Abstract
The main goal of the present paper is to provide a fresh general view on the resurgent research area of nonlocal metamaterials. We explain how theconcept of nonlocality had risen historically in the field of plasma and crystaloptics and highlight the recent interest in using nonlocality as a new type of metamaterials. The paper will develop in increasing complexity the concept of nonlocality starting from general considerations, going through spatial dispersion,and ending up with an abstract but quite broad formulation that unveils the link between general topology and electromagnetic nonlocality in material media. It is shown that electromagnetic nonlocality naturally leads to a Banach (vector) bundle structure serving as an enlarged space on which electromagnetic processes take place. The added structures, essentially fiber space, model the topological microdomains of electromagnetic nonlocality and provide a fine grained picture of field-matter interactions in nonlocal metamaterials. We use standard techniques borrowed from differential topology to construct the Banach bundle structure by paying careful attention to the relevant physics. The electromagnetic response tensor is then reformulated as a bundle homomorphism and the various expressions needed to connect the local topology to global domains are developed. The paper also introduces a discussion of various applications, including elucidating why nonlocal electromagnetic materials often require additional boundary conditions or extra input from microscopic theory in comparison with local electromagnetics.
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