Supplementary Figure S1 Design lookup maps.a, b Numericallycomputed phase distributions of copolar and crosspolar, respectively, transmission coefficient, as a function of nanohole dimensions, assuming an infinite square array with period with normallyincident, polarized planewave illumination at . For better visualization, the phase distributions are “unwrapped”.c, d Corresponding magnitude maps. As a reference, a representative equimagnitude contour () is superposed (dashedmagenta curve) to the crosspolar phase map in panel b.
Overall, this results in about 700,000 degrees of freedom. The MUMPS solver is utilized, with default parameters.
By varying the nanoholesidelengths and (see Figure 1c) around their resonance values, and illuminating the array with a normallyincidentpolarized plane wave at the operational wavelength, we obtain the “lookup” maps shown in Supplementary Figure S1. More specifically, the maps are calculated on a grid of () values, and bilinear interpolation is used for a finer sampling. The transmission coefficients are computed as the scattering parameter pertaining to co and crosspolarized incident and transmitted zerothorder modes.
It can be observed that, different from the copolarized case (Supplementary Figure S1a), the phase map pertaining to the crosspolarized component (Supplementary Figure S1b) spans a full range, which is a necessary condition for designing an arbitrary phase profile. In spite of the inherent efficiency limits in the underlying polarizationconversion mechanism, the desired phase span can be attained for moderate values of the transmissioncoefficient magnitude, as shown by the reference equimagnitude contour () superposed on the crosspolar phase map.
Based on the lookup maps, we can select the nanohole dimensions so as to synthesize a desired phase distribution of the crosspolar transmission coefficient. More specifically, given the targeted phasegradient and an even number of elements in a super cell, we initialize the synthesis procedure by choosing the dimensions of the first element, yielding a phase of the crosspolar transmission coefficient. We then proceed with the subsequent elements by minimizing the cost function
22\* MERGEFORMAT (S)
where denotes the desired phase value, is the argument, is a weight coefficient, and is the maximum value of that allows a full phase excursion within the space. Here, and henceforth the principal value of the phase (within the range) is assumed. The cost function is minimized via a standard NelderMeald (simplex) unconstrained optimization algorithm. The parameters , and are chosen heuristically, via trialanderror.
The remaining elements can be obtained via a rotation of 90° in the plane, which provides a phaseshift on the crosspolar transmission coefficient.^{1}Clearly, the number of free parameters is not sufficient to control both polarization components. Nonetheless, this procedure allows us to obtain a copolar transmission coefficient with sufficiently uniform magnitude and phase distributions over the designed supercells (see Figure 2). Additional degrees of freedom, e.g., the rotation angle of the nanoholes, could be used to attain better control on both polarization components.
The phase gradient can be increased by either reducing , or by increasing the interelement phase difference. However, some limitations exist, since should be large enough to limit the interelement coupling effects, and can be at most equal to in order to guarantee the correct reconstruction of the linear phase profile. The synthesis results summarized in Table 1 implement various combinations of these two parameters.
