# Optical Helicity and Optical Chirality in Free Space and in the Presence of Matter

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## Abstract

**:**

## 1. Introduction

## 2. Rotating and Handed Vector Fields

## 3. Physical Significance of Optical Helicity and Optical Chirality in Free Space

#### Chiral Symmetries in Electromagnetism

## 4. Physical Significance of Optical Helicity and Chirality upon Interaction with Matter

#### 4.1. Observables Derived from Chiral Electromagnetism

## 5. Chiral Light–Matter Interactions in Artificial Nanostructures

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Illustration of parity symmetry for an electric field ($\mathbf{E}$) arising between positive and negative point charges (

**a**) and the magnetic induction field ($\mathbf{B}$) arising from a steady state current (

**b**). (

**c**) Illustration of the distinction between a rotating vector field and a chiral vector field, where rotational motion has a component along the axis of rotation.

**Figure 2.**(

**a**) Illustration of an elastic (

**left**) and inelastic (

**right**) collision of two objects with masses ${m}_{1}$, ${m}_{2}$, moving at velocities ${\mathbf{v}}_{1}$, ${\mathbf{v}}_{2}$ at times ${t}_{1}$, before the collision (

**top**) and ${t}_{2}$, after the collision (

**bottom**), respectively. While the total linear momentum of the system $\mathbf{p}=m\mathbf{v}$ is conserved for the elastic (

**left**) and inelastic collision (

**right**), the mechanical (kinetic) energy is not conserved for the inelastic collision due to energy dissipation. (

**b**) Illustration of the interaction between chiral light and macroscopic matter for a lossless, dual-symmetric medium (

**left**) and a lossy, dispersive medium (

**right**). While both optical helicity and optical chirality conservation hold for the lossless, dual-symmetric case (

**left**), the presence of a lossy, dispersive medium (

**right**) breaks duality symmetry and helicity conservation no longer holds in its physically observable form. In contrast, the conservation law of optical chirality can be formulated to account for dissipative effects in the presence of lossy, dispersive media.

**Figure 3.**(

**a**) Schematic illustration of a gold nanoparticle (spherical geometry, 75 nm radius) interacting with a circularly polarized plane wave. Numerical simulations of the total, volume-integrated optical chirality flux $\overline{\mathcal{F}}={\int}_{V}\nabla \xb7\mathcal{F}{d}^{3}x$ of the gold nanosphere upon excitation with linearly polarized (LP, black), left-handed circularly polarized light (CPL) (LCP, red), and right-handed CPL (RCP, blue). (

**b**) Schematic illustration of a silicon nanoparticle (spherical geometry, 75 nm radius) interacting with a circularly polarized plane wave. Numerical simulations of the total, volume-integrated optical chirality flux $\overline{\mathcal{F}}$ of the silicon nanosphere upon excitation with linearly polarized (LP, black), left-handed CPL (LCP, red), and right-handed CPL (RCP, blue).

**Table 1.**Conservation laws of optical helicity (

**left column**) and optical chirality (

**right column**) in free space with optical helicity density h, optical helicity flux density $\Phi $, optical chirality density $\chi $, optical chirality flux density $\mathbf{F}$, electric vector potential $\mathbf{C}$, magnetic vector potential $\mathbf{A}$, electric field $\mathbf{E}$, and magnetic induction field $\mathbf{B}$. ${\u03f5}_{0}$ and ${\mu}_{0}$ represent the free-space electric permittivity and magnetic permeability.

Optical Helicity Conservation in Free Space | Optical Chirality Conservation in Free Space |
---|---|

$h=\frac{1}{2}\left[\sqrt{\frac{{\u03f5}_{0}}{{\mu}_{0}}}\mathbf{A}\xb7(\nabla \times \mathbf{A})+\sqrt{\frac{{\mu}_{0}}{{\u03f5}_{0}}}\mathbf{C}\xb7(\nabla \times \mathbf{C})\right]$ | $\chi =\frac{{\u03f5}_{0}}{2}\mathbf{E}\xb7(\nabla \times \mathbf{E})+\frac{1}{2{\mu}_{0}}\mathbf{B}\xb7(\nabla \times \mathbf{B})$ |

$\Phi =\frac{1}{2}\left[\sqrt{\frac{{\u03f5}_{0}}{{\mu}_{0}}}\mathbf{A}\times (\nabla \times \mathbf{C})-\frac{1}{c}\mathbf{C}\times (\nabla \times \mathbf{A})\right]$ | $\mathbf{F}=\frac{1}{2}\left[\mathbf{E}\times (\nabla \times \mathbf{B})-\mathbf{B}\times (\nabla \times \mathbf{E})\right]$ |

$\frac{\delta h}{\delta t}+\frac{1}{{\mu}_{0}}\nabla \xb7\Phi =0$ | $\frac{\delta \chi}{\delta t}+\frac{1}{{\mu}_{0}}\nabla \xb7\mathbf{F}=0$ |

**Table 2.**

**Top**: The relationship between force and momentum for linear and rotational motion in classical mechanics, for linear momentum $\mathbf{p}$, force $\mathbf{F}$, angular momentum $\mathbf{L}$, and torque $\mathbf{\tau}$.

**Bottom**: The relationship between the optical helicity density h and the optical chirality density $\chi $ in classical electrodynamics.

Physical Significance | Fundamental | → | Observable |
---|---|---|---|

Classical Mechanics | |||

Linear Motion: $\frac{d\mathbf{p}}{dt}=\mathbf{F}$ | Linear Momentum [$\mathrm{N}\phantom{\rule{4.pt}{0ex}}\mathrm{s}$] | $\stackrel{\frac{d}{dt}}{\to}$ | Force [N] |

Rotational Motion: $\frac{d\mathbf{L}}{dt}=\mathbf{\tau}$ | Angular Momentum [$\mathrm{N}\phantom{\rule{4.pt}{0ex}}\mathrm{m}\phantom{\rule{4.pt}{0ex}}\mathrm{s}$] | $\stackrel{\frac{d}{dt}}{\to}$ | Torque [$\mathrm{N}\phantom{\rule{4.pt}{0ex}}\mathrm{m}$] |

Classical Electrodynamics | |||

Handed Motion: $h\stackrel{\nabla \times}{\to}\chi $ | Optical Helicity Density [$\frac{\mathrm{N}\phantom{\rule{4.pt}{0ex}}\mathrm{m}\phantom{\rule{4.pt}{0ex}}\mathrm{s}}{{\mathrm{m}}^{3}}$] | $\stackrel{\nabla \times}{\to}$ | Optical Chirality Density [$\frac{\mathrm{N}}{{\mathrm{m}}^{3}}$] |

**Table 3.**Parity and time symmetries of the optical chirality density $\chi $ and the optical chirality flux density $\mathbf{F}$.

Physical Quantity | Tensor Rank | Parity Symmetry | Time Symmetry |
---|---|---|---|

Optical Chirality Density $\chi $ | 3 | Odd (pseudoscalar) | Even |

Optical Chirality Flux Density $\mathbf{F}$ | 1 | Even (pseudovector) | Even |

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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.
https://doi.org/10.3390/sym11091113

**AMA Style**

Poulikakos LV, Dionne JA, García-Etxarri A. Optical Helicity and Optical Chirality in Free Space and in the Presence of Matter. *Symmetry*. 2019; 11(9):1113.
https://doi.org/10.3390/sym11091113

**Chicago/Turabian Style**

Poulikakos, Lisa V., Jennifer A. Dionne, and Aitzol García-Etxarri. 2019. "Optical Helicity and Optical Chirality in Free Space and in the Presence of Matter" *Symmetry* 11, no. 9: 1113.
https://doi.org/10.3390/sym11091113