Chapter 11

Liquid Crystals in Metamaterials

Augustine M. Urbas and Dean P. Brown

Materials and Manufacturing Directorate, Air Force Research Laboratory WPAFB, OH, USA

11.1 Introduction

Poised to deliver advances in electromagnetic technologies, the field of metamaterials is rapidly expanding the range of electromagnetic properties available in materials by leveraging structured composites to generate targeted response outside of what is available in conventional materials. While novel and useful electromagnetic properties are the genesis of metamaterials research [1], the field has expanded to include acoustic/mechanical systems [2] and thermal materials [3, 4] as well. Structured electromagnetic materials have a long history in research [5] and this latest incarnation was spurred by the work of Pendry [6] and Smith et al. [7] on creating materials with artificial negative refractive index to demonstrate phenomena predicted by Veselago [8] decades ago. Many excellent references for a general understanding of metamaterials are available that describe the physics and research of these interesting and potentially useful systems in great detail and we direct readers to these references for background information in order to better cover the topic at hand. In short, metamaterial systems typically rely on sub-wavelength features or inclusions, sometimes called meta-atoms, with a structurally defined response, such as a shaped metallic or dielectric inclusion that possesses a resonance, which supplant the fundamental response of the atoms and molecules of the medium. Combined with secondary ordering which modifies the individual properties of the meta-atoms and introduces collective behavior, this meso-scale, local response to incident fields gives rise to a new effective bulk response of the system. When carefully composed, such systems can yield the target electromagnetic materials properties combinations, which are uniquely suited for a specific application. The most illustrative example of a metamaterial, and one of the first things to be called so, is the split ring resonator array. These can be simply a cubic lattice of metallic rings with a cut or gap in each ring. These arrays of shaped metal inclusions used to generate artificial magnetic media even though they are composed of non-magnetic component materials [9, 10]. The overall utility of metamaterials is enhanced by the ease with which gradients can be made by simple structural changes to the individual meta-atoms. This attribute is a fundamental enabler of the co-rising field of transformation optics which aims to change the way in which electromagnetic devices are conceived [11, 12]. Metamaterials also provide unique opportunities for tunable, active, and responsive materials since the fundamental unit, the meta-atom, can be loaded with small, dynamic components [13] or materials [14, 15] in order to change the overall system response. Indeed, the use of liquid crystals has been frequently suggested as a mode of tuning metamaterial response and key demonstrations have already shown significant potential in this area. Tunable metamaterials derived in this way will enable the design of highly nonlinear and dramatically responsive systems that are not possible in conventional materials and they will enable reconfigurable and dynamic antenna and aperture systems for optical and radio frequency (RF) applications.

11.2 Metamaterials Background

Simple metamaterials are composed of meta-atoms or unit cells, such as split rings or metallic wires, in a periodic array. The properties of the meta-atoms define the overall response of the material. In order to modify both permeability (μ = H/B) and permittivity (ε = E/D), susceptibilities for both magnetic and electric fields must be inherent to the meta-atom and material structure. In a basic example, a wire array and a split ring resonator array, both structured to give negative values at the same frequency of the permittivity and permeability, respectively, are combined to yield a negative index [10]. We will look closer at these simple independent media to see the key characteristics and how their properties can be tuned by the incorporation and control of LCs. Wire arrays have long been used as artificial dielectrics for RF frequencies [16]. A simple example is composed of long wires on a regular square lattice. The combination of wire size and spacing tunes both the density and effective mass of electrons in the medium. This in turn defines the new plasma frequency of the wire medium, creating an artificial response for a range of frequencies analogous to a Drude metal with the effective plasma frequency as shown in the equation below, for incident waves with electric field polarized along the wires [17].

(11.1)

A more characteristic example within metamaterials is that of the split ring resonator (SRR), initially proposed as a component of a negative index material by Smith et al [10]. A wide range of designs for resonant inclusions have since been explored, each having unique characteristic and advantages. In many cases, these can still be understood as one or more modified SRR units. In its simplest form the SRR is simply a cut loop as shown in Figure 11.1. As an inclusion, it can be understood as a free standing resonant circuit, by considering the gap as a capacitor with its plates shorted by the inductive ring. The effective inductance and capacitance of this structure combine to give its resonant frequency. The SRR couples to the magnetic component of an incident wave polarized along the axis of the ring. This induces current in the ring and drives oscillating fields whose magnitude depends on the incident frequency relative to the resonance of the ring giving the SRR an effective magnetic susceptibility. In a medium or array as shown in Figure 11.2, the response or effective permeability for a wave propagating through a medium composed of layers stacked out of the page with the magnetic field polarized along the axis of the ring is determined by the fill factor and geometric properties of the split ring units. The effective permeability of the SRR array can be written as:

(11.2)

after the notation of Smith et al. [10] and Pendry et al. [18] where f = πr²/a² is the fill fraction of rings, ω is the incident frequency and ω₀ = 1/LC is the resonant frequency of the SRR unit. Coupling between neighboring meta-atoms [19, 20] and the spatial dispersion [21] play a significant role in the effective response and are widely discussed in the context of metamaterials. The resonant properties of the unit cell or meta-atom are also defined by the local dielectric or magnetic environment of the inclusion. In the split ring unit, for example, the effective value of the capacitive part of the structure is determined by the gap dielectric. If it is filled with a liquid crystal material (LC material, not to be confused with the product of inductance and capacitance), then the resonant frequency of the SRR units is determined by the effective dielectric properties of the LC material and this in turn changes the effective medium properties of the array. This mechanism allows the overall structure to be sensitive to order and orientation of the LC material, or the properties of any material, within the gap which becomes the basis for tunable properties in metamaterials. Initial computational explorations and successful experimental demonstrations have been made specifically addressing liquid crystalline materials as the dynamic component in metamaterials systems.

Metamaterials were initially explored as effective bulk media where extended systems, built up from multiple layers of two dimensionally or three dimensionally patterned systems, are composed so as to deliver a new effective bulk response. This is an elusive goal and typically single effective media values for permeability, permittivity, or index can only be used to describe the behavior of these complex systems over some frequency range, incident angle range or under other specific conditions. Apart from effective bulk systems, in many applications, only a surface is needed to achieve the desired materials response [22]. This is clearly the case in frequency selective surfaces [23] or artificial magnetic conductors [24], which are typically planar periodic arrays of metallic features sometimes paired with and/or connected to an underlying ground plane. These planar systems have been incorporated into the larger understanding of metamaterials. It is no longer sensible to ascribe an effective bulk permittivity or permeability to this single layer, but the systems give useful response for some applications and in particular allow for the easy integration of liquid crystals as tunable media [25].

11.3 RF LC Metamaterials

Several key demonstrations of tunable RF metamaterials have been accomplished through the introduction of liquid crystals into typical RF metamaterials structures. A notable study demonstrated tunable magnetic permeability and in particular negative permeability in a split ring resonator type composite [26] and was followed up with tunable negative index in an Ω particle system [27] and fishnet type structures [28]. For more background on the Ω metamaterials see Simovski and He [29] or Huangfu et al. [30]. Tunable metamaterials at RF frequencies have been demonstrated in other ways, as well, relying on electronic components such as varactors (voltage tunable capacitors), [13, 31, 32] to change the resonant frequency of meta-atoms. Liquid crystals represent simple integration potential, however, and are beginning to be explored for RF applications [33, 34]. In several liquid crystal-based studies, standard metamaterial structures with resonant frequencies in the 10 s–100 GHz range were fabricated with alternating patterned metallic split rings or omega particles on Teflon substrates, which were sandwiched with spacers and infiltrated with liquid crystal materials. This has become a typical LC tunable metamaterial configuration due to the large field concentration between the closely spaced complementary patterns, ability to apply alignment fields and the convenience for introducing liquid crystal in a confined cell. As an example, the experimental structure implemented with split rings is by Zhang et al. [26] shown in Figure 11.3. The complementary pairs of split ring resonators are sandwiched facing one another with a layer of oriented liquid crystal between them in a two-dimensional array of standing pairs. The composite was aligned so that pairs of meta-atoms overlaid one another and were separated by a thin layer of liquid crystal as shown in the diagram. In this configuration the fields coupling between the two complementary resonant elements are presumed to be largely perpendicular to their faces and most intense in the liquid crystal [35]. The orientation of the LC material was controlled with externally applied magnetic fields from permanent magnets. Using this scheme, Zhang et al. were able to provide two distinct orientations to the LC material, with the director either parallel or perpendicular to the planar meta-atoms. This presented a shift between effectively high and effectively low dielectric environments for the electric field coupling of the complementary structures. The coupling of the complementary rings is sensitive to the effective dielectric properties of the LC material between them which range from a low permittivity of 2.25 to a high permittivity of 2.82. Changes in the orientation of the LC director are shown to change the coupling of the rings and modify their effective resonant frequency. As is shown in Figure 11.4, a change of only a few percent in the center frequency of the resonance was experimentally demonstrated when compared with the roughly 20% shift of the dielectric properties of the LC. It is worth noting that the relative volume occupied by the LC was 1/3 of the overall unit cell. As shown above, the effective permittivity at a frequency depends on the separation from the resonance frequency. The shift in resonant frequency from the applied field provides a mechanism for tuning the effective properties of this material. Zhang et al. claimed to have tuned the band of negative permittivity. Though not explicitly shown in the report, it would have occurred very near resonance as was shown in an earlier report from the same group. In that study, a concentric complementary SRR structure exhibited tunable negative permeability but only over a very small frequency range [36]. These reports demonstrate uniform tuning of effective properties. The systems were, however, composed of only one or two layers of a two dimensional array. The use of magnetic fields, while effective, is likely to be impractical for larger systems and in many applications. Moreover, individually addressed, tunable unit cells may ultimately be more useful especially when considered for implementing dynamic transformation optics devices [37, 38].

11.4 RF Tunable “Meta-Surfaces” with LCs

Significant effort has been devoted to investigating tunable meta-surfaces for possible applications and tunable surfaces have received much of that focus. In addition to their inherent utility, this work can provide a “look ahead” at strategies and effects for bulk metamaterials systems. While, in general, liquid crystals have not received much attention in RF applications because of significant losses and response time limitations when compared with using electronic components for dynamic systems, they present a relatively simple and effective route for tunable systems and are being increasingly explored for components and tunable systems [39–41]. The advantages may be specifically suited to large area structures and for higher frequency applications (above 20 GHz) where electronic components become costly and performance is limited. This has made meta-surface-based studies a venue where useful demonstrations of LC tunable RF metamaterial based components and devices have been shown. Initial studies on tunable frequency selective surfaces (FSS) exhibited voltage tunable properties [42, 43]. The frequency selective surfaces used were composed of paired metal films with complementary or matching patterns, such as hole arrays to give resonant transmission or absorption windows at RF frequencies. The paired films were separated by a narrow cavity (~100 μm) that could be filled with LC. In general, FSSs have strong frequency dependent properties that vary considerably over spectral bands. The shape and size of the holes and properties of the surrounding dielectric determine the resonant frequency where transmission occurs. For paired films, coupling effects can also influence the overall properties. In an early study [42], paired FSS slot arrays were sandwiched around an LC cell containing BL037. The continuous FSSs were used as bias electrodes for the LC. When the LC director orientation was changed with applied voltage, the frequency of transmission shifted significantly (about 10%), which was consistent with predictions based on the expected properties of BL037. It is not clear from the reported analysis if the shift in resonant frequency was simply related to the change in local dielectric around individual holes and if a change in coupling between films also played a role.

A subsequent experiment by Wenfei et al. [44, 45] utilized a more subtle aspect of meta-surfaces to create a tunable reflector. The meta-surface in this case was an array of square metal patches on a square two-dimensional lattice spaced over a metal ground plane as shown in Figure 11.5. In this configuration, isolated metal patches act as capacitive elements above the ground plane and possess a resonance based on their area or size. In this case, the patches were connected with thin bias lines to allow for electric field control of the liquid crystal filled between the upper metal film and lower ground plane, but minimally affected the RF properties. Applying a bias field changes the director orientation between the patch array and ground plane, as in a typical liquid crystal cell for display applications. This in turn tunes the effective capacitance of the patch and thereby its resonant frequency. The useful aspect of this, however, is that waves reflected from the surface acquire a phase shift. The phase shift of a given frequency depends on its displacement from the resonant frequency. By tuning the resonant frequency of the structure, the phase shift of the incident wave is changed. In this study, the patches were all biased in common and a single phase shift for the array was measured in a plane wave reflected from the surface. Phase shifts of nearly 165° and 130° were possible in regions of relatively low loss at 102 GHz and 130 GHz for separate structures. The same strategy was again used by Wenfei et al. [46], with a patterned array of diverse, different sized patches to generate a switchable beam pattern. Phase front synthesis requires more dynamically tunable phase control as shown in an example by Gaebler et al. [47]. Their demonstrations of a horn fed reflectarray based on a similar LC tunable meta-surface approach, where individual lines could be biased and subsequently, individual patches using a strategy similar to active LC displays, provided continuous scanning over ±25° along one plane for the line addressed and over a comparable cone angle for the fully addressable system with a bias of less than 15 V. It is important to point out that scanning is accomplished by the introduction of a predefined phase pattern, not simply by increasing bias voltage. More complex patterns can accomplish beam forming and other functions and require more precise control of phase fronts. Individual control of patches and sufficient area allow for complete control of beam shape and direction and ultimately could provide the significant utility, albeit with additional complexity. Losses from the LC material are comparable to those from scattering and dissipation in the metal structures. Significant power performance gains could be made with RF compatible, low loss LCs in this frequency range. These tunable surface arrays have been proposed as satellite-based antennas for RF communications [48]. The combination of large area, light weight and low operating power make them potentially appealing for this application.

11.5 LC Tuning of Meta-Atoms

The inherent leveraging of changes of a small volume of material to influence the effective bulk response of a metamaterial is a key characteristic that makes the intersection of metamaterials and liquid crystals intriguing and promising for research. Liquid crystals, of course, are an archetypal class of tunable materials which have been widely employed in changing the optical properties of devices, typically in displays and filters. The range that the properties of an LC system can be tuned over and the ease with which this is done by either temperature or applied fields, make them a material system of choice for devices and demonstrations of dynamic systems. In general, the bulk changes of dielectric properties that liquid crystals exhibit, either through phase changes (say from isotropic to nematic or between ordered phases) or by director orientation changes, such as from an applied field, are used directly. For example, in a liquid crystal display, polarization rotation is accomplished by a bulk layer of LC with appropriate alignment layers to give a texture through the layer which acts as an optical retarder. In essence, the whole optical path interacts with the LC in order for this device to function. In contrast, for a metamaterial system, the response of the system, e.g. of split rings, is determined by the resonant properties of the metallic meta-atom. For example, the split ring resonator we discussed before, this can be described as a resonant circuit with an inductive component (the ring) and a capacitive component (the split). The value of the capacitance is influenced by the dielectric properties of the material in and around the split. This has been studied in detail by Kowerdziej and coworkers [48–50] and by Zhang and coworkers [35]. Each has performed numerical analyses of different meta-atom designs that were loaded with LCs. The response of the system for different director orientations and as a function of the dielectric properties of the LC with temperature was predicted. Changes in resonance frequency, refractive index, and loss in bulk materials made from these unit cells were also done. Zhang et al. showed how the local fields within the meta-atom required that an accurate model of the dielectric anisotropy and director orientation be used to predict accurate properties. Overall, significant shifts in resonant frequency and effective properties were shown, providing some promise for applications.

The studies of Zhang et al. and Kowerdziej et al. favored complementary meta-atom configurations, similar to those favored in experimental studies. Design studies aimed at enhancing the sensitivity of metamaterial structures to LC loading are still to be done. A simple series of calculations can show the sensitivity of a standard SRR to changes in LC orientation. The SRR design in Figure 11.6 was used for this purpose. A series of calculations was made with the structure immersed in BL006 (experimental data from Utsumi et al. [51] was used) with the director oriented perpendicular to the plane of the ring, along the y-axis. The incident wave was propagated in the z direction with the electric field polarized in the x direction and, as shown in Figure 11.7, the ring was found to have a resonance at 6.8 GHz. A comparison was made between this LC configuration and the LC director changed to the z orientation where the resonant frequency shifted minimally to ~5 MHz higher frequency. In order to contrast this with the sensitivity of the structure to changes in the gap region alone, a calculation was made changing only the gap LC director to the z direction while leaving the ring immersed in the y oriented LC matrix. Interestingly, the resonance shifted to a significantly lower frequency of 6.73 GHz even though a much smaller volume of material was changed. Due to the complex fields surrounding the SRR structure, however, this counterintuitive behavior is not surprising. This shows the tremendous sensitivity to the gap dielectric that can be engineered into such systems, but clearly some care has to be taken to design the systems appropriately.

If we compare the volumetric sensitivities exhibited by the two cases in terms of frequency shift per unit volume-refractive index change there are six orders of magnitude difference between the gap only and all LC case, being 25 GHz/mm³-RIU and 0.06 MHz/mm³-RIU, respectively. Another way to think about this series of calculations is to consider the change in response of the resonant meta-atom to the change in LC director in the gap alone is comparable to the impact of the change due to the LC director orientation everywhere else (though the shifts counteract one another). Clearly there are significant advancements in unit cell design and polling field configuration possible to increase the tuning range and decrease potential losses of LC loaded cells. This may enhance the potential for more dramatic tuning and lower power requirements for tunable metamaterials compared with conventional bulk materials. The indirect loading of the SRR by the LC, which also can be understood as the coupling of the incident field to the LC via the metallic structure, is responsible for the overall change in effective response of the metamaterial. Although liquid crystals provide easy access to highly tunable properties and may be a system of choice for exploration of tunable properties in research and applications, there is nothing unique to LCs about the observed effects and the results are relevant to any tunable material used with a metamaterial. Indeed, in many meta-systems, the influence of the electromagnetic properties of a relatively small volume in the vicinity of the metallic or dielectric structure on the overall response of the structure and therefore the effective response of the composite metamaterial has been the proposed basis for sensor platforms [52], structural evaluations [53], highly nonlinear materials [54, 55], in addition to, tunable systems.

11.6 Optical Metamaterials with LCs

The bulk of research in metamaterials falls in the RF frequency and LC metamaterials are no exception, although in both cases optical frequencies are actively and increasingly explored. Initial LC metamaterials work was mainly in the optical regime, an excellent exploration of which can be found here. In general, however, metamaterials has followed the opposite course in research with initial demonstrations in RF followed by other frequency regimes, typically THz, infrared and then optical. As the frequency of the EM excitation increases, the character of the interaction between the structure and the electromagnetic field becomes more plasmonic in nature. Plasmons are collective oscillations of electron polarization at the interface between a metal and a dielectric [56]. Sub-wavelength metallic structures can exhibit localized surface plasmons at optical frequencies [57]. Plasmonic systems, which can still be effectively modeled with equivalent circuits to some extent, have been the basis for resonant metamaterials systems at optical frequencies and many key demonstrations, such as optical negative index, have been executed [58]. It is interesting to note the scaling and change of meta-atom structure and configuration as higher frequencies are explored. As we discussed above, the resonances within metamaterials can be modeled with equivalent circuits where structural inductances and capacitances are represented by the discrete equivalent circuit elements. The scaling with size of these two quantities and constituent materials properties is different as is highlighted by Soukoulis et al. [59]. At optical frequencies the structures used for negative index are modified to provide the correct balance of inductance and capacitance for the magnetic component and provide the effective permittivity response as well. One of the more popular configurations is the so called fishnet structure composed of a perforated multilayer, metal/dielectric film. The fishnet uniquely combines an electric and magnetic resonance in different structural elements at comparable frequencies as described by Zhang [60]. This structure has been widely explored as a negative index material over a range of frequencies [61]. Tunable fishnet structures for optical devices and modulators have been a long running goal of metamaterials research as well. Indeed, the infiltration of a fishnet structure with liquid crystal to provide tunable response was proposed shortly after the initial publication of the fishnet structure itself [25, 62, 63].

Liquid crystals have almost immediately been incorporated into materials at these higher frequencies in order to demonstrate tunable response due to the lack of other options and characterization difficulties of alternatives. The application of liquid crystals to optical metamaterials has been recently addressed by Diaz and Khoo [64], including excellent coverage of the underlying physics and phenomena. There are new and different challenges that arise here, however. They have both to do with the relatively small feature size of the metamaterials subunits at optical frequencies and with the difficulty in controlling the liquid crystal in regions where its orientation will alter the response of the overall metamaterials system. Bulk optical metamaterials are elusive due to loss and fabrication challenges. In thin or single layer metamaterials systems thermal response (tuning) has been demonstrated in a meta-magnetic system [65] and proposed in various configurations in an LC infiltrated fishnet structure. Electrical response has, to date, only been shown in single layer, ‘meta-surface’ configurations [66]. There is a substantial body of work on the interaction between LC's and plasmonic systems, which will be relevant in analyzing the current results and identifying future challenges.

A comparable study to the RF work above was explored, where LC was used in conjunction with a magnetic resonant grating to provide a tunable magnetic metamaterial, a first step to tunable negative index. This grating structure is, in essence, analogous to the split ring component of a negative index metamaterial at RF frequencies and the study presented by Xiao et al. shows a clear parallel to this work. In brief, the structure in Figure 11.8 is composed of grating lines made of multilayered metal and dielectric materials. There are two metal films within each line separated vertically by a dielectric and capped with additional dielectric layers. For TM illumination, the magnetic component of the incident field excites circulating, anti-parallel currents in the metal films within each stripe yielding a magnetic resonance. A more detailed discussion can be found in the work of Cai et al. [67]. In the study by Xiao et al., the grating was infiltrated with LC and capped in a cell by a superstrate that had an applied PMMA alignment layer, as is shown in the diagram. The alignment layer had been rubbed so that the LC director would be aligned perpendicular to the grating lines, that is, the incident electric field was primarily influenced by the extraordinary permittivity of the LC medium. The cell construction did not include provision for the application of field and so the only method of tuning the LC properties available was through temperature. It is likely that in the confined geometry, the local field profile and the ultimate effect on the director orientation field of the infiltrated LC from the applied field would be complex given the metal–dielectric composition of the film. The change in optical properties would be sensitive to this subtle configuration, but it is not intuitively clear what impact that would have on the properties. With the simpler temperature-based tuning of LC order, however, the structure did show a significant shift of the magnetic resonance center wavelength, about 50 nm shift in the center frequency, as is shown in Figure 11.9, as the sample was raised above the LC clearing temperature. The modeled results showed that the change was consistent with a change in index of 0.15 between the two states that is roughly consistent with the expected response of the structure and the 5CB liquid crystal used in the study. The authors, however, find that the effective extraordinary index of the LC appears to be lower than the literature values. This is attributed to air inclusions within the sample due to the complex structure. As noted above, in such a confined system there is likely to be a complex LC texture due to the structural influences and chemical anchoring changes influencing the LC director field. It is quite possible that the low effective index observed for the nematic state was caused by this director field which may have provided a more complex local orientation and distribution of the permittivity as it interacts with the plasmonic structure of the meta-magnetic grating than was accounted for in the model.

11.7 LC Interaction with Plasmonic Metamaterial Structures

Dynamic plasmonic structures based on the interaction with LCs are a research field of their own that often crosses technically with optical metamaterials. There are a few interesting observations that can be made which highlight challenges that dynamic optical metamaterials using liquid crystals face. Many studies of changes in optical properties of plasmonic systems in response to liquid crystal order and orientation changes have been published. It is interesting to note the complex behavior observed in studies of single plasmonic particles and arrays. While switching of optical properties, typically observed as a change in the peak of the plasmon resonance, is observed roughly consistent with expectations based on the anisotropy of the LC, there are significant enhancements to shifts that have been attributed to field induced density fluctuations in the LC medium. A notable and early study by Muller et al. [68] explored the effect on an isolated spherical gold nanoparticle of immersion in LC and subsequent poling of the LC with an applied electric field. The shift in wavelength of the plasmon resonance peak for two orthogonal polarizations was shown to be in opposite directions, contrary to expectations. This is evidence of more complex field induced director textures around the particle or, as the authors suggest, possible density fluctuations induced by the field. A subsequent study by Chu et al. [69] on nanorods prepared in a similar fashion showed a clearer trend. In this study, while there was a clear difference between polarizations before applying the field, that is a dramatic splitting in the plasmons excited by two incident polarizations was observed [70, 71], which has been attributed to LC texture at the surface of the plasmonic particle. When field was applied, this splitting was dramatically reduced. The authors attributed this to possible density fluctuations, as Muller had, especially the pronounced red-shift of the cross polarized transverse mode.

In the simple model of liquid crystal orientation under applied field, this mode should only feel the low index in both the presence and absence of a poling field, as described by the authors. The reduced splitting in this and Muller's study can, however, be explained by the applied field acting to reduce anisotropy in the surface orientation of LC molecules close to the particle. Near the particle, a substantial component of the local electric field would be perpendicular to the particle surface as indicated in Figure 11.10 by the equipotentials curving around the central gold nanorod. This would direct the near surface orientation of LC director as illustrated by the dotted lines. In Chu's study, the LC near the surface of the particle would have the extraordinary axis oriented perpendicular to the surface. This would remove anisotropy that yields splitting in the plasmon modes. Notably, for the cross polarized transverse mode, it would result in a switch from the low index of the LC medium to the high index, being the dominant dielectric component. It is important to point out that at optical frequencies, the field associated with the surface plasmon decays rapidly away from the interface [72, 73], on the order of the particle size, and that surface orientation of LC molecules around plasmonic particles and resulting optical properties will dominate over bulk orientation. In the liquid crystal, the surface ordering effect extends into the bulk some distance related to the elastic properties of the LC medium and the surface interaction strength, as well as, the direction of electric fields at the particle surface. The complex interplay between surface ordering, orientation, and fields in and around particles is a rich area of study which may prove essentially enabling to optical metamaterials incorporating LCs. The complex interaction of surfaces and fields in the highly structured metallic systems that compose optical metamaterials will provide an interesting challenge for LC tuned plasmonic structures. The high optical field region close to the metallic inclusion is also the region where the LC is likely to see significantly altered applied poling fields and feel the most influence from surface ordering effects. This becomes particularly interesting in systems where the electromagnetic coupling between adjacent particles is mediated by an LC medium separating them. The small dimensions and confinement of LC systems has been studied for the variety of novel textures produced in such geometries. The complex dielectric textures created would certainly influence the coupling between plasmonic meta-atoms, for example, which may play a significant role in novel electromagnetically induced transparency (EIT) or Fano-type structures [74].

11.8 Liquid Crystals in Self-Assembled Metamaterials

Studies of tunable magnetic gratings and the RF SRR arrays represent the most direct intersection between metamaterials as they are basically understood and liquid crystals. This simple crossing of utilities can provide significant potential for applications, but there are also novel approaches that leverage LC properties to uniquely exploit metamaterials concepts. Liquid crystals as dynamic materials are only one attribute that has been widely exploited in research and technology. Another is their use a structuring agent, or a structured system. Of course, their diverse and tunable optical and dielectric properties are directly resulting from the microstructures adopted by the LC molecules, but in some cases, the structure is directly relied on or used as a host to template a guest material. Two key examples are cholesteric LC Bragg reflectors and nano/micro particle template systems. In each case, secondary order arising from the LC texture generates longer length scale patterns. The LC is a critical component for providing the alignment or ordering forces, and the optical properties are secondary. Here too lies tremendous potential for liquid crystals in metamaterial systems. Since metamaterials typically rely on subwavelength inclusions ordered on a length scale also smaller than a wavelength, fabrication becomes a challenge as optical metamaterials move to shorter wavelengths and higher frequencies. Indeed, visible wavelength systems in this regime require inclusions of tens of nm in size spaced at only several times their size (often, very small gaps or separations are required between or inside of unit cells, as well). This becomes a fabrication challenge even under the best circumstances, and large area, three-dimensional patterning is beyond the reach of conventional techniques. LCs have potential for providing self-assembled templates for optical metamaterials in this range, where they could be used to structure functional inclusions in order to realize designed materials and their effective medium response.

Research directed at metamaterials fabricated though LC-mediated self-assembly is emerging rapidly to address the challenges of large area fabrication and small dimensions [1]. A wide range of self-assembly approaches are being considered and liquid crystals have been studied as hosts for structured effective media for some time. In contrast to their typical use for anisotropic electromagnetic and optical properties, in this venue, it is the mechanical and elastic anisotropies that are ultilized. Recently, results on the creation of ordered arrays of nanoparticles facilitated by liquid crystalline surface coatings have shown significant progress toward fabricating artificial dielectric materials at optical frequencies. Draper et al. [75] have recently published a study of inherently liquid crystalline gold nanoparticle materials which incorporate polar mesogenic corona molecules. The spherical particles adopt tactoidal shapes as the LC corona, or surface layer orders around hard central core. Potential for novel mesophase formation in neat material and blends is also discussed. These phases would order the gold nanoparticle cores with oriented and ordered LC phases separating them as shown in Figure 11.11. Significant potential for artificial dielectric properties at optical frequencies is possible via this route. Further, additional work on incorporating magnetic particles yields potential for novel magnetic materials following similar strategies. There is precedence for exploring these structures and initial experimental work on optical properties derived from structured nanoparticle composites. Khatua et al. [76] created closely packed structures from LC-coated gold nanoparticles. The ordered nanoparticles showed a significant, though small, shift in the plasmonic resonance between the nematic and isotropic states of the surrounding LC. Other types of dynamic response from liquid crystal systems, such as photochromism [77], could also be employed in these systems. It is important to note that for this system, both the structuring and tunability were resulting from the LC corona/matrix. The change in plasmon resonance would translate into a shift in the effective properties of the system, though the authors of the study did not analyze the systems as such, others have proposed this [78, 79]. Following work on dipole arrays, the effective properties of this system could be extracted [80]. Significantly, no features indicating collective oscillation or coupling between particles were seen in contrast to a study by Augui and Barnes of lithographically fabricated nanorods [81] likely due to polydispersity in the particles and disorder. In addition, this approach has been applied to magnetic nanoparticle cores as well, which raises the potential of optically magnetic or magnetically tunable optical materials [82]. There are key observations needed in order demonstrate the capability to fabricate desired effective properties, but the potential is clear in these initial studies.

11.9 Chiral Metamaterials

Novel properties in meta-systems can also arise from extremely subwavelength structure in systems without a specific need for order. In particular, chiral metamaterials are an example where molecular scale chirality can influence the effective index of a material for a polarization and even reduce it below unity [83, 84]. In this case the molecular structure imparting the extreme chiral response can be λ/200 or smaller. Again, fabrication challenges for plasmonic chiral metamaterials at visible frequencies arise. Quantum chemical design efforts have yielded progress toward making extremely chiral systems for visible wavelengths. Molecular systems have the potential to provide low loss chiral subunits that can exhibit negative or extreme values of refractive index for one circular polarization [85]. Indeed cholesteric liquid crystal materials exhibit chiral structures, through the optical activity and dielectric anisotropy is too low to yield negative refractive index. However, the coupling of self-assembled liquid crystal based systems and plasmonic particles to make chiral metamaterials is a natural direction for research. Following the strategies above used by Khatua et al. [76], Draper et al. [75], and Demortière et al. [82] in combination with anisotropic particles and chiral host materials is a path with clear potential.

11.10 Conclusion Outlook

While we have identified LCs as having some potential in tunable metamaterials and as a method of fabricating or structuring meta-systems, these approaches are just beginning to be explored. In addition, there is clearly a role which overlaps these two functions [78]. Liquid crystal components of meta-systems could guide assembly of meta-atoms yielding the desired effective response and provide a mechanism by which field tuning, optical response or other dynamic properties are possible. In this chapter, we explored current efforts at the intersection of LCs and metamaterials in the categories outlined above and looked forward to the potential in a variety of areas that have yet to be explored, or for which only basic results have been shown. Our goal in this work is to show were potential has been realized, where new topics exist and what fundamental limitations may play a role in the performance and utility of LC metamaterials. By reviewing current work, it is clear that the intersection of metamaterials and liquid crystals offers a variety of applications in dynamic electromagnetic properties of structured materials across the spectrum. There are many limitations which have already been encountered, as well as, areas where more research and development can be done to maximize the potential of this combination. One of the most critical is the development of novel liquid crystal systems for RF applications. This will alleviate the significant absorption related losses that show up in current experiments. Performance enhancements are possible in the short term through this and the formulation of LC mixtures with high dielectric anisotropy in the relevant frequency ranges. A second significant area is the design of unit cells or meta-atoms that best take advantage of the dielectric properties and changes available from liquid crystals. To date, experiments have focused on loading well-known or typical meta-systems with LCs to observe tuning. Designing unit cells for interaction with LCs is sure to open up a wider range of available properties and will be essential for optical metamaterials where complex interactions can unlock new phenomena. The active research in this area is sure to yield interesting and useful results in the coming years.

References

1. Applications of Metamaterials. In: F. Capolino, Ed., CRC Press, Boca Raton, 2009.

2. M.-H. Lu, L. Feng, and Y.-F. Chen. Phononic crystals and acoustic metamaterials. Mater. Today 2009, 12, 34–42.

3. Z. M. Zhang and B. J. Lee. Chapter 3 Theory of thermal radiation and radiative properties. In: B. K. T. Zhuomin, M. Zhang, and M. Graham, Eds, Experimental Methods in the Physical Sciences, Academic Press, 2009.

4. G. Chen, A. Narayanaswamy, and C. Dames. Engineering nanoscale phonon and photon transport for direct energy conversion. Superlattices Microstruct. 2004, 35, 161–172.

5. Theory and Phenomena of Metamaterials. In: F. Capolino, Ed., CRC Press, Boca Raton, 2009.

6. J. B. Pendry. Negative refraction makes a perfect lens. Phys. Rev. Lett. 2000, 85, 3966.

7. R. A. Shelby, D. R. Smith, and S. Schultz. Experimental verification of a negative index of refraction. Science 2001, 292, 77–79.

8. V. G. Veselago. The electrodynamics of substances with simultaneously negative values of permittivity and permeability. Soviet Physics Uspekhi 1968, 10, 6.

9. A. N. Lagarkov, V. N. Semenenko, V. A. Chistyaev, D. E. Ryabov, S. A. Tretyakov, and C. R. Simovski. Resonance properties of bi-helix media at microwaves. Electromagnetics 1997, 17, 213–237.

10. D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz. Composite medium with simultaneously negative permeability and permittivity. Phys. Rev. Lett. 2000, 84, 4184.

11. U. Leonhardt and T. G. Philbin. Chapter 2 Transformation Optics and the Geometry of Light. In: Progress in Optics. E. Wolf, Ed., Elsevier, 2009.

12. J. Pendry. Metamaterials: Transformation optics. New Scientist 2011, 209, iv–v.

13. B. Kante, A. Ourir, S. N. Burokur, F. Gadot, and A. de Lustrac. Metamaterials for optical and radio communications. Comptes Rendus Physique 2008, 9, 31–40.

14. W. J. Padilla, A. J. Taylor, C. Highstrete, M. Lee, and R. D. Averitt. Dynamical electric and magnetic metamaterial response at terahertz frequencies. Phys. Rev. Lett. 2006, 96, 107401.

15. Y. Huang, X. Zhao, L. Wang, and C. Luo. Tunable left-handed metamaterial based on electrorheological fluids. Prog. Nat. Sci. 2008, 18, 907–911.

16. J. B. Birks, Ed., Progress in Dielectrics. Wiley, New York, 1960.

17. W. Rotman. Plasma simulation by artificial dielectrics and parallel-plate media. Antennas Propag. IRE Trans. on 1962, 10, 82–95.

18. J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart. Magnetism from conductors and enhanced nonlinear phenomena. Microwave Theory Tech. IEEE Trans. on 1999, 47, 2075–2084.

19. C. R. Simovski and S. A. Tretyakov. Local constitutive parameters of metamaterials from an effective-medium perspective. Phys. Rev. B 2007, 75, 195111.

20. D. R. Smith. Analytic expressions for the constitutive parameters of magnetoelectric metamaterials. Phys. Rev. E 2010, 81, 036605.

21. I. Nefedov, A. Viitanen, and S. Tretyakov. On reflection from interfaces with some spatially dispersive metamaterials. J. Magn. Magn. Mater. 2006, 300, 107–110.

22. L. Markley and G. V. Eleftheriades. Meta-screens and near-field antenna-arrays: a new perspective on subwavelength focusing and imaging. Metamaterials 2011, 5, 97–106.

23. B. A. Munk. Frontmatter in Frequency Selective Surfaces. John Wiley & Sons Inc., 2005.

24. R. E. Diaz and S. A. Clavijo. Artificial magnetic conductor. In: Encyclopedia of RF and Microwave Engineering, John Wiley & Sons Inc., 2005.

25. J. A. Bossard, L. Xiaotao, L. Ling, Y. Seokho, D. H. Werner, B. Weiner, T. S. Mayer, P. F. Cristman, A. Diaz, and I. C. Khoo. Tunable frequency selective surfaces and negative-zero-positive index metamaterials based on liquid crystals. Antennas Propag. IEEE Trans. on 2008, 56, 1308–1320.

26. F. Zhang, Q. Zhao, L. Kang, D. P. Gaillot, X. Zhao, J. Zhou, and D. Lippens. Magnetic control of negative permeability metamaterials based on liquid crystals. Appl. Phys. Lett. 2008, 92, 193104–193103.

27. F. Zhang, L. Kang, Q. Zhao, J. Zhou, X. Zhao, and D. Lippens. Magnetically tunable left handed metamaterials by liquid crystal orientation. Opt. Exp. 2009, 17, 4360–4366.

28. F. Zhang, W. Zhang, Q. Zhao, J. Sun, K. Qiu, J. Zhou, and D. Lippens. Electrically controllable fishnet metamaterial based on nematic liquid crystal. Opt. Exp. 2011, 19, 1563–1568.

29. C. R. Simovski and S. He. Frequency range and explicit expressions for negative permittivity and permeability for an isotropic medium formed by a lattice of perfectly conducting [Omega] particles. Phys. Lett. A 2003, 311, 254–263.

30. J. Huangfu, L. Ran, H. Chen, X.-M. Zhang, K. Chen, T. M. Grzegorczyk, and J. A. Kong. Experimental confirmation of negative refractive index of a metamaterial composed of Omega-like metallic patterns. Appl. Phys. Lett. 2004, 84, 1537–1539.

31. S. Larouche and D. R. Smith. A retrieval method for nonlinear metamaterials. Opt. Commun. 2010, 283, 1621–1627.

32. E. Poutrina, S. Larouche, and D. R. Smith. Parametric oscillator based on a single-layer resonant metamaterial. Opt. Commun. 2010, 283, 1640–1646.

33. F. Goelden, A. Lapanik, A. Gaebler, S. Mueller, W. Haase, and R. Jakoby. Systematic investigation of nematic liquid crystal mixtures at 30 GHz. Digest of 2007 IEEE/LEOS Summer Topical Meetings.

34. S. Mueller, M. Koeberle, F. Goelden, A. Penirschke, A. Gaebler, A. Lapanik, W. Haase, and R. Jakoby. W-Band characterization of anisotropic liquid crystals at room temperature. 38th European Microwave Conference.

35. F. Zhang, Q. Zhao, D. P. Gaillot, X. Zhao, and D. Lippens. Numerical investigation of metamaterials infiltrated by liquid crystal. J. Opt. Soc. Am. B 2008, 25, 1920–1925.

36. Q. Zhao, L. Kang, B. Du, B. Li, J. Zhou, H. Tang, X. Liang, and B. Zhang. Electrically tunable negative permeability metamaterials based on nematic liquid crystals. Appl. Phys. Lett. 2007, 90, 011112–011113.

37. N. Kundtz and D. R. Smith. Extreme-angle broadband metamaterial lens. Nat. Mater. 2010, 9, 129–132.

38. H. Chen, C. T. Chan, S. Liu, and Z. Lin. A simple route to a tunable electromagnetic gateway. New J. Phys. 2009, 11, 083012.

39. A. Gaebler, F. Goelden, A. Manabe, M. Goebel, S. Mueller, and R. Jakoby. Investigation of high performance transmission line phase shifters based on liquid crystal. 2009 European Microwave Conference.

40. A. Gaebler, F. Goelden, S. Mueller, and R. Jakoby. Modeling of electrically tunable transmission line phase shifter based on liquid crystal. 2008 Antennas and Propagation Society International Symposium, IEEE.

41. N. Tentillier, F. Krasinski, R. Sauleau, B. Splingart, H. Lhermite, and P. Coquet. A liquid-crystal, tunable, ultra-thin Fabry-Perot resonator in Ka band. Antennas Wireless Propag. Lett. IEEE 2009, 8, 701–704.

42. H. Wenfei, R. Dickie, R. Cahill, H. Gamble, Y. Ismail, V. Fusco, D. Linton, N. Grant, and S. Rea. Liquid crystal tunable mm wave frequency selective surface. Microwave Wireless Comp. Lett. IEEE 2007, 17, 667–669.

43. F. Zhang, Q. Zhao, W. Zhang, J. Sun, J. Zhou, and D. Lippens. Voltage tunable short wire-pair type of metamaterial infiltrated by nematic liquid crystal. Appl. Phys. Lett. 2010, 97, 134103.

44. W. Hu, M. Y. Ismail, R. Cahill, H. S. Gamble, R. Dickie, V. F. Fusco, D. Linton, S. P. Rea, and N. Grant. Tunable liquid crystal reflectarray patch element. Electronics Lett. 2006, 42, 509–511.

45. H. Wenfei, R. Cahill, J. A. Encinar, R. Dickie, H. Gamble, V. Fusco, and N. Grant. Design and measurement of reconfigurable millimeter wave reflectarray cells with nematic liquid crystal. Antennas Propag. IEEE Trans. on 2008, 56, 3112–3117.

46. H. Wenfei, M. Arrebola, R. Cahill, J. A. Encinar, V. Fusco, H. S. Gamble, Y. Alvarez, and F. Las-Heras. 94 GHz Dual-reflector antenna with reflectarray subreflector. Antennas Propag. IEEE Trans. on 2009, 57, 3043–3050.

47. A. Gaebler, A. Moessinger, F. Goelden et al. Liquid crystal-reconfigurable antenna concepts for space applications at microwave and millimeter waves. Int. J. Antennas Propag. 2009.

48. R. Kowerdziej, J. Parka, P. Nyga, and B. Salski. Simulation of a tunable metamaterial with nematic liquid crystal layers. Liq. Cryst. 2011, 38, 377–379.

49. R. Kowerdziej, J. Parka, and P. Nyga. Tunable liquid crystalline metamaterial structure in GHz range. Mol. Cryst. Liq. Cryst. 2011, 545, 91–95.

50. R. Kowerdziej, J. Parka and J. Krupka. Experimental study of thermally controlled metamaterial containing a liquid crystal layer at microwave frequencies. Liq. Cryst. 2011, 38, 743–747.

51. Y. Utsumi, T. Kamei, and R. Naito. Measurements of effective dielectric permittivity of microstrip-line-type liquid crystal devices using inductive coupled ring resonator. Elect. Lett. 2003, 39, 849–850.

52. I. A.I. Al-Naib, C. Jansen, and M. Koch. Thin-film sensing with planar asymmetric metamaterial resonators. Appl. Phys. Lett. 2008, 93, 083507–083503.

53. R. Melik, E. Unal, N. K. Perkgoz, C. Puttlitz, and H. V. Demir. Flexible metamaterials for wireless strain sensing. Appl. Phys. Lett. 2009, 95, 181105–181103.

54. D. Huang, E. Poutrina, and D. R. Smith. Analysis of the power dependent tuning of a varactor-loaded metamaterial at microwave frequencies. Appl. Phys. Lett. 2010, 96, 104104–104103.

55. I. V. Shadrivov, A. B. Kozyrev, D. W. van der Weide, and Y. S. Kivshar. Tunable transmission and harmonic generation in nonlinear metamaterials. Appl. Phys. Lett. 2008, 93, 161903–161903.

56. R. H. Ritchie. Plasma losses by fast electrons in thin films. Phys. Rev. 1957, 106, 874.

57. H. Kuwata, H. Tamaru, K. Esumi, and K. Miyano. Resonant light scattering from metal nanoparticles: practical analysis beyond Rayleigh approximation. Appl. Phys. Lett. 2003, 83, 4625–4627.

58. W. Cai and V. Shalaev. Optical Metamaterials: Fundamentals and Applications. Springer, New York, 2009.

59. C. M. Soukoulis, S. Linden, and M. Wegener. Negative refractive index at optical wavelengths. Science 2007, 315, 47–49.

60. S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck. Experimental demonstration of near-infrared negative-index metamaterials. Phys. Rev. Lett. 2005, 95, 137404.

61. M. Kafesaki, I. Tsiapa, N. Katsarakis, T. Koschny, C. M. Soukoulis, and E. N. Economou. Left-handed metamaterials: The fishnet structure and its variations. Phys. Rev. B 2007, 75, 235114.

62. X. Wang, D.-H. Kwon, D. H. Werner, I.-C. Khoo, A. V. Kildishev, and V. M. Shalaev. Tunable optical negative-index metamaterials employing anisotropic liquid crystals. Appl. Phys. Lett. 2007, 91, 143122.

63. D. H. Werner, D.-H. Kwon, I.-C. Khoo, A. V. Kildishev, and V. M. Shalaev. Liquid crystal clad near-infrared metamaterials with tunable negative-zero-positive refractive indices. Opt. Express 2007, 15, 3342–3347.

64. A. K. Diaz, I. C. Khoo. Liquid-crystalline nanostructured optical metamaterials. In: D. Andrews, G. Scholes, and G. Wiederrecht, (Eds.), Comprehensive Nanoscience and Technology, Oxford Academic Press, 2011.

65. S. Xiao, U. K. Chettiar, A. V. Kildishev, V. Drachev, I. C. Khoo, and V. M. Shalaev. Tunable magnetic response of metamaterials. Appl. Phys. Lett. 2009, 95, 033115.

66. K. Iam Choon, A. Diaz, J. Liou, M. V. Stinger, H. Junbin, and M. Yi. Liquid crystals tunable optical metamaterials. IEEE J. Select. Topic Quantum Elect. 2010, 16, 410–417.

67. W. Cai, U. K. Chettiar, H.-K. Yuan, V. C. de Silva, A. V. Kildishev, V. P. Drachev, and V. M. Shalaev. Metamagnetics with rainbow colors. Opt. Exp. 2007, 15, 3333–3341.

68. J. Muller, C. Sonnichsen, H. von Poschinger, G. von Plessen, T. A. Klar, and J. Feldmann. Electrically controlled light scattering with single metal nanoparticles. Appl. Phys. Lett. 2002, 81, 171–173.

69. K. C. Chu, C. Y. Chao, Y. F. Chen, Y. C. Wu, and C. C. Chen. Electrically controlled surface plasmon resonance frequency of gold nanorods. Appl. Phys. Lett. 2006, 89, 103107.

70. S. Y. Park and D. Stroud. Splitting of surface plasmon frequencies of metal particles in a nematic liquid crystal. Appl. Phys. Lett. 2004, 85, 2920–2922.

71. S. Y. Park and D. Stroud. Surface-enhanced plasmon splitting in a liquid-crystal-coated gold nanoparticle. Phys. Rev. Lett. 2005, 94, 217401.

72. A. V. Whitney, J. W. Elam, S. Zou, A. V. Zinovev, P. C. Stair, G. C. Schatz, and R. P. Van Duyne. Localized surface plasmon resonance nanosensor: a high-resolution distance-dependence study using atomic layer deposition. J. Phys. Chem. B 2005, 109, 20522–20528.

73. P. M. Adam, L. Salomon, F. de Fornel, and J. P. Goudonnet. Determination of the spatial extension of the surface-plasmon evanescent field of a silver film with a photon scanning tunneling microscope. Phys. Rev. B 1993, 48, 2680.

74. B. Luk'yanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong. The Fano resonance in plasmonic nanostructures and metamaterials. Nat. Mater. 2010, 9, 707–715.

75. M. Draper, I. M. Saez, S. J. Cowling, P. Gai, B. Heinrich, B. Donnio, D. Guillon, and J. W. Goodby. Self-assembly and shape morphology of liquid crystalline gold metamaterials. Adv. Funct. Mater. 2011, 21, 1260–1278.

76. S. Khatua, P. Manna, W.-S. Chang, A. Tcherniak, E. Friedlander, E. R. Zubarev, and S. Link. Plasmonic nanoparticles−Liquid crystal composites. J. Phys. Chem. C 2009, 114, 7251–7257.

77. Y. Li, D. Yu, L. Dai, A. Urbas, and Q. Li. Organo-soluble chiral thiol-monolayer-protected gold nanorods. Langmuir 2010, 27, 98–103.

78. Q. Liu, Y. Cui, D. Gardner, X. Li, S. He, and I. I. Smalyukh. Self-alignment of plasmonic gold nanorods in reconfigurable anisotropic fluids for tunable bulk metamaterial applications. Nano Lett. 2010, 10, 1347–1353.

79. A. B. Golovin, J. Xiang, Y. A. Nastishin, and O. D. Lavrentovich. Electrically reconfigurable optical metamaterials based on orientationally ordered dispersions of metal nano-rods in dielectric fluids, SPIE, San Diego, California, USA, 2010.

80. K. Kempa, R. Ruppin, and J. B. Pendry. Electromagnetic response of a point-dipole crystal. Phys. Rev. B 2005, 72, 205103.

81. B. Augui and W. L. Barnes. Collective resonances in gold nanoparticle arrays. Phys. Rev. Lett. 2008, 101, 143902.

82. A. Demortière, S. Buathong, B. P. Pichon, P. Panissod, D. Guillon, S. Bégin-Colin, and B. Donnio. Nematic-like organization of magnetic mesogen-hybridized nanoparticles. Small 2010, 6, 1341–1346.

83. J. B. Pendry. A chiral route to negative refraction. Science 2004, 306, 1353–1355.

84. E. Plum, V. A. Fedotov, A. S. Schwanecke, N. I. Zheludev, and Y. Chen. Giant optical gyrotropy due to electromagnetic coupling. Appl. Phys. Lett. 2007, 90, 223113.

85. A. Baev, M. Samoc, P. N. Prasad, M. Krykunov, and J. Autschbach. A quantum chemical approach to the design of chiral negative index materials. Opt. Exp. 2007, 15, 5730–5741.