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

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 so-called aperture design. Here, the optimization is modeled as a non-linear optimization problem with partial differential equation 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.