"The objective of this proposal is to devise new ways of shaping light and other electromagnetic waves in complex and novel structures. The research will delve into fundamental aspects of electrodynamics, from classical optics to the quantum mechanical dynamics of photons.Over the past twenty years, several new kinds of artificial media for electromagnetic waves were demonstrated, among them photonic crystals, metamaterials, plasmonic materials, and more recently graphene-like photonic lattices. These new kinds of ""electromagnetic media"" call for new methods for beam shaping to facilitate control over the electromagnetic fields propagating within them. Moreover, many novel photonic systems and devices rely on strong coupling between light and matter. Naturally, such systems combine quantum mechanics effects with classical electrodynamics. For example, modern experimental research would greatly benefit from efficient coupling of light to quantum dots, nano-photonic structures embedded in silicon chips, or particular molecules – to manipulate and probe the molecular dynamics. All of these necessitate novel methods for shaping light in new kinds of structures, which cannot be addressed by traditional techniques.In this research plan, I aspire to make fundamental theoretical contributions that, apart from their basic research component, will also contribute to a large variety of present and future applications. I believe this ambitious research will have high impact, advancing the knowledge on waves' propagation in structured media, which is at the forefront of current research.The outgoing phase (first two years) will take place at Stanford, under the supervision of Prof. Shanhui Fan. The return phase (third year) will take place at the Technion, under the supervision of Prof. Gadi Eisenstein. The return department (the Faculty of Electrical Engineering) is highly supportive of my application: the support letter of the Dean is incorporated in the proposal (in B3)."