# Optical Helicity and Chirality: Conservation and Sources
## 4 Macroscopic Sources
### Currents and Charges in Bi-Isotropic Media  
We now consider what happens when a local current density is placed inside a chiral medium. Inserting the constitutive relations (28) into \(\mathbf{\
abla}\times\mathbf{E}=-\hat{\mathbf{B}}\) and \(\mathbf{\
abla}\times\mathbf{H}=\hat{\mathbf{D}}+\mathbf{j}\) leads to:  
\[\dot{\mathbf{A}}=\beta\left(\mathbf{\
abla}\times\mathbf{E}\right)+\frac{1}{ \epsilon}\left(\mathbf{\
abla}\times\mathbf{C}+\alpha\mathbf{H}\right) \tag{39}\]  
and:  
\[\dot{\mathbf{C}}=\mathbf{g}+\beta\left(\mathbf{\
abla}\times\mathbf{H}\right)- \frac{1}{\mu}\left(\mathbf{\
abla}\times\mathbf{A}-\alpha\mathbf{E}\right), \tag{40}\]  
where again, \(\mathbf{\
abla}\times\mathbf{g}\) is equal to the transverse part of the current density. We insert \(\dot{\mathbf{A}}\) and \(\dot{\mathbf{C}}\) into \(\partial_{t}h_{1}\) from (35) and find to \(\mathcal{O}(\beta)\):  
\[\partial_{t}h_{1}+\mathbf{\
abla}\cdot\mathbf{v}=\frac{1}{2}\sqrt{\frac{\mu}{ \epsilon}}\left[\mathbf{g}\cdot\left(\mathbf{\
abla}\times\mathbf{C}\right)+ \mathbf{C}\cdot\left(\mathbf{\
abla}\times\mathbf{g}\right)\right]-\sqrt{\epsilon \mu}\beta\mathbf{j}\cdot\mathbf{E}-\alpha\left(\sqrt{\frac{\epsilon}{\mu}}| \mathbf{E}|^{2}-\sqrt{\frac{\mu}{\epsilon}}|\mathbf{H}|^{2}\right). \tag{41}\]  
This reduces to the result for a medium with no chiroptical or magnetoelectric response when \(\alpha=\beta=0\). The helicity contribution due to the chiral response of the material is identifiable as an energy source, as obtained from the energy continuity equation in the presence of charges [41]. As this term is a scalar, the pseudoscalar nature of \(\beta\) is responsible for this term's acting as a source of helicity. The Tellegen contribution, proportional to \(\alpha\), is identical to the helicity source in an absence of currents or charges, as given in (38). Consider a chiral, reciprocal material described by (28) with \(\alpha=0\). From (41), we would expect that an emitter that in free space emits no net helicity, such as a single oscillating electric dipole, may act as a source of helicity when placed inside a chiral medium. [PERSON] et al. have shown that  
Figure 4: At the interface between a vacuum and a dual-symmetric, lossless chiral medium characterised by \(\epsilon\), \(\mu\) 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. this is so, by explicitly calculating the radiation pattern of a single oscillating electric dipole embedded in a sphere composed of a lossless chiral medium [49]. From the point of view of an observer in the far field, outside the sphere, the radiation pattern from the point electric dipole embedded in the sphere appears identical to that of a point electric and point magnetic dipole oscillating in a vacuum. In particular, the chiral medium is impedance matched with the surrounding vacuum to produce this result, so that the net helicity cannot be attributed to a gradient in \(\epsilon/\mu\) at the vacuum-chiral interface. As the chiral medium considered above and in [49] is lossless, neither can this generation of helicity arise as a result of circular dichroism within the chiral sphere [30]. Furthermore, we know that the electromagnetic helicity of a field is conserved within a dual-symmetric chiral medium, so the helicity \"source\" in this case can only be attributed to the interaction of the embedded current and the chiral medium itself; a result which seems worthy of further investigation.