We propose electrically tunable hybrid metamaterial consisting of special
wire grid immersed into nematic liquid crystal. The plasma-like permittivity of
the structure can be substantially varied due to switching of the liquid
crystal alignment by external voltages applied to the wires. Depending on the
scale of the structure, the effect is available for both microwave and optical
frequency ranges.Comment: 3 page

de Jeu W. H.

M. A. Osipov

M. V. Gorkunov

English

arXiv

Crossref

Tunability of wire-grid metamaterial immersed into nematic liquid crystal

We propose electrically tunable hybrid metamaterial consisting of special wire grid immersed into nematic liquid crystal. The plasma like permittivity of the structure can be substantially varied due to switching of the liquid crystal alignment by external voltages applied to the wires. Depending on the scale of the structure, the effect is available for both microwave and optical frequency range

Gorkunov, M.V.

Osipov, M.A.

University of Strathclyde Institutional Repository

Strathprints Institutional RepositoryGorkunov, M.V. and Osipov, M.A. (2008) Tunability of wire-grid metamaterial immersed into nematicliquid crystal. Journal of Applied Physics, 103 (3). 036101-1. ISSN 0021-8979Strathprints is designed to allow users to access the research output of the University of Strathclyde.Copyright c© and Moral Rights for the papers on this site are retained by the individual authorsand/or other copyright owners. You may not engage in further distribution of the material for anyprofitmaking activities or any commercial gain. You may freely distribute both the url (http://strathprints.strath.ac.uk/) and the content of this paper for research or study, educational, ornot-for-profit purposes without prior permission or charge.Any correspondence concerning this service should be sent to Strathprints administrator:mailto:strathprints@strath.ac.ukhttp://strathprints.strath.ac.uk/  Gorkunov, M.V.* and Osipov, M.A.* (2008) Tunability of wire-grid metamaterial immersed into nematic liquid crystal. Journal of Applied Physics, 103 (3). 036101-1-036101-3. ISSN 0021-8979     http://eprints.cdlr.strath.ac.uk/6476/    This is an author-produced version of a paper published in Journal of Applied Physics, 103 (3). 036101-1-036101-3. ISSN 0021-8979. This version has been peer-reviewed, but does not include the final publisher proof corrections, published layout, or pagination.  Strathprints is designed to allow users to access the research output of the University of Strathclyde. Copyright © and Moral Rights for the papers on this site are retained by the individual authors and/or other copyright owners. You may not engage in further distribution of the material for any profitmaking activities or any commercial gain. You may freely distribute both the url (http://eprints.cdlr.strath.ac.uk) and the content of this paper for research or study, educational, or not-for-profit purposes without prior permission or charge. You may freely distribute the url (http://eprints.cdlr.strath.ac.uk) of the Strathprints website.  Any correspondence concerning this service should be sent to The Strathprints Administrator: eprints@cis.strath.ac.uk  arXiv:0708.4286v1  [cond-mat.mtrl-sci]  31 Aug 2007Tunability of wire-grid metamaterial immersed into nematic liquid crystalM.V. Gorkunov1,2 and M.A. Osipov11Department of Mathematics, University of Strathclyde, Glasgow G1 1XH, UK2Institute of Crystallography, Russian Academy of Sciences, 119333 Moscow, RussiaWe propose electrically tunable hybrid metamaterial consisting of special wire grid immersed intonematic liquid crystal. The plasma-like permittivity of the structure can be substantially varied dueto switching of the liquid crystal alignment by external voltages applied to the wires. Depending onthe scale of the structure, the effect is available for both microwave and optical frequency ranges.Electromagnetic properties of periodic arrays of metalwires, stripes and finely structured particles have at-tracted increasing attention in recent years. The varietyof emerging phenomena extends from the negative refrac-tion and subwavelength light constraining [1] to extraor-dinary transmission [2], which all have been attributed toregular metallic structures with periods smaller or muchsmaller than the wavelength of electromagnetic radiation.As a rule, the unusual properties are characterized by aconsiderable or even anomalously strong frequency dis-persion. Undoubtedly, this opens prospective possibili-ties for tuning and switching, since a small modificationof the system may result in a substantial change of itselectromagnetic response at a given signal frequency.Liquid crystals (LC) inherently have all the necessarycharacteristics to provide tunability to subwavelengthmetal arrays. By switching the orientation of LC mediumbetween metal elements or in metal cavities one can trig-ger substantial changes in the overall properties. Impor-tantly, LCs are transparent and significantly anisotropicin both microwave and optical frequency ranges, in whichsubwavelength metal arrays are usually operating [3, 4].Finally, the metal constituents can be readily used aselectrodes for applying switching voltages.In the past decade, various types of metamaterials havereceived significant attention due to negative values ofpermeability and permittivity (see e.g. the recent reviews[5, 6] and refs. therein). Certain methods are knownhow to provide tunability to metamaterials at microwavefrequencies by inserting electronic semiconductor compo-nents [7, 8, 9]. However, it still remains challenging toachieve this at optical frequencies. In this Letter we de-scribe how nematic LC environment can provide tunabil-ity to the plasma frequency of wire-grid metamaterial.Wire lattices exhibit plasma-like permittivity for elec-tromagnetic radiation with wavelengths much larger thanthe lattice period [10, 11]. In particular, for electro-magnetic waves traveling normally to the grid of infiniteparallel wires and polarized along them (xy-plane of in-cidence and z-polarization of electric field in Fig. 1), theemployed component of the effective local permittivity isεzz(ω) = ε(h)zz (ω)−Ω2ω2. (1)Here ε(h)zz is the zz-component of the permittivity tensorof host medium surrounding the wires, and the parameterΩ is determined by the grid geometry.dyzxaa+Q –Q–Q    +QH         E k+VFIG. 1: (Color online) Schematic of switching of nematic LCsurrounding grid of wires. Note: the size of LC molecules isextremely exaggerated.The so-called plasma-frequency, at which the permit-tivity (1) changes its sign, is ω0 = Ω/(ε(h)zz )1/2. The gridon its own is transparent at frequencies higher than ω0and reflecting at lower frequencies. Being accompaniedby a 3D array of split-ring resonators providing negativepermeability, the structure is a transparent left handedmedium for ω < ω0 and reflects waves with ω > ω0.The presence of ε(h)zz in Eq. (1) suggests an easy wayof tuning the metamaterial by varying the host mediumpermittivity. Nematic LCs provide a natural opportunityto realize this. Anisotropic LC permittivity axes followthe orientation of the director n:ε(h)ij (ω) = ε⊥(ω) δij + εa(ω) ninj. (2)Typically, elongated molecules of nematic LC tend toalign along interfaces due to surface anchoring. The re-sulting LC alignment along the wires is shown in Fig.1 onthe left. If external voltages are applied to the neighbor-ing wires, this will load them with charges of differentsigns. The arising static electric field in the xy-planewill force the LC molecules to orient perpendicularly tothe wires as shown in Fig.1 on the right. As a result,ε(h)zz switches from (ε⊥ + εa) to ε⊥ providing the relativeswitching range of ω0∆ω0ω0≃εa2ε⊥. (3)The dielectric anisotropy of LC is known to be at least ofthe order of several dozens of percent for both microwaves20.0 0.2 0.4 0.6 0.8 1.010-1110-10Pair capacitance per length,c 1[F/m]Wire diameter to lattice constant ratio, d/aFIG. 2: Grid capacitance per pair of wires in vacuum perlength versus the ratio of wire diameter to lattice constant.and light [3, 4]. Therefore, the proposed design providesa possibility of shifting the plasma-frequency by 10–20%.Let us find out the voltage to be applied to the gridfor efficient switching. Consider the quadratic lattice ofcylindrical wires as shown in Fig. 1. Assume that a volt-age V applied across the nearest neighbors loads the wireswith charge per length densities ±Q. The charges are dis-tributed over the surfaces of thin wires. We neglect theangular inhomogeneity of this distribution and assumethe electric field of a wire to be the same as that of acharged wire axis. We also presume that the static di-electric anisotropy of the nematic εsta = εa(0) is small andthe electric field pattern is not perturbed by the inhomo-geneously oriented LC.A single wire stretched along the line x = 0, y = 0 andloaded with the charge Q per unit length produces theelectric field potential ϕ1(ρ) = −Q2piε0εstlog(2ρ/d), whereρ =√x2 + y2, εst = ε⊥(0) is the static permittivity ofthe LC, and the zero of the potential is assigned to thewire surface. In the infinite 2D lattice of alternatinglycharged wires, the total potential outside of the wires isϕ(r) =n,m=∞∑n,m=−∞(−1)n+mϕ1 (ρnm), (4)where ρnm =√(x− na)2 + (y −ma)2 and the coordi-nate origin lies on the axis of a positively charged wire.We have calculated (4) for finite latices. In spite of theabsence of true mathematical convergence of the series,the potential distribution establishes already for a latticeof 10× 10 wires. Further increase of the number of wiresdoes not affect the potential in the middle provided thatthe overall electric neutrality is preserved.Applied voltage equals the difference of the potentialon positively and negatively charged wires, V = ϕ+−ϕ−.The capacitance per length of a grid of large even numberN of wires is C = QN/(2V ). It is convenient to intro-duce the capacitance per pair of wires in the vacuumper length, c1 = 2C/(Nεst), which naturally appears inthe relation Q = εstc1V . Apparently, c1 is a functionof two geometric parameters, a and d, and depends ontheir ratio d/a. The calculated values of c1 are presented−1 −0.5 0 0.5 1−1−0.500.51x/ay/a+++++–– ––FIG. 3: Lines of electric filed produced by 2D grid of alter-natingly charged wires. The wire diameter is set to be 0.1 ofthe lattice periodin Fig. 2. Note that c1 tends to infinity at d ≃ 1.08ainstead of d = a, when the neighboring wire surfacescome in contact. This is the consequence of neglectingthe anisotropy of the charge distribution over wire sur-face. We see that even for thick wires our approximationis reasonably good.Typical calculated electric field pattern is shown inFig. 3. The field amplitude is maximum at the wire sur-faces (where the lines are condensed) and vanishes at themiddle points of the diagonal lines connecting the nearestwires with the same sign of charge.Now we consider the effect of this field on the nematic.If we totally neglect the LC elasticity, the anisotropicmolecules should align along the field lines. The directorn then lies strictly in the xy-plane and the switching isperfect. The finite elasticity of the LC disturbs the idealpicture and the bending pattern of the director differsfrom the field lines pattern. However, this is of minorimportance for the ε(h)zz switching since the director stillstays within the xy-plane.The critical effect of LC elasticity occurs in the vicin-ity of the wires. To ensure the LC orientation along thewires when the voltage is switched off, the anchoring ofLC molecules at wire surfaces must be strong. Whenthe voltage is switched on, the anchoring still forces themolecules at the surfaces to point along the z-axis. Ac-cordingly, the director has to rotate by an angle of pi/2within the transient layer from the wire surface to the LCbulk. Efficient switching occurs when this layer is thincompared to the scale of the structure (lattice constant),i.e., practically important is the limit of strong electricfield and thin transient layer.In this limit, we can neglect the contributions fromother wires to the field near a wire surface. In the cylin-drical coordinates, the only present ρ-component of theelectric field equals Eρ(ρ) =Q2piε0εstρ−1 while the nematicdirector has two components nρ(ρ) and nz(ρ). At the31 2 3 4 5 6 70.00.20.40.60.81.0106421.5cos2???/d1.1FIG. 4: Spatial profiles of cos2 θ determining modulation ofLC permittivity εzz. Corresponding values of v2 are shown.wire surface nρ(d/2) = 0.The free energy of the LC can be presented as the sumF = FK + FE of the elastic deformation energy and thedielectric energy [3], where the latter readsFE = −piε0εsta∫ Rdρ dρ (Eρnρ)2. (5)Here the upper limit R is large enough to assure nz(R) =0 and can be extended to infinity. The simplest formof the LC elastic energy is given by the so-called oneconstant approximation:FK =K2∫dV[(∇ · n)2 + (∇× n)2]. (6)Introducing the director polar angle θ as nρ = sin θ,nz = cos θ we obtain the compact form of the total energyF = piK∫ ∞dρ dρ[(θ′)2 + (1 − v2)sin2 θρ2], (7)where the parameter v2 = V 2c21εsta /(4pi2Kε0) character-izes relative contribution of the voltage-driven term.Minimizing the functional (7) yields the differentialequation for θ(ρ):θ′′ +1ρθ′ + (v2 − 1)sin 2θ2ρ2= 0 (8)with the boundary conditions θ(d/2) = 0 and θ(∞) =pi/2. For v2 > 1 the exact solution reads:θ(ρ) =pi2− 2 arctan[(d/2ρ)√v2−1]. (9)The spatial variation of the zz-component of dielectrictensor (2), is controlled by the factor n2z(ρ) = cos2 θ(ρ).The profiles of the latter are given in Fig. 4 for several val-ues of v2 larger than unity. It is seen that the voltage hasto exceed considerably the critical value to provide theLC reorientation within thin transient layer. For v2 >∼ 2the layer thickness becomes comparable with the wirediameter. For estimates, we take a nematic LC elasticmodulus K ≃ 10−11N, static permittivity εst ≃ 10 andstatic dielectric anisotropy εsta ≃ 1. We also set the wirediameter to be ten times smaller than the lattice con-stant, which according to Fig. 2 yields c1 ≃ 10−11F/m.Then appropriate voltage is estimated asV =2pic1√2Kε0εsta≃ 8.4V. (10)Remarkably, this moderate voltage is independent of thescale of the grid.One notes that the studied LC switching geometry dif-fers qualitatively from the conventional one within flatelectrooptic cells. In our case, the LC bulk is directlyaligned and reoriented by the immersed wires. We be-lieve this to be an additional advantage for microwaveapplications, where the controlling of millimeter (or evencentimeter) thick LC samples by cell surfaces is practi-cally impossible.In summary, we have demonstrated that the plasmafrequency of wire-grid metamaterial immersed into ne-matic LC can be efficiently tuned by 10-20% due to theswitching of LC alignment. Our estimates show that thenecessary voltage applied to the wires is of the order ofseveral Volts. The main conclusions are valid for bothmicrowave and optical frequency ranges.[1] J.B. Pendry, Phys. Rev. Lett. 85, 3966 (2000).[2] W.L. Barnes, A. Dereux, and T.W. Ebbesen, Nature424, 824 (2003).[3] W.H. de Jeu, Physical Properties of Liquid CrystallineMaterials, Gordon and Breach, London, 1980.[4] J.R. Sambles, R.Kelly, and F. Yang, Phil. Trans. R. Soc.A 364, 2733 (2006).[5] V.M. Shalaev, Nature Photonics, 1, 41 (2007).[6] S. Linden, C. Enkrich, G. Dolling et al., IEEE J. of Se-lected Topics in Quant,. Electr., 12, 1097 (2006).[7] M. Lapine, M. Gorkunov, and K.H. Ringhofer, Phys.Rev. E 67, 065601(R) (2003).[8] A.A. Zharov, I.V. Shadrivov, and Yu.S. Kivshar, Phys.Rev. Lett. 91, 037401 (2003).[9] H.-T. Chen, W.J. Padilla, J.M.O. Zide, et al., Nature444, 597 (2006).[10] J.B. Pendry, A.J. Holden, W.J. Stewart, and I. Youngs,Phys.Rev. Lett. 76, 4773 (1996).[11] P.A. Belov, R. Marques, S. I. Maslovski et al., Phys. Rev.B 67 113103 (2003).

We propose electrically tunable hybrid metamaterial consisting of special wire grid immersed into nematic liquid crystal. The plasmalike permittivity of the structure can be substantially varied due to switching of the liquid crystal alignment by external voltages applied to the wires. Depending on the scale of the structure, the effect is available for both microwave and optical frequency range

Tunability of wire-grid metamaterial immersed into nematic liquid crystal

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