**2. Results**

For investigating the role of the geometric shape in nano-optics, it is of interest to identify those planes of highest symmetry of a chiral object: Although there is no mirror plane in a chiral object, a transformation may be identified in which the right- and left-handed *T*-matrices are closest to one another. Rating the closeness is done here by calculating the 2-norm of the difference of these two matrices. Accordingly, we introduce the coefficient *χ*TT which minimizes the difference between the *T*-matrices of mirror images as

$$\chi\_{\rm TT} = \min\_{\left(\Theta, \Phi, d\right)} \left\| \left| T\_l - \mathcal{R}^{-1} (\Theta, \Phi, d) T\_l \mathcal{R} (\Theta, \Phi, d) \right| \right\|\_2. \tag{3}$$

This means that for the mirror plane corresponding to minimal parameters (<sup>Θ</sup>min, Φmin, *d*min) of (3), the optical responses of the two mirror images are as similar as possible. In other words, the mirror images are hardly distinguishable. For an achiral object *χ*TT vanishes since there exists a transformation for which the mirror images are identical.

Obviously, the choice of the norm is not unique and other quantifications of similarity of the mirror images could be defined (see Supplementary Materials for the physical relevance of the 2-norm). A recently introduced coefficient *χ*SV is, for example, based on the singular-value decomposition of the *T*-matrix in the helicity basis [10]. Alternatively, the angular-averaged differential energy extinction *χ*CD due to illuminating with either right- or left-handed circularly polarized plane waves is experimentally accessible as the CD spectrum.

In order to exemplary introduce our formalism and compare it to previous work, we investigate a nano-optical device numerically. The finite element method is employed to accurately simulate the electromagnetic properties due to incident monochromatic light. Within this study we use the commercial FEM package JCMsuite [23]. In postprocessing, the *T*-matrix is computed by decomposing the scattered field into vector spherical wave functions [20] from illumination with 150 plane waves with randomly chosen parameters (see Supplementary Materials).

In Figure 2, we compare simulations of the aforementioned three coefficients quantifying the e.m. chirality for a gold helix as realized experimentally [24]. The helix is constructed on the surface of a cylinder with height 230nm and radius 60nm (see Supplementary Materials). The CD spectrum *χ*CD shows zero values at incident wavelengths of *λ* = 615 nm and *λ* = 1070 nm. If only these wavelengths

were analyzed, one could draw the conclusion that an achiral object is investigated. This contradicts the goal of this study to obtain insights into illumination-*independent* symmetries of the scatterer—for illuminations with *λ* = 615 nm and *λ* = 1070 nm, the scatterer seems to be geometrically achiral which is obviously not the case. Nevertheless, CD makes the chiral geometric nature of the helix visible as a maximum at 823 nm and a minimum at 1452 nm of smaller amplitude. For a helix with an opposite twist—that is, the mirror image—the roles of the extrema are interchanged.

**Figure 2.** Chiral response of a gold nano-helix depending on the incident wavelength *λ*. The angular averaged differential extinction of circularly polarized plane waves *χ*CD (black dotted line) vanishes at 615 nm and 1070 nm which could be interpreted as achirality of the studied object. The electromagnetic chirality coefficient *χ*SV (dashed blue line) is based on the singular values of the *T*-matrix in the helicity basis. Values below 0.1 at 610 nm and 1085 nm indicate nearly achiral optical response. However, the minimal difference *χ*TT (red solid line) between *Tr* and *RTlR*−<sup>1</sup> reveals that the helix is chiral at all wavelengths. Its maxima correspond to those of *χ*CD and are, hence, observable.

On the other hand, the coefficient *χ*SV is normalized by the average interaction strength of the *T*-matrix at each wavelength. This yields a fairly flat spectrum with two narrow minima below 0.1 at the two *λ* for which *χ*CD = 0. These minima are not present in the minimized *χ*TT introduced in (3). However, the maxima of this latter coefficient are in accordance with the experimentally observable CD extrema (*χ*CD). In the long wavelength regime, all three coefficients tend to zero as expected for point-like particles due to vanishing off-diagonal elements in the *T*-matrix.

The minimization in the three-dimensional parameter space in (3) is carried out using Bayesian optimization [25] (see Supplementary Materials). Since the shape of the minimized function highly depends on the actual object, the Bayesian approach is well suited for finding a global minimum. The parameters (<sup>Θ</sup>min, Φmin, *d*min) of the optimized value are related to geometric mirror planes. In Figure 3a, the planes following from the respective transformation *<sup>R</sup>*(<sup>Θ</sup>min, Φmin, *d*min) of the *xy*-plane are plotted for all incident wavelengths from 550 nm to 2.05 μm. The inclination Θ and azimuthal angle Φ are given in the shown coordinate system which is centered at the centroid of the helix.

We identify three distinct classes shown in blue, red and green. These correspond to planes which are parallel and perpendicular to the helix axis, as well as tilted by a small angle Θ from the horizontal position, respectively. The dark grey plane corresponds to the minimal geometric parameters which will be explained in the following paragraphs. Details on the optimization such as challenging flat behaviour for translations from the centroid and, on the obtained minimizing parameters, are given in Supplementary Materials. Note that the minimization required to obtain the illumination-independent coefficient *χ*TT involves significantly higher numerical effort than the simple averaging for *χ*CD for which most information contained in *T* is ignored.

**Figure 3.** (**a**) Transformed *xy*-planes (blue, red, green) corresponding to minimal *χ*TT computed from *T*-matrix of the gold helix (yellow). Planes for all incident wavelenghts *λ* ∈ [550, 2050] nm are shown. The dark grey plane corresponds to minimal *χ*GE. (**b**) Geometric chiral coefficient *<sup>χ</sup>*GE(<sup>Θ</sup>, Φ) for the helix and its mirror image which is rotated around the centroid (grey colormap). The minimal value of 0.57 belongs to the dark grey plane in Figure 3a. Angles of the colored planes are shown by circles.

Next, we compare the findings on the symmetry based on the optical *T*-matrix to those stemming from purely geometric properties. As discussed previously, there is no coefficient which unambiguously rates the geometric chirality of an object. We choose a coefficient *χ*GE based on the overlap of the left- *Ol* and right-handed *Or*(<sup>Θ</sup>, Φ, *d*) object, where the latter results from mirroring *Ol* at the *xy*-plane and transformation with (<sup>Θ</sup>, Φ, *d*). Namely, the volume *V* of the overlap is compared to the volume of the object [13] (see Supplementary Materials):

$$\chi\_{\rm GE}(\Theta, \Phi, d) = 1 - \frac{V\left(O\_l \cap O\_r(\Theta, \Phi, d)\right)}{V(O\_l)}.\tag{4}$$

This coefficient vanishes for achiral objects as required for a degree of chirality [14].

Figure 3b displays the geometric chirality coefficient *<sup>χ</sup>*GE(<sup>Θ</sup>, Φ, 0) for planes rotated around the centroid of the helix as a grey colourmap. Dark regions with large values of *χ*GE indicate a vanishing overlap between the two mirrored helices. Note that for large distances to the origin *d* → <sup>∞</sup>, the mirror images do not overlap and *χ*GE = 1. However, this is always possible no matter if the object is chiral or not. As in the case of *χ*TT, the parameter points of interest of *<sup>χ</sup>*GE(<sup>Θ</sup>, Φ, *d*) are those corresponding to a minimum: The minimum 0.57 in Figure 3b occurs at (180◦, 55◦) and (0◦, 125◦) which show the intrinsic chiral property of the investigated helix. These two minima are equivalent since a finite helix is *C*2 symmetric. The corresponding transformed *xy*-plane is shown in dark grey in Figure 3a.

Alongside the geometric coefficient *χ*GE, the planes identified for the minimized *T*-matrix difference are shown as colored circles in Figure 3b. The colors (red, blue and green) of these circles are

the same colors used for the planes, that is, a direct comparison of the angle parameters is possible. As seen, the planes are ranked according to their Θ values: The perpendicular class 1 (blue) has Θ ∈ [83, 105]◦ The flat planes belonging to class 2 (red) show Θ ∈ [0, 8.5]◦ and Θ ∈ [174, 180]◦ and the tilted class 3 (green) has Θ ∈ [10, 19]◦ and Θ = 170◦.
