Can an angular engineered meta-lens break the conservation of etendue?

Meta-lenses and meta-surfaces have shown great potential for future optical components. Advantages of meta-lenses over conventional optics rely on the practical aspects (thin/light-weight), free phase profile creation (high NA lenses, holography), specific tailored dispersion engineering, polarization engineering and more. <br/><br/> A more overlooked possibility is the angular engineering of meta-atoms. Often simply ignored or stated as angleindependent, angle engineering can be highly interesting for example for wide field of view lenses. As for dispersion engineered meta-atoms, where the phase delay ϕ of a meta-atom can be wavelength dependently defined as ϕ(λ), it is possible to design angular dependent meta-atoms where the meta-atom phase delay is dependent on the incident angle θ, denoted as ϕ(θ. Here we do a thought experiment of creating an idealized perfect collimator, meaning that at any location light from any incident direction comes out perpendicular on the surface. Even if this would be possible for a small area and a small opening angle, etendue would be reduced in the system, which is in direct conflict with the conservation of etendue. With a full wave RCWA simulation of slanted cones and a local periodic based component design within the PlanOpSim software, we can simulate this type of component and estimate the outcoupling angles. We discuss the contradiction between etendue conservation and this thought experiment backed up with a simulation.