USPTO published first patent applications of the secretive Canadian startup Lensvector. The company seems to work on a compact way to do camera lens using liquid crystal (LC) structures. The company site does not give much information, other than saying it's under construction for the last 1.5 years or more.
US20090316097 application talks about a way to make a tunable lens out of concave-shaped LC film. The effective refractive index of the lens is controlled by applying an electric field across the gap with the LC lens. Lensvector proposes a manufacturing method having the improved lens shape:
Application US20090213321 talks about LC flat lens with spatially changing refractive index. The index is controlled by means of spatially modulated electric field.
Application US20100007807 uses magnetic field to control flat gradient LC lens. The lens has transparent spiral structure to create spatially varying magnetic field. The spiral is called "Hidden Electromagnetic Source (HEMS)":