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PHYSICAL REVIEW B 75, 241402共R兲 共2007兲

Highly confined optical modes in nanoscale metal-dielectric multilayers Ivan Avrutsky,1,* Ildar Salakhutdinov,1 Justin Elser,2 and Viktor Podolskiy2 1Department

of Electrical and Computer Engineering, Wayne State University, Detroit, Michigan 48202, USA 2Department of Physics, Oregon State University, Corvallis, Oregon 97331, USA 共Received 1 March 2007; published 7 June 2007兲

We show that a stack of metal-dielectric nanolayers, in addition to the long- and short-range plasmon polaritons, guides also an entire family of modes strongly confined within the multilayer—the bulk plasmon polariton modes. We propose a classification scheme that reflects specific properties of these modes. We report experimental verification of the bulk modes by measuring modal indices in a structure made of three pairs of silica 共⬃29 nm兲 / gold 共⬃25 nm兲 layers. DOI: 10.1103/PhysRevB.75.241402

PACS number共s兲: 73.20.Mf, 42.25.Bs, 42.82.Et, 73.21.Ac

Nanoscale confinement of light is of great interest for applications in sensing, imaging, all-optical signal processing, and computing. Subwavelength confinement attributed to gap plasmon polaritons 共GPPs兲 has been demonstrated in a thin dielectric layer surrounded by metallic claddings.1 Here we present another solution to subdiffraction confinement of light and show that a stack of metal-dielectric nanolayers guides a family of modes strongly confined within the multilayer—the bulk plasmon polariton 共BPP兲 modes. The bulk modes have very short penetration depth into the claddings even if the claddings are made of dielectric materials. Their modal indices 共ratio of the light velocity in vacuum to the phase velocity of a guided mode兲 are typically large in absolute value and may be both positive and negative.2 We propose a classification scheme that reflects specific properties of BPPs. We verify BPPs experimentally by measuring their modal indices in a structure made of three pairs of silica-gold nanolayers. When considering light confinement in a waveguide, the modal index n* 共rather than the group index兲 is of interest because it defines the modal profile. The field penetration length into the cladding with permittivity ␧cl is Lcl = ␭ / 2␲冑n*2 − ␧cl, where ␭ is the vacuum wavelength. For a given ␧cl, the larger modal index leads to shorter penetration length and stronger confinement. This justifies the interest in the high-index modes. In all-dielectric waveguides, the modal index is smaller than the core index, which limits the confinement scale to ⬃␭ / 7 in a silicon-on-insulator waveguide with silicon core 共n ⬇ 3.5兲 and silica or vacuum claddings.3 The surface plasmon polariton4 共SPP兲 propagating along the interface between media with permittivities ␧m and ␧d of different sign 共metal and dielectric兲 is an example of a strongly confined mode. Its modal index n*SPP = 冑␧d␧m / 共␧d + ␧m兲 is slightly above the index of the dielectric. In the visible and near-infrared spectral regions, the permittivity of metal is typically negative and 兩␧m 兩 Ⰷ ␧d, leading to subwavelength field penetration into metal, while the penetration into the dielectric can be as large as several wavelengths. A remarkable exception is the case of resonant SPPs5 when permittivities of materials at different sides of the interface are exactly opposite: ␧m + ␧d → 0 and thus n* → ⬁. In homogeneous media, the field distribution of the resonant SPP is expected to be confined infinitely close to the interface. The actual scale of the field distribution is defined by the applicability of the concept of dielectric 1098-0121/2007/75共24兲/241402共4兲

permittivity—that is, by the discrete atomic structure of materials. At optical frequencies, the resonant SPPs are possible if the dielectric has a huge optical gain.6 A metallic film of thickness tm between dielectric claddings supports SPPs at each metal-dielectric interface. In sufficiently thin films, coupling between individual SPPs leads to the formation of symmetric and antisymmetric film modes known as long- and short-range plasmon polaritons 共LRPs and SRPs兲.7,8 The coupling leads to a splitting of the dispersion characteristics of these modes. The modal index of the * ⬇ 冑␧d + 共␲tm / ␭兲2共共␧d − ␧m兲␧d / ␧m兲2, is reduced LRP, nLRP compared to that of the SPP approaching the cladding index nd ⬇ 冑␧d. Its penetration into the claddings increases accordingly. The modal index of the SRP is higher than that of the * ⬇ 冑␧d + 共␭ / ␲tm兲2共␧d / ␧m兲2, yielding the confineSPP: nSRP ment scale LSRP = tm + 2Lcl ⬇ tm共1 + 兩␧m兩 / ␧d兲. Note that even for nanoscale 共⬃10-nm兲 metal films, both LRPs and SRPs are typically extended to quite a macroscopic scale. The modal indices of SRPs and LRPs in fused silica 共␧d = 1.4442 ⬇ 2.085兲 共Ref. 9兲 gold 共␧m = −114.5+ 11.01i兲 共Ref. 10兲 system at ␭ = 1550 nm as functions of metal thickness are shown in the first column of Fig. 1. The splitting of modal indices resulting from the coupling is an electromagnetic analog of a universal phenomenon appearing in linear wave-dynamic systems: energy-level splitting in quantum mechanics, characteristic frequency splitting in acoustics, splitting of relaxation time in coupled ac circuits, etc. Modes in a metal-dielectric multilayer can be represented as a linear combination of surface modes at individual interfaces, with splitting between modal indices determined by the overlapping of the original wave functions. The system of two metal films supports four modes, which differ by symmetry. Two of these modes are dominated by SPPs propagating at interfaces with the claddings, and their properties are identical to SRPs and LRPs. The other two modes are primarily composed of SPPs propagating inside the gap between the metal films. The mode with antisymmetric magnetic field distribution experiences a cutoff when the gap thickness becomes less than dc ⬇ ␭ / 2冑␧d, and therefore it does not exist in nanoscale structures. The mode with symmetric field distribution, the GPP, survives. For a nanoscale dielectric gap of thickness td between infi* of gap nitely thick metallic claddings, the modal index nGPP plasmon polaritons,

241402-1

©2007 The American Physical Society

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PHYSICAL REVIEW B 75, 241402共R兲 共2007兲

AVRUTSKY et al.

冑 冉 冊 冑冉 冊 1

␭ ␧d

2 ␲td ␧m

␭ ␧d

2

+

␲td ␧m

is inversely proportional to the gap size. Note that the modal index of the gap plasmon polariton greatly exceeds that of the SRP, providing truly nanoscale-mode confinement. Similar modes exist also in a gap between sharp metallic wedges.11 Exact dispersion equations for modes in a gap between metallic claddings were proposed earler.12 As more layers are added to the structure, the collective interaction of SPPs leads to the mutual repulsion of modal indices. The gap plasmon polaritons contribute to the emergence of high-index supermodes strongly confined to the bulk of the layered material. Since the bulk modes originate from repulsion of GPPs, the total number of these modes is equal to the number of dielectric layers. The entire multilayer may also support surface modes at the interfaces with the claddings. In the case of symmetric claddings they are similar to the LRPs and SRPs of a metallic film. The surface modes behave fundamentally different when 兩␧m兩 ⬍ ␧d. Although a single interface does not support a SPP, a thin film of negative-permittivity medium supports two modes with identical field distributions 共TM0兲 but different “handedness”—the relation between the directions of the electric and magnetic fields and wave vector of the mode2,13 共Fig. 1, right兲. One of these modes is a direct “right-handed” analog of a LRP. The other one is its negative-index “lefthanded” counterpart. While the analogy is not complete, the negative-index medium is often called optical antimatter.14 The close-to-cutoff TM0-mode doublet can therefore be considered as an optical mode-antimode pair. A further analogy

2

共␧d − ␧m兲 +

冉 冊 ␭ ␧d

4

4 ␲td ␧m

共1兲

,

is seen in the formation of an antisymmetric left-handed gap plasmon polariton in a dielectric gap between metallic claddings when 兩␧m兩 ⬍ ␧d.2,15,16 Similar to the case of 兩␧m兩 ⬎ ␧d, the absolute value of the modal index for the left-handed waves increases as the metal interfaces get closer. Naturally, such an increase of the absolute value of the modal index is accompanied by stronger mode confinement. Figures 2 and 3 further summarize the modal indices of metal-dielectric composites with 25-nm layers and dielectric claddings, realized in our experiment. While optical losses of the bulk plasmon polariton modes in multilayers are larger than losses of SPPs, these modes are of significant interest to nanophotonics due to extremely strong field confinement. Besides light guiding by subwavelength structures, nanoscale multilayers with appropriately patterned films are promising candidates for the development of metamaterials with negative refractive index,16–21 as well as for the development of nanosensors. The traditional classification scheme for transverse electromagnetic modes in multilayers relies on the number of nodes in the field distributions. Accordingly, the SPP supported by a single interface is a TM0 mode. The LRP and SRP supported by a thin metal film are labeled as TM0 and TM1 modes. The gap plasmon polariton is a TM0 mode. Such a mode labeling scheme is perfect for dielectric waveguides but causes some confusion when applied to nanoscale metal-dielectric structures. It is a bit inconvenient that the same label TM0 is designated to rather different electromagnetic excitations such as surface plasmon polaritons, long-range film plasmon polaritons, and gap plasmon polaritons. Two modes of a metal strip with 兩␧m兩 ⬍ ␧d are both TM0 waves. A further inconvenience is that essentially the same mode—the gap plasmon polariton—is labeled TM0 in the structure with metallic claddings, and it becomes a TM2 1.2 1.0

H||

0.8 0.6

εAu = -114.5 +11.01i εSiO2 = (1.444)2 λ = 1550 nm

0.4

Au

0.2

Losses (cm-1) Effective index

0.0 -100 -80 -60 -40 -20

FIG. 1. 共Color online兲 Evolution of coupled SPP modal indices in the layered materials with dielectric claddings for 兩␧m 兩 ⬎ ␧d 共left兲 and 兩␧m兩 ⬍ ␧d 共bottom right兲. System geometries are sketched in top-right inset, where blue 共gray兲 and yellow 共light-gray兲 regions represent dielectric and metal layers. Graphs at the left correspond to Au/ SiO2 composite considered in this work; graphs at the right represent its “negative index” counterpart with ␧m = −1 + 0.1i and ␧d = 1.4442; ␭ = 1550 nm. First vertical portion of the figure illustrates the appearance of a SRP-LRP doublet as a result of splitting between two SPPs 共left兲 and as a result of mode-antimode formation 共right兲. Other vertical portions illustrate modal indices in a layered structure when a metal layer is being gradually brought closer to the nanolayer stack.

1

0

0.5 0.0 1.0 0.5 0.0

Au 20

-0.5

40

60

-1.0 1.0

80 100

Coordinate (nm)

4

0.5

Silica film Symmetric mode Gold claddings (GPP)

3 2

0 10000

0.0 1.0 0.5 0.0 -1.0 1.0 0.5 0.0

Asymmetric mode

-0.5 1.0

1000

0.5 0.0

Symmetric mode (GPP) 0

100

200

300

n* = 1.4570 Au

Long-range TM plasmon LRP 0 λ = 1550 nm Short-range plasmon SRP TM1

α = 97.7 cm-1

n* = 1.4574 α = 111.2 cm-1 n* = 2.1586 α = 1,844 cm-1

Bulk plasmon mode BPP0 TM

n* = 2.4858 α = 4,516 cm-1

Bulk plasmon mode BPP1

2

TM3

-0.5

Asymmetric mode

1

100

1.0

td = 25 nm n* = 2.4557 α = 3344 cm -1

H||

␧d +

SiO2

* nGPP ⬇

n* = 2.8242 α = 6,904 cm-1

Bulk plasmon mode BPP2

n* = 3.0635 α = 8,438 cm-1

Bulk plasmon mode BPP3

TM4

TM5

-0.5

400

500

600

Dielectric layer thickness (nm)

700

-1.0

-500 -400 -300 -200 -100

0

100 200 300 400 500

Coordinate (nm)

FIG. 2. Field profile 共strength of magnetic field versus coordinate across the structure兲 for a gap plasmon polariton and its modal index and losses as a function of the dielectric layer thickness 共left兲. Profiles of modes supported by a nanoscale multilayer of four silica layers and five gold layers between silica claddings 共right兲.

241402-2

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HIGHLY CONFINED OPTICAL MODES IN NANOSCALE … High-order mode limit

10000

BPP0 = TM2

2.5

2.0 Fundamental mode limit 1.5

SRP = TM1 LRP = TM0

1 2 3 4 5 6 7 8 9 10

Number of Metal Layers

High-order mode limit

8000 BPP1 = TM3

Losses (cm -1)

Effective index

3.0 BPP1 = TM3

6000

BPP0 = TM2

4000 2000 SRP = TM1 0

Fundamental mode limit LRP = TM0

1 2 3 4 5 6 7 8 9 10

Number of Metal Layers

FIG. 3. Effective indices 共left兲 and losses 共right兲 for the modes in silica-gold multilayers. Dashed lines show the limits for the lowest-order and highest-order bulk modes in approximation of infinite number of layers 共2兲.

mode in a structure with dielectric claddings, two thin metallic layers, and a guiding dielectric layer. The TMn label indicates the number of nodes in the modal field distribution. This number, however, is not associated directly with the character of a particular mode guided by a nanoscale multilayer 共smooth, oscillating, confined to the bulk of the multilayer, surface wave, etc.兲. Once the highly confined bulk modes and the filmplasmon-polariton-type modes have distinct properties, it is reasonable to label them differently. In particular, in a structure with a large number of layers, the bulk mode with relatively smooth profile is reasonable to call the fundamental bulk plasmon polariton mode BPP0. In the traditional numbering scheme this is a TM2 mode because it has two nodes close to the interfaces with the claddings. Accordingly, the bulk mode of order n 共BPPn兲 would be labeled as TMn+2 in the traditional classification. The labels such as LRP and SRP should be reserved for the modes with intensity maxima at the interfaces with the claddings. When claddings have different permittivities, the labels LRP and SRP become rather senseless. Instead, titles such as SPP bounded to particular interfaces will be more appropriate. We stress that in a finite-thickness nanolayered film the bulk modes, due to minor penetration into the claddings, are rather independent of the cladding indices. Therefore, all BPPn-mode labeling is also independent of the claddings—a significant advantage over the traditional scheme, which is strongly affected by the cladding indices. The proposed scheme is illustrated in Fig. 2 共right兲 in the example of modes of a composite with Nd = 4 dielectric and Nm = 5 metallic 25-nm-thick layers. The structure supports the long-range plasmon polariton 共LRP= TM0兲, the shortrange plasmon polariton 共SRP= TM1兲, and four bulk plasmon polariton modes 共BPP0, . . . , BPP3 = TM2, . . . , TM5兲. Both LRPs and SRPs show a large penetration into the claddings. In contrast, the bulk plasmon polariton modes are confined within the multilayer with a minor fraction of optical power propagating in the claddings. The fundamental bulk mode BPP0 has a rather smooth field profile. For the highest-order bulk mode, the modal field reveals fast oscillations so that it has opposite signs in neighboring dielectric layers. Note that the modal indices of the BPP modes can be several times higher than the refractive index of the dielectric in the multilayer. The origin of this surprising behavior is in the

PHYSICAL REVIEW B 75, 241402共R兲 共2007兲

coupling-induced repulsion of the modal indices of individual gap plasmon polaritons discussed above. The modal indices of guided modes in a multilayer with alternating 25-nm-thick layers of gold and silica are shown in Fig. 3. As predicted, the number of bulk modes is equal to the number of dielectric layers and the maximal modal index increases with the number of layers increasing. For any given number of layers, modes of higher order have larger losses and larger modal indices. Assuming the number of layers is approaching infinity, N → ⬁, we find the dispersion relation for the highest- and lowest-order bulk plasmon polariton modes 共BPPN and BPP0兲 by setting periodical boundary conditions and requiring that the magnetic field strength have a node in the middle of every metal layer 共BPPN兲 or not have such a node 共BPP0兲:



tanh



␧d + ␧m



tanh

冊 冉

␲tm *2 ␲td *2 nBPP − ␧m tanh nBBP − ␧d N N ␭ ␭



*2 nBPP − ␧m N

*2 nBPP N

␲td *2 nBPP − ␧d 0 ␭

+





␧d ␧m

− ␧d



0

*2 nBPP − ␧d

共2a兲

= 0,

冊冒 冉

*2 nBPP − ␧m



tanh

␲tm *2 nBPP − ␧m 0 ␭



冊 共2b兲

= 0.

0

Equations 共2a兲 and 共2b兲 do not contain the cladding indices or the overall composite thickness, further indicating the “bulk” origin of these modes. In the limit of thin 共nanoscale兲 layers, Eqs. 共2a兲 and 共2b兲 yield the following approximation for the modal indices: * ⬇ nBPP N



␧d −

␭2 ␧d , ␲ 2t dt m ␧ m

* nBPP ⬇ 0



␧d␧m共td + tm兲 . t d␧ m + t m␧ d 共3兲

Note that the wavelength disappears from the expression for the modal index of the fundamental mode and the equation becomes equivalent to the predictions of the effective medium theory. The highest-order mode has a larger modal index than a single-gap plasmon polariton provided that td,m Ⰶ ␭ / ␲冑兩␧m兩. A particular case of the fundamental mode, the one with the smallest losses, was studied22 back in the 1950s. SPPs supported by a structure consisting of few metal and dielectric thin films were studied earlier.23 Nonlocal corrections to the averaged permittivity in metal-dielectric nanocomposites are reported in our recent paper.24 A similar effect has been found in GHz systems with conducting wireshaped inclusions.25 For the gold-silica multilayers with td = tm = 25 nm, * * 兲 = 3.1488 and ␣ = 共4␲ / ␭兲Im共nBPP 兲 Eq. 共2兲 gives Re共nBPP N N −1 = 8959 cm —maximal possible values for the modal index and losses for the high-order bulk mode BPPN. Correspond* 兲 = 2.0128 ing values for the fundamental mode are Re共nBPP 0 * −1 and ␣ = 共4␲ / ␭兲Im共nBPP 兲 = 462.8 cm . These limits are 0 shown in Fig. 3 by horizontal dashed lines.

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PHYSICAL REVIEW B 75, 241402共R兲 共2007兲

AVRUTSKY et al. 1.0

1.0

the substrate is overdamped and the SPP at the interface with air has a vanishing small evanescent coupling with the incident beam. By fitting the experimental data with numerically simulated angular reflection, the best-fit structure has been identified as Si/ 共33 nm silica兲 / 共24 nm gold兲 / 共24 nm silica兲 / 共26 nm gold兲 / 共31 nm silica兲 / 共24 nm gold兲 关on average, 共29 nm silica兲 / 共25 nm gold兲兴, which is close to the deposition target numbers. Figure 4 also shows the modes’ profiles in the manufactured structure calculated using the best-fit data for the layers’ thicknesses. In conclusion, we analyzed TM-polarized modes supported by nanoscale metal-dielectric multilayers. We show that, in addition to the long-range and short-range plasmons supported by thin films, there are high-index guided modes strongly confined within the bulk of the multilayer. The dispersion relation is derived for the highest-order and lowestorder modes in approximation of an infinite number of layers. A classification scheme is proposed for the modes supported in nanoscale metal-dielectric multilayers. Highindex highly confined modes in a structure made of thee pairs of gold 共⬃25 nm兲 / silica 共⬃29 nm兲 layers are experimentally verified.

*Corresponding author. [email protected]

12 T.

Tanaka and M. Tanaka, Appl. Phys. Lett. 82, 1158 共2003兲. 2 A. Alu and N. Engheta, J. Opt. Soc. Am. B 23, 571 共2006兲. 3 Y. A. Vlasov and S. J. McNab, Opt. Express 12, 1622 共2004兲. 4 V. M. Agranovich and D. L. Mills, Surface Polaritons: Electromagnetic waves at surfaces and interfaces 共Elsevier, New York, 1982兲. 5 H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings 共Springer, Berlin, 1988兲. 6 I. Avrutsky, Phys. Rev. B 70, 155416 共2004兲. 7 D. Sarid, Phys. Rev. Lett. 47, 1927 共1981兲. 8 J. J. Burke et al., Phys. Rev. B 33, 5186 共1986兲. 9 E. D. Palik, Handbook of Optical Constants of Solids 共Academic Press, New York, 1988兲. 10 P. B. Johnson and R. W. Christy, Phys. Rev. B 6, 4370 共1972兲. 11 D. F. P. Pile et al., J. Appl. Phys. 100, 013101 共2006兲. 1 K.

0.8

SPP(Air)

0.5

0.6

0.5 0.4 0.2 0.0 0.0

With Si substrate: * n = 1.0050 -1 α = 44.12cm

0.0 1.0

BPP1

BPP0 0.5

1.0

1.5

H||

Reflection (arb. units)

The modes in a nanoscale metal-dielectric multilayer are relatively easy to predict and simulate numerically, but their experimental verification is challenging due to the deep subwavelength confinement and very high optical losses. In this paper we report experimental studies of guided modes in nanoscale metal-dielectric multilayers. To excite the highindex modes, we used the evanescent light-coupling scheme. High-index material 共silicon, nSi = 3.48兲 was used to match the wave vector of light in free space to the wave vector of a guided mode. To access a wider range of modal indices, semicylinder geometry instead of more traditional prism was used. In a similar manner, a high-index semispherical solid immersion lens can be used, but for accurate angular measurement the semicylindrical geometry is preferable. Light from a fiber-coupled tunable 共1490– 1590 nm兲 semiconductor laser 共Photonetics Inc.兲 was collimated using a 10⫻ objective. The laser was tuned to the wavelength of 1550 nm, which was verified by an optical spectrum analyzer 共HewlettPackard HP70951B兲. The spectral width of the laser radiation was below the resolution of the spectrometer 共⬍0.1 nm兲. The angular reflection spectrum was measured, and data were plotted as a function of the product nSi sin共␪兲, where ␪ is the incident angle. In this scale, the intensity minima directly indicate the modal indices of the guided modes excited through the evanescent coupling. The multilayer structure was designed to consist of three pairs of silica 共⬃25 nm兲 / gold 共⬃25 nm兲. The layers were deposited directly on the flat facet of the semicylinder. The gold layers were deposited by electron beam evaporation and silica layers by plasma-enhanced chemical vapor deposition. With two dielectric gaps between three metal layers, the structure supports two BPP modes. The thickness of the layers was chosen to ensure that the effective indices of the bulk modes are comfortably in the measurement range 共n* ⬍ 3.0兲. The first silica layer between silicon and gold is not crucial. Its role is to adjust the evanescent coupling strength for clear observation of guided modes. The experimental data are shown in Fig. 4 on the left. They verify the guided modes with modal indices 2.31 and 2.88 recognized as BPP0 and BPP1. The structure was designed for measuring the modal indices of the bulk modes, while the SPP at the interface with

2.0

2.5

nSisin(θ)

3.0

0.0

1.0 0.5

3.5

0.5

θ

S i A u

SPP-SiO2 (TM1) *

With Si substrate: over-damped

n = 1.4573 -1 α = 44.1cm

With Si substrate: * n = 2.3051 -1 α = 6,094cm

BPP (TM ) 0 2 * n = 2.3032 -1 α = 3,514cm

With Si substrate: * n = 2.8786 -1 α = 9,595cm

BPP1 (TM3) * n = 2.8481 -1 α = 7,164cm

0.0 1.0

SPP-Air (TM0) * n = 1.0050 -1 α = 44.08cm

0.0

S iO 2

Si

SiO

Au

Air

2 -0.5 -500-400-300-200-100 0 100 200 300 400 500 600

Coordinate (nm)

FIG. 4. Experimental reflection data for a silica-gold nanoscale multilayer 共circles兲 and theoretical fit 共solid line兲 indicating BPP0 and BPP1 modes 共left兲. Simulated mode profiles in the experimental structure using the best-fit thicknesses of the layers 共right兲. Solid lines correspond to the structure with high-index silicon substrate. Dashed lines show mode profiles assuming the substrate is silica.

Takano and J. Hamasaki, IEEE J. Quantum Electron. QE-8, 206 共1972兲. 13 I. V. Shadrivov et al., Phys. Rev. E 67, 057602 共2003兲. 14 J. B. Pendry and D. Smith, Phys. Today 57 共6兲, 37 共2004兲. 15 G. Shvets, Phys. Rev. B 67, 035109 共2003兲. 16 H. Shin and S. Fan, Phys. Rev. Lett. 96, 073907 共2006兲. 17 V. A. Podolskiy and E. E. Narimanov, Phys. Rev. B 71, 201101共R兲 共2005兲. 18 J. B. Pendry, Phys. Rev. Lett. 85, 3966 共2000兲. 19 C. Luo et al., Phys. Rev. B 65, 201104 共2002兲. 20 N. Fang et al., Science 308, 534 共2005兲. 21 V. M. Shalaev, Nat. Photon. 1, 41 共2007兲. 22 S. M. Rytov, Sov. Phys. JETP 2, 466 共1956兲. 23 E. N. Economou, Phys. Rev. 182, 539 共1969兲. 24 J. Elser et al., Appl. Phys. Lett. 90, 191109 共2007兲. 25 D. P. Makhnovskiy et al., Phys. Rev. B 64, 134205 共2001兲.

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