Overcoming losses with gain in a negative refractive index metamaterial
Sebastian Wuestner, Andreas Pusch, Kosmas L. Tsakmakidis, Joachim M. Hamm, Ortwin Hess
aa r X i v : . [ phy s i c s . op ti c s ] O c t Overcoming losses with gain in a negative refractive index metamaterial
Sebastian Wuestner, Andreas Pusch, Kosmas L. Tsakmakidis, Joachim M. Hamm, and Ortwin Hess ∗ Advanced Technology Institute and Department of Physics,University of Surrey, Guildford, GU2 7XH, Surrey, United Kingdom (Dated: October 2, 2018)On the basis of a full-vectorial three-dimensional Maxwell-Bloch approach we investigate thepossibility of using gain to overcome losses in a negative refractive index fishnet metamaterial. Weshow that appropriate placing of optically pumped laser dyes (gain) into the metamaterial structureresults in a frequency band where the nonbianisotropic metamaterial becomes amplifying. In thatregion both the real and the imaginary part of the effective refractive index become simultaneouslynegative and the figure of merit diverges at two distinct frequency points.
PACS numbers: 78.67.Pt, 78.20.Ci, 42.25.Bs, 78.45.+h
Negative refractive index metamaterials offer the pos-sibility of revolutionary applications, such as subwave-length focusing [1], invisibility cloaking [2], and “trappedrainbow” stopping of light [3]. The realization of thesematerials has recently advanced from the microwave tothe optical regime [4, 5]. However, at optical wavelengthsmetamaterials suffer from high dissipative losses due tothe metallic nature of their constituent meta-molecules.It is therefore not surprising that overcoming loss restric-tions is currently one of the most important topics inmetamaterials research [6].It has been suggested that, owing to causality, simul-taneous loss-compensation and negative refractive indexmight only be attainable in a very narrow bandwidthwith high losses nearby [7]. In an ongoing discussion sev-eral authors have reasoned that causality-based criteriahave to be applied carefully and do not in general lead tosuch a strict result [8, 9]. This said, the theoretical possi-bility to compensate losses in optical metamaterials doesnot necessarily imply that the gain available from opti-cally active media suffices to achieve this goal. Indeed,bulk gain coefficients are usually an order of magnitudesmaller than the absorption coefficients of metals at op-tical frequencies.A vital clue as to how the aforementioned limita-tion could be overcome came from [10]. There it wasshown that the incorporation of gain in regions of highfield intensity gives rise to an effective gain coefficientthat can exceed its bulk counterpart by potentially or-ders of magnitude. Exploiting this gain-enhancementeffect loss reduction and giant field enhancement havebeen reported for a double-fishnet metamaterial usingfrequency-domain models [11, 12]. The particular modelused in [11] accounts for spatial nonuniformity and sat-uration of the gain. However, the dynamic evolutionof the gain system, nonlinearly pumped by a short in-tense pulse, cannot be described self-consistently withsuch a frequency-domain approach and relies on assump-tions of cw-excitation. A time-domain calculation of gainin two-dimensional electric or magnetic metamaterialshas recently been reported in [13], but therein the ef- fect of the pump field has again not been considered self-consistently.In this Letter, we study the optical response of a neg-ative refractive index (NRI) metamaterial with a gainmedium embedded in the structure and we find that com-plete loss-compensation and even amplification is possi-ble using realistic gain parameters. To this end we use afull-vectorial time-domain approach that manages to self-consistently couple the evolution of the occupation den-sities in the gain medium directly to Maxwell’s equationsin three dimensions [14–16]. Nonlinearity, saturation ofthe gain medium, and spatio-temporal variations of bothabsorption and emission are inherent to our model, avoid-ing the need for external, precalculated inputs.The considered structure is an optical double-fishnetmetamaterial [11, 17, 18] with a square periodicity of p = 280 nm, perforated with rectangular holes of sides a x = 120 nm and a y = 80 nm (see Fig. 1). The additionalgeometrical parameters are h m = 40 nm, h d = 60 nm,and h c = 60 nm. This type of NRI metamaterial, which FIG. 1. (color online). Illustration of the double-fishnet struc-ture with a square unit cell of side-length p highlighted. Thetwo perforated silver films are embedded in a dielectric hostmaterial which holds the dye molecules (translucent). Dimen-sions are given in the text. Pump (red dashed line) and probe(blue solid line) pulses illustrate the pump-probe configura-tion with the electric field polarized along the x direction. has been the topic of intense research (see, e.g., [11, 17–19]), exhibits low absorption compared to other meta-materials in the optical wavelength range. Its relativelylow absorption makes it the most promising structure forcomplete loss-compensation [20].We consider two configurations, passive and active. Inthe passive configuration two silver fishnet films are em-bedded inside a dielectric host that has a real refractiveindex of n h = 1 .
62 (see Fig. 1). The permittivity of sil-ver follows a Drude model corrected by two Lorentzianresonances to match experimental data at visible wave-lengths [21]. In the active configuration we insert Rho-damine 800 dye molecules into the dielectric host andexcite them optically in numerical pump-probe experi-ments. The chosen geometric parameters ensure a goodoverlap of the metamaterial’s resonant response with theemission spectrum of the dye for an electric field polar-ization along the long side a x of the rectangular holes(see Fig. 1).In order to self-consistently calculate the gain dynam-ics in this system the dye molecules are described usinga semiclassical four-level model with two optical dipoletransitions [14–16]. This model is implemented by intro-ducing auxiliary differential equations for the position-and time-dependent polarization densities P i and oc-cupation densities N j into the three-dimensional finite-difference time-domain (FDTD) algorithm. The timeevolution of the polarization densities for the absorption( i = a) and emission ( i = e) lines is then given by ∂ P i ∂t + 2Γ i ∂ P i ∂t + ω ,i P i = − ω i e d i ¯ h ∆ N i · E loc (1)where ω ,i = ( ω i + Γ i ) / are the oscillator frequencies,¯ hω i the electronic transition energies, Γ i the half-widthsof the resonances, and e · d i the dipole strengths. Thedye molecules embedded in the dielectric host experi-ence, in the Lorentz approximation, the local electric field E loc = (cid:2) ( n h + 2) / (cid:3) E and not the average electric field E [22]. Saturation is accounted for by the electric fielddependence of the occupation inversions ∆ N a = N − N and ∆ N e = N − N for absorption and emission re-spectively, which couple to Eq. (1). Their dynamics aregoverned by ∂N ∂t = 1¯ hω a (cid:18) ∂ P a ∂t + Γ a P a (cid:19) · E loc − N τ , (2a) ∂N ∂t = N τ + 1¯ hω e (cid:18) ∂ P e ∂t + Γ e P e (cid:19) · E loc − N τ , (2b) ∂N ∂t = N τ − hω e (cid:18) ∂ P e ∂t + Γ e P e (cid:19) · E loc − N τ , (2c) ∂N ∂t = N τ − hω a (cid:18) ∂ P a ∂t + Γ a P a (cid:19) · E loc . (2d)Nonradiative decay of the occupation densities is quan-tified by the lifetimes τ jk . The Γ i P i -terms stem from the transformation from complex- to real-valued polar-izations ([14], p. 174).Absorption and emission cross-sections, taken from ex-perimental data, are used to calculate the dipole length d i via σ i = (cid:0) ω ,i e d i / ¯ h (cid:1) / ( ǫ cn h Γ i ), with ǫ being thevacuum permittivity and c the vacuum speed of light.The parameters for the four-level system are chosen asfollows (cf. [23]): λ e = 2 πc/ω e = 710 nm, λ a = 680 nm,Γ e = Γ a = 1 / (20 fs), d e = 0 .
09 nm, and d a = 0 . τ = τ = 100 fs and τ = 500 ps. These valuescorrespond to cross-sections σ e = 2 . × − cm and σ a = 3 . × − cm . We set the density of the dyemolecules as N = P j =0 N j = 6 × cm − ≈
10 mMleading to a bulk gain coefficient of approximately g ≈ N · σ e ≈ − at full inversion.In order to study the active configuration we first pumpthe dye molecules with a short, intense pulse of dura-tion 2 ps. After a short delay of 7 ps we probe the struc-ture with a weak broadband pulse of duration 12 fs. Fig-ures 2(a) and (c) show a snapshot of the spatial distribu-tion of the occupation inversion generated by the pumppulse at two perpendicular planes inside the unit cell ofthe active metamaterial. The effective gain coefficient FIG. 2. (color online). (a) Snapshot of the occupation inver-sion ∆ N e in a plane 5 nm below the upper silver fishnet filmjust before probing and (b) the electric field enhancement at710 nm in the same plane; both for a pump-field amplitude of2 . in the structure can be maximized with a good matchingbetween the spatial distribution of the inversion and thatof the plasmon-enhanced electric field amplitude at theemission wavelength λ e [Figs. 2(b) and (d)]. Indeed, wesee from Fig. 2 that such a matching is achieved whenthe pump and the probe have the same electric field po-larization.We remark that the considered planar structure has alow cavity Q -factor ( Q < n = n ′ + in ′′ , first, of thepassive configuration. Note that the metamaterial struc-ture considered in this work is surrounded by air aboveand below the dielectric host; i.e., it is deliberately notplaced on a thin substrate, in order to be symmetric andnonbianisotropic [27]. The spectral variation of n ′ ( n ′′ )in the passive structure is similar to that shown by thecyan solid (dashed) line in Fig. 3(a) (corresponding tothe metamaterial that has dye molecules included but isnot pumped). We find that in this passive case the figureof merit FOM = − n ′ ( λ ) /n ′′ ( λ ) has a maximum value of2 . . . −
714 nm)where the losses in the metamaterial are completely com-pensated. In this region, the active metamaterial exhibits -1012 E ff e c t i v e r e f r a c t i v e i nde x
690 700 710 720 730 740 -20020 P e r m eab ili t y
690 700 710 720 730 740Wavelength (nm)0510 F O M no pumping0.5 kV/cm1.0 kV/cm1.5 kV/cm2.0 kV/cm imaginarypartrealpart increasingpump intensity real partimaginary part (a)(b) FIG. 3. (color online). (a) Real and imaginary part of the re-trieved effective refractive indices of the double-fishnet struc-ture for different pump amplitudes. The peak electric fieldamplitude of the pump increases in steps of 0 . . . a negative absorption [cf. Figure 4(b)] and both the realand imaginary part of the refractive index become simul-taneously negative. Note from Fig. 3(b) that in this casethe FOM diverges at the two wavelength points bound-ing the negative-absorption (amplification) region owingto n ′′ becoming exactly zero at these two wavelengths.To further verify the causal nature of the obtainedeffective parameters we use the method of [28] to cal-culate, based on the Kramers-Kronig relations, the real(imaginary) part of the effective permeability from theimaginary (real) part of the numerically retrieved [26]effective permeability. An example of such a calcula-tion for a pump amplitude of 2 kV/cm (correspondingto the negative-absorption regime) is shown in the insetof Fig. 3(a). The excellent agreement between the results -0.020.000.020.04 n " -0.10.00.10.2 A b s o r p t i on
700 705 710 715 720Wavelength (nm)110100 F O M (a)(b)(c) FIG. 4. (color online). Detailed view of (a) the imaginarypart of the retrieved effective refractive indices n ′′ , (b) theabsorption, and (c) the figures of merit (FOM) for peak pump-field amplitudes close to and above compensation between 1 . . . obtained from the standard retrieval method and thecomplementary Kramers-Kronig approach further con-firms that the extracted parameters do obey causality.Finally, Fig. 4 presents a more detailed look at n ′′ , theabsorption coefficient, and the FOM for pump intensi-ties close to and above complete loss compensation. Wenote that there is a critical amplitude of approximately1 .
85 kV/cm for the pump pulse beyond which the presentmetamaterial configuration becomes amplifying. A fur-ther increase of the pump field up to levels of 2 . . ∗ Corresponding author. [email protected][1] J. B. Pendry, Phys. Rev. Lett. , 3966 (2000).[2] J. B. Pendry, Nature (London) , 579 (2009).[3] K. L. Tsakmakidis, A. D. Boardman, and O. Hess, Na-ture (London) , 397 (2007).[4] V. M. Shalaev, Nat. Photonics , 41 (2007).[5] J. Valentine et al. , Nature (London) , 376 (2008).[6] N. I. Zheludev, Science , 582 (2010).[7] M. I. Stockman, Phys. Rev. Lett. , 177404 (2007).[8] J. Skaar, Phys. Rev. E , 026605 (2006).[9] P. Kinsler and M. W. McCall, Phys. Rev. Lett. ,167401 (2008).[10] N. I. Zheludev et al. , Nat. Photonics , 351 (2008).[11] Y. Sivan et al. , Opt. Express , 24060 (2009).[12] Z.-G. Dong et al. , Phys. Rev. B , 235116 (2009); Appl.Phys. Lett. , 044104 (2010).[13] A. Fang et al. , Phys. Rev. B , 241104 (2009); A. Fang,T. Koschny, and C. M. Soukoulis, J. Opt. , 024013(2010).[14] K. B¨ohringer and O. Hess, Prog. Quantum Electron. ,159 (2008); , 247 (2008).[15] A. Klaedtke, J. Hamm, and O. Hess, Lect. Notes in Phys. , 75 (2004).[16] A. Klaedtke and O. Hess, Opt. Express , 2744 (2006).[17] S. Zhang et al. , Opt. Express , 4922 (2005).[18] S. Zhang et al. , Phys. Rev. Lett. , 037402 (2005).[19] A. Mary et al. , Phys. Rev. Lett. , 103902 (2008).[20] Shortly before the submission of this Letter an importantexperimental work (V. M. Shalaev, CIMTEC 2010, 5 th Forum on New Materials, Montecatini Terme, Italy; June17, 2010) has been presented where negative absorptionin (bianisotropic) metamaterials with n ′ < n ′′ > et al. , Nature (London) , 735 (2010).[21] J. A. McMahon et al. , J. Phys. Chem. C , 2731(2009).[22] P. de Vries and A. Lagendijk, Phys. Rev. Lett. , 1381(1998).[23] P. Sperber et al. , Opt. Quantum. Electron. , 395(1988).[24] M. A. Noginov et al. , Nature (London) , 1110 (2009).[25] E. Gehrig and O. Hess, Spatio-Temporal Dynamicsand Quantum Fluctuations in Semiconductor Lasers (Springer, Heidelberg, 2003) Chap. 7.4.[26] D. R. Smith et al. , Phys. Rev. B , 195104 (2002).[27] C. E. Kriegler et al. , IEEE J. Sel. Top. Quantum Elec-tron. , 367 (2010).[28] J. J. H. Cook, K. L. Tsakmakidis, and O. Hess, J. Opt.A: Pure Appl. Opt.11