# Optical Helicity and Chirality: Conservation and Sources
## 1 Introduction  
The study of the handedness, or chirality, of matter has its roots in the work of [PERSON] [1], [PERSON] [2] and [PERSON] [3, 4] in the early 19 th Century, with the discovery that the polarisation of light rotates upon propagation through certain crystals and molecular solutions. It was [PERSON] who introduced the word chiral to describe such matter, which is non-superimposable upon its mirror image [5], with a chiral object and its mirror image being called enantiomers. In particular, it was realised in these early experiments that if the rotation angle of the polarisation vector through a solution of chiral molecules is given by \(\theta\), then the rotation of the light in the enantiomeric form of the solution is through the angle \(-\theta\). We need not look far to observe that chirality is in fact ubiquitous in nature: our left and right hands and feet are distinct from each other, with the word \"chiral\" itself derived from the Greek word for hand, \(\chi e\mu\)[6]. The weak force, being parity violating, is a striking example of the role of handedness in nature [7], as is the remarkable selectivity evident in biological homochirality [8]: the complex molecules DNA, RNA, as well as the proteins and sugars comprising all living organisms, are indeed chiral. A simple example of a chiral molecule, alongside an achiral counterpart, is shown in Figure 1.  
The response of such chiral matter to the polarisation of light is called natural optical activity [9], and it is with this topic which we are primarily concerned. Of course, light itself can have a chiral structure: left- and right-circularly polarised fields trace out helices with opposite handedness, and accordingly act as a chiral influence on matter which itself exists in two enantiomeric forms. This is manifest in many different effects [6], the most well-known being optical rotation, due to left- and redright-circularly polarised light having a different refractive index in the chiral medium [2], and circular dichroism, arising from the different absorption coefficients for the two polarisations [10]. We therefore look for a way of characterising such polarised fields before and after interaction with an object which will allow us to infer any chiral influence of that object. Light carries an intrinsic angular momentum [11], or spin, which differs in sign for left- and right-circular polarisation, promoting the spin as a natural candidate for our purposes [6]. Being a pseudovector, however, means that thespin has even parity, whereas a little thought on the problem reveals that an odd-parity observable is required to distinguish the interaction of light with two enantiomers of a chiral object, which are parity odd. The answer lies in the projection of the spin in the direction of propagation of the light beam: the optical helicity [12; 13; 14; 15; 16; 17; 18; 19]. We use the well-known method of constructing continuity equations for conserved quantities in electrodynamics [17] to show that the helicity can be used to characterise the interaction of light with different types, and indeed scales, of matter. In Section 2, we review the definition of the electromagnetic helicity in a vacuum, before looking at microscopic sources of helicity in the form of mixed radiating electric-magnetic dipoles in Section 3. We extend the method in Section 4 to examine the conditions under which helicity is conserved in lossless bulk media, before summarising the results in Section 5.