**Supplementary Figure S2 |****Results from the design procedure (MT**_{2}** and MT**_{4}**).****a, b** Numerically-synthesized phase and magnitude distributions, respectively, of the transmission coefficient pertaining to the single nanoholes in the supercell (shown on top) of the MT_{2} design (with parameters as given in Table 1), for the co-polarized (blue square markers) and cross-polarized (red circle markers) components, assuming an infinite periodic array of period , under normally-incident -polarized plane-wave illumination at . Element #1 is chosen as phase reference. Continuous curves are guides to the eye only. **c, d** Same as above, but for MT_{4} design.
We verified that the look-up maps in Supplementary Figure S1, calculated for a specific inter-element spacing (), could also be utilized for designs featuring slightly smaller values of (e.g., MT_{2} and MT_{3}), yielding only slight distortions. Supplementary Figure S2 shows the phase and magnitude profiles pertaining to the MT_{2}and MT_{4} designs (not shown in the main text for brevity). We highlight that, for the MT_{5} design, any value of the nanoholesidelengths would produce the same phase gradient. However, the chosen design was found to provide a good trade-off between small size (so as to reduce ) and high cross-polar transmission coefficient.
The far-field intensity profiles in Figure 4 (as well as Supplementary Figure S7below) are computed in two steps. First, we compute the transmitted field pertaining to a MT of finite width ~ along the -direction, and assumed as infinitely-periodic in the -direction, illuminated by a 1-D Gaussian beam with waist size of impinging from the silica region (so as to partially mimic the modal field of the SMF-28 optical fiber).The structure is terminated by Bloch-type periodic boundary conditions along the -direction, and by *ad hoc*perfectly matched layers (PML) in the remaining directions. The above assumptions, which neglect the non-uniform polarization distribution (along the -direction) of a realistic 2-D beam fiber-optic illumination, are instrumental to maintain the computational burden within affordable limits, while still providing a good estimate of the illumination-tapering effects.
The near-field distribution is computed by means of COMSOL Multiphysics, with the simulated region extending up to a distance of in air and in the fiber region, in order to guarantee the computational affordability. Once again, an adaptive meshing is employed, with maximum element size of in the uniform dielectric regions, 50nm in the air regions of the array, and a minimum number two element per skin-depth in the gold layer,resulting in about 4.5 million degrees of freedom. The PARDISO solver is utilized, with default parameters.As a second step, the computed near-field is propagated to the far-field region using well-known formulae for the radiation from planar apertures (e.g., Sec. 4.1 in Ref. 2). In order to account for the lack of polarization control in the measurements shown in Figure 4 (as well as Supplementary Figure S7 below), the far-field intensity profiles are averaged over the two limiting cases of- and -polarized incidence. Results are normalized with respect to the maximum values.
From the same numerical simulations, we also estimate the efficiency of the designed MT prototypes, i.e., the fraction of incident power that gets transferred to the anomalous beam. To this aim, assuming an-polarized incidence, we compute the flux of the Pointing vector (real-part) associated with the cross-polarized transmitted field through a planar surface at a distance from the metasurface. We then obtain the efficiency by normalizing this quantity by the same flux pertaining to the incident field only, calculated in the absence of the metasurface, and by truncating the fiber region with a PML.
The field maps shown in Figures79are also computed via COMSOL Multiphysics by assuming a periodic 2-D array under normally-incident-polarized plane-wave illumination. Once again, Bloch-type periodic boundary conditions are assumed along the - and -directions, with *ad hoc* PML terminations along the -direction. In this case, the simulated region extends up to a distance of in air and in the fiber region, with maximum mesh-element size of in the uniform dielectric regions and of 20nm in the air regions of the array, and a minimum number two element per skin-depth in the gold layer (resulting in about 2 million degrees of freedom). Also in this case, the MUMPS solver is utilized. Results are normalized with respect to the incident-field amplitude.
For better computational affordability, the reflectivity spectra for these configurations are computed by means of a public-domain numerical code (sourceforge.net/projects/rcwa-2d/files/) that implements a 2-D rigorous coupled wave analysis (RCWA).^{3} Once again, a normally-incident-polarized plane-wave illumination is assumed. Convergence is achieved by using modes. To realistically simulate the experimental reflection setup, the reflectivity pertaining to the zero-th diffraction order is considered, corresponding to light traveling back along the fiber axis. Moreover, in order to roughly mimic a realistic deposition process, the SiO_{x} overlay is assumed as conformal to the MT surface (i.e., filling the nanoholes and covering the unpatterned gold regions).
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