Topology and Material Optimization of Optical Properties of Particulate Products

In recent years, the design of electromagnetic scattering objects with respect to their optical properties has become an interesting field of research and represents one of the main focuses within nanoparticle technology. In particular, there has been a notable increase in interest in the topology and the more generalized material optimization of individual particles and specially arranged assemblies of nanoparticles. These particulate products exhibit intriguing optical properties, which render them attractive for applications in a wide range of research fields. In this work, we predict the electromagnetic scattering behaviour, which is described by the Maxwell’s equations in time-harmonic form, of a range of objects. The solution to this system is approximated by either the discrete dipole approximation (DDA) or transition matrix method (TMM). While in DDA the whole scattering object is discretized into individual cells called dipoles, in TMM the electromagnetic fields of each constituent in the system are expanded in terms of vector spherical wave functions. Therefore, DDA is ideal for predicting scattering properties of arbitrarily shaped individual nanoparticles and small particle clusters, while TMM is also applicable to larger particle assemblies where the individual components may be more distant from each other. In order to compute an improved design in terms of an optical property function, we consider a design optimization approach in which either DDA or TMM is employed to approximate the solution of the time-harmonic Maxwell’s system. The fundamental aspect of the optimization approach is the iterative assignment of a material property from a pre-defined catalogue of admissible materials to each component of the system until a local minimum of the chosen objective function is reached. In the DDA setting, the material property of each dipole is changed, whereas in the TMM framework, the material property of each nanoparticle within the system itself is modified. The design updates are calculated by applying a separable model of the original optimization objective function, which allows the solution of the resulting optimization subproblem to global optimality and thus gives rise to a sequential global programming (SGP) algorithm. We propose various types of separable models, including exact separable models, which remain fully accurate as long as a single design variable is modified. Furthermore, we suggest the use of tight separable approximations of the aforementioned models, which offer a straightforward computational tractability. We demonstrate the strength and efficacy of the DDA-SGP and TMM-SGP approach for an academic proof-of-concept example and apply these methods to address different problems in nanoparticle technology. The application of DDA allows for a significantly higher design resolution than that achievable with finite element based methods. Moreover, the efficiency and accuracy of TMM permits the consideration also of large particle clusters. The combination of these two approaches with structural design algorithms expands the range of numerically tractable and feasible design problems concerning photonic nanostructures and their optical applications.