Metric-Torsion Duality of Optically Chiral Structures

We develop a metric-torsion theory for chiral structures by using a generalized framework of transformation optics. We show that the chirality is uniquely determined by a metric with the local rotational degree of freedom. In analogy to the dislocation continuum, the chirality can be alternatively interpreted as the torsion tensor of a Riemann-Cartan space, which is mimicked by the anholonomy of the orthonormal basis. As a demonstration, we reveal the equivalence of typical three-dimensional chiral metamaterials in the continuum limit. Our theory provides an analytical recipe to design optical chirality.

Figure 1

Schematics of the local anisotropy of $(\mathit{\epsilon },\mathit{\mu })$ (the crossed rods) and the local frames (the triads) at two nearby points in (a) the achiral background and (b) the chiral continuum. The Cartan’s circuits are shown in the right panels.

Figure 2

(a) Schematics of a CLC (right) and the anisotropic background (left). (b),(c) Unit cells of typical 3D chiral metamaterials consisting of spiral ladders and helices (right) and the anisotropic backgrounds (left). The bars are real nanostructures.