# Optical Helicity and Chirality: Conservation and Sources

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## Abstract

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## 1. Introduction

## 2. Review and Motivation

#### 2.1. Integrated Helicity and Local Densities

#### 2.2. Helicity and Chirality

#### 2.3. Quantum Helicity Operator

#### 2.4. Helicity and Duality Symmetry

#### 2.5. Continuity Equations in Free Space

## 3. Microscopic Sources

#### 3.1. Helicity in the Presence of Current and Charge

#### 3.2. Dipole Model of a Helicity Source

## 4. Macroscopic Sources

#### 4.1. Helicity in Achiral, Reciprocal Media

#### 4.2. Helicity in Bi-Isotropic Media

#### Helicity Conservation in a Chiral Medium

#### 4.3. Currents and Charges in Bi-Isotropic Media

## 5. Concluding Remarks

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) The molecule bromochlorofluoromethane is chiral, as the molecule and its mirror image cannot be superimposed, even after rotation. (

**b**) Dichlorofluoromethane, on the other hand, is achiral, as the molecule can be superimposed upon its mirror image after rotation.

**Figure 2.**Oscillating electric $\mathbf{p}\left(t\right)$ and magnetic $\mathbf{m}\left(t\right)$ dipoles aligned along the z axis, with the latter lagging by a phase of $\pi /2$. The far-field pattern is that of right-circularly polarised light in the $+\phantom{\rule{0.166667em}{0ex}}x$ direction and left-circularly polarised light in the $-\phantom{\rule{0.166667em}{0ex}}x$ direction: there is a net flux of right-circular polarisation in the $yz$ plane.

**Figure 3.**Two oscillating electric dipoles, labelled ${\mathbf{p}}_{1}\left(t\right)$ and ${\mathbf{p}}_{2}\left(t\right)$, oscillate along the z and y axes, respectively, with a phase difference of $\pi /2$. The far-field pattern in the $+\phantom{\rule{0.166667em}{0ex}}x$ direction is identical to that produced by the electric-magnetic dipole configuration in Figure 2, but has opposite polarisation in the $-\phantom{\rule{0.166667em}{0ex}}x$ direction: the net helicity flux in the $yz$ plane is zero.

**Figure 4.**At the interface between a vacuum and a dual-symmetric, lossless chiral medium characterised by $\u03f5$, $\mu $ and the chirality parameter $\beta $, both the energy and helicity of an electromagnetic field are conserved. As a consequence, the ratio of the helicity density to energy density, $h/w$, must be preserved across the interface. In the chiral medium, the energy density contains a chiral contribution, as given by ${w}_{1}$ (29), requiring a modification of the helicity density to ${h}_{1}$ (35) such that $h/w={h}_{1}/{w}_{1}$ holds.

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**MDPI and ACS Style**

Crimin, F.; Mackinnon, N.; Götte, J.B.; Barnett, S.M.
Optical Helicity and Chirality: Conservation and Sources. *Appl. Sci.* **2019**, *9*, 828.
https://doi.org/10.3390/app9050828

**AMA Style**

Crimin F, Mackinnon N, Götte JB, Barnett SM.
Optical Helicity and Chirality: Conservation and Sources. *Applied Sciences*. 2019; 9(5):828.
https://doi.org/10.3390/app9050828

**Chicago/Turabian Style**

Crimin, Frances, Neel Mackinnon, Jörg B. Götte, and Stephen M. Barnett.
2019. "Optical Helicity and Chirality: Conservation and Sources" *Applied Sciences* 9, no. 5: 828.
https://doi.org/10.3390/app9050828