On Fractal Electrodynamics

Geometry has played a crucial role in the formulation and understanding of mathematical methods in electromagnetic theory. It was the pioneering work of James Clerk Maxwell and Lord Rayleigh in the nineteenth century that provided the first glimpse into the world of wave interactions with regular Euclidean structures, and led to the solution of canonical scattering problems typical of the early and mid-twentieth century. These problems involved scattering by edges, spheres, and infinite cylinders and the guiding of waves by planar and cylindrical structures of regular cross section. More recently, wave interactions with wedges, ellipsoids, cones, cubes, truncated cylinders, and small groups of these objects have been described using both experimental and numerical methods to attack increasingly complex geometry. In most cases, these more complex objects are either descriptive of simple man-made objects or are elementary approximations to more complicated objects which occur in nature. However, models which accurately display the multitude of scale lengths typical of naturally occurring structures are not well suited to Euclidean description and so fall outside the realm of problems noted above. It is exactly these structures, sometimes denoted variegated, ramified, spiky, tortuous, pathological, wiggly, or wildly irregular, and their interactions with electromagnetic waves which are of interest here.