Title: Transmission efficiency limit for nonlocal metalenses

Abstract: The rapidly advancing capabilities in nanophotonic design are enabling complex functionalities limited mainly by physical bounds. The efficiency of transmission is a major consideration, but its ultimate limit remains unknown for most systems. Here, we introduce a matrix formalism that puts a fundamental bound on the channel-averaged transmission efficiency of any passive multi-channel optical system based only on energy conservation and the desired functionality, independent of the interior structure and material composition. Applying this formalism to diffraction-limited nonlocal metalenses with a wide field of view, we show that the transmission efficiency must decrease with the numerical aperture for the commonly adopted designs with equal entrance and output aperture diameters. We also show that reducing the size of the entrance aperture can raise the efficiency bound. This work reveals a fundamental limit on the transmission efficiency as well as providing guidance for the design of high-efficiency multi-channel optical systems.