# Optical Helicity and Chirality: Conservation and Sources
## 3 Microscopic Sources
### Dipole Model of a Helicity Source  
As an illustration of how charges and currents can act as a source of helicity, we consider a point source consisting of an oscillating electric and a magnetic dipole, with electric and magnetic dipole moments \(\mathbf{p}(t)\) and \(\mathbf{m}(t)\), as has been examined by [PERSON] et al. [28]. This can be thought of as a simple model of a radiating chiral molecule, as the optical activity of chiral molecules ultimately arises from the simultaneous induction of electric and magnetic dipole moments through the electric-dipole magnetic-dipole polarisability tensor \(G\)[9] (see Chapter 3, SS5.4). [PERSON] et al. treat the emission from the dipoles quantum-mechanically, calculating a differential irradiance for left- and right-circularly polarised light by considering the difference in decay rates into the two circular polarisation modes. From this, they obtain an expression for the net emitted helicity. There are three contributions to the total irradiance: one depending on \(|\mathbf{p}|^{2}\), one on \(|\mathbf{m}|^{2}\) and one on \(\mathbf{p}\cdot\mathbf{m}\). The mixed dipole term \(\mathbf{p}\cdot\mathbf{m}\) is the only contribution to the total irradiance that is different for the two enantiomers, and hence contributes to the net emitted helicity. We present here an analysis of helicity emission from the dipole system in the context of the continuity Equation (16), working within classical electromagnetism. Consider an electric dipole oscillating along the \(+z\) axis with dipole moment:  
\[\mathbf{p}(t)=p_{0}\exp[i(\omega t+\phi_{\text{p}})]\hat{\mathbf{z}}, \tag{18}\]  
where we have defined \(p_{0}=q_{0}d\), with \(d\) and \(q_{0}\) the size and charge of the dipole, respectively. The resultant (retarded) vector potential in the far field defined by \(d\ll\lambda\ll r\) is [43]:  
\[\mathbf{A}_{\text{p}}^{\prime}=\Re\left[\frac{i\omega t_{0}p_{0}}{4\pi r}\exp[ i(\omega(t-r/c)+\phi_{\text{p}})]\hat{\mathbf{z}}\right]. \tag{19}\]  
We call this \(\mathbf{A}_{\text{p}}^{\prime}\), as we reserve the symbol \(\mathbf{A}\) for the transverse part of the vector potential. From this expression, \(\mathbf{B}_{\text{p}}\), \(\mathbf{E}_{\text{p}}\), \(\mathbf{C}_{\text{p}}\) and \(\mathbf{A}_{\text{p}}\equiv\mathbf{A}_{\text{p}}^{\prime\text{T}}\) can be found. Similarly, we consider a current oscillating in a loop of radius \(b\) in the \(xy\) plane, \(I(t)=I_{0}\exp[i(\omega t+\phi_{\text{m}})]\). The resultant oscillating dipole moment is given by:  
\[\mathbf{m}(t)=\int I(t)\,d\mathbf{a}=m_{0}\exp[i(\omega t+\phi_{\text{m}})] \hat{\mathbf{z}}, \tag{20}\]  
where \(m_{0}=\pi b^{2}I_{0}\). The vector potential is calculated as [43]:  
\[\mathbf{A}_{\text{m}}=\mathbf{A}_{\text{m}}^{\text{T}}=\Re\left[\frac{i\omega \mu_{0}m_{0}}{4\pi r^{2}c}\left(x\hat{\mathbf{y}}-y\hat{\mathbf{x}}\right) \right]. \tag{21}\]  
Using this to find \(\mathbf{B}_{\text{m}}\), \(\mathbf{E}_{\text{m}}\) and \(\mathbf{C}_{\text{m}}\), we calculate the total helicity density and flux density of the combined electric-magnetic dipole system in the far-field:  
\[h_{\text{pm}}=\frac{e_{0}}{c}\left(\frac{\mu_{0}^{2}m_{0}p_{0}\omega^{3}}{16 \pi^{2}r^{4}}\left(x^{2}+y^{2}\right)\sin(\phi_{\text{m}}-\phi_{\text{p}}) \right), \tag{22}\]\[\mathbf{v}_{\mathrm{pm}}=\epsilon_{0}\left(\frac{\mu_{0}^{2}m_{0}p_{0}\omega^{3}}{16 \pi^{2}r^{5}}\left(x^{2}+y^{2}\right)\sin(\phi_{\mathrm{m}}-\phi_{\mathrm{P}}) \right)\mathbf{r}. \tag{23}\]  
From (22), \(\partial_{t}h_{\mathrm{pm}}=0\), and we use [PERSON]'s theorem to calculate the net helicity flux of the combined dipole system:  
\[\int\mathbf{\
abla}\cdot\mathbf{v}_{\mathrm{pm}}\,d^{3}\mathbf{r}=\frac{\epsilon_ {0}\mu_{0}^{2}m_{0}p_{0}\omega^{3}}{6\pi}\sin(\phi_{\mathrm{m}}-\phi_{\mathrm{ P}}). \tag{24}\]  
We can identify this with the source term on the right-hand side of (16).