The Uniqueness Theorem for Nonlocal Hydrodynamic Media

TL;DR

This paper establishes a fundamental uniqueness theorem for nonlocal electromagnetic media modeled by the Hydrodynamic Drude Model and its extensions, providing a theoretical foundation for the use of Additional Boundary Conditions in simulations.

Contribution

It offers a mathematically relaxed proof of the uniqueness theorem for nonlocal media, clarifying material-response requirements and supporting the theory with numerical examples.

Findings

Proves the necessity of Additional Boundary Conditions in nonlocal electromagnetics.

Provides clear material-response criteria for uniqueness in nonlocal models.

Supports theoretical results with semianalytical and numerical examples.

Abstract

We investigate a fundamental electromagnetic theorem, namely the uniqueness theorem, in the context of nonlocal electromagnetics, as simulated by a popular semiclassical model, the Hydrodynamic Drude Model (HDM) and extensions thereof such as the Generalized Nonlocal Optical Response (GNOR). The derivations and proofs presented here give a theoretical foundation to the use of the Additional Boundary Conditions (ABCs), whose necessity is recognized and underlined in virtually all implementations and applications of HDM. Our proofs follow a mathematically relaxed style, borrowing from the literature of established electromagnetics textbooks that study the matter from an engineering perspective. Through this simpler route we deduce clear and intuitive material-response requirements for uniqueness to hold, while using a familiar parlance in a topic that is mostly studied through a physics perspective. Two numerical examples that examine the problem from either a semianalytical or a purely numerical viewpoint support our findings.