Geometric optics and the wave equation on manifolds with corners
نویسنده
چکیده
According to geometric optics, light propagates in straight lines (in homogeneous media), reflects/refracts from surfaces according to Snell’s law: energy and tangential momentum are conserved. Thus, when reflecting from a hypersurface (which has codimension one) one gets the usual law of incident and reflected rays enclosing an equal angle to the normal to the surface. Indeed, conservation of tangential momentum and kinetic energy implies that of the magnitude of the normal component. When reflecting from a higher codimension (≥ 2) corner, the law is unchanged (momentum tangential to the corner and energy are conserved) – but now this allows each incident ray to generate a whole cone of reflected rays, see Figures 1-2. On the other hand, light is a form of electromagnetic radiation, satisfying Maxwell’s equations – which in turn implies that each component of the electromagnetic field (in free space) satisfies the wave equation,
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