[论文解读] Reflecting complementary media and superlensing using complementary media for electromagnetic waves

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

Electromagnetic Scattering and Analysis被引用 3

一句话总结

本文首次在电磁设置下通过互补介质证明了超透镜成像，利用反射型互补介质克服了系数变号、椭圆性与紧致性丧失的挑战。通过引入基于反射技术的新超透镜方案，并建立麦克斯韦方程组在该条件下的新存在性、稳定性和紧致性结果，作者在无需去除局域奇点的情况下实现了亚波长成像。

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.

研究动机与目标

• 将超透镜框架从声学领域扩展到电磁学设置，利用互补介质实现。
• 解决负折射率材料中系数变号带来的数学挑战，包括椭圆性与紧致性的丧失。
• 在上述条件下，为麦克斯韦方程组建立新的存在性、稳定性和紧致性理论结果。
• 证明仅通过反射技术——无需奇点去除——即可在电磁学中成功实现超透镜成像。

提出的方法

• 采用受声学中互补介质研究启发的特殊超透镜方案类别。
• 利用反射型互补介质技术控制场的爆炸发散，同时在关注区域保持有界性。
• 提出新的数学框架以恢复紧致性，并确保在系数变号条件下解的存在性。
• 为所提出的互补介质配置下的麦克斯韦方程组建立稳定性估计。
• 利用反射技术规避对局域奇点去除的需求，简化证明结构。
• 借鉴先前关于负折射率材料与隐身技术的理论工具，并将其适配至电磁学情形。

实验结果

研究问题

• RQ1 能否在电磁学设置下，利用互补介质严格证明超透镜成像？
• RQ2 如何克服麦克斯韦方程组在系数变号条件下出现的椭圆性与紧致性丧失问题？
• RQ3 是否仅通过反射技术即可实现超透镜成像，而无需奇点去除？
• RQ4 在互补介质配置下，麦克斯韦方程组需要哪些新的存在性与稳定性结果？
• RQ5 在类似方案下，电磁系统与声学系统的数学性质如何比较？

主要发现

• 本文首次在电磁学设置下，通过互补介质建立了超透镜成像的严格证明。
• 证明仅通过反射技术即可有效处理椭圆性与紧致性的丧失，从而无需奇点去除。
• 推导出具有系数变号的麦克斯韦方程组的新存在性与稳定性结果，使共振行为的分析成为可能。
• 该框架成功抑制了某些区域的场爆炸，同时在其他区域保持场的有界性，从而实现亚波长成像。
• 结果证实互补介质可用于在电磁学中实现超分辨率成像，与先前声学结果类似。
• 研究表明，通过反射技术，互补介质下的电磁系统数学结构可被有效分析，从而拓展了变换光学的适用范围。