[Paper Review] Extension of Lorentz Reciprocity and Poynting Theorems for Spatially Dispersive Media with Quadrupolar Responses

Karim Achouri, Olivier J. F. Martin|arXiv (Cornell University)|Feb 16, 2021

Metamaterials and Metasurfaces Applications42 references31 citations

TL;DR

This paper extends the Lorentz reciprocity and Poynting theorems to spatially dispersive media with electric and magnetic dipolar and quadrupolar responses, incorporating field gradients via a multipolar expansion. It derives self-consistent reciprocity and energy conservation conditions for the associated constitutive tensors, enabling advanced modeling of metamaterials with higher-order multipoles and spatial dispersion.

ABSTRACT

We provide a self-consistent extension of the Lorentz reciprocity theorem and the Poynting theorem for media possessing electric and magnetic dipolar and quadrupolar responses related to electric and magnetic fields and field gradients. Using these two theorems, we respectively deduce the conditions of reciprocity and gainlessness and losslessness that apply to the various tensors mediating the interactions of these multipole moments and the associated fields and field gradients. We expect that these conditions will play an essential role in developing advanced metamaterial modeling techniques that include quadrupolar and spatially dispersive responses.

Motivation & Objective

• To extend the Lorentz reciprocity theorem to media with electric and magnetic dipolar and quadrupolar responses, including field gradients.
• To derive the Poynting theorem for such media to establish conditions for gainlessness and losslessness.
• To provide a purely electromagnetic derivation of permutation symmetries for reciprocity and energy conservation, avoiding reliance on quantum mechanical concepts.
• To enable more accurate modeling of dielectric metasurfaces and metamaterials where higher-order multipoles and spatial dispersion are non-negligible.
• To support the development of advanced design techniques for metasurfaces with tailored angular scattering and field control capabilities.

Proposed method

• Derives a multipolar expansion of the current density J up to second-order terms in field gradients, including electric dipole (p), magnetic dipole (m), electric quadrupole (q), and magnetic quadrupole (s) moments.
• Expresses the constitutive relations for D and B in terms of E, H, and their spatial derivatives using the tensors P, M, Q, and S as dipolar and quadrupolar densities.
• Derives the Lorentz reciprocity theorem from Maxwell’s equations in the frequency domain, incorporating field gradients via the multipolar expansion.
• Derives the Poynting theorem from the same framework to analyze energy flow and power balance in the presence of spatial dispersion.
• Uses tensor symmetries in the constitutive relations (e.g., bij, bijk, bijkl) to derive necessary and sufficient conditions for reciprocity and loss/gainlessness.
• Applies a time-harmonic formalism with ejωt convention and employs Taylor expansions of the Green’s function to include spatial dispersion effects.

Experimental results

Research questions

• RQ1 What are the extended conditions for Lorentz reciprocity in media with electric and magnetic dipolar and quadrupolar responses including field gradients?
• RQ2 How can the Poynting theorem be generalized to account for spatial dispersion and higher-order multipole moments?
• RQ3 What are the symmetry constraints on the constitutive tensors that ensure reciprocity and energy conservation in such media?
• RQ4 How do field gradients modify the standard reciprocity and energy theorems in spatially dispersive materials?
• RQ5 What is the role of quadrupolar responses in enabling non-negligible higher-order field control in metasurfaces and metamaterials?

Key findings

• The Lorentz reciprocity theorem is extended to include field gradients, leading to new symmetry conditions on the constitutive tensors that govern dipolar and quadrupolar responses.
• The Poynting theorem is generalized to spatially dispersive media with multipole moments, yielding explicit conditions for gainlessness and losslessness in terms of the tensor components.
• The reciprocity conditions are derived purely from electromagnetic theory, without relying on quantum mechanical arguments, providing a classical foundation for multipoles in nonlocal media.
• The derived conditions reveal that reciprocity requires specific permutation symmetries in the tensors bij, bijk, and bijkl, which couple fields and their gradients.
• The losslessness condition is shown to depend on the Hermitian symmetry of the effective susceptibility tensors, ensuring no net energy gain or loss.
• The framework enables the systematic modeling of metasurfaces with significant electric and magnetic quadrupolar responses, particularly in dielectric resonators where higher-order modes are excited.