WIAS Preprint No. 2377, (2017)

A PDE-constrained optimization approach for topology optimization of strained photonic devices


  • Adam, Lukáš
  • Hintermüller, Michael
    ORCID: 0000-0001-9471-2479
  • Surowiec, Thomas M.

2010 Mathematics Subject Classification

  • 49J20 35Q93 35Q74 90C46


  • semiconductor lasers, germanium, topology optimization, optimization with PDE constraints, elasticity, phase-field




Recent studies have demonstrated the potential of using tensile-strained, doped Germanium as a means of developing an integrated light source for (amongst other things) future microprocessors. In this work, a multi-material phase-field approach to determine the optimal material configuration within a so-called Germanium-on-Silicon microbridge is considered. Here, an ``optimal" configuration is one in which the strain in a predetermined minimal optical cavity within the Germanium is maximized according to an appropriately chosen objective functional. Due to manufacturing requirements, the emphasis here is on the cross-section of the device; i.e. a socalled aperture design. Here, the optimization is modeled as a non-linear optimization problem with partial differential equation (PDE) and manufacturing constraints. The resulting problem is analyzed and solved numerically. The theory portion includes a proof of existence of an optimal topology, differential sensitivity analysis of the displacement with respect to the topology, and the derivation of first and second-order optimality conditions. For the numerical experiments, an array of first and second-order solution algorithms in function-space are adapted to the current setting, tested, and compared. The numerical examples yield designs for which a significant increase in strain (as compared to an intuitive empirical design) is observed.

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