Beyond optical chirality density: tensor-based description of electromagnetic chirality

Abstract

Optical chirality density is widely used as a scalar measure of the chiral properties of electromagnetic fields and their interaction with matter. However, in anisotropic and structured media, a single scalar quantity is generally insufficient to capture the full complexity of chiral field-matter coupling. In this work, we go beyond the conventional optical chirality density and introduce a set of tensor measures of electromagnetic chirality based on the Lipkin formalism. These tensor quantities provide a richer and more physically transparent description of chiral electromagnetic fields, particularly in an anisotropic environment. The physical meaning of individual tensor components is discussed, and their role in characterizing different aspects of electromagnetic chirality is clarified. The proposed approach reveals multiple, complementary measures of field chirality that naturally emerge in anisotropic cases and are directly relevant to the interaction of structured electromagnetic fields with matter.

Figures (3)

• Figure 1: Explicit view of Z-tensor. The red color denotes the matrix $Z^{ij0}$. The components with a purple color indicate the optical chirality density and corresponding chiral flux components.
• Figure 2: Explicit view of Z-tensor. From left to right, the index $k=0,1,2,3$ in $Z^{ijk}$. The red color denotes the matrix $Z^{ij0}$. The components with a purple color indicate the optical chirality density and corresponding chiral flux components. The blue color quantities denote the new terms $\Xi^{ij0}$ and $\Xi^{ijk}$ that satisfy the equation of continuity $\partial_k\Xi^{ijk}=0$.
• Figure 3: Emergence of purely anisotropic optical chirality under LH–RH superposition: calculated time-averaged Lipkin tensor density components $\overline{Z^{\mu\nu0}}$ for (a) a left-handed circularly polarized wave propagating along the $z$ axis; (b) a right-handed circularly polarized wave propagating in the $XZ$ plane at an angle $\pi/4$ with respect to the $z$ axis; and (c) the superposition of the two monochromatic waves. Indices $\mu=\nu=0$ are indicated in the upper-left corner of the tables. The index $\mu$ corresponds to rows and $\nu$ to columns. The scalar optical chirality density $\overline{Z^{000}}$ vanishes for the total field, while the diagonal spatial components $\overline{Z^{aa0}}$ remain finite, revealing a purely anisotropic chiral field structure.