Over the last few decades, nano-fabrication techniques have evolved to such precision that novel optical material responses can be tailor-made to challenging specifications. This development has opened a new avenue to research areas such as nano-plasmonics and metamaterials, which investigate the behaviour of light on the nano-scale and promise a range of exciting new applications. A valuable tool for a better understanding of these systems are numerical simulations. However, the correct description of the phenomena in this new regime is however challenging as it inherently involves theories of multiple length and time scales, often including non-linear phenomena. This thesis discusses several simulation techniques and investigates their suitability for describing such systems. The focus lies on the Finite-Difference-Time-Domain technique (FDTD) - a method originally developed for the study of radio and microwave electro-magnetics. Unlike frequency-domain solvers, the FDTD method can naturally describe non-linear dynamics due to its time-domain nature. However, two main challenges must be overcome to extend the FDTD method to the nano-plasmonic regime: first, an accurate time-domain description of frequency-dependent, dispersive materials such as metals must be found and second - due to its lack of adaptive meshing - the FDTD method struggles to computationally efficiently calculate systems with features over a wide range of length scales. Both challenges are discussed in detail, along with suggestions on how to overcome them. In the second part of the thesis, the FDTD method is applied to the investigate the behaviour of a non-linear, nano-plasmonic dimer system - two infinitely long and nearly touching silver nanowires, coated with an active material (such as Rh800 dye). Although there have been many studies treating the purely plasmonic system - the silver nanowires - and the non-linear system - the dye coating - individually, describing the combined system with computer simulations has been challenging due to the reasons mentioned above. This system is therefore not only physically highly relevant but also offers a benchmark for the proposed numerical methods in this thesis.
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