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Double-Nonlinear Metamaterials De View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by The Australian National University APPLIED PHYSICS LETTERS 97, 231114 ͑2010͒ Double-nonlinear metamaterials ͒ Rongcao Yang1,2,a and Ilya V. Shadrivov1 1Nonlinear Physics Centre, Research School of Physics and Engineering, Australian National University, Canberra, Australian Capital Territory 0200, Australia 2College of Physics and Electronics Engineering, Shanxi University, Taiyuan 030006, People’s Republic of China ͑Received 6 September 2010; accepted 16 November 2010; published online 10 December 2010͒ We study a double-nonlinear metamaterial composed of a mixture of both nonlinear electric and nonlinear magnetic resonators. We predict multistable behavior in such metamaterial with the possibility to control the effective index of refraction by the electromagnetic field intensity. In contrast to the structure with just one type of nonlinear inclusions, our composite material may switch properties between the transparent negative index and the transparent positive index regimes. © 2010 American Institute of Physics. ͓doi:10.1063/1.3525172͔ Left-handed materials were first theoretically studied by capacitance model for the ER, see Fig. 1͑b͒. We assume that 1 2 20 Pafomov and Veselago, and later they were experimentally the nonlinear capacitance Cn has the form demonstrated in the form of composite metamaterials by Smith et al.3 First, metamaterials with negative refraction were composed of periodic arrays of split-ring resonators 2 2 4 r ␧ ͉U ͉ ͑SRRs͒ and thin conducting wires. By now, a number of C ͑U ͒ = e Deͩ1+␣ n ͪ, ͑1͒ n n Ј e 2 various designs of metamaterials have been suggested for 4dge Uc realizing the negative refractive index.5–9 where U and U are, respectively, voltage across the capaci- Recent studies have shown that nonlinear metamaterials n c tor, and characteristic voltage of the nonlinear capacitor, dЈ exhibit a range of intensity-dependent phenomena, such ge is the central slit size, ␣ = Ϯ1 describes focusing or defo- as bistability,10,11 harmonic generation,12 and parametric e cusing types of nonlinear response, ␧ is the linear dielectric amplification.13,14 The nonlinear metamaterials can be made De permittivity of the dielectric infilling the capacitor, re is the by embedding the SRRs in the nonlinear dielectric host me- radius of the capacitor plates. The current I in the effective dium or by inserting a varactor diode within the gap of circuit is governed by the equation SRRs.10,15 It was shown that the magnetic or electric reso- nances in nonlinear magnetic or nonlinear electric metama- terials can be dynamically tuned by varying external fields.16–18 Moreover, it was recently demonstrated that the dI L + U + U + RI =−EЈd , ͑2͒ same physical mechanism leads to the enhancement of non- dt n ge linearity in optical metamaterials.19 All nonlinear metamaterials studied so far are composed where U is the voltage across the linear capacitor and dge is of either nonlinear electric or nonlinear magnetic resonators. the size of the side slits of the ER. EЈ is the local micro- In this letter, we consider a three-dimensional composite scopic electric field creating the electromotive force in the structure with both nonlinear electric and nonlinear magnetic resonator, which is related to the external macroscopic elec- inclusions, double-nonlinear metamaterial. For the magnetic tric field E in cubic lattice through the Lorenz–Lorentz rela- 20 Ј ␲ / 2 Ј/ ␻ components, we choose the nonlinear SRRs studied earlier, tion E =E+4 P 3, where P=2nedgeE i Z is the electric ␻ ͑ ␻ ͒−1 ͑ ␻ ͒−1 while for electric inclusions, we develop a model based on polarization of the unit volume, Z=i L+ i C + i Cn nonlinear electric resonators ͑ERs͒, which were recently +R,ne is the density of the ERs. Then, the expression for the studied experimentally.18 We predict that such nonlinear effective electric permittivity of the composite can be ob- metamaterial can be engineered to exhibit tunable nonlinear tained in the form response with a control over both electric and magnetic non- linearities. Our results suggest that it is possible to realize metamaterial with the index of refraction changing from L positive to negative values, while metamaterial remains E mostly transparent. The schematic of the unit cell of nonlinear ERs is shown de C Cn in Fig. 1͑a͒, where the nonlinear response is generated by the nonlinear element inserted into the central slit of the ER. We R assume that the unit cell size de of the structure is much Cn smaller than the wavelength of the propagating electromag- (a) (b) netic wave, and we use an equivalent resistance-inductance- FIG. 1. ͑Color online͒͑a͒ Schematic of a unit cell of the nonlinear electric metamaterial and ͑b͒ the equivalent circuit with the parameters used in the ͒ a Electronic mail: [email protected]. derivations. 0003-6951/2010/97͑23͒/231114/3/$30.0097, 231114-1 © 2010 American Institute of Physics Downloaded 23 May 2011 to 130.56.107.38. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissions 231114-2 R. Yang and I. V. Shadrivov Appl. Phys. Lett. 97, 231114 ͑2010͒ 0.5 10 10 2 ) ) 0 eff 5 eff 5 ε ε −0.5 ( 1.5 0 0 Re( −1 Re (a) (b) (a) (b) −1.5 −5 −5 1 0 0.005 0.01 0 1 2 0 1 2 3 0 0.02 0.04 0 0 0 0 ) ) eff eff ε ε −5 ( −5 −0.5 Im Im( (c) (d) (c) −0.05 (d) −10 −10 0 0.02 0.04 0 0.005 0.01 0 1 2 0 1 2 3 2 2 −4 2 2 2 2 2 2 −3 |E| /E (x10 ) |E| /E |E| /E |E| /E (x10 ) c c c c ͑ ͒ FIG. 2. ͑Color online͒ Re͑␧ ͓͒͑a͒ and ͑b͔͒ and Im͑␧ ͓͒͑c͒ and ͑d͔͒ of the FIG. 3. Color online Same as in Fig. 2, but for a defocusing nonlinearity eff eff ␣ ͓͑ ͒ ͑ ͔͒ ⍀ ␥ ͓͑ ͒ ͑ ͔͒ ⍀ ␥ ␣ ͓͑ ͒ ͑ ͔͒ ⍀ e =−1. a and c e =1.05, e =0.01; b and d e =0.95, e =0.01. electric metamaterial for a focusing nonlinearity e =1. a and c e ␥ ͓͑ ͒ ͑ ͔͒ ⍀ ␥ =1.03, e =0.01; b and d e =0.95, e =0.01. ⍀ Ͼ tance. Then, for e 1, the real and imaginary parts of ef- fective electric permittivity of the composite structure in- 2 Fe ␧ ͉͑E͉ ͒ =1+ , ͑3͒ 2 eff ␻2 ͉͑ ͉2͒ ␻2 ␻⌫ / crease with ͉E͉ , as shown in Figs. 2͑a͒ and 2͑c͒. From Fig. E − + i e + Fe 3 0NLe ͑ ͒ ͑␧ ͒ 2 a , we see that Re eff may be positive or negative de- ␻2 ͉͑ ͉2͒ −1͓ / / ͉͑ ͉2͔͒ pending on the intensity of electric field. For ⍀ Ͻ1, X is a where 0NL E =L 1 C+1 Cn E is the eigenfre- e e ⌫ ͑ ͒/ ␴ ͉E͉2 ͑ ͒ quency of nonlinear oscillations, e = 2ae +3h 2L Sr is the three-valued function of in Eq. 4 , leading to a three- valued dependencies of effective electric permittivity on ͉E͉2 damping coefficient, ae is the length of the side slits of the ␴ ͓see Figs. 2͑b͒ and 2͑d͔͒. This nonlinear behavior is generic resonator, is the conductivity of the metal wires, Sr is the to many nonlinear resonant systems, including the split-ring effective area of the wire cross-section; and assuming circu- 10 lar cross-section, S Ϸ␲r2, for ␦ Ͼr , and S Ϸ␲␦ ͑2r −␦ ͒, resonators for nonlinear magnetic metamaterials. r e e e r e e e ͑ ␣ ͒ ␦ Ͻ ␦ /ͱ ␲␴␻ For a defocusing nonlinearity i.e., e =−1 , the nonlin- for e re, where e =c 2 is the skin-layer thickness. Here F =8␲d2 /Lh3, C=␧ a ␦ /2␲d and L=͓ln͑h/r ͒ ear electric eigenfrequency Xe increases with the increasing e ge De e e ge e intensity of the electric field, leading to the nonlinear permit- +2 ln͑2h/r ͒+Y͔h/c2 with Y =−ln͑1+ͱ5͒/2−2 ln͑2+ͱ5͒ e tivity behavior shown in Fig. 3. For ⍀ Ͼ1, Re͑␧ ͒ can ͱ / h ͑ ͒ e eff +2 5−21 4, where is the height of the ER. Equation 3 switch from negative to positive values with the change of shows that the effective electric permittivity depends on the ͉E͉2 ͓see Fig. 3͑a͔͒. However, for ⍀ Ͻ1, Re͑␧ ͒ is always intensity of the external electric field due to the intensity- e eff positive ͓see Fig. 3͑b͔͒, which is easy to understand from the dependent resonant frequency. The physical origin of the moving resonance picture: the frequency of the wave is to nonlinear behavior of the dielectric permittivity is identical the left of the resonance, and the resonance keeps moving to to that of magnetic permeability.10,20 The relation between the right with increase of intensity, ensuring a positive sign the resonant frequency and the electric field can be implicitly of the dielectric permittivity. obtained from Eqs. ͑1͒–͑3͒ as follows: It is natural to extend the concept to a metamaterial with simultaneously nonlinear electric and nonlinear magnetic re- 1 ͑1−X2͓͒͑X2 − ⍀2 + M͒2 + ⍀2␥2͔ ͉E͉2 = ␣ ͉E ͉2 e e e e e , sponses, which are expected to lead to nontrivial nonlinear e c ͑ ͒ ͑ 2 ͒2 1−Cr Xe − Cr phenomena and tunability of metamaterials. Here, we com- ͑4͒ bine the metamaterials of nonlinear electric structure studied above and nonlinear magnetic structure considered in the / where Ec =Uc dge is the characteristic electric field of non- Ref.
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