NEGATIVE REFRACTION

Refraction is one of the most fundamental phenomena in nature. It gives rise to such well-known effects as the apparent bending of objects partly immersed in water, rainbows, mirages, green flashes, and haloes. Refraction is also utilized in many existing optical instruments, including microscopes, telescopes, and eyeglasses. All these phenomena and applications rely on conventional or “positive” refraction. What would the world look like if the sign of refraction were reversed?

The law of refraction predicts that an electromagnetic wave, crossing the interface between two materials with refractive indices n1 and n2, changes its trajectory, depending on the difference in the refractive indices, such that n1 sin θ1 = n2 sin θ2, where θ1 and θ2 are the angles from the normal of the incident and refracted waves. The direction of the refracted wave depends on the sign of n2 (assuming n1> 0). The refraction is referred to as positive when n2 > 0 (Fig. 1a) and as negative when n2<0 (Fig. 1b). While positive refraction is a well-known phenomenon, a negative index of refraction leads to many unusual and often surprising effects. For example, Fig. 1c and d show calculated images of a metal rod in a glass filled with regular water and in a glass filled with negative-index water.

Left-handed world. The refractive index is one of the basic characteristics of electromagnetic wave propagation in continuous media. It is closely related to two fundamental physical parameters that characterize material properties, the dielectric permittivity ε and the magnetic permeabilityμ, through the equation n=± √εμ. While nearly all transparent conventional materials have positive ε and μ, corresponding to positive n, there are no fundamental physical reasons prohibiting materials from possessing simultaneously negative ε and μ, and as a result negative . Although not found in nature, such materials were recently created artificially and were named “metamaterials.”

A detailed theoretical study of electromagnetic wave propagation in materials with simultaneously negative ε and μ was performed by Victor Veselago in 1967. Maxwell’s equations, which relate the electric field E, the magnetic field H, and the wave vector k, predict that E, H, and k form a “left-handed” set and the sign of the refractive index is negative if both ε and μ are negative, and a “right-handed” set if both ε and μ are positive. The former class of materials is often referred to as left-handed materials or negative index materials (NIMs), while the latter class is referred to as right-handed materials or positive-index materials (PIMs). At the same time, the direction of the Poynting vector S, which defines the direction of the energy flow, is the same in positive-index and negative-index materials. Thus, the Poynting vector is antiparallel to the k-vector in negative-index materials and is parallel to the k-vector in positive-index materials. The opposite directionality of the k-vector and the Poynting vector is often taken as the most general definition of negative-index materials. Therefore, the negative refraction illustrated in Fig. 1 is a direct result of the opposite directionality of k and S and of the continuity of the tangential components of the wave vector at the interface between the two media.

Although the term “left-handed materials” was originally coined to describe materials with simultaneously negative ε, μ, and n, currently it is used in a broader context to include other optical structures that possess antiparallel k-vectors and Poynting vectors and support negative refraction. Examples of such materials include photonic crystals, anisotropic waveguides, organic and uniaxial gyrotropic crystals and a thin film on a metal substrate, and nanotransmission lines. Negative refraction has been demonstrated at microwave frequencies in a metamaterial wedge and in the visible frequency range at the interface between a bimetal Au-Si3N4-Ag waveguide and a conventional Ag-Si3N4-Ag slotwaveguide using plasmons. Negative refraction at optical frequencieswas demonstrated in photonic crystals. Although many unusual phenomena associated with the negative index of refraction can be observed in photonic crystals, the main limitation of photonic crystals is that the size of their characteristic features is comparable to the wavelength of light. On the contrary, optical metamaterals with feature sizes much smaller than the wavelength of light are predicted to enable many truly remarkable phenomena. However, currently optical metamaterials are available only in the form of subwavelength thin films, thus permitting the measurement of a phase advance but not of negative refraction per se.

Besides negative refraction, negative-index materials have been predicted to give rise to a wide variety of extraordinary linear and nonlinear optical phenomena, including reversed Cerenkov radiation, the reversed Doppler effect, backward phase-matched second-harmonic generation and optical parametric amplification, lasing without a cavity, bistability, and gap solitons in PIM-NIM couplers with no external feedback.

Superresolution: from “super” to “hyper” lens. A very unusual property of negative-index materials gives rise to the possibility of imaging using a flat slab of negative-index material with n = −1 surrounded by a conventional medium with n = 1. Moreover, under the appropriate conditions this slab not only focuses propagating field components but also recovers the evanescent field components, which decay exponentially with distance from the source (Fig. 2a), through the excitation of a plasmon resonance on the surfaces of the negative-index material. These evanescent field components, which are responsible for imaging of the high-frequency and correspondingly small-scale features of the object, cannot be restored by conventional lenses, inevitably limiting their resolution. Thus, at least in the ideal (lossless) case, an imaging system based on a slab of negative-index material, named a “superlens” by John B. Pendry, has the potential for significantly improving resolution in the image plane. Unfortunately, a superlensing effect is extremely sensitive to losses in the negative-index-material slab. While the superlens is likely to be useful in numerous nearfield applications, including biomedical imaging and nanolithography, superresolution in the far field is challenging.

Recently, a promising solution, a hyperlens, was proposed independently by Nader Engheta and Evgenii Narimanov. Instead of reamplifying and refocusing the evanescent field components as Pendry’s superlens does, a hyperlens converts those evanescent waves into propagating waves (Fig. 2b). Once all the components are propagating waves, they can easily be imaged by a conventional lens (microscope) in the far field (Fig. 2c). The only remaining limitation of a hyperlens is that the object plane must be situated very close to the hyperlens surface.

Optical metamaterials. Many of the predicted extraordinary properties of negative-index materials would not have been possible without rapid progress in the design and fabrication of optical metamaterials. As mentioned above, no materials in nature possess negative ε and μ in the same range of frequencies. While dielectric permittivity of some existing materials is negative at certain frequencies, no isotropic materials with negative μ are known. Moreover, magnetism is usually weak at optical frequencies so that μ ≈ 1.

On the contrary, metamaterials are built of artificial or “meta” atoms, which are resonant structures such as split-ring resonators and paired metal nanorods or nanostrips. The meta-atoms can be engineered and arranged such that their ε,μ, and n are positive, negative, or even zero at any selected frequency. The first optical metamaterials with a negative index of refraction have been demonstrated using paired nano rods, and independently by another group using paired dielectric voids in metal.

While these first experiments confirmed the possibility of the realization of a negative index of refraction at optical frequencies, the negative-index materials were realized only in the form of subwavelength films and possessed significant losses. Some essential requirements for practical negative-index-material designs include minimized losses or a large ratio of the real and imaginary parts of n, often taken as a figure of merit; a broad bandwidth over which both ε and μ are negative; optimized impedance matching; and realization of three-dimensional negative-index materials. Using a self-supporting fishnet structure consisting of rectangular dielectric voids in parallel metal films, a figure of merit of 3 has been demonstrated at a wavelength λ = 1.4 μm. This structure represents the current state-of-the-art for optical negative-index materials. Recently, the first three-layered negative index material with a figure of merit of 2.5 at λ =1.41 μm was also reported.

Refractive index engineering. While one of the original motivations behind the development of metamaterials was the demonstration of negative-index materials, metamaterial technology has stimulated rapid progress in an entirely new and exciting branch of modern optics, transformation optics, which is based on the idea of mapping a coordinate transformation to a set of material parameters, ε and μ. Metamaterials allow precise control over these material parameters and, more generally, enable refractive index engineering. Such mapping turned out to be particularly useful for cloaking applications and facilitated the first experimental demonstration of cloaking of a copper cylinder at microwave frequencies. In that experiment the object was concealed by a cylindrical metamaterial cloak built using splitring resonators. The coordinate transformation and a schematic of the first nonmagnetic cloak operating at optical frequencies, as proposed theoretically, are illustrated in Fig. 3a. Figure 3b shows the results of numerical simulations of cloaking of an ideal metallic cylinder. Currently, in both microwave and optical cloak designs, the effect has been achieved at only one frequency. Obviously, broadband cloaking would be desirable for most practical applications, and further research is therefore required.

Finally, cloaking is only one realization of the great potential of transformation optics in conjunction with metamaterials. Other promising applications include field concentrators and a variety of reflectionless devices. Metamaterials are bringing new degrees of freedom for designing structures with almost any desired optical properties, thus presenting enormous opportunities for a wide range of applications relying on refractive index engineering.

BIBLIOGRAPHY

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$ouradip Banerjee