Nonlocal current-response theory of structured-light dichroism

Abstract: We develop a microscopic theory of optical absorption and structured-light dichroism in a nonlocal minimal-coupling framework. Starting from the minimal-coupling Hamiltonian, we express absorption as a bilinear functional of the electromagnetic vector potential and the nonlocal current response, providing a general description of light--matter coupling in optical vortex beams and other inhomogeneous fields. Dichroic signals are identified as helicity-odd projections of the nonlocal response kernel, and the response is resolved into symmetry, tensorial, and mode-space sectors. For single helical modes, the theory yields diagonal OAM-resolved contributions together with the corresponding selection rules and symmetry constraints. For mixed modes, interference between distinct OAM components provides access to off-diagonal coherence of the nonlocal kernel, with a tensor structure determined by the polarization composition of the field. The theory also clarifies how local structures associated with symmetric spatial dispersion, optical chirality, and tensorial anisotropy emerge as gradient-level manifestations of the underlying nonlocal response.

• The paper introduces a comprehensive microscopic formulation of structured-light dichroism using a nonlocal minimal-coupling Hamiltonian to reveal how dichroic signals emerge.
• It decomposes the nonlocal current response into irreducible tensor components, linking axial and tensor sectors to both diagonal and off-diagonal optical absorption processes.
• The work bridges local gradient limits with nonlocal optical chirality measures, offering a unified framework for advanced chiral spectroscopy and nanophotonic applications.

Nonlocal Current-Response Theory of Structured-Light Dichroism

Overview

This paper presents a rigorous, microscopic formulation of structured-light dichroism, unifying the treatment of circular dichroism (CD), helical dichroism (HD), and helical circular dichroism (HCD) within a fully nonlocal minimal-coupling framework. The central contribution is the characterization of light–matter interaction, especially in the context of optical vortex beams, through a bilinear functional involving the electromagnetic vector potential and the nonlocal current response kernel. The authors demonstrate how dichroic signals emerge as specific helicity-odd projections of this nonlocal kernel, and systematically resolve the response into symmetry, tensorial, and mode-space sectors, with explicit implications for diagonal (OAM-resolved) and off-diagonal (mode-coherent) absorption processes.

Theoretical Framework

The starting point is the minimal-coupling Hamiltonian, $$\hat{H}_{\mathrm{int}} = -\int d\mathbf{r}\, \hat{\mathbf{j}}(\mathbf{r}) \cdot \hat{\mathbf{A}}(\mathbf{r}, t)$$, which treats spatial nonlocality in the electromagnetic interaction explicitly. Optical absorption is formulated as a bilinear contraction between the vector potential and a two-point nonlocal current-current response kernel, $\mathcal{J}_{ab}(\mathbf{r}, \mathbf{r}'; \omega)$. This structure remains intact for both linear and nonlinear regimes, with higher-order response functions naturally generalizing the kernel to multivariate functionals in the applied field.

Significantly, dichroism is operationalized through differences under reversal of the helicity indices: spin ($\sigma = \pm 1$), orbital angular momentum ($\ell \in \mathbb{Z}$), or both. The analysis differentiates among CD ($\Delta_{\mathrm{CD}}$), HD ($\Delta_{\mathrm{HD}}$), and HCD ($\Delta_{\mathrm{HCD}}$), corresponding to reversals of $\sigma$, $\ell$, or $(\sigma, \ell)$, respectively.

Symmetry and Tensorial Decomposition

The nonlocal response is systematically decomposed into irreducible tensor components of ranks $\mathcal{J}_{ab}(\mathbf{r}, \mathbf{r}'; \omega)$0 (scalar, axial vector, and symmetric traceless tensor). The dichroic optical bilinear—which isolates helicity-odd content—is likewise decomposed, revealing that:

• CD signals (spin reversal) are purely axial-vector in the single-mode (OAM-eigenstate) case.
• HD and HCD signals generically involve all tensor ranks, with distinct angular dependences, manifest through azimuthal correlations $\mathcal{J}_{ab}(\mathbf{r}, \mathbf{r}'; \omega)$1 and $\mathcal{J}_{ab}(\mathbf{r}, \mathbf{r}'; \omega)$2 for different tensor sectors.

Such decomposition elucidates the connection between experimental observables and microscopic current structures. In particular, the axial-vector sector is tightly linked to antisymmetric, circulating-current contributions, aligning with traditional notions of molecular chirality and optical activity, while rank-2 sectors encode anisotropic response channels accessible only via structured illumination.

Mode-Space Structure: Diagonal and Off-Diagonal Response

For single helical modes, absorption projects onto OAM-diagonal components of the nonlocal kernel. When considering superpositions or mixed OAM modes, the formalism uncovers access to coherences (off-diagonal kernel elements) between different angular-momentum sectors. The possibility of experimentally probing these off-diagonal sectors is dependent on the field's polarization composition; for instance, opposite circular polarizations ($\mathcal{J}_{ab}(\mathbf{r}, \mathbf{r}'; \omega)$3) in mixed modes isolate purely rank-2 response channels—a result with concrete implications for manipulating or detecting tensorial matter responses using polarization-textured or skyrmionic beams.

The authors also clarify symmetry constraints under continuous and discrete rotation, dictating selection rules for the observability of these off-diagonal coherence effects.

Local Gradient Limit and Physical Interpretation

Expansion of the optical bilinear in local gradients bridges the fully nonlocal theory to familiar gradient-based measures of optical chirality and spatial dispersion (e.g., Lipkin's "zilch" and its generalizations). The scalar, axial-vector, and tensor contributions at the local level correspond to spatially variant field structures (helicity, chirality, and anisotropy) and underpin various regimes of structured-light–matter interaction, particularly for tightly focused, strongly inhomogeneous, or nonparaxial fields.

This analysis reveals that local chiroptical measures (e.g., optical chirality density) are fundamentally gradient-level approximations of the more general nonlocal response addressed here.

Experimental Relevance and Extensions

The theory provides a framework encompassing both diagonal (pure-mode) and off-diagonal (mode-coherent) dichroic signals, unifying diverse experimental observations in molecular, nanophotonic, and metasurface systems. It predicts that strong HD can arise from intrinsic diagonal inversion-odd response, while mode mixing or imperfect beam purity may activate otherwise symmetry-forbidden off-diagonal contributions.

The minimal-coupling and nonlocal current-response form retains explicit mode profile dependence and is naturally extendable to nonlinear spectroscopies, vectorial and multipolar interactions, and higher-order field effects, providing a systematically improvable basis for future theoretical and computational treatments.

Conclusion

This work formulates a comprehensive, mode-explicit description of structured-light dichroism rooted in the nonlocal minimal-coupling Hamiltonian and current-response kernel. By resolving dichroic signals into symmetry, tensorial, and mode-space channels, the analysis clarifies physical selection rules and provides a unified picture that encompasses both linear and nonlinear, local and nonlocal chiroptical phenomena. The results delineate pathways for probing inversion-odd, anisotropic, and coherence-sensitive matter responses, with direct relevance to advanced chiral spectroscopy, nanophotonics, and quantum materials research. Future directions include quantitative modeling of the response kernel for specific materials and the extension to complex, topological, or strongly correlated systems.