Multidimensional measures of electromagnetic chirality and their conformal invariance

Abstract

Proper assignment of left- and right-handed labels to general chiral objects is known to be a theoretically unfeasible problem. Attempts to utilize a pseudoscalar function to distinguish enantiomers face two unavoidable difficulties: false chiral zeros and unhanded chiral states. In here, we demonstrate how both of these problems can be solved in the context of light-matter interactions. First, we introduce a two-dimensional quantity called complex electromagnetic chirality that solves the problem of false chiral zeros. Next, we define an infinite-dimensional pseudovector called chirality signature that completely quantifies the multidimensional nature of electromagnetic chirality, does not have false global chiral zeros, and allows to continuously distinguish any pair of enantiomers because it does not produce unhanded chiral states. We prove that the introduced measures are invariant under the largest group of symmetries of Maxwell’s equations – the conformal group. The complete, continuous, and conformally invariant quantification of electromagnetic chirality provided by the chirality signature distinguishes it as a particularly suitable tool for the study of chirality and its applications.

Maximizing the electromagnetic chirality of thin metallic nanowires at optical frequencies

“…If this is not the case, then the object is em-chiral. This notion of em-chirality is consistent with the traditional geometric concept of chirality, but in contrast to the geometric chirality of an object, its em-chirality can be quantified directly in terms of the object's interaction with electromagnetic waves using em-chirality measures [7,16,45]. These em-chirality measures are bounded from below by zero and from above by the total interaction cross-section of the scattering object, which makes them well-suited objective functionals for a shape optimization.…”

“…In this section we briefly summarize the concept of electromagnetic chirality. For further details we refer to [7,16,45]. An electric plane wave with direction of propagation θ ∈ S 2 and polarization A ∈ C 3 , which must satisfy A • θ = 0, is described by the matrix E i ( • ; θ) ∈ C 3×3 defined by…”