Supporting Information An Optical “Janus” Device for Integrated Photonics

Device design using quasi-conformal mapping Figure 1 in the main text specifies how the boundaries between the virtual and the physical spaces of our device are transformed into each other. In principle, any coordinate map satisfying such a transformation can be used to construct the actual permittivity profile. However, this would lead in general to a very anisotropic permittivity for the structure. Here we utilized the procedures specified in Ref. 25: By using the Modified Liao Functional for numerically generating the coordinate map, a quasi-conformal map is obtained in which a square cell in the virtual space is transformed to a rectangle with least anisotropy. This small amount of anisotropy is then neglected so that only isotropic permittivities are used in our Janus device. Such a map generates an effective permittivity ranging from 0.6 to 5 for the 250-nm-thick silicon waveguide mode with the specified lateral dimensions of the device. In order to realize this profile by etching spatially varying air holes in the silicon slab we have to neglect the values below 2 around the four corners, which are the only areas where the permittivity drops below 2. These regions with the highest density of the holes would result in an overlap of the finite holes. However, the light beams rarely accesses this spatial region so that these parts can be omitted without losing functionality. A detailed description about the generation of the hole coordinates for a certain permittivity profile can be found in Ref. 1. The abrupt change of the permittivity at the device boundaries leads to reflectivity of ≈5% for each boundary due to the impedance mismatch to the waveguide index. Such a