# Role of Geometric Shape in Chiral Optics

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Results

## 3. Discussion

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Kelvin, W.T.B. Baltimore Lectures on Molecular Dynamics and the Wave Theory of Light; CJ Clay and Sons: Baltimore, MD, USA, 1904. [Google Scholar]
- Tang, Y.; Cohen, A.E. Optical chirality and its interaction with matter. Phys. Rev. Lett.
**2010**, 104, 163901. [Google Scholar] [CrossRef] - Nieto-Vesperinas, M. Optical theorem for the conservation of electromagnetic helicity: Significance for molecular energy transfer and enantiomeric discrimination by circular dichroism. Phys. Rev. A
**2015**, 92, 023813. [Google Scholar] [CrossRef][Green Version] - McPeak, K.M.; van Engers, C.D.; Bianchi, S.; Rossinelli, A.; Poulikakos, L.V.; Bernard, L.; Herrmann, S.; Kim, D.K.; Burger, S.; Blome, M.; et al. Ultraviolet Plasmonic Chirality from Colloidal Aluminum Nanoparticles Exhibiting Charge-Selective Protein Detection. Adv. Mater.
**2015**, 27, 6244–6250. [Google Scholar] [CrossRef] [PubMed][Green Version] - Poulikakos, L.V.; Dionne, J.A.; García-Etxarri, A. Optical Helicity and Optical Chirality in Free Space and in the Presence of Matter. Symmetry
**2019**, 11, 1113. [Google Scholar] [CrossRef][Green Version] - Bohren, C.F.; Huffman, D.R. Absorption and Scattering of Light by Small Particles; John Wiley & Sons: Hoboken, NJ, USA, 1983. [Google Scholar]
- Plum, E.; Fedotov, V.; Zheludev, N. Optical activity in extrinsically chiral metamaterial. Appl. Phys. Lett.
**2008**, 93, 191911. [Google Scholar] [CrossRef] - Kramer, C.; Schäferling, M.; Weiss, T.; Giessen, H.; Brixner, T. Analytic optimization of near-field optical chirality enhancement. ACS Photonics
**2017**, 4, 396–406. [Google Scholar] [CrossRef] - Gutsche, P.; Nieto-Vesperinas, M. Optical Chirality of Time-Harmonic Wavefields for Classification of Scatterer. Sci. Rep.
**2018**, 8, 9416. [Google Scholar] [CrossRef] - Fernandez-Corbaton, I.; Fruhnert, M.; Rockstuhl, C. Objects of maximum electromagnetic chirality. Phys. Rev. X
**2016**, 6, 031013. [Google Scholar] [CrossRef] - Fowler, P.W. Quantification of chirality: Attempting the impossible. Symmetry Cult. Sci.
**2005**, 16, 321–334. [Google Scholar] - Efrati, E.; Irvine, W.T. Orientation-dependent handedness and chiral design. Phys. Rev. X
**2014**, 4, 011003. [Google Scholar] [CrossRef][Green Version] - Gilat, G. On quantifying chirality-obstacles and problems towards unification. J. Math. Chem.
**1994**, 15, 197–205. [Google Scholar] [CrossRef] - Buda, A.B.; Mislow, K. A Hausdorff chirality measure. J. Am. Chem. Soc.
**1992**, 114, 6006–6012. [Google Scholar] [CrossRef] - Rassat, A.; Fowler, P.W. Is there a “most chiral tetrahedron”? Chemistry
**2004**, 10, 6575–6580. [Google Scholar] [CrossRef] [PubMed] - Romanov, A.V.; Konokhova, A.I.; Yastrebova, E.S.; Gilev, K.V.; Strokotov, D.I.; Maltsev, V.P.; Yurkin, M.A. Sensitive detection and estimation of particle non-sphericity from the complex Fourier spectrum of its light-scattering profile. J. Quant. Spectrosc. Radiat. Transf.
**2019**, 235, 317–331. [Google Scholar] [CrossRef] - Yurkin, M.A. Symmetry relations for the Mueller scattering matrix integrated over the azimuthal angle. J. Quant. Spectrosc. Radiat. Transf.
**2013**, 131, 82–87. [Google Scholar] [CrossRef] - Mishchenko, M.; Travis, L.; Lacis, A. Scattering, Absorption, and Emission of Light by Small Particles; Cambridge University Press: Cambridge, UK, 2002. [Google Scholar]
- Jackson, J.D. Classical Electrodynamics, 3rd ed.; John Wiley and Sons: Hoboken, NJ, USA, 1998. [Google Scholar]
- Garcia Santiago, X.; Hammerschmidt, M.; Burger, S.; Rockstuhl, C.; Fernandez Corbaton, I.; Zschiedrich, L. Decomposition of scattered electromagnetic fields into vector spherical wave functions on surfaces with general shapes. Phys. Rev. B
**2019**, 99, 045406. [Google Scholar] [CrossRef][Green Version] - Stein, S. Addition theorems for spherical wave functions. Q. Appl. Math.
**1961**, 19, 15–24. [Google Scholar] [CrossRef][Green Version] - Lindell, I.V.; Sihvola, A.H. Electromagnetic Wave in Chiral and Bi-Isotropic Media; Artech House: London, UK, 1994. [Google Scholar]
- Pomplun, J.; Burger, S.; Zschiedrich, L.; Schmidt, F. Adaptive finite element method for simulation of optical nano structures. Phys. Status Solidi B
**2007**, 244, 3419–3434. [Google Scholar] [CrossRef][Green Version] - Wozniak, P.; De Leon, I.; Höflich, K.; Haverkamp, C.; Christiansen, S.; Leuchs, G.; Banzer, P. Chiroptical response of a single plasmonic nanohelix. Opt. Express
**2018**, 26, 19275–19293. [Google Scholar] [CrossRef] - Schneider, P.I.; Santiago, X.G.; Rockstuhl, C.; Burger, S. Global optimization of complex optical structures using Bayesian optimization based on Gaussian processes. Proc. SPIE
**2017**, 10335, 103350O. [Google Scholar] - Zambrana-Puyalto, X.; Bonod, N. Tailoring the chirality of light emission with spherical Si-based antennas. Nanoscale
**2016**, 8, 10441–10452. [Google Scholar] [CrossRef] [PubMed][Green Version] - Nakata, Y.; Urade, Y.; Nakanishi, T. Geometric Structure behind Duality and Manifestation of Self-Duality from Electrical Circuits to Metamaterials. Symmetry
**2019**, 11, 1336. [Google Scholar] [CrossRef][Green Version]

**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.

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**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