Metamaterials 9 J. Sun, N.M. Litchinitser University at Buffalo, The State University of New York, NY, USA 9.1 Introduction We know from basic physics and chemistry that all materials consist of atoms and mol- ecules. The atoms and their relative compositions determine many physical properties of matter, including its optical properties. For example, in metals, valence electrons are weakly bound to the nuclei; as a result, they can freely move inside of the material, which enables conduction. Another example is the property of birefringence in crys- tals, which originates from anisotropy in the binding forces between the atoms, form- ing a crystal. For thousands of years, people studied various materials to use them in a wide range of applications, spanning from architecture, transportation, and medicine to elec- tronics, optics, and space exploration. Eventually, we have reached a stage when natu- rally available materials can no longer meet the demands of fast-developing fundamental science and applications. For instance, the best resolution of a conven- tional optical microscope, which once revolutionized the entire field of natural sciences by allowing the visualization of microscale objects, is on the order of 100 nm because of the fundamental diffraction limit. Therefore, to image nanoscale struc- tures, one has to use electron-based microscopy that enables orders of magnitude higher resolution, but can only be used for conducting materials. The crucial ques- tion that scientists today are asking is whether it is possible to overcome the limi- tations imposed by the natural materials and components used in optical microscopy and develop an all-optical technique for nanoscale imaging. Although several near-field techniques have been developed for the point-by-point nanoscale imaging of planar structures, far-field, large-scale, high-resolution imaging remained a challenge until now. The emergence of metamaterials is likely to pro- vide a viable solution to this long-standing problem. Metamaterials are rationally designed artificial materials that gain their properties from their structures and carefully engineered unit cells (meta-atoms) rather than from the properties of their constitutive materials. In Greek language, meta means “beyond.” It should be noted that the optical properties of another class of engineered materialsdphotonic crystalsdare also somewhat defined by their structures. However, there is a fundamental difference between these two classes of engineered materials: the length scales of their unit cells. Although the period and the unit cell size of photonic crystals are comparable to the wavelength of light, the size of the meta-atom is much smaller than the wavelength of light. Thus, metamaterials can be considered as effective media, whereas photonic crystals cannot be. Fundamentals and Applications of Nanophotonics. http://dx.doi.org/10.1016/B978-1-78242-464-2.00009-9 Copyright © 2016 Elsevier Ltd. All rights reserved. 254 Fundamentals and Applications of Nanophotonics In the following sections, we will show several examples of metamaterials enabling very unusual electromagnetic (EM) properties and functionalities, such as negative, near-zero, and indefinite permittivity or permeability indices; backward waves and backward phase matching in nonlinear optics; imaging below the diffraction limit; perfect absorption; and cloaking. 9.2 Types of metamaterials The prediction of a possibility of negative refractive index and flat lenses based on negative index materials (NIMs) stimulated the development of the entire field of metamaterials. NIMs were first studied in detail by Russian physicist Victor Veselago in 1968 [1]. An important property of the wave propagation in NIMs is that wave and Poynting vectors (or phase and energy velocities) are antiparallel. Today, this property is often considered to be the most general definition of NIM. Using Maxwell’s equa- tions for the medium with negative dielectric permittivity and magnetic permeability (and, as a result, negative refractive index), Veselago predicted that the right-handed triplet of vectors E (electric field), H (magnetic field), and k (wave vector) in conven- tional, positive index material (PIM) changes to the left-handed triplet in NIM, leading to the negative refraction of a light beam [1]. In other words, when an EM wave prop- agates from PIMs to NIMs, negative refraction occurs at the interface, such that both incident and refracted waves are on the same side of the normal to the interface. To date, negative refraction was predicted and realized in three different physical systems: double-negative or “left-handed” metamaterials (with negative permittivity and perme- ability) [2e9], so-called hyperbolic metamaterials with anisotropic permittivity or permeability [10e13], and near the bandgap edge in photonic crystals [14e16]. 9.2.1 Double-negative metamaterials Following the original Veselago’s work [1], we consider EM wave propagation in NIMs using the set of Maxwell equations. We assume a plane wave with the frequency u is incident from air with refractive index n1 ¼ 1 onto the double-negative NIM with refractive index n2 < 0. The electric field E and magnetic field H can be described by E ¼ E0 exp½iðkr À utÞ and H ¼ H0 exp½iðkr À utÞ, respectively, where k is the wave vector. Substituting E and H fields into Maxwell’s equations and taking into account the constitutive relations um V  E ¼ i H; (9.1a) c uε V  H ¼i E; (9.1b) c D ¼ εE; (9.1c) B ¼ mH; (9.1d) Metamaterials 255 we obtain um k  E ¼ H; (9.2a) c uε k  H ¼ E; (9.2b) c where ε and m are relative dielectric permittivity and magnetic permeability, respec- tively. From Eqns (9.2a) and (9.2b), one can see that E, H, and k form a left-handed triplet when both ε and m are negative, which explains why the NIMs sometimes are referred to as “left-handed materials.” Because the Poynting vector is determined by S ¼ E  H*, the wave vector of the beam is unparallel to the direction of the Poynting vector in NIMs, which means that the directions of the phase velocity and the energy velocity are opposite to each other. ε The refractive index is the squarepffiffiffiffiffi root of the m and, in this case, we take the negative value of the square root n ¼ εm. According to Snell’s law, the refraction angle is also negative: sinqr ¼ sinqi/n, which means that the incident beam and the refracted beam are on the same side of the normal. fl Let uspffiffiffiffiffi brie y discuss why one has to choose the negative value of the square root in n ¼ εm when ε < 0 and m < 0. Assuming that both ε and m are complex numbers, that ε ¼ εr þ iεi, and that m ¼ mr þ imi, the refractive index can be written as n ¼ nr þ ik. Then, the relation between them should be: 2 ¼ð þ Þ2 ¼ðε þ ε Þð þ Þ n nr ik r i i mr imi 2 À 2 þ ¼ðε À ε Þþ ðε þ ε Þ nr k 2inrk rmr imi i rmi imr (9.3) 2 À 2 ¼ ε À ε nr k rmr imi ¼ ε þ ε 2nrk rmi imr where εr < 0, εi > 0 and mr < 0, mi > 0. Therefore, (εrmi þ εimr) is negative. For materials with no gain, the extinction value k should be positive. According to 2nrk ¼ εrmi þ εimr < 0, the value of n is negative under the condition of εr < 0, εi > 0, and mr < 0, mi > 0. The phenomenon of negative refraction can also be explained using equi-frequency contours (EFCs). Let us consider a transverse magnetic (TM) plane wave with frequency u and the wave vector k incident on an NIM with ε < 0 and m < 0, such that the wave vector k is along the zx-plane and the H field is along the y-axis. Accord- ing to Eqn (9.2b), the E field inside of the NIM is given by c E ¼ k H bu À k H bu exp i k bu x þ k bu z À ut ; (9.4) uε z 0 xx x 0 zz x xx z zz H ¼ H0buyy exp i kxbuxxx þ kzbuzzz À ut ; (9.5) 256 Fundamentals and Applications of Nanophotonics where buxx, buyy, and buzz are the unit vectors along the x-, y-, and z-axes, respectively. Then, substituting Eqns (9.4) and (9.5) into Eqn (9.2a), we obtain c k2H þ k2H bu exp i k bu x þ k bu z À ut uε z 0 x 0 yy x xx z zz um ¼ H bu exp i k bu x þ k bu z À ut (9.6) c 0 yy x xx z zz Finally, the dispersion relation for the EFC can be written as k2 k2 u2 z þ x ¼ (9.7) εm εm c2 The lateral component of the wave vector kz can be determined from Snell’s law ¼ ¼ as kz pkffiffiffiffiffii sinqi kr sinqr. Now, letpffiffiffiffiffi us compare wave propagation in a PIM (n2 ¼ εm) and an NIM (n2 ¼ εm). þ The wave vector kx in the PIM is given by vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u ! u 2 2 þ t u k k ¼ jn j À z : (9.8) x 2 c2 εm À The wave vector kx in the NIM is given by vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u ! u 2 2 À t u k þ k ¼jn j À z ¼k : (9.9) x 2 c2 εm x The refraction angle of the wave vector qr ¼ atan(kx/kz) in the NIM is negative À þ (qr ¼qr ) with respect to the PIM, as shown in Figure 9.1(a). In addition, the Àð þ; Þ wave vector corresponding to the refracted wave kr kx kz in the NIM bends to the opposite side of the normal to the interface, as compared with that in the PIM.