian Engström (University of Umeå)
Functions whose values are unbounded linear operators describe a large number of processes in optics. The applications include photonic crystals with material properties of Drude-Lorentz type and computations of scattering resonances.
In this talk we present new enclosures of the numerical range of an operator functions T. Contrary to the numerical range of T, the presented enclosure can be computed exactly given only the numerical ranges of the operator coefficients. Moreover, we establish variational principles for rational functions whose values are self-adjoint operators. The new enclosure and the variational principles are then applied to photonic crystal applications.
In the case of finite photonic crystals we compute scattering resonances and discuss pros and cons with computations based on the Lippmann-Schwinger equation, a Dirichlet-to-Neumann map, and a perfectly matched layer.