## 3.1 Experiments

In Fig. 2a, we show the wavelength-incidence angle map when the sample is excited with linearly p-polarized light, and the QWP angle in the output is set to zero (linearly polarized output). We notice transmission minima features, strongly dependent on the incidence angle, which correspond to coupling of the light to surface plasmon polaritons (Petronijevic et al. 2017). When the sample is fixed at the angle of θ = 0°, the transmission intensity is moderately high for longer wavelengths, with specific transmission dip around 700 nm. Then, as the sample angle is gradually altered, the transmission decreases significantly. Note that here only the zeroth order of transmission is resolved, while other techniques such as photo-deflection can be used to monitor the diffraction effects in such asymmetric samples (Leahu et al. 2021). At 800 nm, we choose three angles of incidence (indicated by stars), and further study how the transmitted intensity changes with QWP rotation.

Hence, we fix the excitation wavelength at 800 nm, and we rotate the QWP, at three chosen points of incidence. Figure 2b shows the transmission intensity dependance on the QWP rotation at -25°, 0° and 25° angles of incidence, where all the results are normalized to the respective maximum value; -45° and 45° of the QWP angle correspond to LCP and RCP, respectively. We note that light is transmitted to LCP more than to RCP, with the difference of around 0.15. This means that the sample introduces a disbalance in the outgoing optical field polarization, which does not exit the sample with the linear polarization, even though it is excited with p-polarized light. Moreover, LCP to RCP difference does not invert with the inversion of the incidence angle, which would be a feature of extrinsic chirality (Leahu et al. 2017). Next, we perform the complete spectral characterization at constant QWP angles corresponding to LCP and RCP. Figure 2c-d shows that over a broadband range, and at various incidence angles, light gets transmitted to LCP more than to RCP. All the transmission maps are normalized to the signal obtained with QWP at 0° without the sample.

Next, we plot the difference of the signal transmitted to the LCP and RCP, defining it as \(\varDelta T={T}_{toLCP}-{T}_{toRCP}\); the results are shown in Fig. 3a. In the shortest wavelength range (700–720 nm), around normal incidence, the light gets transmitted into linear polarization, as the difference between LCP and RCP drops to zero; going to the oblique incidence, \(\varDelta T\) increases. On the other side, at all near-infrared ranges, transmission to LCP prevails, and the difference decreases for oblique incidence. There are no specific resonant features in\(\varDelta T\); interestingly, this sample showed only a low extinction CD, which was attributed to the unoptimized geometric parameters in ref. (Petronijevic et al. 2020a). In Fig. 3b, we display spectra at oblique incidence θ = 25° for a full rotation of QWP of -90° to 90°; results are normalized to the spectrum of the sample at θ = 25° with QWP at 0°. A stronger transmission at RCP with respect to LCP appears throughout the whole spectrum.

## 3.2 Simulations

To gain more insight into the origin of the measured asymmetry, we further perform numerical simulations using the 3D Finite Difference Time Domain (FDTD) method by Lumerical commercial software (Lumerical Solutions, Inc). The chiro-optical properties of elliptic Ag-NHAs were numerically investigated by defining a rectangular unit cell (𝑎 and 𝑎√3, where 𝑎 = 518𝑛𝑚), representative of the hexagonal nanoholes base cell, with Bloch boundary conditions in the xy-plane, and perfectly matched layer boundary conditions in the z-direction. In the upper part of Fig. 4a we show the *xy* cross-section of the simulated cell. A semi-infinite rectangle is first defined to simulate the glass substrate with a refractive index of 1,5 covered with 55 nm thick silver film (the complex refractive index of Silver is taken from the ellipsometry measurements in ref. (Petronijevic et al. 2020b). The nanoellipse hole in the metal layer is defined with long and short diameters \({D}_{l}\) and \({D}_{s}\), and with the tilt \(\varphi\) of the long diameter with respect the hexagonal symmetry axis; this shape “etches” the Ag layer at positions forming hexagonal symmetry. In the bottom part of Fig. 4a, we show 3D sketch of the simulation set-up which mimics the experimental set-up. We excite the sample from the air with a linearly (p) polarized source; note that we set this polarization to be always in the x-direction of the simulation set-up, while we can change the nanoellipse tilt angle\(\varphi\). Therefore, the linearly polarized light excites the ENHA, and the transmitted field impinges on the polarization ellipse analysis group, which resolves the polarization and ellipticity of the outcoming beam (the script can be found in “polarztn_ellipse” details of the solver). As in the experiment, we focus only on the zeroth order. The result of this analysis is the plot of the polarization ellipse, which is defined by the major and minor axes, angle \(\psi\) (which describes the tilt of the major axis with respect to the p-polarized direction), and handedness. Another feature investigated by this simulation set-up is the absorption CD; in this case, to simulate circular polarization, two linear x- and y-polarized plane-wave sources were made with a phase difference of 90° and − 90° corresponding to RCP and LCP polarizations, respectively. The total absorption by the silver ENHA is calculated by integrating the absorption density \({\rho }_{abs}\) over the unit cell volume.

We start by simulating the ENHA with \({D}_{s}=356nm\), and \({D}_{l}=380nm\), as such low difference in the ellipse axes corresponds to the real sample investigated here. In Fig. 4b we show the output polarization ellipse, i.e. the ellipse described by the electric field vector in time, when the ENHA is excited at 800 nm, and \(\varphi\) is varied. For \(\varphi =0^\circ\), only linear dichroism exists in such sample; therefore, the x-polarized incident light keeps this polarization when transmitted to the glass. However, a tilting of the sample to \(\varphi =10^\circ\), and \(\varphi =20^\circ\) introduces an increasing ellipticity. Next, we set \(\varphi =28^\circ\) (Petronijevic et al. 2020a), and we investigate the influence of the changing \({D}_{l}\) starting from a circular NHA (\({D}_{s}={D}_{l}=356nm\)). As expected, the circular shape does not change the polarization, while \({D}_{l}=404nm\) introduces a considerable ellipticity of the output polarization, Fig. 4c.

We next investigate the wavelength dependence of the ellipse parameters which will lead to the optimization of the sample geometry. At normal incidence, the polarization ratio is defined as the ratio of the power transmitted to the x- and y-directed polarizations, respectively (i.e. the ratio between p- and s-polarizations). High polarization ratios mean that the ENHA does not change the incident polarization, while the polarization ratio approaching 1 means that the ENHA transmits linear to circular polarization. In Fig. 5a, we show the extracted values of the polarization ratio for different ENHA. For the unoptimized ENHA, as in our sample, the parameters are \(\varphi =28^\circ\), \({D}_{l}=380nm\), and \({D}_{s}=356nm\). The minimum polarization ratio (~ 7) lies around 690 nm, just below our measurement range, and strongly increases to 15 around 700 nm. This agrees well with the \({\Delta }T\) drop in the short wavelength range (no considerable difference between LCP and RCP in the output). Next, we perform the first optimization to lower the polarization ratio minimum; we use the ENHA analysis from ref. (Petronijevic et al. 2020b; Petronijevic et al. 2021b), where absorption CD of ENHA was studied in detail. In those works, ENHA showed strong chiral behavior when diameters scaled as \({D}_{l}=a\), and \({D}_{s}=280a/500\), while the tilt angle was kept at \(\varphi =15^\circ\), and the metal thickness was 100 nm. Scaled with \(a=518nm\), we calculated the polarization ratio for such strongly elliptical ENHA. The first optimization leads to the polarization drop to 1.5 at 690 nm (blue line in Fig. 5a). Note that this scaling of parameters was first investigated in Au-ENHA in (Petronijevic et al. 2020b), and then applied to Ag-ENHA (Petronijevic et al. 2020b). Therefore, we perform Optimization 2 for Ag to shift the optimal lowest polarization to the range of our measurements. To this end, we optimize absorption CD in terms of \({D}_{l}\), \({D}_{s}\), \(\varphi\) and Ag thickness in the near-infrared range. The resulting parameters are \({D}_{l}=513nm\), \({D}_{s}=280nm\), \(\varphi =15^\circ\), and the Ag thickness of 55 nm. The optimum polarization ratio is as low as 1.1 at 750 nm (red line in Fig. 5(a)). Away from the resonant features (where also absorption CD drops), the polarization ratio increases, meaning that the polarization state of the input beam does not get significantly altered.

In Fig. 5b-d we show the resulting polarization ellipses for the three Ag-ENHA, at the wavelengths of the polarization ratio minima. As expected, the unoptimized sample gives the elliptical polarization mostly in x-direction when excited by x-polarized light at 690 nm, Fig. 5b. On the other hand, Optimization 1 leads to the transmission of x-polarized light to a left elliptical polarization which approaches LCP, Fig. 5c. Finally, Optimization 2 moves the polarization ratio minimum to 750 nm, when the linearly polarized excitation is transmitted to almost perfect LCP. We conclude that the strongly elliptical shape and proper in-plane tilt cause this simple geometry to work as a resonant circular polarizer within tens of nm of metal, Fig. 5d.