[Paper Review] Reflecting complementary media and superlensing using complementary media for electromagnetic waves

Hoài-Minh Nguyên|arXiv (Cornell University)|Nov 25, 2015

Electromagnetic Scattering and Analysis3 citations

TL;DR

This paper presents the first proof of superlensing using complementary media in the electromagnetic setting, leveraging reflecting complementary media to overcome challenges from sign-changing coefficients and loss of ellipticity/compactness. By introducing a novel superlensing scheme based on the reflecting technique and establishing new existence, stability, and compactness results for Maxwell's equations, the authors achieve subwavelength imaging without requiring the removal of localized singularities.

ABSTRACT

Negative index materials were first investigated theoretically by Veselago in \cite{Veselago} and were confirmed experimentally by Shelby, Smith, and Schultz in \cite{ShelbySmithSchultz}. Mathematically, the study of NIMs faces two difficulties. First, the equations modeling NIMs have sign changing coefficients; hence the ellipticity and the compactness are lost in general. Secondly, the localized resonance, the fields blow up in some regions and remain bounded in some others as the loss goes to 0, might appear. The study of negative index materials has attracted a lot attention in the scientific community thanks to their many possible applications. One of them is superlensing initiated by Veselago in \cite{Veselago}. The proof of superlensing using complementary media for arbitrary objects was given in \cite{Ng-Superlensing} in the acoustic setting. The superlensing schemes used in \cite{Ng-Superlensing} were guided by the notion of reflecting complementary media introduced and studied in \cite{Ng-Complementary} and the proof in \cite{Ng-Superlensing} used the reflecting and the removing localized singularity techniques introduced in \cite{Ng-Complementary} and \cite{Ng-Superlensing, Ng-Negative-cloaking} respectively. In this paper, we provide a background for reflecting complementary media and present the first proof of superlensing using complementary media in the electromagnetic setting. Using a special class of superlensing schemes inspired by \cite{Ng-Complementary, Ng-Superlensing}, we are able to implement only the reflecting technique in the proof of superlensing to handle the lost of the ellipticity and the compactness. To successfully extend the ideas from \cite{Ng-Complementary, Ng-Superlensing}, we establish new results on the compactness, existence, and stability for the Maxwell equations.

Motivation & Objective

• To extend the superlensing framework from the acoustic to the electromagnetic setting using complementary media.
• To address the mathematical challenges posed by sign-changing coefficients in negative index materials, including loss of ellipticity and compactness.
• To establish new theoretical results on existence, stability, and compactness for Maxwell's equations under these conditions.
• To demonstrate that the reflecting technique alone—without the need for singularity removal—can successfully achieve superlensing in electromagnetism.

Proposed method

• Adopting a special class of superlensing schemes inspired by prior work on complementary media in acoustics.
• Employing the reflecting complementary media technique to manage field blow-up and maintain boundedness in regions of interest.
• Introducing a new mathematical framework to restore compactness and ensure existence of solutions despite sign-changing coefficients.
• Establishing stability estimates for the Maxwell equations under the proposed complementary media configuration.
• Using the reflecting technique to circumvent the need for localized singularity removal, simplifying the proof structure.
• Leveraging theoretical tools from previous works on negative-index materials and cloaking to adapt them to the electromagnetic case.

Experimental results

Research questions

• RQ1 Can superlensing be rigorously proven in the electromagnetic setting using complementary media?
• RQ2 How can the loss of ellipticity and compactness in Maxwell's equations with sign-changing coefficients be overcome?
• RQ3 Is it possible to achieve superlensing using only the reflecting technique, without requiring singularity removal?
• RQ4 What new existence and stability results are required for Maxwell's equations under complementary media configurations?
• RQ5 How do the mathematical properties of the electromagnetic system compare to those in the acoustic case under similar schemes?

Key findings

• The paper establishes the first rigorous proof of superlensing using complementary media in the electromagnetic setting.
• It demonstrates that the reflecting technique alone is sufficient to handle the loss of ellipticity and compactness, eliminating the need for singularity removal.
• New existence and stability results for Maxwell's equations with sign-changing coefficients are derived, enabling the analysis of resonant behavior.
• The framework successfully suppresses field blow-up in certain regions while maintaining bounded fields in others, enabling subwavelength imaging.
• The results confirm that complementary media can be used to achieve super-resolution imaging in electromagnetism, analogous to prior acoustic results.
• The mathematical structure of the electromagnetic system under complementary media is shown to be amenable to analysis through the reflecting technique, extending the scope of transformation optics.