Liquid crystals, photonic crystals, metamaterials, and transformation optics.

R ecent advances of modern optics have been made possible with the development of unique materials, most notably liquid crystals (LCs), photonic crystals (PCs), and metamaterials (MMs). LCs have already completed a revolution in the way we present information nowadays, enabling an entire industry of flat panel liquid crystal displays (LCDs). PCs (1) and MMs (2) are further behind in terms of broad commercialization, but the change they produce in our understanding of how matter can control light is no less revolutionary, fueling dreams that only recently were science fiction, such as subwavelength imaging and focusing, invisibility cloaking, black hole-like light trapping, and more. PCs and MMs are formed by building units of a size s intermediate between the molecular scale m = (1–3) nm and the optical wavelength λ and cannot be simply synthesized as small organic molecules that form LCs. Design, manufacturing, and control of properties of PCs and MMs at the scale m < s ≤ λ is the major challenge. There are a variety of ways by which the LCs can help in the development of PCs and MMs; one of the latest advances is presented in PNAS by Ravnik et al. (3). A PC is a periodic structure that modulates the refractive index at the scale s ≈ λ. The structure is designed to form “photonic bandgaps” (i.e., a range of wave vectors for which light cannot propagate) (1). To provide the full bandgap in all directions, the PC should be 3D, which represents a challenge. In addition, the PC often needs to have structural defects, such as points (to trap light) or dislocations (to guide light). One of the approaches is self-assembly of small particles, typically spheres, from water solutions into 3D colloidal crystals. In the MM, an elementary “metaatom” is an artificial combination of dielectric and metal elements, such as split-ring resonators or paired metal nanorods, of the typical size s = (10–100) nm (2). Because s << λ, the closely packed metaatoms still form a “material” that is continuous at the scale λ, and thus can be characterized by an effective refractive index n. Combining a noble metal with a large negative real part of the dielectric permittivity and a regular dielectric of positive permittivity, one can achieve practically any value of n, including n < 0 (2). The MMs are produced mostly by top-down techniques, such as electron beam lithography. Because s is submicron, a large MM is hard to manufacture. Another challenge is to make the MM reconfigurable/switchable. A still higher level of complexity is required to produce MMs and PCs for transformation optics applications, in which the density, shape, and arrangements of s-elements vary from point to point over the scales much larger than λ to produce a variable n (4). In LCs, the small rod-like molecules are free to move around as in a regular fluid, but they remain locally parallel to each other, thus establishing an orientational order along a nonpolar axis b n 1⁄4 − bn called the director. In the simplest case of the so-called “nematic” LC, there is no other type of long-range order and bn is the optic axis. Because of the anisotropy of dielectric permittivity at low and high (optical) frequencies, the nematic LCs are widely used as tunable component infiltrating PCs (5, 6) and MMs (7). The degree and direction of the orientational order of the nematic filler can be controlled by a variety of means (temperature, pressure, and electromagnetic fields), thus allowing one to tune the PC and MM hosts dynamically. In addition to the simple nematic structure, the rich world of LCs offers phases with spatial modulation of density and molecular orientation, periodic in one (smectics and cholesterics), two (columnar phases), or three (blue and cubic phases) dimensions. The period is often in the range of 10 nm to 1 μm relevant to PCs and MMs. In particular, the LC blue phases with a period p ∼ 10 nm represent a self-assembled 3D photonic bandgap material by themselves (8, 9). The blue phases are formed by chiral molecules. Chirality forces the neighboring LC molecules to be slightly twisted with respect to each other. Let b n0 be a local director (Fig. 1A). Moving away from