# Optical Helicity and Chirality: Conservation and Sources
## 3 Microscopic Sources
### Dipole Model of a Helicity Source
Noting \(\int\mathbf{g}\cdot\left(\mathbf{\
abla}\times\mathbf{C}\right)d^{3}\mathbf{r}= \int\mathbf{C}\cdot\left(\mathbf{\
abla}\times\mathbf{g}\right)d^{3}\mathbf{r}\)[17], we obtain:  
\[\int\mathbf{g}\cdot\left(\mathbf{\
abla}\times\mathbf{C}\right)d^{3}\mathbf{r}= \frac{\epsilon_{0}u_{0}m_{0}p_{0}\omega^{3}}{6\pi c}\sin(\phi_{\mathrm{m}}- \phi_{\mathrm{P}}), \tag{25}\]  
which is maximised for a phase difference of \(\pm\pi/2\) between the dipoles. Left- and right-circularly polarised light can also be produced by a pair of orthogonally-aligned oscillating electric or magnetic dipoles. We consider the former configuration: \(\mathbf{p}_{1}(t)=|\mathbf{p}_{1}(t)|\mathbf{\hat{z}}\) and \(\mathbf{p}_{2}(t)=|\mathbf{p}_{2}(t)|\mathbf{\hat{y}}\), with a phase difference of \(\phi_{\mathrm{P}_{2}}-\phi_{\mathrm{P}_{1}}\) between the oscillations. The resultant helicity flux density in the far-field is:  
\[\mathbf{v}_{\mathrm{PP}}=\frac{\epsilon_{0}c}{2}\left(\frac{\mu_{0}^{2}p_{0}^ {2}\omega^{3}}{16\pi^{2}r^{5}}\sin(\phi_{\mathrm{P}_{2}}-\phi_{\mathrm{P}_{1}} )xr\right)\mathbf{r}, \tag{26}\]  
again maximised for \(\phi_{\mathrm{P}_{2}}-\phi_{\mathrm{P}_{1}}=\pi/2\). The helicity density of this coupled dipole system is again time independent, so that the \"source term\" is found in analogy with the above calculation. We find \(\int\mathbf{\
abla}\cdot\mathbf{v}_{\mathrm{PP}}\,d^{3}\mathbf{r}\)=0: the coupled electric-electric dipole system does not describe a source of helicity. These results follow intuitively by considering Figures 2 and 3, where the far-field patterns of the electric-magnetic and electric-electric dipole systems in the \(yz\) plane have been drawn. A \(\pi/2\) phase difference between the dipole oscillations results in right-circular polarisation in the \(+\,x\) direction in both dipole systems. In the \(-\,x\) direction, however, we obtain left-circular polarisation for the electric-magnetic dipole pair and right-circular polarisation for the electric-electric dipoles. Thus, we obtain a net flux of helicity for the electric-magnetic dipole system only. The results of this calculation are intimately related to the respective parity of the electric-magnetic and electric-electric dipole systems. Considering again Figure 2, a parity transformation reverses the direction of the electric dipole only, so that the field patterns in both the \(+\,x\) and the \(-\,x\) direction are reversed, and a net flux of left-circular polarisation is produced. A parity transformation \(\mathbf{\hat{P}}\) thus interconverts the two \"enantiomeric\" configurations of the electric-magnetic dipole systems and produces a negative sign in the integrated source term in (25). The two enantiomeric forms can therefore be described simply by the aligned and anti-aligned configurations (or equivalently, as a positive or negative phase difference between the two oscillations in either configuration), which can be distinguished in the far-field \(yz\) plane by an excess of right- or left-circular polarisation, respectively. For the electric-electric dipoles in Figure 3, however, both dipoles are reversed under \(\mathbf{\hat{P}}\), so that the field patterns in \(\pm\,x\) directions remain unchanged. As the electric-electric system is parity-even, but left- and right-circularly polarised fields are reversed under a parity transformation, it follows that the electric-electric dipole system cannot produce an excess of either polarisation and hence cannot be used to describe a source of helicity.