Optical Magnetism and Negative Refraction in Plasmonic Metamaterials

$Optical Magnetism and Negative Refraction in Plasmonic Metamaterials$

Solid State Communications 146 (2008) 208–220 www.elsevier.com/locate/ssc

Optical magnetism and negative refraction in plasmonic metamaterials

Yaroslav A. Urzhumov, Gennady Shvets∗

Department of Physics, The University of Texas at Austin, Austin, TX 78712, United States

Received 31 July 2007; accepted 16 August 2007 by the Guest Editors Available online 12 February 2008

Abstract

In this review we describe the challenges and opportunities for creating magnetically active metamaterials in the optical part of the spectrum. The emphasis is on the sub-wavelength periodic metamaterials whose unit cell is much smaller than the optical wavelength. The conceptual differences between microwave and optical metamaterials are demonstrated. We also describe several theoretical techniques used for calculating the effective parameters of plasmonic metamaterials: the effective dielectric permittivity eff(ω) and magnetic permeability µeff(ω). Several examples of negative permittivity and negative permeability plasmonic metamaterials are used to illustrate the theory. c 2008 Elsevier Ltd. All rights reserved.

PACS: 42.70.-a; 41.20.Gz; 78.67.Bf

Keywords: A. Nanostructures; C. Crystal structure and characterization; D. Optical properties

1. Introduction negative permeability µeff < 0) and a continuous thin metal wire [11] (responsible for the negative permittivity eff < 0). A new area of electromagnetics has recently emerged: Remarkably, even in the first microwave realizations of the NIM electromagnetic metamaterials. The emergence of this new field its unit cell was strongly sub-wavelength: a/λ ≈ 1/7, where happened in response to the demand in materials with the a is the lattice constant and λ is the vacuum wavelength. In electromagnetic properties that are not available in naturally fact, the condition of a λ must be satisfied in order for occurring media. One of the best known properties unattainable the effective description using eff and µeff to be sensible. If without significant metamaterials engineering is a negative the electromagnetic structure consists of larger unit cells with refractive index. The main challenge in making a negative a ≥ λ/2nd , where nd is the refractive index of a substrate index metamaterial (NIM) is that both the effective dielectric onto which metallic elements are deposited (e.g. Duroid in permittivity eff and magnetic permeability µeff must be some of the recent microwave experiments [8]), they cannot negative [1]. Numerous applications exist for NIMs in every be described by the averaged quantities such as permittivity spectral range, from microwave to optical. Those include and permeability. It is the high λ/a ratio that distinguishes “perfect” lenses, transmission lines, antennas, electromagnetic a true meta-material from its more common cousin, photonic cloaking, and many others [2–5]. Recent theoretical [6,7] and crystal [12,13]. experimental [8] work demonstrated that for some applications Developing NIMs for optical frequencies, however, has such as electromagnetic cloaking it may not even be necessary proven to be much more challenging than for microwaves. to have a negative index: just controlling the effective magnetic Microwave structures can be made extremely sub-wavelength permeability suffices. using several standard microwave approaches to making a First realizations of NIMs were made in the microwave sub-wavelength resonator: enhancement of the resonator’s part of the spectrum [9]. The unit cell consisted of a capacitance by making its aspect ratios (e.g., ratio of the metallic split-ring resonator (SRRs) [10] (responsible for the SRR’s radius and gap size) high, inserting high-permittivity materials into SRR’s gap, etc. These microwave techniques are ∗ Corresponding author. briefly illustrated in Section 1.1 using a simple SRR shown in E-mail address: [email protected] (G. Shvets). Fig. 1. Simply scaling down NIMs from microwave to optical

Fig. 1. Magnetic field distribution inside a lattice of split-ring resonators. Resonator parameters: periods ax = ay = 1.2 mm, ring size W = 0.8 mm, ring thickness T = 80 µm, gap height H = 0.44 mm, gap width G = 80 µm. The gap is filled with a high-permittivity dielectric d = 4. At the magnetic cutoff (µeff = 0) shown here the vacuum wavelength λ = 1.57 cm. wavelengths does not work for two reasons. First, to develop a of the most serious issues in the plasmonic regime: as it turns λ/10 unit cell requires much smaller (typically, another factor out, even a small amount of losses can drastically reduce the 10) sub-cellular features such as metallic line widths and gaps. magnetic response and prevent access to the µeff < 0 regime. For λ = 1 µm that corresponds to 10 nm features which are Presently, the only theoretical method of characterizing plas- presently too difficult to fabricate. For example, the classic SRR monic metamaterials is by carrying out fully electromagnetic has been scaled down to λ = 3 µm, but further wavelength scattering simulations, obtaining complex transmission (t) and reduction using the same design paradigm seems unpractical. reflection (r) coefficients, and then calculating the effective pa- Second, as the metal line width approaches the typical skin rameters eff and µeff of the metamaterial from r and t [24,25]. depth lsk ≈ 25 nm, metal no longer behaves as an impenetrable Such direct approach is lacking the intuitive appeal and rigor of perfect electric conductor (PEC). Optical fields penetrate into the earlier microwave work that provided semi-analytic expres- the metal, and the response of the structure becomes plasmonic. sions for both eff [11] and µeff [10]. Moreover, the extracted These difficulties have not deterred the researchers from parameters of a periodic structure exhibit various artifacts such trying to fabricate and experimentally test magnetically-active as anti-resonances [26] that make their interpretation even less and even negative index structures in the infrared [14–17] intuitive. and even visible [18,19] spectral regions. Because fabricating Recent progress has been made in rigorously calculating intricate metallic resonators on a nanoscale is not feasible, the quasi-static dielectric permittivity qs of plasmonic much simpler magnetic resonances such as pairs of metallic nanostructures [22,23,27] exhibiting optical magnetism. In fact, strips or wires [14,16,17] or metallic nanoposts [18] have been the frequency dependence of qs(ω) of an arbitrary periodic used in the experiments. Unfortunately, so far there has been nanostructure can be reduced to a set of several numbers [28, no success in producing sub-wavelength magnetically-active 29] (frequencies and strengths of various electric dipole metamaterials in the optical range satisfying a < λ/2nd , where resonances) that can be obtained by solving a generalized nd is the refractive index of a dielectric substrate or filler eigenvalue equation. However, there has been a very limited onto which magnetic materials are deposited. The reason for progress in calculating the magnetic permittivity of such this is very simple [20]: in the absence of plasmonic effects, structures. Only for several specific structures has µeff(ω) simple geometric resonators (such as pairs of metal strip or been calculated [30–32], usually under highly restrictive wires) resonate at the wavelength λ ∼ 2nd L, where L is the assumptions. characteristic size of the resonator. In other words, “simple” It would be highly desirable if a simple formula expressing metallic resonators are not sub-λ resonators. µeff(ω) in terms of several easily computable parameters could Fortunately, metallic resonators can be miniaturized using be derived. The increased complexity of new devices based plasmonic effects [20–23]. In the optical regime metals can on magnetic metamaterials [8] further highlights the need for no longer be described as perfect electric conductors. Instead, such rapid and intuitive determination of µeff(ω). Simply put, they are best described by a frequency-dependent plasmonic it is important to obtain rigorous expressions for µeff(ω) that dielectric permittivity (ω) ≡ 0 + i00. For low-loss metals are similar in their form and complexity to those available for 00 0 0 such as silver, | | and −1. Therefore, there qs(ω). In addition, the formulas for qs(ω) themselves need is a significant field penetration into metallic structures that further refinement. For example, when the size of a plasmonic are thinner√ than or comparable with the skin depth lsk = nanostructures becomes a sizable fraction of the wavelength (as λ/2π −0 ≈ 25 nm. In metals (ω) is determined by the small as λ/6), the assumption of eff ≈ qs loses accuracy. Drude response of the free electron to the optical fields. When One of the reasons is that the electric dipole resonances acquire the kinetic energy of the oscillating free electrons becomes a considerable electromagnetic red shift [33] that needs to be comparable to the energy of the electric field, one can refer accounted for. The main objective of this paper is to derive, in to the structure as being operated in a plasmonic regime. One the limit of a λ, accurate formulas for eff(ω) and µeff(ω). 210 Y.A. Urzhumov, G. Shvets / Solid State Communications 146 (2008) 208–220

1.1. Why is optical magnetism difficult to achieve? We now investigate if this structure can be naturally scaled down to optical wavelengths. Because high-d dielectrics are Before trying to understand how one can make a sub- not available at the optical frequencies, it is assumed that wavelength magnetic material in the optical part of the the SRR’s gap is air-filled. Instead of using PEC boundary spectrum, it is useful to review how magnetism is accomplished conditions at the metal surface, metal is modelled as a in microwave metamaterials. Consider one of the most basic lossless negative- dielectric with (ω) taken from the standard design elements of negative index metamaterials: split ring tables [35]. By scanning the value of the plasmonic resonator (SRR). The particular design shown in Fig. 1 has (physically equivalent to scanning the frequency), we have been recently used [8] for making an invisibility cloak. For calculated the corresponding size of the SRR that has the same simplicity, we’ve simulated an infinite array of two-dimensional geometric proportions as the one shown in Fig. 1. Losses have (infinite in the z-direction) SRRs, with all sizes given in the been neglected for this calculation: ≡ 0. Nontrivial solutions caption to Fig. 1. Metal resistivity has been neglected and of Eq. (1) corresponding to the magnetic cutoff were found PEC boundary conditions used at the metal surface. The non- only for < res ≡ −330 corresponding to λ > λres ≡ vanishing components of the electromagnetic field are H , E , z x 3 µm for silver. As will be shown below, res corresponds and Ey. The following eigenvalue equation has been solved: to the electrostatic resonance of an SRR. The electrostatic potential φ corresponding to the electrostatic resonance and the 1 ω2 −∇E · ∇E H = H , corresponding electric field EE = −∇φ are shown in Fig. 2(a). z 2 z (1) c Magnetic field distribution for λ = 3.44 µm is presented in where (x, y) is the spatially nonuniform dielectric permittiv- Fig. 1. Clearly, magnetic field penetrates deep into the metal. ity. Finite Elements Frequency Domain (FEFD) code COM- This specific wavelength has been chosen because for λ = SOL Multiphysics [34] was used for solving Eq. (1). 3.44 µm the ratio of the wavelength to the SRR’s width is the Periodic boundary conditions have been applied to the cell same as for the microwave design: 1/20. One may be tempted to boundaries. Therefore, the calculated eigenfrequency f = assume that we’ve demonstrated a successful down-scaling (by ω/2π = c/λ = 19 GHz (λ = 1.56 cm) corresponds to a factor 4400) of a NIM from microwave to optical frequencies, the magnetic cutoff µeff( f0) = 0. Assuming that the wave and that there are no conceptual differences between the λ = is propagating in the x-direction, the dispersion relation for 1.56 cm and λ = 3.44 µm. To see why this is not the case, the entire propagation band can be computed by setting the consider the implications of the plasmonic of metals at optical phase-shifted boundary conditions between the x = −ax /2 and frequencies. x = +ax /2 sides of the unit cell. In this particular case, the For simplicity assume a collisionless Drude model for metal: 2 propagation band extends from f = f0 upwards in frequency (ω) = b − ωp/ω(ω + iγ ) with γ = 0. Then the total energy and corresponds to µeff ≥ 0 and eff > 0. The µeff < 0 region is density just below f . Because the ratio of the SRR size to wavelength 0 Z 2 E2 ! is approximately 1/20, and because its magnetic response is so 2 Hz E ∂(ω) Utot = d x + strong, it can be characterized as a sub-wavelength magnetic Vm +Vv 8π 8π ∂ω resonator. Z E2 Z E2 2 Z E2 2 E 2 E ωp 2 E Why is this structure subwavelength? The√ natural resonance = d x + b d x + d x 8π 8π ω2 8π frequency of a resonator scales as 1/ L RCR. Therefore, Vv Vm Vm its size can be reduced from its “natural” λ/2 value by Z H 2 + 2 z either increasing its inductance L R or capacitance CR. For d x , (2) + 8π this particular design the largest increase comes from the Vm Vv capacitance which is increased by a factor d H/G ≈ 36. Note where Vv and Vm are the vacuum and metallic volumes, that both the narrow gap and high permittivity of the dielectric respectively. The physical meaning of the four terms in Eq. (2) placed in the gap enhance the capacitance and reduce the is as follows: the first two terms represent the energy of the resonant frequency. The role of the dielectric filling is verified electric field UE , the third one represents the kinetic energy Uk by an additional numerical simulation (not shown) in which the of plasma electrons, and the fourth one represents the magnetic d = 4 dielectric filler is removed from the gap. The resonant energy Um. Because both the kinetic and magnetic energies 2 2 2 E E 2 −2 frequency increases by a factor 1.8 confirming that most of (with Hz = c /ω |∇⊥ × E| ) scale as ω , these two terms the capacitance comes from the gap region. Note that at the can be grouped into UL = Um + Uk. Again, at the resonance 2 2 resonant frequency L RhI i = hQ i/CR, the average electric the “capacitive” energy UE matches the “inductive” energy UL . and magnetic energies are equal to each other. This intuitive For the most relevant cases of || 1 the following rules result is confirmed by the numerical simulations. The fact that can be established for plasmonic structures: (a) most of the the average magnetic energy constitutes a significant fraction capacitive energy UE resides in the vacuum region (outside of of the total energy (one-half) explains the strong magnetic the plasmonic material); (b) most of the magnetic energy Um response of the structure. The results of this dissipation-free also resides in the vacuum region. Of course, the same is true for simulation are not significantly affected by the finite metallic the microwave structures. The main difference between the two losses because microwaves penetrate by only a fraction of a types of structures comes from the kinetic energy Uk present in micron into a typical metal (e.g., 0.45 µm into copper). plasmonic but absent from microwave structures. Y.A. Urzhumov, G. Shvets / Solid State Communications 146 (2008) 208–220 211

Fig. 2. Potential distribution φ inside a lattice of (a) split-ring resonators (SRRs), (b) split rings (SRs), and (c) metal strips separated by a metal ﬁlm corresponding E to electrostatic resonances responsible for the magnetic response. Arrows: E = −∇φ. Electrostatic resonances occur at (a) res ≡ −330 (λ = 3 µm) for SRR, (b) res ≡ −82 (λ = 1.5 µm) for SR, and (c) res ≡ −8.8 (λ = 0.5 µm) assuming that the plasmonic material is silver.

It is instructive to compare the values of UE , Um, and Uk of narrowing the propagation band. Even small losses tend to for the plasmonic structure. From the results of the numerical destroy such bands. To get a qualitative estimate of the role of simulation shown in Fig. 1(b) it is found that UE = 0.5Utot, resistive losses we assume a finite γ in the Drude model. The Uk = 0.32Ut , and Um = 0.18Ut . As noted earlier, for group velocity can be roughly estimated as vg/c ∼ Um/Ut . the microwave SRR UE = Um = 0.5Ut . Therefore, the The propagation band is assumed to be destroyed by losses if distinction between the scaled-down plasmonic structure and its the transit time across a single cell is longer than the decay time −1 −1 microwave counterpart is the contribution of the kinetic energy γ , or ax /vg > γ . This results in the following condition Uk of the Drude electrons vs that of the magnetic energy Um. In for achieving optical magnetism in a lossy system: the present example, Uk exceeds Um by almost a factor 2. Thus γ 00 hωa i−1 it is fair to say that the present structure operates in a stronly ≈ x + 0 < 1 Tp . (3) plasmonic regime. This is quite remarkable, given that the ω c operating wavelength λ = 3.44 µm is fairly long. To quantify This condition is trivially satisfied for any sub-wavelength the plasmonic effects in a magnetic structure, we introduce structure as long as Tp is of order unity. It becomes increasingly the plasmonic parameter Tp = Uk/Um. The structure can be difficult to satisfy for the structures operated in a strongly said to operate in a strongly plasmonic regime when T > 1. plasmonic regime as will be shown below. Note that in the Plasmonic SRR can be made even more sub-wavelength by absence of losses strong magnetic response of a sub-wavelength reducing the operating wavelength to λ = λres. Of course, this SRR can be achieved for any λ > λres. happens at the expense of the magnetic energy: the plasmonic Several instructive conclusions can be drawn from the above parameter Tp diverges as the structure shrinks and the operating examples of the large (microwave) and small (mid-infrared) wavelength approaches that of the electrostatic resonance. At rings. First, for given geometry of the unit cell there exists the the electrostatic resonance Tp = ∞ and UE = Uk. The analogy shortest wavelength λres for which a strong magnetic response between the plasmonic energy and inductance energy has been is expected. This wavelength corresponds to the electrostatic suggested earlier [36]. resonance of the structure, and for the SRR geometry shown in The importance of having a reasonably small plasmonic Fig. 1 λres = 3 µm. Strong optical response can be obtained parameter for making a magnetic resonator becomes apparent for any λ > λres, with the actual dimensions of the ring when losses are taken into account. Achieving µeff = 0 adjusted accordingly. The positive result here is that even for (qualitative threshold for a strong magnetic response) sets a a very sub-wavelength ring (λ/20) the plasmonic parameter threshold for the magnetic energy Um in the resonator. The total Tp ≈ 2 is modest and, therefore, finite losses are not affecting energy Utot = 2(1 + Tp)Um increases with Tp. This reduces the magnetic response. The negative result is that the standard the group velocity of the propagating mode, with the effect SRR design cannot be used for obtaining magnetic response 212 Y.A. Urzhumov, G. Shvets / Solid State Communications 146 (2008) 208–220 for visible/near-infrared frequencies because the electrostatic intricate SRR designs correspond to extremely large negative resonance occurs in mid-infrared. values of res, they do not exhibit a strong magnetic response The rule of thumb is that the more elaborate is the design in the visible/near-infrared parts of the spectrum. Therefore, of the plasmonic structure, the longer is the wavelength of the the transition from SRRs (see Fig. 2(a)) to simple pairs of electrostatic resonance. For example, for a simplified version strips (see Fig. 2(c)) is not just a matter of fabricational of the SRR (no vertical capacitor-forming strips, similar to the convenience: it is physically necessary for making optical one used earlier [37]) it is found that the electrostatic resonance NIMs. The price one pays for using the simplified structures is occurs at λres = 1.5 µm (corresponding res = −82). The that they are no longer sub-wavelength for λ λres. Therefore, electrostatic potential φ and the corresponding electric field one is forced to operate close to λres. The price for that is EE = −∇φ are shown in Fig. 2(b). Using a typical dielectric the significant plasmonic effects that enhance the plasmonic substrate or a filler with d = 2.25 would reduce the resonant parameter Tp and make the bandwidth of the magnetic response permittivity to res = −186 (corresponding to λres = 2.25 µm very narrow. That makes these near-resonant structures highly for Ag). At the same time, this design simplification comes at susceptible to losses according to Eq. (3). Thus, plasmonic a cost: for very long wavelengths (microwave) such split ring effects play two roles in optical magnetism. Their positive role (SR) is only moderately sub-wavelength, with λ/W = 5.2. is in turning simple metallic structures into sub-wavelength Again, using a filler with d = 2.25 would result in λ/W = 7.8. resonators. Their negative role is in enhancing the effects of If a more sub-wavelength resonator is required, one needs to losses. operate in the vicinity of λres. The drawback of operating too Thus, the role of an optical NIM designer is to choose close to λres is a high plasmonic parameter Tp and, therefore, a structure that (a) has a λres in the visble/mid-IR, and susceptibility to resistive losses. (b) is sufficiently sub-wavelength in the PEC limit (long λ) Of course, the structure can be simplified even further: that the structure does not need to be operated too close from a split ring to a pair of metal strips [15,20] or a to λres. The electrostatic resonance at λres is the natural pair of metal strips separated by a thin metal film [27]. starting point for computing the optical response of the sub- The advantage of this structural simplification is that the wavelength metamaterial. In the rest of the paper we present magnetically-active plasmonic resonance is pushed even further an analytic perturbation theory of the electric and magnetic into the visible/near-infrared part of the spectrum: λres = responses of a plasmonic nanostructure in the optical frequency 0.5 µm without a dielectric filler and λres = 0.7 µm with the range that uses the electrostatic resonances as the expansion d = 2.25 filler. The potential distribution for the Strip Pair- set. Section 2 describes several techniques of calculating the One Film (SPOF) structure is shown in Fig. 2(c). Because of the effective dielectric permittivity eff(ω) ≈ qs(ω) in the quasi- promise of the SPOF for NIM development, we have scaled the static regime. Section 3 describes the quasi-static calculation of structure in nanometers. Of course, the results of electrostatic the effective magnetic permeability µeff(ω). simulations can be plotted with an arbitrary spatial scale (as was done in Fig. 2((a), (b)) because there is no spatial scale 2. Effective quasi-static dielectric permittivity of a plas- in electrostatics. In the PEC limit (long wavelength) the ratio monic metamaterial of the wavelength to period was found to be λ/ax = 1.85 at the magnetic cutoff for the SPOF structure without a filler, Several theoretical techniques are available for calculating and λ/ax = 2.8 with the d = 2.25 filler. Therefore, the qs(ω) of plasmonic nanostructures: (a) the “capacitor” SPOF structure presents a unique opportunity for developing a approach described in Section 2.1: voltage is imposed strongly sub-wavelength (λ/10 − λ/6) optical (visible to near- across the unit cell of a structure, and the effective IR) NIM. As was recently shown [27], the addition of a thin capacitance is evaluated; (b) the method of homogenizing film modifies the electric response of the double-strip structure electrostatic equations using a multi-scale expansion described and turns this magnetically-active metamaterial into a true sub- in Section 2.2, with the two independent spatial variables: the wavelength (ax = λ/6) NIM. The simple qualitative remarks intra-cell (small scale) variable ξE and the inter-cell (large scale) presented above explain why. Note that no optical NIM with variable XE; (c) and the method of electrostatic eigenvalues, and the cell size smaller that λ/2.5 has ever been experimentally its extension to plasmonic structures with continuous plasmonic demonstrated [14–17]. phase described in Section 2.3. We summarize this section by noting that optical magnetic resonators are conceptually very different from their microwave 2.1. The “capacitor” model counterparts. Intricate designs of microwave resonators (such as SRRs) are simply inappropriate for optical frequencies The most simple and intuitive method of introducing because they lose their strong response for λ < λres. This is the effective dielectric permittivity of a complex periodic a rigorous quantitative result: for a given resonator geometry, plasmonic metamaterial is to imagine what happens when a there is no strong magnetic response for the wavelengths shorter single cell of such structure is immersed in a uniform electric than the one corresponding to the electrostatic resonance. field. For simplicity, all calculations in Section 2 will be λres enters the electrostatic theory parametrically, through the limited to a two-dimensional case, i.e. the system is assumed dependence of the metal’s dielectric permittivity: (λ) < uniform along the z-axis; generalizations to three dimensions res ≡ (λres) is necessary for the strong response. Because are straightforward. To further simplify our calculations, it Y.A. Urzhumov, G. Shvets / Solid State Communications 146 (2008) 208–220 213 will be assumed that the unit cell is a rectangle of the density. The total dipole moment of a unit cell pE = R dxdy PE size a × a in the xy plane: −a /2 < x < a /2 and E = −1 E x y x x (where P 4π E is the polarization density) is linearly −a /2 < y < a /2. The unit cell is assumed to consist E y y proportional to the external electric field E0. In a homogeneous of a plasmonic inclusion with a complex frequency-dependent medium with anisotropic p = 1 i j − δi j E a a , where dielectric permittivity (ω) embedded into a dielectric host with i 4π qs 0 j x y i j the dielectric permittivity d . The plasmonic inclusion may qs is the effective permittivity tensor. Therefore, qs can be intersect the unit cell’s boundary. We assume that the structure defined by requiring that has an inversion symmetry and two axes of reflection (x and y). Z = i j i j In that case the effective permittivity tensor is diagonal: qs ax ay(qs − δ )E0 j ≡ dxdy( − 1)Ei . (6) xx yy diag(qs , qs ). E R Applying a constant electric field E0 =e Ex E0x +e Ey E0y Because dxdyEi = ax ay E0i , this dipole density definition of is equivalent to solving the Poisson equation for the potential qs simplifies to φ ≡ φ E : E0 Z Z i j (k) ≡ (k) = (k) ax ayqs E0 j dxdyDi dxdyEi , (7) ∇E · (∇E φ) = 0 (4) (k) ≡ on a rectangular domain ABCD (where AB and CD are where the external field E0 j E0δ jk applied to the unit cell parallel to y, BC and AD are parallel to x). The external produces the total internal field EE(k), and k = 1, 2. The internal E electric field E0 determines the boundary conditions satisfied electric fields are computed by solving Eq. (4) subject to its by φ: (a) φ(x + ax , y) = φ(x, y) − E0x ax , where (x, y) E(k) E0 -dependent boundary conditions. Eq. (7) is equivalent to belongs to AB, and (b) φ(x, y + ay) = φ(x, y) − E yay, 0 the capacitance-based definitions of qs given by Eq. (5). Owing where (x, y) belongs to AD. We now view the unit cell of a to an identity R dxdyD = H dsx (DE ·En), where nE is the unit metamaterial as a tiny capacitor immersed in a uniform electric i i normal to the closed contour of integration (unit cell boundary), field which is created by the voltage applied between its plates. xx Eq. (7) is expressed through a contour integral of the first kind: For calculating qs assume that the voltage V0 = E0x ax is applied between its sides AB and CD, and that E = 0. From 1 I 0y i j (k) = E (k) ·E qs E0 j dsxi (D n). (8) the potential distribution φ(x, y) the required surface charge ax ay density on the “capacitor plate” AB is σ (y) = (nE · DE) = This contour integral indeed reduces to the surface charge −d ∂x φ(x = −ax /2, y). The total charge (per unit length in z) R +ay /2 on the capacitor plates. For example, for x-polarized external on the capacitor is Q = − dyσ (y). The opposite capacitor H E ay /2 field (k = 1) we obtain dsx(D ·En) = (−ax /2)(−Q) + plate CD is oppositely charged. The unit length capacitance of (a /2)Q = a Q, thereby completing the proof of equivalence ≡ xx = x x this capacitor, C qs ay/ax is thus given by C Q/V0, or between definitions (5) and (6). All formulas of this Section can

Z +ay /2 be generalized to 3D. Also, note that Eq. (8) can be used for xx d ax qs = − dy∂x φ(x = −ax /2, y) determining all (diagonal and off-diagonal) elements of tensor E0ay −a /2 y qs of an arbitrary nanostructure (with or without inversion a /2 a−1 R y dyD (x = −a /2, y) symmetry of a unit cell), because the number of unknown y −ay /2 x x ≡ . (5) components of qs is equal to the number of equations. a−1 R ax /2 dx E (x, y = −a /2) x −ax /2 x y yy 2.2. Effective medium description through electrostatic homog- The qs component is determined similarly by applying the enization voltage V0 = E0yay between the capacitor plates AD xx yy and BC. Both qs and qs depend parametrically on the While the capacitance model developed in Section 2.1 is frequency ω because of the (ω) dependence of the plasmonic simple and intuitive, it needs to be justified in the context of permittivity. Therefore, extracting the frequency-dependent the rigorous homogenization theory. In this section we review components of qs tensor involves scanning the frequency ω, a multi-scale approach [39,40] to calculating the effective repeatedly solving (4) with the described boundary conditions, permittivity of a metamaterial and show its equivalence with the and applying the capacitance-based definition of qs given by capacitance model. Also known as the homogenization theory Eq. (5). We refer to this technique of extracting the electrostatic of differential operators with periodic coefficients [39,40], it eff-tensor as the “frequency scan” technique. As it turns out, is the most vigorous approach to homogenizing a periodic there is a faster and more physically appealing approach to metamaterial with a unit cell size a being much smaller than i j calculating eff(ω), described in Section 2.3. Note that the that of the typical variation scale Λ of the dominant electrostatic definition of the effective permittivity given by Eq. (5) is potential Φ. As in the previous sections, the key assumption equivalent to the one introduced earlier by Pendry et al. in [10] leading to the electrostatic approximation is that ωa/c which was derived [38] from the integral form of Maxwell’s ωΛ/c 1. Under this set of assumptions, the frequency ω equations. enters only as a parameter determining the plasmonic dielectric The capacitor model can be shown to be equivalent to permittivity (ω). The basis of the method is the two-scale E another intuitive definition of qs based on the dipole moment expansion. Let X be the macroscopic coordinates enumerating 214 Y.A. Urzhumov, G. Shvets / Solid State Communications 146 (2008) 208–220 the cells (large spatial scale Λ), and ξE the local coordinates 2.3. Eigenvalue expansion approach inside the unit cell (small spatial scale a). The potential φ(XE, ξ)E and the local permittivity (ξ)E are periodic in ξE; the latter can The frequency scan technique described in Section 2.1 be restricted to one cell. is a simple yet time consuming approach to calculating Using τ = a/Λ as the small parameter, we expand the quasistatic response of sub-wavelength metamaterials. 2 3 φ(x) = φ0(X, ξ) + τφ1(X, ξ) + τ φ2(X, ξ) + O(τ ) Electrostatic eigenvalue (EE) approach [22,28,41] enables and use ∇x = ∇ξ + τ∇X to solve the Laplace equation calculate this response for a wide range of frequencies by ∇x (x)∇x φ(x) = 0 order-by-order in τ. The goal of evaluating the position and strength of the electric dipole homogenization theory is to show that there exists a active plasmonic resonances in that range. As we shall see macroscopic potential φmacro(X) that obeys Laplace–Poisson from examples below, there are only a few eigenmodes that equation in a certain homogeneous, possibly anisotropic, contribute to qs, making the EE approach extremely efficient. medium. In doing so, both the rigorous definitions of φmacro(X) Additional theoretical insights (such as the Hermitian nature i j of qs that is not evident from Eq. (7)) can be gained from and eff are discovered. This goal is achieved by expanding Poisson equation in powers of τ. Terms with different powers the EE approach. In addition to reviewing some of the known of τ must vanish independently, resulting in three equations: facts about the EE approach [28,29,41] to calculating qs, we extend the original theory of plasmon resonances to include ∇ξ (ξ)∇ξ φ0(X, ξ) = 0, (9) plasmonic metamaterials with continuous plasmonic phase. Such structures have become increasingly important in the field ∇ξ (ξ)∇ξ φ1(X, ξ) = − ∇ξ (ξ) + (ξ)∇ξ ∇X φ0(X, ξ), (10) of negative index materials (NIMs) since the introduction of ∇ ∇ + ∇ + ∇ ∇ X (ξ) X φ0(X, ξ) (ξ) ξ ξ (ξ) X φ1(X, ξ) the so-called fishnet structure [14,17], as well as the SPOF + ∇ξ (ξ)∇ξ φ2(X, ξ) = 0. (11) structure [27]. One way of obtaining eigenvalue expansions is the Eq. (9) is satisfied by φ0(X, ξ) ≡ φ0(X), where the role of generalized eigenvalue differential equation (GEDE) [22,41]. the macroscopic potential is played by φ0(X). Next, φ1(X, ξ) Another, essentially equivalent, method is based on a surface is expressed through the macroscopic gradients of φ0: integral eigenvalue equation [33,42]. The steps of the GEDE approach are briefly described here, with the details appearing ∂φ0(X) φ (X, ξ) = − φ(i)(ξ), (12) elsewhere [22,23]. We assume that a periodic nanostructure 1 ∂ X sc i consists of two dielectric, non-magnetic materials: one with (i) where φsc (ξ) (i = 1, 2 in 2D) are periodic basis functions a frequency-dependent permittivity (ω) < 0 and another with permittivity d . Local permittivity of such a structure satisfying the local Poisson equation: h i E = − 1 E E = is (r, ω) d 1 s(ω) θ(r) , where θ(r) 1 inside the E E (i) E ∇ξ · (ξ)∇ξ φsc (ξ) = ∇ξ (ξ) ·Eei , (13) plasmonic material and θ(rE) = 0 elsewhere, and s(ω) = −1 (1 − (ω)/d ) is the frequency label. where eEi is the ith basis vector of Cartesian coordinate (i) First, the GEDE system. Note that the periodic potentials φsc are linearly h i related to the φ E defined by Eq. (4) through φ E ≡ E E 2 E0 E0 ∇ · θ(rE)∇φn = sn∇ φn (15) (i) E E E E0i φ (ξ) − E0 · ξ /|E0|. The difference between Eqs. (13) sc is solved for the real eigenvalues s . Spectral properties of the E E n and (4) is that the macroscopic electric field E0 ≡ −∇X φ0 GEDE are discussed in detail in [29,41] and references therein. is explicitly included in the rhs in (13) and embedded in the Second, the solution of Eq. (4) is expressed as an eigenmode boundary conditions in (4). expansion [41] Finally, the macroscopic equation for φ0(X) is obtained X sn (φn, φ0) by substituting φ1 from Eq. (12) into Eq. (11) and averaging φ(rE) = φ (rE) + φ (rE), (16) i j 0 n s(ω) − sn (φn, φn) over the local variable ξ: ∇Xi qs∇X j φmacro(X) = 0, where n φ (X) ≡ φ (X) ≡ hφ(X, ξ)i and macro 0 where the scalar product is defined as (φ, ψ) = R dxdyθ∇E φ∗ · i j ( j) ∇E = − qs = h(ξ)δi j − (ξ)∇ξi φsc (ξ)i ψ, and φ0 E0x represents x-polarized external field. Third, the quasistatic permittivity is calculated by substituting ≡ h ∇ − ( j) i; (ξ) ξi (ξ j φsc (ξ)) (14) φ(rE) from Eq. (16) into any of the equivalent definitions of angle brackets denote averaging over the unit cell, hFi ≡ qs. For example, the dipole moment definition (6) leads to the R 2 R 2 following analytical expression for qs: Fd ξ/ d ξ. It can be shown that this definition of qs is equivalent to the one obtained from the capacitor model. i j ! f i j E(i) = ∇E − (i) + i j 0 X fn Indeed, note that E (ξ) ξ ( φsc (ξ) ξi ) is the total qs(ω) = d δi j − − , (17) E s(ω) s(ω) − sn electric field excited by external electric field E0 =e Ei with n>0 i j the unit amplitude. Therefore, Eq. (14) is equivalent to qs = i j −1 where fn = A (φn, xi )(φn, x j )/(φn, φn) are the electric h ( j) i (ξ)Ei (ξ) , in exact agreement with Eq. (7). dipole strengths of the nth resonance, Y.A. Urzhumov, G. Shvets / Solid State Communications 146 (2008) 208–220 215

i j = − X i j f0 (A p/A)δi j fn (18) n>0 is the measure of the electric response of the continuous plasma R phase, and A p = dxdyθ(rE) is the area of the plasmonic phase contained within the area A = ax ay of a unit cell. From the i j expression for fn we note that only dipole-active resonances having a non-vanishing dipole moment (φn, x j ) contribute to the dielectric permittivity. Examples of such resonances are shown in Fig. 2((a),(b)) for the SRR and SR. But the electrostatic resonance of the SPOF structure shown in Fig. 2(c) is not dipole-active. It does not contribute to qs but, as will be shown in Section 3.1, contributes to the magnetic permeability. Application of the equivalent capacitance definition given by i j Eq. (8) leads to the same Eq. (17) with the same fn . However yy i j Fig. 3. (Color online) Effective dielectric permittivity of the SPOF f is obtained in a different (and more instructive) form: eff 0 structure (with the film in yz plane) calculated using two methods: I electromagnetic scattering through a single layer (solid and dotted curves), and i j 1 X (φn, x j ) ∂φn = i j − quasi-static formula (17) (dashed and dash-dotted). Red-shifted green dashed f0 q dsθxi , (19) A n (φn, φn) ∂n curve differs from the black dashed one by the frequency shift (32) discussed in Section 3.2. Structure parameters: periods ax = 100 nm, ay = 75 nm, strip ∂x j i j = 1 H width w = 50 nm, strip thickness ts = 15 nm, film thickness d f = 5 nm, strip where ∂/∂n is the normal derivative, q A dsθxi ∂n , and contour integration is carried out over the boundary of a separation in a pair h = 15 nm; plasmonic component: silver (Drude model); immersion medium: = 1. unit cell. Note that, if there is no continuous plasmonic phase d inside the unit cell of a nanostructure, its boundaries can be ≡ chosen such that they are not intersected by the plasmonic compared with the em eff extracted from the first-principles i j electromagnetic scattering simulations [24–26] of a single layer inclusion and, therefore, f ≡ 0. Combining Eqs. (18) and 0 of SPOF. Electromagnetic simulations are performed using (19) results in a generalized sum rule for plasmonic oscillators FEFD method implemented in the software package COMSOL in nanostructures that contain a continuous plasmonic phase. Multiphysics [34]. Overall, agreement between and is To illustrate this method, we chose the two-dimensional em qs very good everywhere except near the strong absorption line SPOF [27] shown in Fig. 2(c). The real and imaginary parts associated with electric resonance at ≈ 700 nm. Inside that of the yy-component of corresponding to electric field qs band (680–720 nm) the shape of strongly deviates from along the film are plotted as dashed and dash-dotted lines em Lorentzian. We speculate that this irregularity of em is related on Fig. 3, respectively. The green and black dashed lines q yy ≡ = [ yy] show Re qs with and without the retardation correction to to the large phase shift per cell θ kx ax Re eff ωax /c the frequency of the plasmon resonances. It is generally neglected in the quasi-static approximation based on periodic known [33] that frequencies of the optical resonances of finite- electrostatic potentials. More accurate description of eff should sized nanoparticles (with the typical spatial dimension w) include spatial dispersion [43]. Development of an adequate are red-shifted from their electrostatic values because of the theory of this phenomenon in plasmonic crystals is under retardation effects proportional to η2 ≡ ω2w2/c2. As shown way. in Section 3.2, these shifts can be expressed as corrections to (0) (2) (0) the frequency labels sn: sn = sn + sn , where sn are the 3. Perturbation theory of optical response of plasmonic (2) nanostructures electrostatic resonances and sn are the retardation corrections computed in Section 3.2. To obtain any meaningful comparison between the electromagnetically-extracted values of eff and the In Section 2 we have described several theoretical ≈ electrostatic qs, these corrections must be included even for the approaches to calculating eff qs for sub-wavelength structures as small as λ/10. nanostructures. A much more challenging problem is attacked In the chosen range of frequencies, there are only two dipolar in this Section: computing the magnetic permeability µeff(ω). resonances that contribute to yy; quasistatic curves in Fig. 3 One of the intriguing results is that µeff(ω) can be strongly are computed from the Eq. (17) with the following numerical dependent on the propagation direction for low-symmetry yy = structures such as the recently described metal strip pairs and coefficients: conduction pole residue f0 0.043, electric yy = yy = SPOFs [20,27,44]. resonance strengths f1 0.0045, f2 0.0005, and pole = (0) + (2) (0) = (0) = positions sn sn sn with s1 0.0426, s2 0.1630 3.1. From electrostatics to optical magnetism (2) = − (2) = − and s1 0.007, s2 0.004. The other component of qs has no resonances between λ = 500 − 800 nm and In this section, we present a sketch of a perturbation xx remains approximately constant, qs ≈ 1.2 in the frequency theory that uses electrostatic plasmon eigenfunctions, which yy range covered by Fig. 3. Quasistatic calculations of qs are are known to provide a complete orthogonal basis in 216 Y.A. Urzhumov, G. Shvets / Solid State Communications 146 (2008) 208–220 the appropriate linear space [29,41], as the starting point. Before proceeding to calculating the magnetic moment, note E Unlike earlier two-dimensional treatments [45], this theory is that the requirement h jM i = 0 is satisfied with the following E E applicable to both two- and three-dimensional structures. In decomposition of currents: jE = −iω(eff − 1)E/(4π), E E this section, calculations are presented for three-dimensional jM = −iω( − eff)E/(4π), as long as eff satisfies equality R E R E plasmonic crystals; expressions for two-dimensional crystals eff dV E = dV E. One can verify that this definition of are derived by replacing the measure of integration dV → eff is the same as the one given by the total electric dipole E E azdxdy, where az is an arbitrary “period” in the z-direction, moment of a unit cell, because h ji = h jE i. Moreover, eff = 2 which can be set to a unit length. The expansion parameter qs + O(η ), where qs is given by Eq. (17). E of this theory is the dimensionless retardation parameter The scattered electric field Esc is decomposed into the E E E E η = ωa/c. The goal of this Section is to obtain a self- potential and solenoidal parts, Esc = Epot + Esol = −∇Φsc + E E E E consistent analytic expression for µeff of a periodic plasmonic ik0 Asc, where ∇·Asc = 0 and k0 ≡ ω/c. Note that Asc is related E E E E nanostructure in the form of an eigenmode expansion. to the scattered magnetic field: Hsc = (µeff − 1)Hin + ∇ × Asc. E In conventional atomic crystals, low-frequency magnetism It can be demonstrated Asc is first order in η, making the E E stems mostly from internal and orbital magnetic moments contribution of Asc to Esc second order in η. Therefore, the 2 of electrons. In dielectric structures, including plasmonic lowest-order (η ) expression for µeff − 1 can be found without metamaterials, magnetism originates from polarization and directly computing AE or magnetic fields. E sc conduction currents; total local current equals j = −iω( − The potential part of EE is determined from ∇E · DE = 0, E sc 1)E/4π where the local complex permittivity also contains resulting in conductivity. These currents determine both eff and µeff; E E E E E E E E E their contributions to these parameters can be determined ∇Epot = −∇Ein − ∇Esol ≡ − Ein + ik0 Asc · ∇, (20) unambiguously by splitting the total local current into “electric” E E E E 2 and “magnetic” parts: j = jE + jM . The electric dipole where k0 Asc can be neglected to order η . For completeness, R E E moment of a unit cell is defined as pE = dV PE , where we note that Asc is computed, to the lowest order in η, as E E PE = i/ω jE ; effective permittivity is defined by expression −∇2 AE − k ∇E Φ(0) = k − EE = i j − i j sc i 0 sc i 0(qs ) in. (21) pi V eff δ E0 j /4π, where V is the volume of E E Using the approach similar to the one taken in Ref. [45], a unit cell. Splitting of the local current into jE and jM P E Φsc is expanded as Φsc(X, x) = cn(X)φn(x), where φn becomes unambiguous, if we require that jM do not contribute n E R E are electrostatic eigenfunctions of the GEDE with periodic to the electric dipole moment pE, i.e. h jM i ≡ dV jM = boundary conditions. Function Φsc(X, x) is periodic in the 0. The effective magnetic permeability µeff is defined by a “local” coordinate x and depends upon the “macroscopic” similar expression, m = V µi j − δi j H /4π, in which i eff 0 j coordinate X as the macroscopic fields, i.e. ∝ eik X under the magnetic dipole moment is calculated as mE = R dV ME , assumptions of our ansatz. This eigenmode expansion is E = 1 [E × E ] justified because the full set of φ (x) functions is a complete where M 2c r jM is the magnetic polarization density. n Optical magnetism appears in the ω2 order of this perturbation basis in the space of periodic solutions to Laplace equation [46]. E The coupling coefficient c between EE (X, x) = EE eik X eikx theory [20,32]. Since jM already has one factor of ω, to n in 0 E and the nth plasmon eigenmode are found by applying the EE determine µeff in the lowest order it suffices to calculate E with the first-order electromagnetic corrections. approach described in Section 2.3 to Eq. (20): To isolate the role of electrostatic resonances, it is convenient R E E s (∇φn)EinθdV to decompose electric and magnetic fields into “incident” and c = − n + O(η2). n R 2 (22) E E E E E E s(ω) − sn (∇φn) θdV “scattered”, i.e. E = Ein + Esc and H = Hin + Hsc, E E such that Esc and Hsc vanish in a homogeneous structure. c X = c ik X E Coefficients n( ) n(0)e absorb the phase shift per To achieve this, we use the plane wave ansatz: Ein = E E cell; it is sufficient to calculate them in the very first cell, i.e. at E ik·Er E = E ik·Er E E ikx E0e and Hin H0e . After the effective medium X = 0. Expanding the plane wave Ein = E0e in the powers parameters and µ are expressed through kE, we use (0) (1) eff eff of k up to the first order, we obtain cn = cn + cn , where the dispersion√ relation√ of transverse waves in a homogenized 1 √ √ sn medium: k = eff µeffω/c to determine eff(ω) and µeff(ω). (m) m E cn = − (i eff µeffk0) E0 This calculation is reminiscent of the Maxwell Garnett (MG) m! s(ω) − sn effective medium theory, where individual particles to be R E m (∇φn)x θdV homogenized are assumed to be immersed inside an effective × , m = 0, 1. (23) R (∇φ )2θdV medium with some unknown (and self-consistently determined) n E E eff and µeff. To simplify the calculation, we assume that k kx ˆ, The scattered field Esc is now used to calculate mz and, E k ˆ E k ˆ ≡ zz E0 y and H0 z. Therefore, µeff µeff. Additionally, consequently, µeff. For simplicity, we assume that the structure because the incident fields satisfy Maxwell’s equations in the has a center of inversion that coincides with the center of E −1 E homogenized medium, we have H0 = Ze [Enk × E0], where mass of a unit cell. Choosing it as the origin of coordinate √ √ E E Ze = µeff/ eff is effective impedance and nEk = k/|k| is the system, we have hErθi = 0 and hEri = 0. This eliminates R E direction of the phase velocity. the term ( − 1)rE × E0dV , and also guarantees that the Y.A. Urzhumov, G. Shvets / Solid State Communications 146 (2008) 208–220 217 E dipole eigenmodes φn in Φsc excited by uniform field E0 do magnetic polarizability of a sphere is not carry magnetic moment, as explained below. Under these −iη − ∝ 2 χM 3ie assumptions, µeff 1 k0. The currents excited directly by = E R3 2η3 Ein produce a diamagnetic contribution to µeff: η mη − 1 mη η − ( − 1 ) η mη/η cos sin m cos sin 1 m2 sin sin k2µ Z Z × , 0 eff 2 d 2 + i + − 1 ∆µ = (d − ) x dV − x θdV , (24) sin mη cos mη i(1 2 ) sin mη/η diam 2V eff s m m (28) where V is the volume of a unit cell. Quasistatic currents due to = 2 = EE ≈ −∇E Φ give the plasmon resonance part of the magnetic where η k0 R and m (ω). Eq. (28) can be simplified in sc sc | 2| permeability: two limiting cases: (a) η 1 and arbitrary m, and (b) m 1 and arbitrary η. The former expansion results in F zz 1 = − 2 X n + 2 X zz ∆µplasm k0d µeff k0d µeff Kn , (a) 1 s(ω) − sn s χ /R3 = (m2 − 1)η2 + O(η4), (29) n n M 30 (25) in agreement with Eq. (27). The latter expansion results in where χ (b) 3ie−iη(η cos η − sin η) F zz = K zz + ( / − 1)Gzz (26) M = . (30) n n eff d n R3 2η3(η + i) is the magnetic strength of nth resonance, For k0 R → 0 we recover the textbook result for magnetization (b) = − 3 R ∂φn R E ·[E × ∇ ] of a perfectly conducting small sphere: χM 1/2R . 1 x ∂y θ dV (ez r φn θ) dV (a) K zz = Therefore, χ overestimates the magnetic polarizability of the n R 2 M 2 V (∇φn) θdV sphere whenever η 1 but |mη| > 1. The quasi-static theory R ∂φn R E ·[E × ∇ ] developed here suffers from the same limitation. Nevertheless, 1 x ∂y θ dV (ez r φn ) dV and Gzz = . the analytic expression for the effective magnetic permeability n 2 V R ∇ 2 V ( φn) θd µeff(ω) of a plasmonic nanostructure given by Eq. (27) answers several fundamental questions outlined below. Finally, permeability is determined from µeff − 1 = ∆µ + ∆µ : diam plasm 3.1.1. Which plasmon resonances are magnetic? 1 F zz F zz It has been observed earlier [20,23] that in the structures − = zz − 2 0 − 2 X n 1 zz µd k0d k0d , (27) with a sufficiently high spatial symmetry only some plasmon µ s(ω) s(ω) − sn eff n resonances may have a non-vanishing magnetic strength given 2 2 by Eq. (26). Such eigenmodes are sometimes referred to as where µzz = k2( − )h x i and F zz = h x θi − P K zz . d 0 d eff 2 0 2 n n magnetic plasmon resonances (MPR) [32]. For example, if the Here the sum over electrostatic resonances possessing a finite R structure has an inversion center, its electrostatic eigenfunctions magnetic moment (eEz ·[Er × ( − eff)∇φn]) dV represents φn can be either even or odd with respect to spatial inversion. the resonant contribution to magnetic permeability, while the It follows from the definitions of f i j and F zz that even modes term proportional to F zz represents the expulsion of the n n 0 have a vanishing electric strength while the odd modes have a magnetic field from the plasmonic inclusion. Eq. (27) can be vanishing magnetic strength. If inversion center is the only non- generalized for an arbitrary propagation direction kE. trivial element of symmetry, i.e. the structure’s symmetry group Eq. (27) should be used with caution at very large negative is C , all even modes are magnetic resonances (F 6= 0) and dielectric contrasts, i.e. when 0 < s(ω) 1. In this regime, i n odd modes — electric resonances ( f 6= 0). For example, the plasmonic inclusions behave as good conductors. Expulsion n electrostatic resonance of the SPOF structure shown in Fig. 2(c) of magnetic field from plasmonic phase becomes so strong, has a finite magnetic strength proportional to η2. HE that, in fact, scattered magnetic field sc is nearly equal in In structures with higher symmetries, electric and magnetic magnitude (and opposite in sign) to the incident magnetic field E E eigenmodes can be identified by their irreducible representa- Hin. Present theory treats Hsc as perturbation and is therefore tion. It can be shown that electric strength f xx may only be inaccurate in this regime. This limitation manifests in Eq. (27) n non-zero if the corresponding potential φn transforms under the as an unphysical pole at s = 0. Its origin can be seen from Mie yy same representation as the coordinate x (and similarly for fn theory of a dielectric sphere. and f zz). For the magnetic strength F zz to be non-vanishing, For a sphere, all plasmon resonances have zero magnetic n n the potential φn must transform as an operator of rotation zz = zz = h x2 i = 1 5 strength, Fn 0, and F0 2 θ 30 4π R /V , where around z-axis (Rz), which is a component of a pseudovector. V R3 is the volume of a very large domain in which a Standard character tables of the point groups of symmetry [47] sphere is contained. Our formula (27) gives 1 − 1/µeff ≈ provide the information necessary for assigning magnetic or − ≡ = − 2 zz µeff 1 4πχM /V k0 F0 /s(ω), where χM is the electric dipole activity to various resonances. magnetic polarizability of a plasmonic sphere. For d = 1 we In the structures without an inversion center, some 3 = 1 − 2 obtain χM /R 30 ( 1)(k0 R) . The exact Mie result for the eigenmodes may transform as both x and Rz, meaning that 218 Y.A. Urzhumov, G. Shvets / Solid State Communications 146 (2008) 208–220

xx zz they contribute to both eff and µeff. In addition, they provide magneto-electric coupling between Ey and Hz. Whenever such modes exist, the structure is termed bi-anisotropic: it may not be described with eff(ω) and µeff(ω) tensors alone [48]. SRR and SR are examples of such bi-anisotropic structures [49], and their electrostatic resonances shown in Fig. 2(a), (b) contribute zz xx to both µ and qs .

3.1.2. Does the magnetic permeability depend on the propagation direction? zz The magnetic strength Fn , given by Eq. (26), may depend on the orientation of the wave vector kE (or electric field E E E0) in the plane orthogonal to H0. Consequently, magnetic zz permeability µeff near an MPR can be anisotropic. This happens because the magnetic strength is a product of two factors: (i) coupling of the incident plane wave to a plasmon mode, and (ii) magnetic moment contained in the excited mode. While the E E latter is independent of the direction of k or E0, the former coupling is determined by the inhomogeneity of the electric E ikE·Er field E0e and, therefore, depends upon the orientation of E E the orthogonal pair (k, E0). In other words, the components of the µeff tensor of a plasmonic metamaterial are in general dependent on the propagation direction kE/k. We use a pair of plasmonic strips (SPOF without the film) to illustrate the anisotropy of scalar µeff in optical metamaterials. This structure has attracted a great deal of attention as a magnetic component of NIMs [20,44]. Its two-dimensional point group of symmetry is C2v; electric plasmon resonances in this structure correspond to dipole-like eigenpotentials φn, and magnetic resonances are electric quadrupoles [20]. Quasi- zz Fig. 4. (Color online) Effective magnetic permeability µeff of the strip pair static values of the effective permeability from Eq. (27) (labeled structure calculated using two methods: electromagnetic scattering through a as µqs) are compared with those (labeled as µem) extracted single layer (solid and dotted curves), and the quasi-static Eq. (27) (dashed from the single-layer electromagnetic scattering simulations. and dash-dotted). Green dashed curve incorporates the retardation frequency The two illumination geometries and the results are shown in shift given by Eq. (32). Black dashed curve is not corrected by the retardation Fig. 4. Unit cell dimensions are given in the caption to Fig. 4. frequency shift. Orientation of the strips and the incident electric field are shown in the insets. Structure parameters: periods a = a = 100 nm, strip width: Both quasistatic and electromagnetic results plotted in Fig. 4 x y w = 50 nm, strip thickness ts = 15 nm, strip separation in the pair: h = 15 nm; demonstrate that the strength the magnetic plasmon resonance Strips are assumed to be made of silver and embedded in vacuum (d = 1). excitation depends strongly upon the direction of the wave E vector k of the incident wave. Maximum deviation of µeff from 3.2. Electromagnetic red shifts of plasmonic resonances unity is an order of magnitude larger when kE is perpendicular to the strips. The high degree of anisotropy of µeff is related to Eq. (27) predicts that the frequency of magnetic plasmon the relatively low symmetry group (C2v) of the two plasmonic resonance ωn, determined from s(ωn) = sn, is the strips. frequency of magnetic cut-off, µeff = 0. From Fig. 4(a) we For plasmonic nanostructures with the higher degree of observe that the frequency of the magnetic cut-off determined zz symmetry (C3v, C4v, and C6v) the anisotropy of µeff may from electromagnetic simulations (solid curve) is red-shifted disappear. For such structures it can be shown that the coupling with respect to the frequency of corresponding electrostatic zz E E coefficients in Fn for kkx ˆ and kky ˆ are equal in magnitude resonance (black dashed curve). It can also be seen from Fig. 3 R ∂φn = − R ∂φn but differ in sign: x ∂y θdV y ∂x θdV . Therefore, that electric dipole resonances in FEFD simulations are red- shifted with respect to their electrostatic positions. Both shifts R x ∂φn θdV = 1 R x ∂φn − y ∂φn θdV , and the magnetic ∂y 2 ∂y ∂x can be explained as the retardation-induced corrections to the zz zz zz strength Fn as well as other parameters (µd and F0 ) become positions of the purely electrostatic plasmon resonances [33]. isotropic. Thus, for sufficiently symmetric crystals it is possible These frequency shifts are always red and can be understood to introduce an isotropic µeff, independent of the direction physically as follows. Quasi-static currents associated with of applied electric field. A two-dimensional example of a electric fields of electrostatic resonances induce magnetic fields plasmonic metamaterial with isotropic scalar µeff is a square via Ampere’s law. These magnetic fields generate secondary array of plasmonic nanorods shown earlier [22,23,50] to exhibit electric fields according to the Faraday’s law. The latter an isotropic negative refractive index. contribute to the Poisson equation ∇E · EE = 0, causing shifts Y.A. Urzhumov, G. Shvets / Solid State Communications 146 (2008) 208–220 219

1 0 in electrostatic eigenvalues. While such frequency shifts have −2π|y−y |/ax 0 − ln[1 − 2e cos(2π|x − x |/ax ) been calculated earlier for isolated plasmonic nanoparticles, 4π 0 they have never been calculated for periodic plasmonic + e−4π|y−y |/ax ]. (34) metamaterials. Retardation shifts are computed using the perturbation The function (34) is periodic only in the x-direction. It is 0 theory developed in Section 3. To the lowest order in η and therefore applicable only for |y−y | ay, i.e. when plasmonic in the close vicinity of the nth plasmonic resonance (i.e., at inclusions are much thinner in the y-direction than the period (0) (0) (w a ). For periodic metamaterials based on “current s ≈ sn , where sn is the purely electrostatic eigenvalue of the y y Eq. (15)), the vector potential induced by the nth electrostatic loops” (strip pairs, horse shoes, etc.), interaction between E(1) R 0 consecutive layers of resonators is usually insignificant, and resonance is found from Eq. (21): Asc (rE) = ik0 dV G(rE − the function (34) provides reasonable approximation. When the 0 0 0 1 0 rE )(rE )∇φn(rE ), where ≈ d 1 − θ , and G(rE −r E ) is condition wx ax is satisfied, one can use expression (34) s(0) n x ↔ y a ↔ a the modified Green’s function of Poisson equation with periodic with interchanged variables , x y. When both boundary conditions originally calculated [51] in the context dimensions are small, wx,y ax,y, a symmetrized (in x, y) version of Eq. (34) is used. of the solid state physics. Thus computed vector potential AE(1) sc Retardation frequency shifts of selected electric and contributes to Poisson equation: magnetic resonances are illustrated on Figs. 3 and 4, Z ! respectively. Frequency shifts are calculated using Eq. (32) E E 2 E 0 0 1 E ∇θ∇Φ + k0d ∇θ · dV G(r, r ) 1 − θ ∇Φ with the Green’s function given by Eq. (34). It is apparent s(0) n that the quasi-static Eqs. (17) and (27) with frequency 2 (0) (2) = sn∇ Φ. (31) corrections corresponding to sn = sn + sn (green dashed curves on Figs. 3 and 4) are in much better agreement This is a generalized linear eigenvalue problem with with the electromagnetic and µ than the unperturbed integro-differential operator. Treating the integral term as a eff eff electrostatic eigenvalues s(0) (black dashed curves). In perturbation, corrections to electrostatic eigenvalues s(0) can be n n subsequent publications we will show analytically how these shown to be: eigenvalue corrections appear in the resonant denominators I (2) 2 (0) (sn − s) of the driven problem. sn = k0d sn dSφn(r)nE ! 4. Summary Z 1 × G(rE −r E0) 1 − θ(rE0) ∇E φ (rE0)dV 0 (0) n sn In this review we have discussed several physics issues im- Z −1 portant for designing sub-wavelength plasmonic metamaterials 2 × |∇φn| θdV , (32) in the optical part of the spectrum. Several techniques for calculating the effective dielectric permittivity eff ≈ qs in the where H dS is a surface integral over a closed surface S of the quasi-static limit are discussed. We have presented a sketch plasmonic inclusion (which reduces to a contour integral for of a homogenization theory that encompasses both electric (0) and magnetic response of plasmonic nanostructures. As sug- 2D crystals). The renormalized sn is calculated as sn = sn + gested by electrostatic nature of plasmon resonances, the theory s(2). Volume integral in Eq. (32) can be reduced to a surface n starts from the electrostatic description and takes retardation integration over the surface S by introducing an auxiliary vector and magnetic phenomena into account perturbatively. Formu- aE(rE, rE0) = −∇E R G(rE −r E00)G(rE00 −r E0)dV 00: las for the frequencies and strengths of all electric and magnetic I I resonances are universally applicable to any periodic metallo- s(2) = k2 dSφ (rE)nE · dS0aE(rE, rE0) × [En0 × ∇φ (rE0)] n 0 d n n dielectric nanostructures operating in the plasmonic regime. −1 The theory is supported by first-principle electromagnetic sim- I ∂φn × φn dS , (33) ulations. Our results demonstrate that in 2D plasmonic crystals ∂n without rotational symmetry, such as strip pair arrays, scalar zz where the normal derivative ∂φn/∂n is evaluated on the magnetic permeability µeff depends on the orientation of elec- plasmonic side of surface S. A particular case of this formula tric field (or the wavenumber). has been previously reported for isolated three-dimensional particles [33]. Acknowledgements Despite substantial progress in calculations of periodic This work is supported by the ARO MURI W911NF-04- Green’s functions [51], closed-form expressions for double- or 01-0203, AFOSR MURI FA9550-06-01-0279, the DARPA triple-periodic G in two or three dimensions are not known. contract HR0011-05-C-0068, and by the NSF’s NIRT 0709323. However, there exists one simple yet exact result for a 2D Green’s function in the limit ay ax [51]: References 0 2 0 1 (y − y ) 0 ay [1] V.G. Veselago, The electrodynamics of substances with simultaneously G2(rE −r E ) = − |y − y | + 2ax ay 6 negative values of and µ, Soviet Phys. – Uspekhi 10 (1968) 509. 220 Y.A. Urzhumov, G. Shvets / Solid State Communications 146 (2008) 208–220

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