# Optical chirality and its possible hierarchy of generalizations

Optical chirality refers to a constant of motion of the electromagnetic field, which measures in some sense how chiral a light field is. Specifically, the pseudoscalar quantity $$C=\frac{\varepsilon_0}{2}\mathbf{E}\cdot \nabla\times\mathbf{E}+\frac{1}{2\mu_0}\mathbf{B}\cdot \nabla\times\mathbf{B} \tag1$$ obeys the continuity equation $$\frac{\partial C}{\partial t}+\frac{1}{2\mu_0}\nabla\cdot\left(\mathbf{E}\times\nabla\times\mathbf{B}-\mathbf{B}\times\nabla\times\mathbf{E}\right)=0$$ in free space. It was re-discovered by Yiqiao Tang and Adam E. Cohen, in

Yiqiao Tang and Adam E. Cohen. Optical Chirality and Its Interaction with Matter. Phys. Rev. Lett. 104, 163901 (2010); Harvard eprint.

after having been discovered, puzzled over, called 'zilch' for lack of a better name, and forgotten in the 1960s.

This quantity is useful because it is a direct measure of how strongly many chiral biological molecules will interact with a chiral electromagnetic wave, which is an important tool of biochemistry. This rediscovery is a huge step forward, but as Tang and Cohen note, it cannot be the whole story:

Similarly, there cannot be a single measure of electromagnetic chirality appropriate to all EM fields. There exist chiral fields for which $C$ as defined in Eq. (1) is zero. Indeed, the field of any static, chiral configuration of point charges is chiral, yet by Eq. (1), $C=0$.

(This is trivially seen as both curls vanish in the static case.) In response to this, Tang and Cohen offer some conjectures:

The optical chirality of Eq. (1) may be part of a hierarchy of bilinear chiral measures that involve higher spatial derivatives of the electric and magnetic fields [22]. We speculate that all linear chiral light-matter interactions can be described by sums of products of material chiralities and time-even pseudoscalar optical chiralities.

In addition to this, there is the possibility of non-linear chiral interactions, which involve the product of three or more force fields.

To come, finally, to my question: what is the current status of these conjectures? Are there descriptions of higher-order tensors (involving higher spatial derivatives) or nonlinear terms (involving more than two force fields), which are also chirally sensitive? Is there some ordered hierarchy which contains them? Is there some sort of completeness result that guarantees said hierarchy contains 'all' the relevant quantities?

• For a single-photon plane-wave field, $C=\tfrac{\varepsilon_0}{2}\mathbf{E}\cdot \nabla\times\mathbf{E}+\tfrac{1}{2\mu_0}\mathbf{B}\cdot \nabla\times\mathbf{B}$ is indeed proportional to the spin. However, this does not imply that all chiral interactions of a molecule with an EM field are governed by this $C$, as shown by Anco and Pohjanpelto, so there are other independent measures of chirality. – Emilio Pisanty Oct 21 '15 at 13:55
• As to what the higher-order chiralities mean, consider a right circularly polarized wave at wavelength $\lambda$ and amplitude $\mathbf E_0$, co-propagating with a left-circular wave at half the wavelength and half the intensity. The field is globally chiral but at any given point $C$ oscillates sinusoidally about zero, so it cannot give rise to any chiral interactions. Larger molecules, however (or more sensitive experiments) can probe the second derivatives of the field, which rack up an extra factor of $k$ and tilt the scales in favour of the left-circular wave. – Emilio Pisanty Oct 21 '15 at 14:08
• I'm completely unsure what you mean by two circularly polarized waves with different frequencies do not interfere with each other, as the molecule observes the local electric field which is the coherent sum of both contributions. – Emilio Pisanty Oct 21 '15 at 14:40