Reflecting complementary media and superlensing using complementary media for electromagnetic waves

Negative index materials were first investigated theoretically by Veselago in [40] and were confirmed experimentally by Shelby, Smith, and Schultz in [39]. 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 [40]. The proof of superlensing using complementary media for arbitrary objects was given in [23] in the acoustic setting. The superlensing schemes used in [23] were guided by the notion of reflecting complementary media introduced and studied in [22] and the proof in [23] used the reflecting and the removing localized singularity techniques introduced in [22] and [23, 26] 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 [22, 23], 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 [22, 23], we establish new results on the compactness, existence, and stability for the Maxwell equations.