True optical spacial derivatives for plasma density measurements

This paper shows analytically and numerically that a vortex plate coupled to a neutral density filter can deliver a true optical derivative when placed at the focal plane of a $2f$ lens pair. This technique turns spatial variations in intensity into an intensity, which square root is the spatial derivative of the initial intensity variation. More surprisingly, it also turns any spatial variations in phase into an intensity, which square root is the spatial derivative of the initial phase variation. Since the optical derivative drops the DC component of the signal, it is possible to measure the full electron plasma turbulence spectrum optically, without using any interferometer.