What do forbidden light-ray fields look like?

Ray-optically, optical components change a light-ray field on a surface immediately in front of the component into a different light-ray field on a surface behind the component. In the ray-optics limit of wave optics, the incident and outgoing light-ray directions are given by the gradient of the phase of the incident and outgoing light field, respectively. But as the curl of any gradient is zero, the curl of the light-ray field also has to be zero. The above statement about zero curl is true in the absence of discontinuities in the wave field. But exactly such discontinuities are easily introduced into light, for example by passing it through a glass plate with discontinuous thickness. This is our justification for giving up on the global continuity of the wave front, thereby compromising the quality of the field (which now suffers from diffraction effects due to the discontinuities) but also allowing light-ray fields that appear to be (but are not actually) possessing non-zero curl and thereby significantly extending the possibilities of optical design. Here we discuss how the value of the curl can be seen in a light-ray field. As curl is related to spatial derivatives, the curl of a light-ray field can be determined from the way in which light-ray direction changes when the observer moves. We demonstrate experimental results obtained with light-ray fields with zero and apparently non-zero curl.