# Optical Helicity and Chirality: Conservation and Sources
## 2 Review and Motivation
### Continuity Equations in Free Space  
As mentioned above, helicity is a conserved quantity in vacuum. This fact can be expressed using a local continuity equation, relating the time derivative of the helicity density at a point to the helicity flux through an infinitesimal volume surrounding that point. Taking the time derivative of the helicity density (1):  
\[\partial_{t}h =\frac{1}{2}\left[\sqrt{\frac{\epsilon_{0}}{\mu_{0}}}\left( \mathbf{\hat{A}}\cdot\mathbf{B}+\mathbf{A}\cdot\mathbf{\hat{B}}\right)-\sqrt{ \frac{\mu_{0}}{\epsilon_{0}}}\left(\mathbf{\hat{C}}\cdot\mathbf{D}+\mathbf{C }\cdot\mathbf{\hat{D}}\right)\right]\] \[=\frac{1}{2}\left[\sqrt{\frac{\epsilon_{0}}{\mu_{0}}}\left(- \mathbf{E}\cdot\left(\mathbf{\
abla}\times\mathbf{A}\right)-\mathbf{A}\cdot\left( \mathbf{\
abla}\times\mathbf{E}\right)\right)-\sqrt{\frac{\mu_{0}}{\epsilon_{0} }}\left(\mathbf{H}\cdot\left(\mathbf{\
abla}\times\mathbf{C}\right)+\mathbf{C} \cdot\left(\mathbf{\
abla}\times\mathbf{H}\right)\right)\right], \tag{11}\]  
where, from the free-space Maxwell equations, we have used \(\mathbf{E}=-\mathbf{A}\) and \(\mathbf{H}=-\mathbf{\hat{C}}\), with the dotted notation indicating the time derivative of the fields. Using the vector identity \(\mathbf{\
abla}\cdot\left(\mathbf{E}\times\mathbf{A}\right)=\mathbf{A}\cdot\left( \mathbf{\
abla}\times\mathbf{E}\right)-\mathbf{E}\cdot\left(\mathbf{\
abla}\times \mathbf{A}\right)\) and \(\mathbf{\
abla}\cdot\left(\mathbf{H}\times\mathbf{C}\right)=\mathbf{C}\cdot \left(\mathbf{\
abla}\times\mathbf{H}\right)-\mathbf{H}\cdot\left(\mathbf{\
abla} \times\mathbf{C}\right)\), this is rearranged to produce:  
\[\partial_{t}h+\frac{1}{2}\left[\sqrt{\frac{\epsilon_{0}}{\mu_{0}}}\mathbf{\
abla }\cdot\left(\mathbf{E}\times\mathbf{A}\right)+\sqrt{\frac{\mu}{\epsilon}}\mathbf{ \
abla}\cdot\left(\mathbf{H}\times\mathbf{C}\right)\right]=-\sqrt{\frac{ \epsilon_{0}}{\mu_{0}}}\mathbf{E}\cdot\mathbf{B}+\sqrt{\frac{\mu_{0}}{ \epsilon_{0}}}\mathbf{H}\cdot\mathbf{D}. \tag{12}\]  
Inserting the relations \(\mathbf{B}=\mu_{0}\mathbf{H}\) and \(\mathbf{D}=\epsilon_{0}\mathbf{E}\) and using \(\mathbf{\
abla}\left(\mu_{0}/\epsilon_{0}\right)=0\) leads to the free-space helicity continuity Equation [14; 15; 17]:  
\[\partial_{t}h+\mathbf{\
abla}\cdot\mathbf{v}=0, \tag{13}\]where the helicity flux density \(\mathbf{v}\) is defined as:  
\[\mathbf{v}=\frac{1}{2}\left[\sqrt{\frac{\varepsilon_{0}}{\mu_{0}}}\mathbf{E} \times\mathbf{A}+\sqrt{\frac{\mu_{0}}{\varepsilon_{0}}}\mathbf{H}\times\mathbf{ C}\right]. \tag{14}\]  
Thus, the conservation of helicity in a vacuum is explicitly demonstrated. Note that, thanks to the use of the definition (1), helicity is shown to be locally conserved; a stronger result than if only the integrated quantity were conserved. A pleasing analogy can be drawn between this characteristic of the helicity density and the one of the electromagnetic energy density: the total energy can be written as the volume integral of \(\varepsilon_{0}/2\left(\mathbf{E}^{2}+c^{2}\mathbf{B}^{2}\right)\), or indeed as the integral over all space of either the \(\mathbf{E}^{2}\) or \(\mathbf{B}^{2}\) contributions. Only the energy density formed by the combined electric and magnetic contributions, however, is conserved locally [41].  
There is an appealingly simple relationship between the helicity flux density and the spin density. The latter is often written as \(\varepsilon_{0}\mathbf{E}\times\mathbf{A}\), but it can also be written in the manifestly duplex-symmetric form:  
\[\mathbf{s}=\frac{1}{2}\Big{[}\varepsilon_{0}\mathbf{E}\times\mathbf{A}+ \mathbf{B}\times\mathbf{C}\Big{]}, \tag{15}\]  
which immediately establishes the relation \(\mathbf{s}=\mathbf{v}/c\). The relationship between these quantities and the helicity density is reminiscent of that between the energy density, the energy flux density and the momentum density, with Poynting's vector playing the role of both of the latter two quantities. For helicity and spin, \(\mathbf{s}\) (or \(\mathbf{v}\)) plays an analogous double role.