Optimal structures for focusing and energy accumulation: mathematical models and intuition

Metamaterials and composite structures are able to manipulate waves and focus fields and currents in desirable directions. Designs based on spatial and temporal variation of material properties create structures forcing fluxes into specified parts of the domain or concentrating energy into arrays of progressively sharpening pulses. The paper discusses examples of focusing structures, mathematical and intuitive considerations that influence optimal design theory. The optimality requirement introduces zones of optimal composites with variable microgeometry. The observed absence of classical solutions motivates the extension of the class of optimal partitions to composites. Such materials also provide a solution to the problem of optimal design of a thermal lens focusing thermal fluxes when the incoming fluxes are not completely known in advance. An extension of designs to dynamical materials such as space-time checkerboard composites introduces metamaterials with additional capabilities that control the accumulation of energy in the propagating waves. The discussed mathematical methods of focusing and suitable properties alternation targeted on optimality are illustrated by physical examples.