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Geometrical optics describes, with good accuracy, the propagation ofhigh-frequency plane waves through an electromagnetic medium. Under suchapproximation, the behaviour of the electromagnetic fields is characterised byjust three quantities: the temporal frequency $\omega$, the spatial wave(co)vector $k$, and the polarisation (co)vector $a$. Numerous key properties ofa given optical medium are determined by the Fresnel surface, which is thevisual counterpart of the equation relating $\omega$ and $k$. For instance, thepropagation of electromagnetic waves in a uniaxial crystal, such as calcite, isrepresented by two light-cones. Kummer, whilst analysing quadratic linecomplexes as models for light rays in an optical apparatus, discovered in theframework of projective geometry a quartic surface that is linked to theFresnel one. Given an arbitrary dispersionless linear (meta)material or vacuum,we aim to establish whether the resulting Fresnel surface is equivalent to, oris more general than, a Kummer surface.

Fresnel versus Kummer surfaces: geometrical optics in dispersionless linear (meta)materials and vacuum