## 2 Theory and Meta-Atom Design  
Depending on the geometry, a high refractive indexed meta-atom with sufficient height can manipulate the phase, polarization, and amplitude of the incident electromagnetic wave. Figure 1a shows the sketch of the proposed meta-atom. A Nb\({}_{2}\)O\({}_{5}\) nanobar with an orientation-rotating angle \(\theta\) is patterned on a silica substrate to obtain the desired phase manipulation. As the nanostructure in the designed meta-atom is a simple rectangular nanobar, the transmission coefficients of this structure can be represented using the Jones matrix [30]:  
\[\hat{t}(\theta)\ =\ \left[ \begin{array}{cc}t_{x}cos^{2}\theta+t_{y}sin^{2} \theta&(t_{x}-t_{y})sin\theta cos\theta\\ (t_{x}-t_{y})sin\theta cos\theta&t_{y}cos^{2}\theta+t_{x}sin^{2}\theta\end{array} \right] \tag{1}\]  
where \(t_{x}\) and \(t_{y}\) are the complex transmission coefficients when the polarization of an incident wave is \(x\)-polarized and \(y\)-polarized, respectively. Given the right-handed circular-polarized (RCP) light \(\left(E_{i}\ =\ (\hat{e}_{x}+i\hat{e}_{y})/\sqrt{2}\right)\) of incidence, the transmitted electric field \(E_{t}\) can be written as follows [31]:  
\[E_{t}\ =\ \hat{t}(\theta)\bullet E_{i}\ =\ \frac{t_{x}+t_{y}}{2}E_{RCP}+\frac{t_{x}-t_ {y}}{2}e^{2\theta}E_{LCP} \tag{2}\]  
Figure 1: The optical properties of the Nb\({}_{2}\)O\({}_{5}\)-based meta-atom. (**a**) Sketch of the Nb\({}_{2}\)O\({}_{5}\) nanobar with a rotation angle of \(\theta\) on the silica substrate. This meta-atom is designed to convert the incident right-handed circular polarization (RCP) plane wave into the transmitted left-handed circular polarization (LCP) wave with an additional phase modulation of \(e^{2\theta}\). (**b**) Simulated conversion efficiency \(\eta\) for the nanobar with optimized height H = 340 nm, period P = 200 nm, and various lengths L and widths W under the RCP wave incidence at 355 nm. (**c**) The transmission (orange line), co-polarized (RCP, blue line), and cross-polarized (LCP, red line) efficiencies versus rotation angle \(\theta\) of the Nb\({}_{2}\)O\({}_{5}\) nanobar arrays under the RCP wave incidence at 355 nm. (**d**) The transmitted co-polarized and cross-polarized efficiencies of the optimized meta-atom arrays are investigated within a broadband wavelength range. The first term in Equation (2) represents a co-polarized transmitted wave component with the same helicity as the incident light. The second term represents a cross-polarized transmitted wave with the opposite helicity and an additional Pancharatnam-Berry (PB) phase manipulation of \(e^{2\theta}\). Therefore, through rotating the anisotropic nanobar from 0 to \(\pi\), the full \(2\pi\) phase shift of the transmitted cross-polarized wave component can be achieved. The meta-atom is under the incidence of the RCP plane wave at a wavelength at 355 nm. The simulation of the structure is implemented by using the finite-difference time-domain (FDTD) method through the commercial software Lumerical FDTD. The computational area is set to be \(200\times 200\times 200\) nm in the \(x,y,\) and \(z\) directions, respectively. Periodic boundary conditions are used in the \(x\) and \(y\) directions of a unit cell. In contrast, the perfectly matched layer (PML) boundaries are applied along the \(z\) direction to eliminate unwanted reflections. The optical constant of niobium pentoxide is taken from [PERSON]'s report [32], and the optical constant of the silica substrate is taken from the work of [PERSON] [33]. The length (L) and width (W) are swept to maintain a high cross-polarized efficiency, which is defined as the ratio of the transmitted light with cross-polarization to the total incident light. Figure 1b shows the simulated efficiency changes with the length (L) varying from 110 to 170 nm and the width (W) from 40 to 100 nm. The height (H) of the nanobar and the period (P) of the square meta-atom are fixed to the optimized values of 340 nm and 200 nm, respectively.