  The study of optical systems with asymmetric responses has grown
significantly due to their broad application potential in various fields. In
particular, Janus metasurfaces are notable for their ability to control light
asymmetrically at the pixel level within thin films. However, previous
demonstrations were restricted to the partial control of asymmetric
transmission for a limited set of input polarizations, focusing primarily on
scalar functionalities. Here, we introduce optical metasurfaces consisting of
bi-layer silicon nanostructures that can achieve a fully generalized form of
asymmetric transmission for any possible input polarization. Their designs owe
much to our theoretical model of asymmetric optical transmission in reciprocal
systems that explicates the relationship between front- and back-side Jones
matrices for general cases, revealing a fundamental correlation between the
polarization-direction channels of opposing sides of incidence. To practically
circumvent this constraint, we propose a method to partition transmission
space, enabling the realization of four distinct vector functionalities within
the target volume. As a proof of concept, we experimentally demonstrate
polarization-direction-multiplexed Janus vector holograms generating four
vector holograms. When integrated with computational vector polarizer arrays,
this approach facilitates optical encryption with a high level of obscurity. We
anticipate that our mathematical framework and novel material systems for
generalized asymmetric transmission may pave the way for applications such as
optical computations, sensing, and imaging.
