Refraction of C-line vortices

The refraction and reflection of a plane electromagnetic wave at an interface is governed by the well-known Fresnel coefficients. The incident wave, as a single plane wave, has a Fourier transform, which is a delta-function in wavevector space k. And the transmitted and reflected waves are similar with their delta-functions on different k's. We study instead the refraction and reflection of the natural elaboration of a plane wave; a 'differentiated plane wave'; the first derivative of a plane wave with respect to its direction, whose Fourier transform is, in addition, the derivative of a delta-function. The motivation is that unlike a plane wave this wave has features, such as C-line vortices, and, being so primitive, exact analytical expressions and universal forms are obtained. The refracted and reflected waves are also differentiated plane waves. This construction is loosely connected with the analysis of beam shifts. The wave field geometry is naturally expressed in terms of its C-lines, or vortices of circular polarization, suitably defined. For a single such wave, there are two straight C-lines parallel to the propagation direction, one right-handed and one left-handed. When superposed with another such wave (the reflected wave), differently polarized and in a different direction, each straight C-line becomes a three-dimensional coiled shape on a hyperboloid surface. Finally, separately, for scalar waves, a local 'law of vortex refraction' is derived, independent of any specific model of the incident wave.