The diffraction of monsters from complex apertures
The research presented in this MSc thesis is concerned with understanding the way incoming electromagnetic waves are diffracted by various different types of scattering screens. These classes of problem are fundamental in the laser optics community, and within the arena of wave physics more generally. They are also of potential interest to applied mathematics researchers, particularly those concerned with describing scattering through boundary integral equations. Analysis begins with revisiting Lamb’s ingenious solution to the classic knife-edge problem, known for over a century. Maxwell’s equations are solved for an incoming plane wave (subjectto appropriate boundary conditions on a perfectly-conducting semi-infinite screen of negligible thickness) and this building-block calculation is then generalized to allow for incident andscattered waveforms that have multiscale characters. A candidate model used throughout is the Weierstrass function. In its original form, this function is well known to be continuous everywherebut differentiable nowhere and it was dubbed a ‘monster’ by Charles Hermite. Two distinct families of solution are derived which, for the first time, provide a fairly rigorous description of pre-fractal electromagnetic waves scattering from a single knife-edge. Subsequent investigations consider plane waves scattering from conducting screens that can have a multi-scale character, such as a pre-fractal Cantor set (that is, a diffraction grating modelled onany finite iteration of the Cantor set). Previous related studies have been concerned predominantly with regimes wherein the outgoing waves are observed in the far field (that is, at large distancesfrom the screen), and where recourse has typically been made to the scalar approximation. Here, a more general formulation is developed that is based on Rayleigh-Sommerfeld diffractionintegrals, and the scattered waves are connected more directly to Maxwell’s equations in terms of calculating the corresponding magnetic-field components. Finally, the first steps are taken toward modelling a physical scenario where an incident prefractal wave scatters from a pre-fractal Cantor set. Research into this regime is still ongoing, largely due to the computationally-expensive nature of the required calculations.
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