Metallo-dielectric core-shell nanospheres as building blocks for optical 3D isotropic negative-index metamaterials
R. Paniagua-Domínguez, F. López-Tejeira, R. Marqués, J. A. Sánchez-Gil
MMetallo-dielectric core-shell nanospheres as buildingblocks for optical 3D isotropic negative-indexmetamaterials
R. Paniagua-Dom´ınguez , F. L´opez-Tejeira , R. Marqu´es andJ. A. S´anchez-Gil Instituto de Estructura de la Materia, Consejo Superior de InvestigacionesCient´ıficas, Serrano 121, 28006 Madrid, Spain Departamento de Electr´onica y Electromagnetismo, Universidad de Sevilla, 41012,Sevilla, SpainE-mail: [email protected]
Abstract.
Materials showing electromagnetic properties that are not attainable innaturally occurring media, the so called metamaterials, have been lately, and stillare, among the most active fields in optical and materials physics and engineering.Among those properties, one of the most attractive is the sub-diffraction resolvingcapability predicted for media having index of refraction of -1. Here we propose a fully3D, isotropic metamaterial with strong electric and magnetic responses in the opticalregime, based on spherical metallo-dielectric core-shell nanospheres. The magneticresponse stems from the lowest, magnetic-dipole resonance of the dielectric shell withhigh refractive index, and can be tuned to coincide with the plasmon resonance ofthe metal core, responsible for the electric response. Since the response does notoriginate from coupling between structures, no particular periodic arrangement needsto be imposed. Moreover, due to the geometry of the constituents, the metamaterial isintrinsically isotropic and polarization independent. It could be realized with currentfabrication techniques with materials such as Silver (core) and Silicon or Germanium(shell). For these particular realistic designs, the metamaterials present negative indexin the range within 1 . − . µ m.PACS numbers: 42.25.Bs, 78.67.Pt, 42.25.Fx, 78.20.Ci a r X i v : . [ phy s i c s . op ti c s ] N ov
1. Introduction
Once the possibility of building a negative-index metamaterial (NIM) was proven in themicrowave regime [1], extraordinary effort has been made to obtain analogous behavioursfor increasingly higher frequencies up to the visible range of the electromagneticspectrum [2, 3]. The major obstacle found when trying to translate ideas to higherfrequencies was how to achieve a strong diamagnetic response in the systems designed.Many attempts to tackle the problem were simple miniaturizations of the canonicaldesigns employed in those first metamaterials operating in microwaves. As an example,the split-ring configurations, or slight variations of it, were reduced up to the nanoscaleto obtain such a magnetic response. Apart from the fundamental limitations inheritedfrom those designs, e.g. anisotropy, and the increasingly complexity in the fabricationprocedures, several drawbacks have been found in this process, some of them beingconsequence of the different behaviour of metals when excited with higher frequencywaves [4]. Precisely this different behaviour also inspired some authors to search fordifferent configurations intended to exploit the plasmonic response to obtain artificialmagnetism. In many of them artificial magnetism is due to coupling between differentplasmonic structures, their drawback thus being the high losses within metallic parts [5].In most cases, the proposed designs are restricted to operate under certain polarizationand incidence conditions [6]-[8], or are not truly three-dimensional materials [9, 10].Moreover, in many cases, the behaviour of the proposed designs stems from couplingbetween the different constituents, thus making particular arrangements necessary. Asa consequence, spatial dispersion effects often appear due to propagation of wavesin the lattice. Lately, some approaches based on plasmonic waveguides supportingnegative-index modes have pushed the frequencies in which left-handed (LH) behaviouris obtained well within the visible spectrum. However, propagation inside such materialwould be limited to the propagation length of the plasmon in the waveguide, thusmaking the design a single layer device, lacking truly three-dimensionality [9, 10].In addition, various attempts have been made towards NIM by exploiting magneticresonances occurring in structures made of high permittivity materials [11]-[13]. Someof them combine these structures with secondary structures providing the electricalresponse [14]-[17] or embed them in a metallic host medium [18], thus having inherenthigh losses. Here we report a design that tackle many of the previously mentionedlimitations. A totally three-dimensional isotropic negative-index metamaterial operatingat optical frequencies, and whose response is due to every isolated “meta-atom“.Therefore, no particular arrangement of them is needed. We study the possiblity ofusing a spherical core-shell configuration to obtain with one single structure both electricand magnetic responses. The core, being metallic, is responsible for the electricalresponse, while the shell, made of a high permittivity dielectric, provides the strongdiamagnetic response. An extension of Mie theory is exploited to rigorously determinethe scattering properties and resonances of the whole spherical core-shell configuration,which essentially determine the effective material properties [19]. Calculations for core-shell structures built up with realistic materials (Ag@Si and Ag@Ge) demonstratethe possibility to obtain NIM operating at 1.2 µ m-1.55 µ m. Since both responsesare attained directly from one single constituent, no particular arrangement of theinclusions is needed. In order to extract the effective parameters of the metamaterial,we will assume both a random and a simple cubic distribution. In the case ofa random distribution Lorentz-Lorenz theory is applied, leading to simultaneouslynegative effective permeability and permittivity for several filling fractions. In thecase of a simple cubic lattice arrangement, finite-element-method is applied to carryout numerical simulations, which fully account for interaction between the periodicallyarranged constituents. Effective material constants are then extracted through standardS-parameter [20, 21] retrieval procedure, and tested to fulfill causality and passivity,thus confirming a true double-negative index. Due to the spherical symmetry of theconstituents, the metamaterial response will be essentially isotropic and polarizationindependent.
2. Optical properties of metallo-dielectric core-shell meta-atoms
Let us examine the scattering of a plane electromagnetic wave (wavelength λ ) froma spherical core-shell particle without any approximation, which indeed can be doneanalytically as an extension of Mie theory, obtained first by Aden and Kerker [22].Figure 1(a) depicts the geometry of one of the basic constituents or “meta-atoms”: ahigh permittivity ( (cid:15) ) dielectric shell is considered with outer radius R (cid:28) λ and thickness T , (cid:15) c and (cid:15) being the dielectric constant of the core and the surrounding medium,respectively. The scattering and extinction efficiencies can be expressed in terms of thematerial and geometrical parameters through the scattering coefficients a l and b l (whichrepresent, respectively, the different electric and magnetic multipolar contributions) as: Q sca = 2 y ∞ (cid:88) l =1 (2 l + 1)( | a l | + | b l | ) (1) Q ext = 2 y ∞ (cid:88) l =1 (2 l + 1) (cid:60) ( a l + b l ) , (2)where y = kR . The mentioned scattering coefficients can be written in terms of thethe spherical Bessel functions of the first ( j l ( x )) and second ( y l ( x )) class and depend on (cid:15) c /(cid:15) , (cid:15)/(cid:15) , R in and R . (cid:60) denotes the real part. Their explicit form is: a l = ψ l ( y )[ ψ (cid:48) l ( ny ) − A l χ (cid:48) l ( ny )] − nψ (cid:48) l ( y )[ ψ l ( ny ) − A l χ l ( ny )] ξ l ( y )[ ψ (cid:48) l ( ny ) − A l χ (cid:48) l ( ny )] − nξ (cid:48) l ( y )[ ψ l ( ny ) − A l χ l ( ny )] (3) b l = nψ l ( y )[ ψ (cid:48) l ( ny ) − B l χ (cid:48) l ( ny )] − ψ (cid:48) l ( y )[ ψ l ( ny ) − B l χ l ( ny )] nξ l ( y )[ ψ (cid:48) l ( ny ) − B l χ (cid:48) l ( ny )] − ξ (cid:48) l ( y )[ ψ l ( ny ) − B l χ l ( ny )] . (4)The Ricatti-Bessel functions introduced are ψ l ( z ) = zj l ( z ), χ l ( z ) = − zy l ( z ) and ξ l ( z ) = zh (1) l ( z ), where h (1) l ( z ) = j l ( z ) + iy l ( z ) is the spherical Hankel function of thefirst class. The coefficients A l and B l are: A l = nψ l ( nx ) ψ (cid:48) l ( n c x ) − m c ψ (cid:48) l ( ny ) ψ l ( n c x ) nχ l ( nx ) ψ (cid:48) l ( nx ) − n c χ (cid:48) l ( ny ) ψ l ( n c x ) (5) B l = nψ l ( n c x ) ψ (cid:48) l ( nx ) − n c ψ l ( ny ) ψ (cid:48) l ( n c x ) nχ (cid:48) l ( nx ) ψ l ( n c x ) − n c ψ (cid:48) l ( n c y ) χ l ( nx ) , (6)where x = kR in , n = (cid:15)/(cid:15) and n c = (cid:15) c /(cid:15) . It is inmediate to realize that all informationabout material and geometrical properties of the core is contained in these A l and B l coefficients. To better understand the physics behind, we first characterize the magnetic resonanceof a nanoshell for a real dielectric material with high refractive index. We choose Silicon,Si, the refractive index of which can be considered constant, n = (cid:112) (cid:15)/(cid:15) ∼ .
5, andlossless, within the near-IR range λ = 1 µm − µm . In figure 1(b) the contribution isplotted to the total scattering efficiency of the magnetic dipolar term ( b ) as a functionof R in and the incidence wavelength. It is the dominant contribution, as can be seen infigure 1(c) for the specific case of R in = 45 nm. A resonance can be clearly seen, thewavelength of which corresponds to that of the Si compact sphere [23] when R in → T (cid:46) R/
3) and decreases.Thus the magnetic resonance of the Si nanoshell can be tuned within a certain range ofwavelengths larger than that of the Si sphere resonance.The behaviour of the EM fields at the magnetic resonance has been calculated byfull-wave (finite-element method) numerical simulations [24]. The result is depicted infigure 1(d) and (e). It can be observed that the electric displacement field rotates inplanes parallel to the equator, thus inducing a strong magnetic moment. This patternclearly shows that the electrical displacement current − iωε E is strongly confined insidethe shell, and rotates along the φ direction around the incident H-field. Such behaviourreveals that the resonance is qualitatively quite similar to the first resonance of adielectric ring, already reported [25], following an LC model with an inductance L ∝ µ R associated to the circulation of the displacement current, and a capacitance C ∝ εR associated to the electric energy confined in the shell. At resonance, a strong magneticmoment along the direction of the incident H-field is generated in the same qualitativeway as in the dielectric ring analyzed [25]. Next we analyze the mutual influence of a metallic core and a dielectric shell. Withregard to the plasmon resonance of the metallic core (i.e., collective oscillations ofconduction electrons), it is well known that for small spheres ( R in (cid:28) λ ) of dielectricpermittivity (cid:15) c ( ω ), embedded in a medium with dielectric constant (cid:15) , the induced dipolemoment is resonant at the frequency ω LSP R such that (cid:15) c ( ω LSP R ) = − (cid:15) [26]. Therefore, Figure 1. (a) The geometry of the problem. (b) Dipolar magnetic contribution tothe total scattering efficiency of a Silicon (Si) nanoshell with outer radius R = 170 nmas a function of the inner radius ( R in ) and wavelength of the incident light. (c)Total scattering efficiency, together with the dipolar electric ( a ) and magnetic ( b )contributions to the scattering efficiency for inner radius R in = 45 nm (indicated in(b) by a dashed white line). (d-e) Near-field plots at the magnetic resonance for theSi shell of (c). (d) Norm of electric field. (e) Out-of-plane component (only non-zerocomponent of the incident magnetic field) of H , together with the electric displacementfield in white arrows. when the metallic sphere is coated with a thick dielectric layer of permittivity (cid:15) , thequantity driving the resonance condition is not (cid:15) c itself but, instead, the ratio betweenthe permittivities of the core and the coating. That is, for a small metal core, theresonance occurs when (cid:15) c /(cid:15) ∼ − (cid:15) Agr ∼ − (cid:15) Si . This happens at a wavelength about 720 nm. Nevertheless, we have tokeep in mind that, strictly, this would only be a good approximation for very small Figure 2.
Optical properties of Ag@Si and Ag@Ge core-shell nanospheres. (a-c) Dipolar electric contribution (a), dipolar magnetic contribution (b), and totalscattering efficiency (c) of a Ag@Si core-shell nanosphere with outer radius R = 170 nmas a function of the inner radius ( R in ) and wavelength of the incident light. (d)Extinction and scattering efficiencies, together with the dipolar electric ( a ) andmagnetic ( b ) contributions to the scattering efficiency for inner radius R in = 47 nm(indicated in (c) by a dashed white line). (e-g) Dipolar electric contribution (e),dipolar magnetic contribution (f), and total scattering efficiency (g) of a Ag@Ge core-shell nanosphere with outer radius R = 190 nm as a function of the inner radius( R in ) and wavelength of the incident light. (h) Extinction and scattering efficiencies,together with the dipolar electric ( a ) and magnetic ( b ) contributions to the scatteringefficiency for inner radius R in = 55 nm (indicated in (g) by a dashed white line). metallic particles with relatively thick coatings. Since we want to push the electricresonance to the IR to make it coincide with a magnetic one stemming from the Si shell,we can take increasingly larger radius for the core. When the size of a metallic particleis increased, there is a redshift in the resonance wavelength that can be explained interms of depolarization effects [26].Figures 2(a)-(c) depicts both, the electric dipolar and the magnetic dipolarcontributions to the scattering efficiency, together with the total scattering efficiencyfor a Ag@Si core-shell system of outer radius R = 170 nm as a function of R in andthe incidence wavelength. It can be clearly seen an overlap between the electric andmagnetic resonances. It happens within 1150 nm and 1300 nm and for inner radiusbetween R in = 40 −
50 nm. We have explicitly plotted the case R in = 47 nm (figure2(d)). An interesting feature that can be observed is that the magnetic resonancedisappears as the inner radius increases. It can be explained by the fact that theelectric displacement field in the core rotates in the opposite direction as it does inthe shell, thus reducing the total magnetic moment generated. Another effect can beseen in the electric dipole contribution. As the inner radius increases the resonance Figure 3.
Near-field plots for a Ag@Si shell with R = 170 nm and T = 123 nm.(a) Norm of the electric field in Log scale at combined electromagnetic resonanceand when the system is excited by a purely electric excitation (up-right, E) and bya purely magnetic one (down-right, H). Colour scale, incidence and polarization arepreserved in all figures. (b) Out-of-plane component of H , together with the electricdisplacement ( D ) field in black arrows at combined electromagnetic resonance andwhen the system is excited by a purely electric excitation (up-right, E) or a purelymagnetic one (down-right, H). Colour scale, incidence and polarization are preservedin all figures. broadens and redshifts as expected. Interestingly, this behaviour changes when thethickness is comparable with R in , and the resonance starts to blueshift. This effect canbe attributed to the fact that, as the dielectric shell becomes thinner, the resonancewavelength tends to that of a sphere without coating, thus blueshifting. Finally, itis important to note that the specific wavelengths at which the electric and magneticresonances overlap are determined by the geometrical parameters of the structure, aswell as by the specific materials used. Therefore, it is possible to tune the wavelengthof overlapping by appropriate choice of these parameters. This gives the design a greatdegree of freedom, and make it plausible to operate at different frequencies. As anexample, a Ag@Ge system with R = 190 nm is presented (figures 2(e)-(h)). In this caseresonances overlap within 1500 nm and 1620 nm for R in = 50 −
60 nm. The particluarcase with R in = 55 nm is plotted in figure 2(h).In figure 3, the near-field pattern at the combined magnetic (shell) and electric(core) resonance is shown for the Ag@Si system. Also plotted are the responses ofthe system to a purely electric and magnetic excitation. These were obtained byplacing a perfect mirror at a distance 3 λ/ λ , respectively, from the center of thestructure. The distinctive behaviour of both contributions is preserved in the combinedelectromagnetic resonance, revealed through the rotating field confined within the Sishell (as in figure 1(e)), together with the dipolar LSPR resonance of the Ag core. Notethat, indeed, the electric displacement in the core and the shell rotates in oppositedirections and that the magnetic near field pattern, figure 3(b), can be explained as acombination of the electric and magnetic contributions. Near-field patterns for core-shellwith Ge covers are similar to those shown here.
3. Calculation of effective parameters
What do we expect for a material consisting of such core-shell nanostructures? Ingeneral it is highly non-trivial to extract the effective constitutive parameters of ametamaterial. Here, two different methods are applied to obtain the effective parametersof a metamaterial composed by the core-shell structures presented. In the first onewe will assume a random arrangement of the constituents, while in the second ametamaterial made of a cubic arrangement of them will be studied, both leading toa negative-index behaviour within certain wavelengths.
For composites made of a cubic or random arrangement of dipolar particles, Lorentz-Lorenz theory is widely used [11, 28], leading to the well known Clausius-Mossottiformulas relating the effective permittivity and permeability with the polarizabilities ofthe particles and the filling fraction f = (4 / πN R , where N is the number of particlesper unit volume: (cid:15) eff − (cid:15) (cid:15) eff + 2 (cid:15) = f α E πR , µ eff − µ µ eff + 2 µ = f α M πR (7)where α E and α M are the electric and magnetic polarizabilities of the spherical particles.For the core-shell structures considered R/λ ∼ /
7. Therefore, we expect them to bewell within the approximation considered in the theory, thus behaving essentially aselectric and magnetic point dipoles. The electric and magnetic polarizabilities, α E and α M , respectively, are directly proportional to the scattering coefficients a and b [factor i ( k / π ) − ]. In figure 4(a) and (b), we have plotted the polarizabilities for a core-shellconfiguration of R in = 47 nm and R = 170 nm. It is clear from the graph that thereis a spectral region where negative electric and magnetic polarizabilities are obtained.Interestingly, as mentioned before, no particular arrangement for the constituents isnecessary to build the metamaterial, since the resonant electric and magnetic responsesarise from each core-shell structure separately. Thus, assuming a random distributionwe can compute the effective parameters from (7) for different filling fractions. Figures4(c)-(e) depict the calculated effective permittivity, permeability, and refractive indexfor metamaterials made up by these Ag@Si core-shells with filling fractions f = 1 / f = 0 . f = 2 /
3. The metamaterial has simultaneously negative permittivityand permeability for filling fractions higher than f = 1 /
3. For a filling fraction of f = 2 /
3, the system, moreover, has | n eff | ∼
1, although this relatively high fillingfraction would be in the limit of validity of Clausius-Mossotti formulas. In order toquantify the losses of the system one can compute the so called figure of merit, definedas f.o.m. = |(cid:60) ( n eff ) | / (cid:61) ( n neff ). The computed f.o.m in the left-handed spectralregion for this filling fraction is depicted by the black curve, reaching a maximumvalue of f.o.m. ∼ .
71, corresponding to (cid:60) ( n eff ) ∼ − .
8. If Ge is used instead ofSi, the NIM behaviour starts with lower filling fractions, due to a stronger electricand magnetic responses (see figure 5 and compare the polarizabilities with those of
Figure 4. (a)-(e). Electric and magnetic polarizabilities for a Ag@Si core-shell with R in = 47 nm and R = 170 nm and effective parameters for a metamaterial composedof a random arrangement of these structures. (a) Real (red) and imaginary (blue)parts of the electric polarizability. (b) Real (red) and imaginary (blue) parts of themagnetic polarizability. (c)-(e) Real (red) and imaginary (blue) parts of the effectivepermittivity (c), permeability (d) and refractive index (e) of a metamaterial withseveral filling fractions: f = 1 / f = 0 . f = 2 / f.o.m. for the highest filling fractionin the left-handed spectral region. figure 4). Superlensing capabilities are predicted for filling fractions lower than f = 0 . f.o.m. for the highest filling fraction is plotted as a blackcurve. In this case it reaches a maximum value of f.o.m. ∼ .
75, corresponding to (cid:60) ( n eff ) ∼ − .
5. Although being far from the best f.o.m. values reported for double-fishnet metamaterials ( f.o.m. ∼
3) [7, 29], the predicted values are reasonably good,with the obvious advantage of isotropy of this proposal.0
Figure 5. (a)-(e) Electric and magnetic polarizabilities for a Ag@Ge core-shell with R in = 55 nm and R = 190 nm and effective parameters for a metamaterial composedof a random arrangement of these structures. (a) Real (red) and imaginary (blue)parts of the electric polarizability. (b) Real (red) and imaginary (blue) parts of themagnetic polarizability. (c)-(e) Real (red) and imaginary (blue) parts of the effectivepermittivity (c), permeability (d) and refractive index (e) of a metamaterial withseveral filling fractions: f = 0 .
25 (dotted), f = 0 . f = 0 . f.o.m. for the highest filling fractionin the left-handed apectral region. To further test the metamaterial design, we consider now core-shell nanospheresarranged periodically in the vertices of a cubic lattice. We take Germanium as thehigh permittivity coating and choose the geometrical parameters as R = 190 nm and R in = 55 nm. We consider the case in which the cubic lattice has a period d = 385 nm(corresponding to a filling fraction f ∼ . Figure 6. (a) Real and imaginary parts of the retrieved effective index of refraction of ametamaterial made by Ag@Ge core-shell nanospheres of R = 190 nm and R in = 55 nmarranged in the vertices of a cubic lattice with period d = 385 nm, corresponding toa filling fraction f ∼ .
5. (b) Detail of the normalized near electric field norm (colorscale, red →
1, blue →
0) and electric displacement field (indicated by white arrows) inthe periodic structure. of the simulation is 1.28 to 1.7 µ m and the wave impinges at normal incidence. Theeffective material parameters are retrieved from the complex reflection and transmissioncoefficients through the usual equations found in the literature [20, 21]. While theimaginary part is unambiguously computed from the simulation, once a sufficiently highnumber of unit cells in the propagation direction have been considered, the real partof the index of refraction must be determined with much more care. Concretely, onemust pay special attention to ensure that the retrieved refractive index fulfills causalityand passivity. In this way we have obtained two bounding values for the real part ofthe index, so it fulfills the two mentioned requirements. For a detailed description onhow the index was obtained, see section Methods. In figure 6, the obtained bounds forthe refractive index are plotted. With this retrieval technique, a negative index spectralregion is predicted. In the more conservative prediction, corresponding to the higherbound, the real part of the index reaches a minimum value of n eff ∼ − .
77, with a f.o.m. ∼ .
88, corresponding to a wavelength of 1.347 µ m. Although the values for theindex are lower than those predicted by Clausius-Mossotti, the negative index frequencyband is present and confirms the core-shell design as meta-atom candidate for buildinga negative index metamaterial. It needs to be pointed out that we are far from thelimit of filling fractions that can be obtained with non-overlapping spheres (cannonball2pile). Since for interparticle distances as low as 5 nm, we found that coupling does notprevent the appearance of the magnetic resonance, we expect to reach higher negativevalues of the index in other periodic configurations with higher filling fractions. Furtherstudy of periodic configurations with other homogenization approaches is intended forfuture work.
4. Concluding remarks and fabrication possibilities
Obtaining a three-dimensional isotropic metamaterial having negative index of refractionin the optical part of the spectrum has been one of the major challenges for scientistand engineers devoted to electromagnetism in the last decade. Here we presented a newdesign based on core-shell nanospheres that operates in the near-infrared, which tacklemany of the previously found problems, namely isotropy, polarization Independence andlack of three-dimensionality. In our system, the effective response of the metamaterialis due to every isolated “meta-atom”. Therefore, no particular arrangement of theconstituents is needed. Specifically, we demonstrated with realistic materials that,for a random arrangement, the system achieves double-negative index of refraction fordifferent filling fractions and that super-resolution is possible. We also tested the validitywhen a very simple periodic realization is assumed. Although the achieved values in thelatter case are worse than predicted by Clausis-Mossotti for the same filling fractions,we haven’t explored here some other periodic configurations that are expected to givea stronger response.With regard to building these metamaterials, current Silicon fabrication techniquesallow the realization of complex nanostructures such as hollow nanospheres [30] andopals [31]. In some of the processes, the starting point are Silica (SiO ) nanostructures,as is the case in [31], in which Si opals are fabricated by magnesiothermic reductionof SiO opals. Since silver nanospheres have been successfully covered with SiO invariable thicknesses [32], there are plausible ways to realize the metamaterial proposedhere, at least with Si covers. Concerning the fabrication of Ge shells instead, a layer ofa different material, suitable to grow it, can be added between the core and the shell.This layer, if thin, would not affect excessively the physical response of the system,opening the possibility of fabrication of this system as well. The underlying physicalprinciples can of course be exploited at lower (far-IR and terahertz) frequencies, atwhich some dielectric materials exhibit very large refractive indices and certain materials(e.g. polar crystals, doped semiconductors) behave as plasmonic metals. Therefore, theresults presented pave the way towards potential isotropic three-dimensional opticalmetamaterials designed on the basis of the physics underlying the doubly-resonantmetallo-dielectric configuration.3
5. Methods
Extinction and scattering cross sections based on the extended Mie theory werecalculated with Wolfram Mathematica 8. The near-field plots in figures 1 and 3 werecalculated using the RF module of COMSOL Multiphysics v4.0a. The computationaldomain consisted of four concentric spheres which defined four subdomains. The radiiof the spheres were R in , R , 2 R and 2 . R . From inner to outer, the subdomainsrepresented the core, the shell and the embedding medium (air). The last domainwas set to a spherical PML, which absorbed all scattered radiation. The incidentradiation was defined as a plane wave. The mesh was constructed with the softwarebuilt-in algorithm, which generates a free mesh consisting on tetrahedral elements. Themaximum element edge size was set to 15 nm in the whole core-shell structure, with agrowth rate of 1.35, meaning that elements adjacent to a given one should not be biggerthan 1.35 times the size of it. For the PML and air domains, the maximum elementedge size was 50 nm. All mesh sizes are below the value recommended in the programspecifications, which sets a maximum edge size of 1 /
10 of the effective wavelength for acorrect meshing. Finally, for the simulation of wave propagation along a N-unit-cell thickslab, COMSOL was also used. In this case, the computational domain consisted of twoconcentric spheres representing the core-shells, and a right rectangular prism of widthand height d = 385 nm, and length L = N d . Bloch periodic boundary conditions wereset in the directions perpendicular to the propagation and two ports activated in thedirection of propagation that allow to compute the reflection and transmission complexcoefficients. In these cases the maximum element edge size was set to 40 nm in the core-shell subdomains, 50 nm in the ports boundaries, and 100 nm in the boundaries whereperiodic boundary conditions were applied. A correct solution of the problem requiresan identical meshing for the pairs of boundaries where periodic boundary conditionsare applied. In all cases PARDISO solver was used. As an example, for 9-unit-cell slabsimulation, the total mesh consisted of 149064 elements, and the calculation involved948768 degrees of freedom, requiring almost 18 GB of memory.
The effective refractive index for the metamaterial structure, shown in figure 6, werecalculated from complex transmission and reflection coefficients following [21]. Asmention before, we performed simulations of wave propagation across infinite slabsof different thicknesses. After a sufficient number of unit cells is considered we getconvergence of the results. We needed to consider up to 9 unit cells in the directionof propagation to get convergence. Then, we performed the simulation for 11 and 13unit-cell-thick slabs.It is well known that, when extracting the effective index through transmissionand reflection coefficients, an ambiguity arises related to the branches of the complexlogarithm function. This ambiguity only affects the determination of the real part ofthe index, the imaginary one being univocally defined. It is a common assumption4
Figure 7.
A very large number of branches of the real part of the retrieved refractiveindex for 9 (red asterisks), 11 (black squares) and 13 (blue circles) unit-cell-thickmetamaterial slabs, composed by Ag@Ge core-shell nanospheres of R = 190 nm and R in = 55 nm arranged in the vertices of a cubic lattice with period d = 385 nm,corresponding to a filling fraction f ∼ .
5. The branches that are independent ofthe thickness of the slab are indicated by a surrounding ellipse. The dashed blueline represents the real part obtained from the imaginary part by Kramers-Kronigrelations, plus a constant factor to fit the only thickness-independent index showinggood agreement. that the real part of the index can be found plotting different branches for differentthicknesses. The physical index is assumed to be the one independent of the thicknessof the slab. However, this assumption may be ambiguous as well. The reason is that, ifone plots a sufficiently high number of branches one will find more than one unique indexindependent of the thickness. In figure 7 this situation is shown. We have plotted a verylarge number of branches corresponding to the Ag@Ge core-shell periodic configurationof section 3.2. It can be clearly seen that there is not only one case in which the indexis independent of the thickness. Therefore, although this requirement of thickness-independence of the retrieved index is clearly physically acceptable, we should not takeit as a sufficient condition.In fact, we should impose more conditions to the retrieved index. It must, no doubt,fulfill the basic requirement of causality, expressed mathematically by the Kramers-Kronig relations. Since, fortunately, we can unambiguously determine the imaginary5part of the index, it is possible to apply Kramers-Kronig relations to get the real partof the index [33]. We need to point out that, actually, by this technique one can onlycompute the corresponding real part up to a constant factor. The value of this factor,interpreted physically as the value of the index when the frequency tends to infinity,needs to be determined somehow else. The question is, then, if any of the indices thatdo not depend on thickness also fullfils causality. In figure 7, the real part of the index,retrieved through Kramers-Kronig relations applied to the imaginary part, is plotted ontop of the only thickness-independent index for which the fitting is in good agreement(the constant factor added is 6.35). It is apparent that the agreement is quite good.It should be noticed that this index is the only one for which the fitting is good, thusfulfilling the the causality constrain.However, one more constrain must be imposed in order to get an acceptable effectiveindex. The resulting effective medium should be passive. This condition demands nospontaneous generation of energy, and mathematically relates the real and imaginaryparts of the effective index, with the real and imaginary parts of the effective impedanceof the slab. This impedance can be computed also from the complex reflection andtransmission coefficients [21]. Applying this condition we find that none of the indicesretrieved directly from the transmission and reflection complex coefficients fulfill bothcausality and passivity. Nevertheless, one can still compute a physically acceptableeffective index of refraction. If one uses the passivity condition, it is possible to imposeupper and lower bounds to the constant factor in Kramers-Kronig relations. An indexwith a real part given directly by Kramers-Kronig relations, applied to the univocallycomputed imaginary part, and a constant factor between the two bounds will necessarilyfulfill both conditions. We found that, in fact, the upper and the lower bounds imposedby passivity are quite close in value, being a restrictive condition on the possible valuesof the index of refraction. The two boundary values for the index are plotted in figure6(a).
Acknowledgments
The authors acknowledge support both from the Spain Ministerio de Cienciae Innovaci´on through the Consolider-Ingenio project EMET (CSD2008-00066)and NANOPLAS (FIS2009-11264), and from the Comunidad de Madrid (grantMICROSERES P2009/TIC-1476). R. Paniagua-Dom´ınguez acknowledges support fromCSIC through a JAE-Pre grant. We are indebted to an anonymous referee for helpfulsuggestions on the convergence of the COMSOL results in figure 6.
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