Finding passive, reciprocal metasurfaces for arbitrary wave transformations

We give a general design method for finding the passive, reciprocal surface impedance tensor required to enact any wave transformation. We do this through characterising the surface in terms of a tensorial surface impedance, showing that a large family of impedance distributions can be found that perform an identical wave transformation. Even when the conditions of reciprocity and passivity are imposed, there still remain many solutions to the design problem.} We exploit this as a design method for metasurfaces, giving two examples where the metasurface rotates the input polarization and reshapes the output field, showing we can parameterize the set of equivalent reciprocal metasurfaces in terms of a single complex parameter. \hl{In addition, through allowing dissipation and gain within the response, the surface can have many different functionalities in the orthogonal polarization, opening up a new route for the design of multiplexed metasurfaces.