# Role of Geometric Shape in Chiral Optics

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## Abstract

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## 1. Introduction

## 2. Results

## 3. Discussion

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**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 ${T}_{r}$ coincides with ${T}_{l}$ 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 ${T}_{l}$ and $R{T}_{r}{R}^{-1}$ 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.

**Figure 2.**Chiral response of a gold nano-helix depending on the incident wavelength $\lambda $. The angular averaged differential extinction of circularly polarized plane waves ${\chi}_{\mathrm{CD}}$ (black dotted line) vanishes at $615\phantom{\rule{0.166667em}{0ex}}\mathrm{nm}$ and $1070\phantom{\rule{0.166667em}{0ex}}\mathrm{nm}$ which could be interpreted as achirality of the studied object. The electromagnetic chirality coefficient ${\chi}_{\mathrm{SV}}$ (dashed blue line) is based on the singular values of the T-matrix in the helicity basis. Values below $0.1$ at $610\phantom{\rule{0.166667em}{0ex}}\mathrm{nm}$ and $1085\phantom{\rule{0.166667em}{0ex}}\mathrm{nm}$ indicate nearly achiral optical response. However, the minimal difference ${\chi}_{\mathrm{TT}}$ (red solid line) between ${T}_{r}$ and $R{T}_{l}{R}^{-1}$ reveals that the helix is chiral at all wavelengths. Its maxima correspond to those of ${\chi}_{\mathrm{CD}}$ and are, hence, observable.

**Figure 3.**(

**a**) Transformed $xy$-planes (blue, red, green) corresponding to minimal ${\chi}_{\mathrm{TT}}$ computed from T-matrix of the gold helix (yellow). Planes for all incident wavelenghts $\lambda \in [550,2050]\phantom{\rule{0.166667em}{0ex}}\mathrm{nm}$ are shown. The dark grey plane corresponds to minimal ${\chi}_{\mathrm{GE}}$. (

**b**) Geometric chiral coefficient ${\chi}_{\mathrm{GE}}(\mathsf{\Theta},\mathsf{\Phi})$ 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.

**Figure 4.**Wavelength-dependent classification of symmetry planes of T-matrix (top). Absolute value of averaged diagonal T-matrix entries corresponding to induced electric dipoles (solid), magnetic dipoles (dashed) and electric quadrupoles (dotted). The classes 3 (green), 2 (red) and 1 (blue) belong to decreasing wavelengths. Changes in symmetry of T are due to higher order multipoles.

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**MDPI and ACS Style**

Gutsche, P.; Garcia-Santiago, X.; Schneider, P.-I.; McPeak, K.M.; Nieto-Vesperinas, M.; Burger, S.
Role of Geometric Shape in Chiral Optics. *Symmetry* **2020**, *12*, 158.
https://doi.org/10.3390/sym12010158

**AMA Style**

Gutsche P, Garcia-Santiago X, Schneider P-I, McPeak KM, Nieto-Vesperinas M, Burger S.
Role of Geometric Shape in Chiral Optics. *Symmetry*. 2020; 12(1):158.
https://doi.org/10.3390/sym12010158

**Chicago/Turabian Style**

Gutsche, Philipp, Xavier Garcia-Santiago, Philipp-Immanuel Schneider, Kevin M. McPeak, Manuel Nieto-Vesperinas, and Sven Burger.
2020. "Role of Geometric Shape in Chiral Optics" *Symmetry* 12, no. 1: 158.
https://doi.org/10.3390/sym12010158