Structural tunability in metamaterials
Mikhail Lapine, David Powell, Maxim Gorkunov, Ilya Shadrivov, Ricardo Marqués, Yuri Kivshar
aa r X i v : . [ phy s i c s . op ti c s ] J u l Structural tunability in metamaterials
Mikhail Lapine,
1, 2, ∗ David Powell, Maxim Gorkunov, Ilya Shadrivov, Ricardo Marqu´es, and Yuri Kivshar Nonlinear Physics Center, Research School of Physics and Engineering, Australian National University, Australia Dpto. Electronica y Electromagnetismo, Facultad de Fisica, Universidad de Sevilla, Spain Institute of Crystallography, Russian Academy of Sciences, Moscow, Russia
We propose a novel approach for efficient tuning of the transmission characteristics of metamate-rials through a continuous adjustment of the lattice structure, and confirm it experimentally in themicrowave range. The concept is rather general and applicable to various metamaterials as long asthe effective medium description is valid. The demonstrated continuous tuning of metamaterial re-sponse is highly desirable for a number of emerging applications of metamaterials including sensors,filters, switches, realizable in a wide frequency range.
Metamaterials are prominent for the exceptional op-portunities they offer in tailoring macroscopic proper-ties through appropriate choice and arrangement of theirstructural elements [1, 2]. In this way, it is not only possi-ble to design a metamaterial for a required functionality,but also to implement further adjustment capabilities atthe level of assembly. This makes metamaterials differentfrom conventional materials and opens exciting opportu-nities of multi-functionality via tunability.Tunable metamaterials imply the ability to continu-ously change their properties through an external influ-ence or signal with the intrinsic mechanism of tunabil-ity. The key means of tuning resonant metamaterials,naturally, lies in affecting the system so as to changethe parameters of the resonance. As a consequence, thecharacteristics of metamaterial can be varied, enabling,for instance, tunable transmission.The first approach to realize tunable metamaterialsbased on nonlinear properties [3] has already been provenexperimentally [4, 5], and further methods have been sug-gested, e.g. based on reconfigurability of liquid crystals[6]. However, such methods become increasingly difficultto implement at higher frequencies.In this Letter, we put forward an approach which relieson the structural tuning of the entire metamaterial, andis, conceptually, independent of the specific realization aswell as scalable to any frequency provided that macro-scopic requirements for metamaterials are observed.We explain the general principle of the proposed tuningmethod through a simple analogy. Indeed, the propertiesof crystals are known to be determined by the natureof constituent atoms as well as by the geometry of thecrystal lattice, so in natural materials the collective re-sponse of atoms determines the overall response to exter-nal fields [7]. In natural materials however, possibilitiesto tune their properties dynamically are limited to natu-rally available crystals and yield relatively weak effects,such as electro-/magnetostriction, photorefraction, etc.In contrast, metamaterials offer a unique opportunityto design and vary the structure enabling a desired re-sponse function and a convenient mechanism for tunabil-ity. More importantly, the range of tunability for a givenproperty can be much broader than in natural materi-als, as the lattice effects can be made much strongerthrough higher efficiency of collective effects in the lat-tice, achieved by an appropriate design.To demonstrate the efficiency of this approach, we con- sider an anisotropic metamaterial based on resonant ele-ments suitable for providing artificial magnetism, such assplit-ring resonators of various kind, as shown in Fig. 1.For sufficiently dense arrays, the interaction between suchelements differs considerably from a dipole approxima-tion, and the specific procedure to calculate the effectivepermeability was developed earlier [8]; the latter con-verges correctly to a Clausius-Mossotti approximation inthe limit of a sparse lattice. Consequently, the effect ofmutual coupling is enhanced dramatically as comparedto conventional materials, and therefore it is particularlysuitable to demonstrate the efficiency of lattice tuning.Accordingly, if all the characteristic dimensions (lat-tice constants and element size) are much smaller thanthe wavelength, we can describe a regular lattice of suchelements by the resonant effective permeability µ ( ω ) = 1 − Aω ω − ω r + i Γ ω (1)with the resonant frequency ω r = ω o (cid:18) L Σ L + µ o νS L (cid:19) − / (2)determined by both the properties of individual elements,such as the resonance frequency of a single element ω o ,their geometry (which defines self-inductance L and ef-fective cross-section S ), and concentration ν , as well astheir arrangement. The latter effect is determined bymutual interaction between the elements, which in mostpractically realizable cases is defined by L Σ = L + µ o r Σ , (3) y xz a ab FIG. 1: Schematic of the staggered lattice shift with a lateraldisplacement of every second metamaterial layer. δ a R e s onan c e ( ω / ω o ) (a) 0 0.1 0.2 0.3 0.4 0.500.20.40.60.81 Lattice shift, δ a T r an s m i ss i on , R e f l e c t i on (b) FIG. 2: (a) Theoretical shift of the resonance frequency forcontinuous (dashed) and staggered (solid) lattice shift strat-egy; (b) Calculated transmission and reflection through ametamaterial slab (one wavelength thick) depending on lat-tice shift (staggered) at ω = 0 . ω o .. where the so-called lattice sum Σ can be calculated fora given geometry of elements and their arrangementthrough mutual inductance X n ′ = n L nn ′ ( ω ) = − iωµ o r · Σ , (4)between all the elements in a physically small volumewhere the average macroscopic field is evaluated [8].Note that the collective behavior of a number of ele-ments in the lattice plays a crucial role, so that in densearrays the mutual interaction cannot be reduced to theapproximation of nearest neighbors adopted in recent rig-orous models which account for spatial dispersion [9].The most straightforward lattice tuning approach is tovary the lattice constant b . We have shown [8] that theresonance frequency can be remarkably shifted this way,and confirmed this with microwave experiments [10]. Ac-cordingly, a slab of metamaterial can be tuned betweentransmission, absorption and reflection back to transmis-sion. A clear disadvantage of this method is that varying b implies a corresponding significant change in the overalldimension of the metamaterial along z , which is undesir-able for applications.Here we propose another method of structural tuning,by means of a periodic lateral displacement of layers inthe xy plane, so that the resonators become shifted along x ( y , or both) by a fraction of the lattice constant δa per each b distance from a reference layer with respectto the original position. This decreases the overall mu-tual inductance in the system (Eq. (3)) and leads to agradual increase in resonant frequency, with a maximaleffect archived for displacement by 0.5 a [see Fig. 2(a)].Clearly, further shift is equivalent to smaller shift valuesuntil the lattice exactly reproduces itself for the shift by a . As a consequence, the resonance of the medium canbe “moved” across a signal frequency, leading to a dras-tic change in transmission characteristics [see Fig. 2(b)].It is clear that for practical applications it is not evennecessary to exploit the whole range of lateral shift — inthe above example it is sufficient to operate between 0 . a and 0 . a where most of the transition occurs.Within the effective medium paradigm, a continuousshift of each layer with respect to the previous one ap-pears to provide maximal efficiency. For finite samples, however, this poses certain disadvantages. Indeed, forsmall b/a ratio (which produces a stronger effect) evena small δa shift would imply a remarkable inclinationof the sample interface. This would lead to undesirableshape distortion and cause excitation of additional stand-ing waves. Preliminary experiments with this kind oftuning have shown that the system generally features thepredicted behavior, however is rather unrepeatable withregards to excitation and measurement methods, so theperformance cannot be reliably assessed.To overcome this difficulty, we consider a staggeredlateral shift as shown in Fig. 1, when every second layeris shifted while the rest of the structure remains at theoriginal position. This configuration leads to a slightlydifferent efficiency pattern [compare the two curves inFig. 2(a)], but is equally useful; obviously, the two tuningstrategies converge to the identical result for 0 . a shift,as the lattice patterns shifted with either method coin-cide in this case. For finite samples, the staggered shiftis advantageous as it keeps the sample interface straightand parallel to the axis of resonators at all times, whileslight regular distortion of the interface shape is not ex-pected to deteriorate the performance, provided that theoverall number of elements is sufficiently large so thatsurface effects are negligible.For the experimental verification, we opted for asmall reconfigurable system, built up of single-split rings(2.25 mm mean radius, 0.5 mm strip width, 1 mm gap)printed with period a = 7 mm on 1.5 mm thick circuitboards. We have 5 resonators in the propagation direc-tion x and only one period along y ; 30 boards are stackedtogether in z direction with the minimum possible latticeconstant b = 1 . Frequency, ω / ω o M agne t i c f i e l d ( no r m a li z ed ) FIG. 3: Numerically calculated magnetic field beneath a fi-nite metamaterial slab (5 × ×
30 elements) for a plane-waveincidence. Curves with dips from left to right correspond toincreasing lattice shift from 0 to 0.5 a . Lattice shift, δ a R e s onan c e ( ω / ω o ) FIG. 4: Comparison between theoretical results and exper-imental data for the resonance frequency shift: Effectivemedium approach (solid); finite model (circles); experimen-tal results (squares).
Schwarz ZVB network analyzer) were performed for var-ious lattice shifts in WR-229 rectangular waveguide.Note that the above system of 5 × ×
30 resonators can-not be described by an effective medium approach, as thenumber of elements is small. Also, the system is not suf-ficiently subwavelength for a quasi-static approach to beused and spatial dispersion becomes remarkable [12, 13].For this reason, we also perform semi-analytical calcu-lations for the corresponding finite structures, with allthe mutual inductances included (Eq. (4)), taking retar-dation effects into account. Although the particular res-onance values obtained this way (Fig. 3), are differentfrom those which would be observed in a medium, theoverall effect of lattice tuning was qualitatively the sameand predicts excellent performance (Fig. 4).The experimental transmission spectra are shown inFig. 5, demonstrating dramatic tuning of the resonancefrequency. Furthermore, comparison of the experimen-tal resonance shift with the theoretical predictions shows(Fig. 4) that the experimental system demonstrates evenhigher efficiency. This effect can be explained by account-ing for the mutual capacitance between resonators, ne-glected in our theoretical calculations. Indeed, for thebroadside-like configuration of rings, mutual capacitancebetween them is distributed along the whole circumfer-ence [11]. Clearly, when the resonators are laterally dis-placed, the mutual capacitance decreases, so that thiseffect is added up to the increase of resonance frequencyimposed by decreased inductive coupling.The examples analyzed above illustrate the practicalfeasibility of the proposed tuning concept. Particular de-tails and tuning patterns may differ depending on thespecific structural elements used to create metamaterials,however it is clear that the remarkable resonance shift canbe realized over a wide range of alternative geometries,including numerous resonator varieties and even fishnetstructures which are more popular for higher frequen-cies. Remarkably, the proposed tuning mechanism is notspecific for the microwave range used in the above ex-amples: this can be scaled in size and frequency as longas metamaterial description in terms of effective medium
Frequency, GHz T r an s m i ss i on FIG. 5: Experimental transmission in a waveguide with meta-material slab at different shifts. Curves with dips from left toright correspond to increasing lattice shift. is applicable. And on the practical side, tremendous ef-ficiency of the structural tuning can be used in a hostof applications such as sensors, filters, switches and allkinds of devices where prompt and sensitive response tochanging conditions is required.In conclusion, we have proposed and confirmed exper-imentally a novel concept for tunability of metamaterialsthrough a continuous adjustment of the lattice structure.This work was supported by the Australian ResearchCouncil. M.L. acknowledges hospitality of NonlinearPhysics Center and a support of the Spanish Junta deAndalusia (P06-TIC-01368). M.G. acknowledges supportfrom the Russian Academy of Sciences (OFN Programm“Physics of new materials and structures”). ∗ Corresponding author:[email protected][1] M. Lapine and S. Tretykov, IET Microwaves Antennas &Propagation , 3 (2007).[2] A. Sihvola, Metamaterials , 2 (2007).[3] M. Gorkunov and M. Lapine, Phys. Rev. B , 235109(2004).[4] D. A. Powell, I. V. Shadrivov, Yu. S. Kivshar, and M. V.Gorkunov, Appl. Phys. Lett. , 144107 (2007).[5] I. V. Shadrivov, A. B. Kozyrev, D. W. van der Weide, andYu. S. Kivshar, Appl. Phys. Lett. , 161903 (2008).[6] M. V. Gorkunov and M. A. Osipov, J. Appl. Phys. ,036101 (2008).[7] L. D. Landau and E. M. Lifshitz, Electrodynamics ofContinuous Media (Nauka, Moscow, 1984).[8] M. Gorkunov, M. Lapine, E. Shamonina, and K. H.Ringhofer, Eur. Phys. J. B , 263 (2002).[9] J. D. Baena, L. Jelinek, R. Marqu´es, and M. Silveirinha,Phys. Rev. A , 013842 (2008).[10] I. V. Shadrivov, D. A. Powell, S. K. Morrison, Yu. S.Kivshar, and G. N. Milford, Appl. Phys. Lett. , 201919(2007).[11] R. Marqu´es, F. Mesa, J. Martel, and F. Medina, IEEETrans. Anten. Propag. , 2572 (2003).[12] V. M. Agranovich and Yu. N. Gartstein, Metamaterials , 1 (2009).[13] C. Simovski, Metamaterials2