**1. Introduction**

It is usually a simple task to tell by eye whether an object is chiral or not: Achiral objects are superimposable onto their mirror image and, accordingly, they possess a mirror plane [1]. Recently, chiral scatterers have gained significant interest in nano-optics due to their potential to enhance the weak optical signal of chiral molecules [2–4]. Especially, the quantities of optical chirality and optical helicity as well as their relation to duality symmetry are subjects of current research [5]. The most established experimental technique in this field is the analysis of the circular dichroism (CD) spectrum which equals the differential energy extinction due to the illumination by right- and left-handed circularly polarized light [6].

In order to observe such chiral electromagnetic response, it seems to be obvious that geometrically chiral scatterers are required. However, it has been shown that extrinsic chirality, that is, a chiral configuration of the illumination and geometric parameters, yields comparable effects as intrinsically chiral objects [7]. By tuning the far-field polarization of the illumination, large chiral near-fields may even be generated in the viscinity of achiral objects [8]. In CD measurements, randomly orientied molecules are investigated, which can be classified by their *T*-matrix [9]. The latter has been used for quantifying the electromagnetic (e.m.) chirality, based on a novel definition of it [10].

However, the quantification of the geometric chirality is an elusive task [11] and even the unambiguous association of the terms right- and left-handed enantiomer of a general object is impossible [12]. Different coefficients attempting to rate the chirality of an object are based on the maximal overlap of two mirror images [13] as well as the Hausdorff distance [14]. The choice of a specific coefficient determines the most chiral object [15], that is, there is no natural choice for quantifying geometric chirality. This also holds for the various figures of merit estimating the e.m. chirality. Similar correlations between geometric and optical properties are investigated with respect to the non-sphericity of arbitrary scatterers [16].

In this study, we start by transferring the simple procedure of finding a mirror plane to optics. Such symmetries are present in different mathematical descriptions as for example, block structures in the Mueller scattering matrix [17]. Here, we analyse the *T*-matrix and its associated geometric mirror symmetries by employing translation and rotation theorems of vector spherial harmonics. We illustrate this concept with numerical simulations of an experimentally realized gold helix. Different quantifications of the e.m. chirality are compared. Furthermore, the symmetry planes found in the optical response by our method are correlated to those of geometric origin. It is shown that the complex optical response, including higher order multipoles, yields mirror planes in the *T*-matrix which are not directly related to geometric symmetries.

In the following, we would like to briefly introduce the theory behind the methods described in this study. Further information may be found in Supplementary Materials.

The most general description of an isolated optical scatterer is the well-known *T*-matrix [18]. It relates an arbitrary incident field with the scattered field caused by the scattering object. The optical response to *any* incident field is included in the *T*-matrix. Accordingly, the following analysis of *T* is independent of *specific* illumination parameters such as the direction, polarization and beam shape. The goal of this study is to obtain insights into illumination-*independent* symmetries of the scatterer.

Usually, both the incident as well as the scattered field are given in the basis of vector spherical harmonics for computations with the *T*-matrix [19] (see Supplementary Materials). Physically observable quantities such as the scattered energy, the absorption, as well as the flux of optical chirality are readily computed from *T* [9]. In numerical simulations, *T* may be computed with high accuracy [20]. Knowing the response of the left-handed object *Tl* enables the analytic computation of the response of its mirror image *Tr*:

$$T\_I = \mathcal{M}\_{xy}^{-1} T\_I \mathcal{M}\_{xy} \tag{1}$$

where we choose mirroring in the *xy*-plane M*xy* without loss of generality (see Supplementary Materials for further details on notation). Note that the terminology of right-*Tr* and left-handed *Tl* is ambiguous, as pointed out before, and may be interchanged.

Since we aim to investigate arbitrary mirror planes, we note that an arbitrary plane is given by the three spherical coordinates of its normal: the inclination Θ and the azimuthal angle Φ, as well as the distance *d* from the origin. We define the according transformation *<sup>R</sup>*(<sup>Θ</sup>, Φ, *d*) acting on the object as

$$\mathcal{R}(\Theta, \Phi, d) = \mathcal{T}(\Theta, \Phi, d)\mathcal{R}\_z(\Phi)\mathcal{R}\_y(\Theta), \tag{2}$$

where T (<sup>Θ</sup>, Φ, *d*) is the translation of the *T*-matrix in the direction given by the angles and the distance and R*z*(Φ) and <sup>R</sup>*y*(Θ) are the rotations around the *z*- and *y*-axis, respectively [21] (see Supplementary Materials).

For a geometrically achiral object (see Figure 1a) there exists at least one transformation *<sup>R</sup>*(<sup>Θ</sup>, Φ, *d*) such that *Tl* = *<sup>R</sup>*(<sup>Θ</sup>, Φ, *<sup>d</sup>*)*TrR*−<sup>1</sup>(<sup>Θ</sup>, Φ, *d*). On the other hand, the lack of a geometric mirror plane of a chiral object Figure 1b implies that there exists no such transformation and that *Tl* and *Tr* do not coincide for any set of transformation parameters (<sup>Θ</sup>, Φ, *d*). Note that this does not generally hold in the long wavelength limit, that is, the incident wavelength being much larger than the dimension of the scatterer, due to chiral dispersion [22].

**Figure 1.** (**a**) The mirror image of an achiral object overlaps with its original after proper translations and rotations. This implies that the original *T*-matrix *Tr* coincides with *Tl* of the mirror object after the corresponding transformations *R*. (**b**) A chiral object and its mirror image are not congruent. If the object is much smaller than the incident wavelength, it usually exists a transformation *R* after which *Tl* and *RTrR*−<sup>1</sup> are equal. Note that the achiral isosceles triangle in (**a**) possess a mirror plane in 2D and that the asymmetric triangle in (**b**) is chiral only in 2D.
