# Negative Index Materials: Some Mathematical Perspectives

## Abstract

Negative index materials are artificial structures whose refractive index has a negative value over some frequency range. These materials were postulated and investigated theoretically by Veselago in 1964 and were confirmed experimentally by Shelby, Smith, and Schultz in 2001. New fabrication techniques now allow for the construction of negative index materials at scales that are interesting for applications, which has made them a very active topic of investigation. In this paper, we report various mathematical results on the properties of negative index materials and their applications. The topics discussed herein include superlensing using complementary media, cloaking using complementary media, cloaking an object via anomalous localized resonance, and the well-posedness and the finite speed propagation in media consisting of dispersive metamaterials. Some of the results have been refined and have simpler proofs than the original ones.

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## Notes

1. 1.

A0 plays the role of A in (1.2).

2. 2.

This inequality can be obtained from the following representation of v in $$B_{R_{3}} \setminus B_{R_{1}}$$:

$$v(r, \vartheta) = a_{0} + b_{0} \ln r + \sum\limits_{n = 1}^{\infty} \sum\limits_{\pm} (a_{n, \pm} r^{n} + b_{n, \pm} r^{-n} ) e^{\pm i n \vartheta} \text{ in} \ B_{R_{3}} \setminus B_{R_{1}}.$$

3. 3.

H− 1(Ω) denotes the dual space of$${H^{1}_{0}}({\Omega })$$.

4. 4.

Recall that χDdenotes the characteristic function of a subset D of$$\mathbb R^{d}$$.

5. 5.

Here $$\mathbb C$$ denotes the set of complex numbers.

6. 6.

Here ⋅ stands for the Euclidean scalar product in $$\mathbb C^{6}$$.

7. 7.

Here for a 3 × 3 matrix A, we denote A ≤ 0 if Axx ≤ 0 for all $$x \in \mathbb R^{3}$$.

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## Acknowledgements

This paper is an extended version of the lecture given by the author at VIASM annual meeting in 2017 at Vietnam Institute for Advanced Study in Mathematics. The author warmly thanks the institute for the hospitality.

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Correspondence to Hoai-Minh Nguyen.

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Lecture at the Annual Meeting 2017 of the Vietnam Institute for Advanced Study in Mathematics

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Nguyen, H. Negative Index Materials: Some Mathematical Perspectives. Acta Math Vietnam 44, 325–349 (2019). https://doi.org/10.1007/s40306-018-0258-z

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### Keywords

• Superlensing
• Cloaking
• Finite speed propagation
• Complementary media
• Negative index metamaterials.

• 35B34
• 35B35
• 35J05
• 35Q60.