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Variational modelling for integrated optical devices

TU Vienna
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Variational modelling is the use of a variational structure of a problem to simplify the model or to find approximations of the solutions in a consistent way. In both cases the consistent use of the variational structure consists in restricting the relevant functionals to smaller sets and consider the Euler-Lagrange equation on the restricted set instead of on the original set. One type of restriction may be to specialise the set of phenomena. To find approximate solutions, parameterized manifolds of functions are used to restrict the functional; either low-dimensional manifolds of appropriate 'trial'-functions, or high-dimensional linear subspaces for numerical discretizations. In these notes another type of restriction will be discussed. We describe how typical problems for all-optical devices in integrated optics have to be considered on unbounded domains. The variational structure is then exploited to confine the problem to a finite domain by restriction to functions that satisfy, or approximate, the equations on the exterior domain. For a typical reflection problem this leads to boundary conditions that are 'transparent' for a-priorily unknown radiation and transmittance, but allow a prescribed influx of light into the structure.

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