# Optical Helicity and Chirality: Conservation and Sources
## 2 Review and Motivation
### Helicity and Chirality  
The helicity is closely related to the \"chirality\", another conserved quantity of the free electromagnetic field, introduced by [PERSON] [29] as one of a class of conserved quantities called \"zilches\". It is defined as:  
\[\chi=\frac{\epsilon_{0}}{2}\Big{[}\mathbf{E}\cdot(\mathbf{\
abla}\times\mathbf{E })+c^{2}\mathbf{B}\cdot(\mathbf{\
abla}\times\mathbf{B})\,\Big{]}. \tag{2}\]  
This quantity has also been studied as a means of describing the interaction of light with chiral matter. [PERSON] and [PERSON] [30] demonstrated the physical significance of the chirality, showing that the differential excitation rate between the two enantiomers of a chiral molecule in a monochromatic optical field is proportional to the chirality of the field in which they are immersed. Since that work, the chirality density has been applied to the analysis of a number of scenarios, including that of fields near metamaterial surfaces, where configurations have been suggested in which the ratio of chirality density to energy density is greater than in circularly polarised light. This \"superchirality\" has been proposed as a means of enhancing the enantioselectivity of some chiroptical techniques [31; 32],although the importance of such \"superchirality\" in explaining the reported enhancements has been challenged [33]. Superchirality in the works of [PERSON] and [PERSON] [31] is associated with regions of destructive interference and hence a reduction in the energy density; however, it is also possible to generate bright regions of superchirality [34]. In monochromatic fields, the helicity and chirality densities are proportional to one another [14]. However, they have different frequency dependencies, and so, in a general time-dependent field, no strict proportionality holds. A simple and striking illustration of the non-equivalence of helicity and chirality is provided by considering the superposition of two circular plane waves of opposite handedness and different frequencies; in this case, the two quantities have opposite signs. For right- and left-circular polarisations, the helicity is proportional to \(\pm 1/\omega\) and the chirality to \(\pm\omega\). This might immediately suggest that the two measures give opposite signs: if the right-handed wave has a higher frequency than the left-handed, then the magnitude of the chirality of the right-handed wave will be greater than that of the left, but the magnitude of the helicity will be greater for the left than the right. Of course, the helicity and chirality are both quadratic in the fields, so the above represents little more than a plausibility argument; explicit calculation, however, reveals that they do indeed have opposite signs. If we take our two plane waves to be travelling in the positive \(z\) direction, then choosing vector potentials \(\mathbf{A}=-\int\mathbf{E}\,dt\) and \(\mathbf{C}=-\frac{1}{c^{2}}\)\(\mathbf{J}\)\(\mathbf{B}\,dt\) and using the real parts of the fields and potentials in (2) and (1) gives:  
\[\chi=\frac{2\epsilon_{0}E_{0}^{2}}{c}(\omega_{1}-\omega_{2})\cos^{2}\left( \frac{\omega_{1}-\omega_{2}}{2}\eta\right) \tag{3}\]  
and:  
\[h=-\frac{2\epsilon_{0}E_{0}^{2}}{\omega_{1}\omega_{2}}(\omega_{1}-\omega_{2} )\cos^{2}\left(\frac{\omega_{1}-\omega_{2}}{2}\eta\right), \tag{4}\]  
where \(\omega_{1}\) is the frequency of the right-handed wave, \(\omega_{2}\) that of the left-handed, \(E_{0}\) the peak electric field strength of each wave, and \(\eta\equiv t-z/c\). We see that the helicity and chirality densities are here both proportional to the energy density at all times, but with opposite signs and different dependencies on the two frequencies. Given the sign difference, it seems reasonable to ask which of the helicity and chirality corresponds to the intuitive \"sense\" of the rotation in the example field given here.