*2.2.1 Waveguiding effect*

To gain a better insight into the phase realization mechanism, we calculated the phase imparted solely by the wave-guiding effect [5]. This phase is given by

$$
\phi\_{\rm WG} = \frac{2\pi}{\mathcal{A}\_d} n\_{\rm eff} H \tag{2}
$$

where H is the height of nanopost. The effective index *n*eff of the fundamental mode (HE11) can be calculated from the model the step-index circular waveguide. **Figure 2(e)** indicates that the phase profile from this model follows the calculated result from the finite difference time domain (FDTD) simulation of the nanopost unit-cell on the glass substrate. The good agreement indicates that the confinement of the fundamental mode increases with the waveguide diameter. The average absolute phase difference between the wave-guiding effect and the full-wave analysis is less than π/6. This indicates that the wave-guiding effect is the mechanism dominating considering the practical phase realization. Therefore, as shown in the **Figure 2** below, the phase library can cover 2π by changing the diameter of the cylinder.

### *2.2.2 Resonance modes*

Another major approach of metasurface is the V-shaped nano-antenna array, which consists of series of two pillared Ag strips connected at the ends with the designed angles. The V-shaped antenna unit has the unique property of doubleresonating effect, which includes symmetric and asymmetric components. When a vertical polarized light incidents along x-direction to the nanoantenna array, the incident field can be divided into component Es and component Ea which are perpendicular and parallel to antenna axis, respectively. Component Es excites plasmonic symmetric mode while component Ea excites asymmetric one. The doubleresonance mode enables V-shaped antenna to provide phase changes of 2π with the large amplitude [8]. The **Figure 3** shows that the resonance mode is changed by opening of the V shape to cover 2π.
