# Optical Helicity and Chirality: Conservation and Sources
## 4 Macroscopic Sources  
### Helicity in Achiral, Reciprocal Media  
We have seen how helicity can be produced by a microscopic source (a dipole), but it is also possible to discuss the generation of helicity within the framework of macroscopic electromagnetism in dielectric media. It is perhaps surprising that even in the presence of matter, there are some situations in which the helicity of an arbitrary electromagnetic field is still conserved. The conditions under which the electromagnetic helicity is conserved within media have been studied in recent years [16; 18; 19]. [PERSON] et al. [16] consider the propagation of helicity in isotropic, lossless linear media, and these results are extended by [PERSON] and [PERSON] in [18] to include anisotropic and general linear media. [PERSON] et al. [19] further consider helicity in a dispersive, lossy medium, while the electromagnetic chirality is examined in such media by [PERSON] and [PERSON] [44].  
In the following, we consider a linear, lossless and isotropic medium. If the medium is comprised of distinct, homogeneous regions labelled by \(i\), helicity is conserved so long as \(\epsilon_{i}/\mu_{i}\) remains constant for all \(i\)[16]. Following a similar method to that used in Section 2.5, we can derive a continuity equation  
Figure 3: Two oscillating electric dipoles, labelled \(\mathbf{p}_{1}(t)\) and \(\mathbf{p}_{2}(t)\), oscillate along the \(z\) and \(y\) axes, respectively, with a phase difference of \(\pi/2\). The far-field pattern in the \(+x\) direction is identical to that produced by the electric-magnetic dipole configuration in Figure 2, but has opposite polarisation in the \(-x\) direction: the net helicity flux in the \(yz\) plane is zero.  
Figure 2: Oscillating electric \(\mathbf{p}(t)\) and magnetic \(\mathbf{m}(t)\) 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 \(+x\) direction and left-circularly polarised light in the \(-x\) direction: there is a net flux of right-circular polarisation in the \(yz\) plane.  
for helicity in such media. We use the definitions of \(h\) (1) and \(\mathbf{v}\) (14), along with the constitutive relations \(\mathbf{D}=\epsilon\mathbf{E}\) and \(\mathbf{B}=\mu\mathbf{H}\), to obtain:  
\[\partial_{t}h+\mathbf{\
abla}\cdot\mathbf{v}=\frac{1}{2}\Bigg{[}\mathbf{\
abla}\Bigg{(} \sqrt{\frac{\epsilon}{\mu}}\Bigg{)}\cdot\Big{(}\mathbf{E}\times\mathbf{A} \Big{)}+\mathbf{\
abla}\Bigg{(}\sqrt{\frac{\mu}{\epsilon}}\Bigg{)}\cdot\Big{(} \mathbf{H}\times\mathbf{C}\Big{)}\Bigg{]}, \tag{27}\]  
where we have allowed for the possibility that \(\epsilon\) and \(\mu\) are functions of position. This explicitly shows the conservation of helicity when \(\mathbf{\
abla}\left(\epsilon/\mu\right)=0\), which can be seen as a continuous statement of the result for stratified media presented in [16].  
We extend the results of [17] discussed in Section 3.1, to examine the effects of inserting a local current density \(\mathbf{j}\) into a medium described by the constitutive relations \(\mathbf{D}=\epsilon\mathbf{E}\) and \(\mathbf{B}=\mu\mathbf{H}\). We find a source term analogous to the right-hand side of (16), but with the replacement of \(\epsilon_{0}\to\epsilon\) and \(\mu_{0}\to\mu\). Moreover, this type of source cannot be associated with that produced by a gradient of \(\epsilon/\mu\), as given by the right-hand side of (27).