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Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 14 — Jul. 15, 2013
  • pp: 16514–16527
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Spatial dispersion and nonlocal effective permittivity for periodic layered metamaterials

Ruey-Lin Chern  »View Author Affiliations


Optics Express, Vol. 21, Issue 14, pp. 16514-16527 (2013)
http://dx.doi.org/10.1364/OE.21.016514


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Abstract

The feature of spatial dispersion in periodic layered metamaterials is theoretically investigated. An effective medium model is proposed to derive the nonlocal effective permittivity tensor, which exhibits drastic variations in the wave vector domain. Strong spatial dispersion is found in the frequency range where surface plasmon polaritons are excited. In particular, the nonlocal effect gives rise to additional waves that are identified as the bonding or antibonding modes with symmetric or antisymmetric surface charge alignments. Spatial dispersion is also manifest on the parabolic-like dispersion, a non-standard type of dispersion in the medium. The associated negative refraction and backward wave occur even when the effective permittivity components are all positive, which is considered a property not available in the local medium.

© 2013 OSA

1. Introduction

For periodic structures like crystals, spatial dispersion is important when the coupling of fields between adjacent unit cells cannot be ignored. This is usually the case when the wavelength of electromagnetic wave approaches the lattice period, although for some structures (e.g. wire mediums) strong spatial dispersion appears even in the long wavelength [28

28. P. A. Belov, R. Marques, S. I. Maslovski, I. S. Nefedov, M. Silveirinha, C. R. Simovski, and S. A. Tretyakov, “Strong spatial dispersion in wire media in the very large wavelength limit,” Phys. Rev. B 67, 113103 (2003) [CrossRef] .

, 29

29. C. R. Simovski and P. A. Belov, “Low-frequency spatial dispersion in wire media,” Phys. Rev. E 70, 046616 (2004) [CrossRef] .

]. Periodic metal-dielectric layered structures are considered a simple yet nontrivial example of the nonlocal medium. On the one hand, spatial dispersion is likely to occur due to the strong interactions of fields between adjacent layers inside the structure. In the optical regime, the plasmonic effect may serve as the source of spatial dispersion [30

30. F. J. Garcia de Abajo, “Nonlocal effects in the plasmons of strongly interacting nanoparticles, dimers, and waveguides,” J. Phys. Chem. C 112, 17983–17987 (2008) [CrossRef] .

]. On the other hand, the dispersion relation of the underlying structure is analytical [31

31. L. Brillouin, Wave Propagation in Periodic Structures (Dover, 1953).

] and can be exploited to derive the effective permittivity that reveals the spatial dispersion or nonlocal effect in an explicit manner [32

32. J. Elser, V. A. Podolskiy, I. Salakhutdinov, and I. Avrutsky, “Nonlocal effects in effective-medium response of nanolayered metamaterials,” Appl. Phys. Lett. 90, 191109 (2007) [CrossRef] .

34

34. A. V. Chebykin, A. A. Orlov, C. R. Simovski, Y. S. Kivshar, and P. A. Belov, “Nonlocal effective parameters of multilayered metal-dielectric metamaterials,” Phys. Rev. B 86, 115420 (2012) [CrossRef] .

]. The layered structures also exhibit extraordinary features such as subwavelength imaging [35

35. B. Wood, J. B. Pendry, and D. P. Tsai, “Directed subwavelength imaging using a layered metal-dielectric system,” Phys. Rev. B 74, 115116 (2006) [CrossRef] .

, 36

36. A. Fang, T. Koschny, and C. M. Soukoulis, “Optical anisotropic metamaterials: Negative refraction and focusing,” Phys. Rev. B 79, 245127 (2009) [CrossRef] .

] and negative refraction [37

37. X. Fan, G. P. Wang, J. C. W. Lee, and C. T. Chan, “All-angle broadband negative refraction of metal waveguide arrays in the visible range: theoretical analysis and numerical demonstration,” Phys. Rev. Lett. 97, 073901 (2006) [CrossRef] [PubMed] .

, 38

38. E. Verhagen, R. de Waele, L. Kuipers, and A. Polman, “Three-dimensional negative index of refraction at optical frequencies by coupling plasmonic waveguides,” Phys. Rev. Lett. 105, 223901 (2010) [CrossRef] .

].

In this study, the author investigates the feature of spatial dispersion in periodic layered meta-materials, with emphasis on the nonlocal effective permittivity. An effective medium model is proposed to derive the nonlocal effective permittivity tensor of the layered structure, with the wave vector dependence. The effective permittivity exhibits drastic variations in the wave vector domain, showing a typical feature of nonlocal resonance. In particular, strong spatial dispersion is found in the frequency range where surface plasmon polaritons are excited. Due to the nonlocal effect, additional waves appear in the medium and are identified as the bonding or antibonding modes with symmetric or antisymmetric surface charge alignments [39

39. E. N. Economou, “Surface plasmons in thin films,” Phys. Rev. 182, 539–554 (1969) [CrossRef] .

]. The additional wave can be a forward wave with negative refraction or a backward wave with ordinary refraction, but with a very different feature from that in a local anisotropic medium with opposite signs of the permittivity components [40

40. P. A. Belov, “Backward waves and negative refraction in uniaxial dielectrics with negative dielectric permittivity along the anisotropy axis,” Microw. Opt. Technol. Lett. 37, 259–263 (2003) [CrossRef] .

]. Due to the nonlocal effect, negative refraction and backward wave occur even when the effective permittivity components are all positive.

This paper is organized as follows: In Sec. 2, the effective medium model for the periodic layered metamaterials is proposed, from which the nonlocal effective permittivity tensor is derived. In Secs. 3 and 4, the properties of the in-plane and out-of-plane effective permittivities, respectively, along with four illustrated examples, are discussed. In Sec. 5, spatial dispersion in the underlying layered structure is summarized. Finally, a conclusion of the present study is given in Sec. 6.

2. Effective medium model

Consider a periodic layered structure consisting of two alternating nonmagnetic materials stacked along the x direction, one with the dielectric constant ε1 and thickness a1, and the other with ε2 and a2, as schematically shown in Fig. 1. Assume that the electromagnetic wave is of the form ei(k·rωt). Let the wave vector lie on the xz plane, that is, k = (kx, 0, kz), without loss of generality. The dispersion relation of the periodic layered structure is given by [31

31. L. Brillouin, Wave Propagation in Periodic Structures (Dover, 1953).

]
cos(kxa)=cos(q1a1)cos(q2a2)12(ε2q1ε1q2+ε1q2ε2q1)sin(q1a1)sin(q2a2),
(1)
cos(kxa)=cos(q1a1)cos(q2a2)12(q1q2+q2q1)sin(q1a1)sin(q2a2)
(2)
for TM and TE polarizations, respectively, where q1=ε1k02kz2, q2=ε2k02kz2, a = a1 + a2, and k0 = ω/c. Here, TM and TE refer to transverse magnetic, where E = (Ex, 0, Ez) and H = (0, Hy, 0), and transverse electric, where E = (0, Ey, 0) and H = (Hx, 0, Hz), respectively.

Fig. 1 Schematic diagram of the periodic layered structure consisting of two alternating materials.

2.1. Nonlocal effective permittivity tensor

The basic idea of the effective medium model is to extract the effective permittivity tensor from the analytical dispersion relations of the structure. The essential work is to expand the dispersion relations to higher (than the second) order of the wave vector components. Suppose that the lattice period is much less than the wavelength, aλ = 2π/k0, so that the layered structure can be regarded as an effective medium, characterized by the effective permittivity tensor:
ε_eff=[εxeff000εyeff000εzeff],
(3)
where εxeff, εyeff, and εzeff are the effective permittivity components. The dispersion relations for the effective medium are given by
kx2εzeff+kz2εxeff=k02,
(4)
kx2+kz2=εyeffk02
(5)
for TM and TE polarizations, respectively.

Expanding Eqs. (1) and (2) into power series of kx, kz, and k0 up to fourth order, truncating the higher order terms, and rearranging the expansions in the forms of Eqs. (4) and (5), respectively, we have
εzeff=εz0α12k02a21112kx2a2,
(6)
εxeff=εx0α12k02a21+εx0εz0(β12kz2a2γ6k02a2),
(7)
εyeff=εy0(1+16kz2a2)+a212k02(kx4kz4)α12k02a2,
(8)
where
εy0=εz0=f1ε1+f2ε2,
(9)
εx0=ε1ε2f2ε1+f1ε2,
(10)
are the quasistatic effective permittivities based on the Maxwell-Garnett mixing rule [41

41. J. C. Maxwell-Garnett, “Colours in metal glasses and in metallic films,” Philos. Trans. R. Soc. London 203, 385 (1904) [CrossRef] .

], and
α=[f12ε1+(1f12)ε2][(1f22)ε1+f22ε2],
(11)
β=1ε1ε2[(12f1f2)ε1+2f1f2ε2][2f1f2ε1+(12f1f2)ε2],
(12)
γ=1ε1ε2[f13f2ε13+f1(12f12f2+f23)ε12ε2+f2(12f1f22+f13)ε1ε22+f1f23ε23].
(13)
are the coefficients determined by the fractions of two materials in the unit cell, f1 = a1/a and f2 = a2/a, and the material parameters, ε1 and ε2.

Note that εzeff and εyeff are no longer equal as in the quasistatic case, although the layered structure makes no difference between y and z directions. The medium properties, therefore, depend on the polarization. Note also that kx2n and kz2n (n = 1, 2) appear in the effective permittivities [cf. Eqs. (6) and (8)], meaning that the underlying effective medium is spatially dispersive or nonlocal. The nonlocal effective permittivities differ from the quasistatic ones by the factors depending on the structure and material parameters. In addition, there are no kx and kz terms in the effective permittivities, which is a consequence of inversion symmetry[42

42. R. M. Hornreich and S. Shtrikman, “Theory of gyrotropic birefringence,” Phys. Rev. 171, 1065–1074 (1968) [CrossRef] .

] of the underlying structure, an invariance property of a system when the coordinates are inverted.

The above procedure can be performed to higher orders of the wave vector components to deliver ever more accurate results, at the expense of resolving more involved coefficients. To the author’s experience, expanding the dispersion relations to eighth power of the wave vector components can yield sufficiently accurate effective permittivity tensor that recovers the dispersion of the original structure within a few percent of error. The nonlocal effective permittivities, no matter based on how higher orders of expansions, are characterized by the intrinsic frequencies of the constituent materials.

To extract the effective parameters from the dispersion relations is considered a somewhat intuitive approach. From the viewpoint of wave propagation, a homogeneous medium with the effective parameters is approximately equivalent to a finite structure with the same dispersion if the characteristic length of the structure (that is, the period) is much less than the wavelength: aλ. The benefit of the present approach is to express the effective permittivities in compact formulas. A more rigorous approach to the effective parameters may resort to homogenization based on the field averaging [33

33. A. V. Chebykin, A. A. Orlov, A. V. Vozianova, S. I. Maslovski, Y. S. Kivshar, and P. A. Belov, “Nonlocal effective medium model for multilayered metal-dielectric metamaterials,” Phys. Rev. B 84, 115438 (2011) [CrossRef] .

, 34

34. A. V. Chebykin, A. A. Orlov, C. R. Simovski, Y. S. Kivshar, and P. A. Belov, “Nonlocal effective parameters of multilayered metal-dielectric metamaterials,” Phys. Rev. B 86, 115420 (2012) [CrossRef] .

]. In this approach, the effective permittivities are represented by analytical yet lengthy terms. The dispersion relation also appears as an important factor in the formulation.

2.2. Characteristic frequencies

Let the materials 1 and 2 in the layered structure be the dielectric and metal, respectively. Using the Drude model (with the loss neglected) for the metal: ε2=1ωp2/ω2, where ωp is the plasma frequency of the metal, the quasistatic effective permittivities are given as
εz0=εy0=(f1ε1+f2)(1ω02ω2),
(14)
εx0=(εf2ε1+f1)(ω2ωp2ω2ω2),
(15)
where
ω0=ωpε1(f1/f2)+1,
(16)
ω=ωpε1(f2/f1)+1
(17)
are the zero frequency of εz0 (and εy0) and the pole frequency of εx0, respectively. Note that ω0 (ω) is reduced (raised) as the ratio f1/f2 increases. The zero frequency of εx0 is ωp, which is the same as that of ε2. In the quasistatic limit, the layered structure is regarded as a local anisotropic medium, which behaves like a plasmonic material in the parallel (to the metal-dielectric interface) direction, with a reduced plasma frequency ω0 [cf. Eq. (14)], and like an ionic crystal in the perpendicular direction, with a transverse resonance frequency ω and a longitudinal resonant frequency ωp [cf. Eq. (15)].

Note in Eq. (6) that εzeff has a zero frequency close to ω0, and εxeff has two zero frequencies: one is ω0 and the other is close to ωp. In addition, there are two pole frequency branches of εxeff: one is located at lower frequencies, with ω1 being the lowest frequency (at kz = 0), and the other is located at higher frequencies, with ω2 being the highest frequency (also at kz = 0). They are close to either ω0 or ω, depending on the fraction of dielectric (or metal) in the unit cell. Figure 2(a) is an example of the variations of ω1 and ω2, along with ω0 and ω, with respect to f1.

Fig. 2 Variations of (a) the pole frequencies ω1 and ω2 and (b) the cutoff frequency ω3 with respect to the dielectric fraction f1 (= 1 − f2) in the unit cell.

Concerning the electromagnetic waves in the periodic metal-dielectric layered structure, there are three length scales: the lattice period a, the wavelength λ = 2π/k0, and the plasma wavelength λp = 2πc/ωp. In order for the effective medium to be valid, a/λ = (a/λp)(ω/ωp) should be, in principle, much smaller than unity. In practice, this condition is attained when a/λp is small and ω is below ωp. The parameters used in this article will be carefully arranged so that a/λ is considered small enough, yet still feasible in fabrication with the modern nanotechnology.

3. In-plane effective permittivities

Figure 3 is an example of εzeff and εxeff as the functions of ω and kx or kz for the periodic metal-dielectric layered structure with f1 > f2, based on the effective medium model (cf. Sec. 2) with the expansions of the wave vector components to eighth order. Another example of εzeff and εxeff for f1 < f2 is shown in Fig. 4. In either case, the effective permittivity εzeff [Figs. 3(a) and 4(a)] is basically characterized by the quasistatic effective permittivity εz0 [cf. Eq. (9)] and modified by the perpendicular (to the metal-dielectric interface) wave vector component kx as well as the frequency ω. For periodic structures, the largest value of |kx| is determined by the lattice period: π/a, and εzeff deviates slightly from εz0 as |kx| increases. The nonlocal effect exhibited by εzeff is usually weak.

Fig. 3 In-plane effective permittivity (a) εzeff and (b) εxeff for the periodic metal-dielectric layered structure with ε1 = 3, ε2=1ωp2/ω2, a/λp = 0.2, and f1 = 0.75 (f2 = 0.25).
Fig. 4 In-plane effective permittivity (a) εzeff and (b) εxeff for the periodic metal-dielectric layered structure with ε1 = 3, ε2=1ωp2/ω2, a/λp = 0.2, and f1 = 0.25 (f2 = 0.75).

The effective permittivity εxeff [Figs. 3(b) and 4(b)], on the other hand, shows a strong non-local effect. Being dependent on the parallel wave vector component kz, which is not restricted by the geometry, εxeff differs very much from the quasistatic counterpart εx0. This feature is more evident near the poles of εxeff (extending from the poles of εx0 at kz = 0). For a given frequency, εxeff may experience a drastic change from positive to negative infinity, or the other way around, across a certain value of |kz|, which is considered a typical feature of nonlocal resonance that occurs in a spatially dispersive medium. At kz = 0, the onset of resonance occurs at ω1 and ω2. A similar feature can be observed in the quasistatic effective permittivity εx0 (with the resonance at ω). As kz ≠ 0, the resonance frequencies gradually move toward each other. In the metal-dielectric layered structure, such resonance, as will be shown later, is attributable to the excitation of surface plasmon polaritons.

In the present problem, spatial dispersion is coupled with frequency (temporal) dispersion. Basic properties of the in-plane effective permittivities are categorized into four frequency ranges:

3.1. Weak spatial dispersion range: 0 < ω < ω1

In this range, εzeff<0 and εxeff>0 [cf. Figs. 3 and 4], and the dispersion relation is hyperbolic-like. Here, ω1 is the lowest frequency of the lower pole branch of εxeff, which is close to ω0 for f1 > f2 and close to ω for f1 < f2 [cf. Fig. 2(a)]. The dispersion relation [cf. Eq. (4)] is dominated by the kz2 and kx2 terms, with opposite signs of the attached coefficients (the one with kx2 being negative). Let the xy plane be an interface between vacuum and the effective medium. For a wave incident from vacuum, with the wave vector lying on the xz plane, a forward wave with negative refraction will occur in the medium. This feature is similar as in a uniaxially anisotropic medium with opposite signs of the permittivity components, the one normal to the interface (between vacuum and the medium) being negative [40

40. P. A. Belov, “Backward waves and negative refraction in uniaxial dielectrics with negative dielectric permittivity along the anisotropy axis,” Microw. Opt. Technol. Lett. 37, 259–263 (2003) [CrossRef] .

].

3.2. Strong spatial dispersion range: ω1 < ω < ω2

In this range, εzeff>0 and εxeff changes from positive to negative infinity by experiencing a pole for f1 > f2 [cf. Fig. 3]. Here, ω2 is the highest frequency of the higher pole branch of εxeff, which is close to ω for f1 > f2 [cf. Fig. 2(a)]. At small |kz|, both εzeff and εxeff are positive and the dispersion relation is elliptic-like. At large |kz|, εxeff is dominated by the kz4 term and the dispersion relation becomes parabolic-like, which is a non-standard type of dispersion in the medium. This is the frequency range where spatial dispersion or nonlocal effect tends to be significant. In this situation, the two different relations coexist, leading to a mixed type of dispersion. Accordingly, there are two eigenwaves. This can be seen by using Eq. (6) in Eq. (4) to give
kz4a4+2β(6εz0εx0γk02a2)kz2a2+1β(12kx2a2kx4a412εz0k02a2+αk04a4)=0.
(18)
For a given kx, both of the allowed kz, either positive or negative, can be real. For a wave incident from vacuum onto the layered structure, the splitting of wave in the structure is expected to occur [24

24. A. A. Orlov, P. M. Voroshilov, P. A. Belov, and Y. S. Kivshar, “Engineered optical nonlocality in nanostructured metamaterials,” Phys. Rev. B 84, 045424 (2011) [CrossRef] .

]. For f1 < f2, on the other hand, εzeff<0 and εxeff changes from negative to positive infinity by experiencing a pole [cf. Fig. 4]. In this case, ω2 is close to ω0 [cf. Fig. 2(a)]. At small |kz|, both εzeff and εxeff are negative and the elliptic-like dispersion does not exist. For a given kx, only one allowed kz, either positive or negative, is real. Therefore, there is only one eigenwave, with the parabolic-like dispersion. Regarding the wave propagation characteristics, this range is further divided into two subranges:

3.2.1. Below surface plasma frequency: ω1 < ω < ωsp

Example 1: f1 > f2. Figure 5(a) is an example showing the equifrequency contours, along with the wave vectors and Poynting vectors, for a larger dielectric fraction. The contour is characterized by a quartic curve in the xy plane:
x2(1εx2)a2+y2b2=1,
(22)
where a and b are constants, and ε is a small positive number. The parabolic-like character of the quartic curve comes from the x4 term at larger x, which corresponds to the nonlocal effect in the present problem. The elliptic-like character, on the other hand, is dominant at smaller x. Note that the contour normal (denoted by the green arrow) with the parabolic-like dispersion is oriented toward the same side of the incident wave vector (with respect to the interface normal), leading to negative refraction. The corresponding eigenmode, plotted in Fig. 6(a), shows a typical feature of surface plasmon polariton, with the fields highly concentrated on the metal-dielectric interfaces. In particular, this mode has a symmetric alignment of surface charges and is identified as the bonding mode. The colors of surface charges in the figure have been exaggerated to show more clearly the alignment pattern. Note also that this mode is located (marked by the red dot) near the pole of εxeff (denoted by the vertical gray line) [cf. Fig. 5(a)]. The strong nonlocal effect is manifest on the Lorentzian resonance character of the effective permittivity εxeff, as shown in Fig. 7(a). As |kz| increases, there is a large discrepancy between εxeff and εx0 (indicated by the dashed line). For the eigenmode with negative refraction, both εzeff and εxeff (marked by the red dot) are positive. This is considered a property not available in the local anisotropic medium [40

40. P. A. Belov, “Backward waves and negative refraction in uniaxial dielectrics with negative dielectric permittivity along the anisotropy axis,” Microw. Opt. Technol. Lett. 37, 259–263 (2003) [CrossRef] .

].

Fig. 5 Equifrequency contours at (a) ω/ωp = 0.39 and (b) ω/ωp = 0.6 for the periodic metal-dielectric layered structure with ε1 = 3, ε2=1ωp2/ω2, f1 = 0.75 (f2 = 0.25), and a/λp = 0.2. Gray circles are equifrequency contours in vacuum. Gray and black arrows are wave vectors in vacuum and the layered structure, respectively, at θ = 40°.
Fig. 6 Electric field and surface charge patterns for the eigenmodes associated with the parabolic-like dispersion depicted in Fig. 5 for (a) ω/ωp = 0.39 (bonding mode) and (b) ω/ωp = 0.6 (antibonding mode). Red and blue colors correspond to positive and negative charges, respectively.
Fig. 7 Effective permittivity εxeff at (a) ω/ωp = 0.39 and (b) ω/ωp = 0.6 for the periodic metal-dielectric layered structure with the same parameters as in Fig. 5.

Example 2: f1 < f2. Figure 8(a) is an example for a smaller dielectric fraction. In this case, only one eigenwave with negative refraction exists. The character of the wave, however, is different from the counterpart for f1 > f2 [cf. Fig. 5(a)]. The equifrequency contour is characterized by a similar quartic curve in the xy plane as in Eq. (22), but with the minus sign on the right side:
x2(1εx2)a2+y2b2=1.
(23)
The parabolic-like character remains, while the elliptic-like character disappears. The corresponding eigenmode also shows a typical feature of surface plasmon polariton, but with an antisymmetric alignment of surface charges on the metal-dielectric interfaces, which is identified as the antibonding mode, as shown in Fig. 9(a). The feature of negative refraction is consistent with that in a uniaxially anisotropic medium with opposite signs of the permittivity components: εzeff<0 and εxeff>0[40

40. P. A. Belov, “Backward waves and negative refraction in uniaxial dielectrics with negative dielectric permittivity along the anisotropy axis,” Microw. Opt. Technol. Lett. 37, 259–263 (2003) [CrossRef] .

], as in the range where 0 < ω < ω1 (cf. Sec. 3.1). The spatial dispersion associated with this mode, however, is still strong. The effective permittivity εxeff in Fig. 10(a) also shows a Lorentzian resonance character, but with a reverse sense from the counterpart for f1 > f2 [cf. Fig. 7(a)]. The eigenmode is also located near the pole, where εxeff changes rapidly.

Fig. 8 Equifrequency contours at (a) ω/ωp = 0.38 and (b) ω/ωp = 0.62 for the periodic metal-dielectric layered structure with ε1 = 3, ε2=1ωp2/ω2, f1 = 0.25 (f2 = 0.75), and a/λp = 0.2. Gray circles are equifrequency contours in vacuum. Gray and black arrows are wave vectors in vacuum and the layered structure, respectively, at θ = 40°.
Fig. 9 Electric field and surface charge patterns for the eigenmodes associated with the parabolic-like dispersion depicted in Fig. 8 for (a) ω/ωp = 0.38 (antibonding mode) and (b) ω/ωp = 0.62 (bonding mode). Red and blue colors correspond to positive and negative charges, respectively.
Fig. 10 Effective permittivity εxeff at (a) ω/ωp = 0.38 and (b) ω/ωp = 0.62 for the periodic metal-dielectric layered structure with the same parameters as in Fig. 8.

3.2.2. Above surface plasma frequency: ωsp < ω < ω2

In this range, the wave associated with the elliptic-like dispersion, if any, is the same as in the range where ω1 < ω < ωsp (cf. Sec. 3.2.1). The wave associated with the parabolic-like dispersion, however, is very much different. Two illustrative examples with different dielectric (metal) fractions in the unit cell are given below:

Example 3: f1 > f2. Figure 5(b) is an example showing the equifrequency contours, along with the wave vectors and Poynting vectors, for a larger dielectric fraction. Due to the anomalous frequency dispersion, kz for the parabolic-like dispersion is chosen to be negative, so that the energy flows away from the interface (+z direction). Otherwise, the principle of causality will be violated. The wave with the parabolic-like dispersion is therefore a backward wave with ordinary refraction, rather than the forward wave with negative refraction as in the range below ωsp [cf. Fig. 5(a)]. The eigenmode in Fig. 6(b) is shown to be an antibonding mode of surface plasmon polariton, with an antisymmetric alignment of surface charges on the metal-dielectric interfaces. Note that the antibonding mode has a higher frequency than the respective bonding mode [cf. Fig. 6(a)]. This feature is similar as in a metal film in vacuum [39

39. E. N. Economou, “Surface plasmons in thin films,” Phys. Rev. 182, 539–554 (1969) [CrossRef] .

]. The effective permittivity εxeff in Fig. 7(b) looks alike as in Fig. 7(a) in the wave vector domain. They are, however, different in the frequency domain. By adding a small increment of frequency δω, the patterns of εxeff move in opposite directions between the two cases [compare the gray lines in Figs. 7(a) and 7(b)]. The opposite trend in the frequency domain is responsible for the distinction between negative refraction (forward wave) and ordinary refraction (backward wave). For the eigenmode with backward wave, both εzeff and εxeff (marked by the red dot) are positive. This is also considered a property not available in the local anisotropic medium [40

40. P. A. Belov, “Backward waves and negative refraction in uniaxial dielectrics with negative dielectric permittivity along the anisotropy axis,” Microw. Opt. Technol. Lett. 37, 259–263 (2003) [CrossRef] .

].

Example 4: f1 < f2. Figure 8(b) is an example for a smaller dielectric fraction. In this case, there is only one eigenwave with the parabolic-like dispersion, as in the range below ωsp [cf. Fig. 8(a)]. The eigenwave, however, is a backward wave with ordinary refraction, as in the range above ωsp for f1 > f2 [cf. Fig. 5(b)]. The character of the wave is different from either case. The eigenmode in Fig. 9(b) is shown to be a bonding mode of surface plasmon polariton, with a symmetric alignment of surface charges on the metal-dielectric interfaces. Note that the bonding mode has a higher frequency than the respective antibonding mode [cf. Fig. 9(a)]. This feature is similar as in an insulating film between two semi-infinite metals [39

39. E. N. Economou, “Surface plasmons in thin films,” Phys. Rev. 182, 539–554 (1969) [CrossRef] .

]. The feature of backward wave is also not available in a uniaxially anisotropic medium with opposite signs of the permittivity components, the one tangential to the interface (between vacuum and the medium) being negative [40

40. P. A. Belov, “Backward waves and negative refraction in uniaxial dielectrics with negative dielectric permittivity along the anisotropy axis,” Microw. Opt. Technol. Lett. 37, 259–263 (2003) [CrossRef] .

]. In the present case, the tangential effective permittivity component is positive ( εxeff>0), while the normal component is negative ( εzeff<0). The spatial dispersion is strong as in the range below ωsp. The effective permittivity εxeff in Fig. 10(b) also shows the Lorentzian resonance character as in Fig. 10(a), but with an opposite trend with increasing the frequency [compare the gray lines in Figs. 10(a) and 10(b)].

3.3. Weak spatial dispersion range: ω2 < ω < ωp

In this range, εzeff>0 and εxeff<0 [cf. Figs. 3 and 4], and the dispersion relation is hyperbolic-like. The dispersion curve is dominated by the kz2 and kx2 terms, with opposite signs of the attached coefficients (the one with kz2 being negative). The eigenwave is a backward wave with ordinary refraction, which is similar as in the range ωsp < ω < ω2 (cf. Sec. 3.2.2). The spatial dispersion, however, is weak in this range. The effective permittivities do not change much with the wave vector components.

3.4. Weak spatial dispersion range: ω > ωp

In this range, εzeff>0 and εxeff>0 [cf. Figs. 3 and 4], and the dispersion relation is elliptic-like. The dispersion curve is also dominated by the kz2 and kx2 terms, both with positive attached coefficients. In this range, the effective medium behaves like an ordinary anisotropic dielectric. The eigenwave is a forward wave with ordinary refraction. The spatial dispersion is also weak.

4. Out-of-plane effective permittivity

Unlike the in-plane effective permittivities εzeff and εxeff, where the properties are very different between f1 > f2 and f1 < f2, the out-of-plane effective permittivity εyeff shows similar features for either case. For smaller |kz| and |kx|, εyeff is basically characterized by the quasistatic effective permittivity εy0, with a slight modification factor [cf. Eq. (8)]. In this situation, the effective medium behaves like an isotropic plasmonic metal with a reduced plasma frequency ω0. This frequency is very close to the cutoff frequency ω3 for TE polarization, the lowest frequency for TE waves to propagate in the effective medium [cf. Fig. 2(b)]. For larger |kz| and |kx|, εyeff may deviate much from εy0. Basic properties of the out-of-plane effective permittivity are divided into two frequency ranges:

4.1. Below cutoff frequency: ω < ω3

Figure 11(a) is an example of εyeff as the functions of kz and kx, based on the effective medium model [cf. Sec. 2] for ω < ω3. Note that εyeff is negative at smaller |kz| and |kx|, which becomes more negative as |kz| increases and changes to positive as |kx| increases (the contour of εyeff=0 is plotted in red color). Since the frequency is below the cutoff frequency ω3, there are no waves allowed to propagate in the effective medium. The spatial dispersion or nonlocal effect is not relevant.

Fig. 11 Out-of-plane effective permittivity εyeff for the periodic metal-dielectric layered structure with ε1 = 3, ε2=1ωp2/ω2, a/λp = 0.2, and f1 = 0.5 (f2 = 0.5) for (a) ω/ωp = 0.4 and (b) ω/ωp = 0.6. Red curves correspond to εyeff=0.

4.2. Above cutoff frequency: ω > ω3

Figure 11(b) is an example of εyeff for ω > ω3. Note that εyeff is positive at smaller |kz| and |kx|, which may change to negative as |kz| increases and becomes more positive as |kx| increases. Unlike the in-plane effective permittivities, where the nonlocal effect comes mainly from kz, the nonlocal effect associated with the out-of-plane effective permittivity basically comes from the dependence on kx. This effect is usually weak as |kx| is bound by π/a. The equifrequency contours can be characterized by a quartic curve in the xy plane:
x2(1+εx2)+y2(1εy2)=r2,
(24)
where r is a constant and ε is a small positive number. At smaller x and y, the contours are nearly circles, while at larger x and y, the contours are a bit of distorted along the y axis. The dispersion relation is elliptic-like, with a normal frequency dispersion. The eigenwave is a forward wave with ordinary refraction. The spatial dispersion exhibited by εyeff is not significant.

It is worthy of noting that although the effective permittivity εyeff may change substantially with the wave vector components, the spatial dispersion or nonlocal effect is either not significant or irrelevant. This is because εyeff always becomes negative at larger |kz| and thus no waves exist there. The effect due to kx, however, is weak, as |kx| is bound by π/a.

5. Summary on spatial dispersion

6. Concluding remarks

Acknowledgments

This work was supported in part by National Science Council of the Republic of China under Contract No. NSC 99-2221-E-002-121-MY3.

References and links

1.

J. J. Hopfield and D. G. Thomas, “Theoretical and experimental effects of spatial dispersion on the optical properties of crystals,” Phys. Rev. 132, 563–572 (1963) [CrossRef] .

2.

L. D. Landau, E. M. Lifshitz, and L. P. Pitaevski, Electrodynamics of Continuous Media, 2nd ed. (Butterworth-Heinenan, 1984).

3.

V. M. Agranovich and V. L. Ginzburg, Crystal Optics with Spatial Dispersion, and Excitons (Springer-Verlag, 1984) [CrossRef] .

4.

J. J. Hopfield, “Theory of the contribution of excitons to the complex dielectric constant of crystals,” Phys. Rev. 112, 1555–1567 (1958) [CrossRef] .

5.

S. I. Pekar, “Theory of electromagnetic waves in a crystal with excitons,” J. Phys. Chem. Solids 5, 11–22 (1958) [CrossRef] .

6.

R. Ruppin, “Reflectivity of a nonlocal dielectric with an excitonic surface potential,” Phys. Rev. B 29, 2232–2237 (1984) [CrossRef] .

7.

B. Chen and D. F. Nelson, “Wave propagation of exciton polaritons by a wave-vector-space method,” Phys. Rev. B 48, 15372–15389 (1993) [CrossRef] .

8.

G. E. H. Reuter and E. H. Sondheimer, “The theory of the anomalous skin effect in metals,” Proc. R. Soc. A-Math. Phys. Eng. Sci. 195, 336–364 (1948) [CrossRef] .

9.

K. L. Kliewer and R. Fuchs, “Anomalous skin effect for specular electron scattering and optical experiments at non-normal angles of incidence,” Phys. Rev. 172, 607–624 (1968) [CrossRef] .

10.

R. Ruppin, “Optical properties of a spatially dispersive cylinder,” J. Opt. Soc. Am. B 6, 1559–1563 (1989) [CrossRef] .

11.

R. Ruppin, “Extinction properties of thin metallic nanowires,” Opt. Comm. 190, 205–209 (2001) [CrossRef] .

12.

A. A. Maradudin and D. L. Mills, “Effect of spatial dispersion on the properties of a semi-infinite dielectric,” Phys. Rev. B 7, 2787–2810 (1973) [CrossRef] .

13.

A. R. Melnyk and M. J. Harrison, “Resonant excitation of plasmons in thin films by elecromagnetic waves,” Phys. Rev. Lett. 21, 85–88 (1968) [CrossRef] .

14.

W. E. Jones, K. L. Kliewer, and R. Fuchs, “Nonlocal theory of the optical properties of thin metallic films,” Phys. Rev. 178, 1201–1203 (1969) [CrossRef] .

15.

A. R. Melnyk and M. J. Harrison, “Theory of optical excitation of plasmons in metals,” Phys. Rev. B 2, 835–850 (1970) [CrossRef] .

16.

R. Ruppin, “Optical properties of small metal spheres,” Phys. Rev. B 11, 2871–2876 (1975) [CrossRef] .

17.

V. Yannopapas, “Non-local optical response of two-dimensional arrays of metallic nanoparticles,” J. Phys.-Condes. Matter 20, 325211 (2008) [CrossRef] .

18.

R. J. Pollard, A. Murphy, W. R. Hendren, P. R. Evans, R. Atkinson, G. A. Wurtz, A. V. Zayats, and V. A. Podolskiy, “Optical nonlocalities and additional waves in epsilon-near-zero metamaterials,” Phys. Rev. Lett. 102, 127405 (2009) [CrossRef] [PubMed] .

19.

J. M. McMahon, S. K. Gray, and G. C. Schatz, “Nonlocal optical response of metal nanostructures with arbitrary shape,” Phys. Rev. Lett. 103, 097403 (2009) [CrossRef] [PubMed] .

20.

J. M. McMahon, S. K. Gray, and G. C. Schatz, “Optical properties of nanowire dimers with a spatially nonlocal dielectric function,” Nano Lett. 10, 3473–3481 (2010) [CrossRef] [PubMed] .

21.

S. Raza, G. Toscano, A.-P. Jauho, M. Wubs, and N. A. Mortensen, “Unusual resonances in nanoplasmonic structures due to nonlocal response,” Phys. Rev. B 84, 121412 (2011) [CrossRef] .

22.

B. Gompf, J. Braun, T. Weiss, H. Giessen, M. Dressel, and U. Hubner, “Periodic nanostructures: spatial dispersion mimics chirality,” Phys. Rev. Lett. 106, 185501 (2011) [CrossRef] [PubMed] .

23.

S. I. Pekar and O. D. Kocherga, Crystal Optics and Additional Light Waves (Benjamin/Cummings, 1983).

24.

A. A. Orlov, P. M. Voroshilov, P. A. Belov, and Y. S. Kivshar, “Engineered optical nonlocality in nanostructured metamaterials,” Phys. Rev. B 84, 045424 (2011) [CrossRef] .

25.

M. F. Bishop and A. A. Maradudin, “Energy flow in a semi-infinite spatially dispersive absorbing dielectric,” Phys. Rev. B 14, 3384–3393 (1976) [CrossRef] .

26.

D. F. Nelson, “Generalizing the Poynting vector,” Phys. Rev. Lett. 76, 4713–4716 (1996) [CrossRef] [PubMed] .

27.

M. A. Vladimir and N. G. Yu, “Spatial dispersion and negative refraction of light,” Phys. Usp. 49, 1029 (2006) [CrossRef] .

28.

P. A. Belov, R. Marques, S. I. Maslovski, I. S. Nefedov, M. Silveirinha, C. R. Simovski, and S. A. Tretyakov, “Strong spatial dispersion in wire media in the very large wavelength limit,” Phys. Rev. B 67, 113103 (2003) [CrossRef] .

29.

C. R. Simovski and P. A. Belov, “Low-frequency spatial dispersion in wire media,” Phys. Rev. E 70, 046616 (2004) [CrossRef] .

30.

F. J. Garcia de Abajo, “Nonlocal effects in the plasmons of strongly interacting nanoparticles, dimers, and waveguides,” J. Phys. Chem. C 112, 17983–17987 (2008) [CrossRef] .

31.

L. Brillouin, Wave Propagation in Periodic Structures (Dover, 1953).

32.

J. Elser, V. A. Podolskiy, I. Salakhutdinov, and I. Avrutsky, “Nonlocal effects in effective-medium response of nanolayered metamaterials,” Appl. Phys. Lett. 90, 191109 (2007) [CrossRef] .

33.

A. V. Chebykin, A. A. Orlov, A. V. Vozianova, S. I. Maslovski, Y. S. Kivshar, and P. A. Belov, “Nonlocal effective medium model for multilayered metal-dielectric metamaterials,” Phys. Rev. B 84, 115438 (2011) [CrossRef] .

34.

A. V. Chebykin, A. A. Orlov, C. R. Simovski, Y. S. Kivshar, and P. A. Belov, “Nonlocal effective parameters of multilayered metal-dielectric metamaterials,” Phys. Rev. B 86, 115420 (2012) [CrossRef] .

35.

B. Wood, J. B. Pendry, and D. P. Tsai, “Directed subwavelength imaging using a layered metal-dielectric system,” Phys. Rev. B 74, 115116 (2006) [CrossRef] .

36.

A. Fang, T. Koschny, and C. M. Soukoulis, “Optical anisotropic metamaterials: Negative refraction and focusing,” Phys. Rev. B 79, 245127 (2009) [CrossRef] .

37.

X. Fan, G. P. Wang, J. C. W. Lee, and C. T. Chan, “All-angle broadband negative refraction of metal waveguide arrays in the visible range: theoretical analysis and numerical demonstration,” Phys. Rev. Lett. 97, 073901 (2006) [CrossRef] [PubMed] .

38.

E. Verhagen, R. de Waele, L. Kuipers, and A. Polman, “Three-dimensional negative index of refraction at optical frequencies by coupling plasmonic waveguides,” Phys. Rev. Lett. 105, 223901 (2010) [CrossRef] .

39.

E. N. Economou, “Surface plasmons in thin films,” Phys. Rev. 182, 539–554 (1969) [CrossRef] .

40.

P. A. Belov, “Backward waves and negative refraction in uniaxial dielectrics with negative dielectric permittivity along the anisotropy axis,” Microw. Opt. Technol. Lett. 37, 259–263 (2003) [CrossRef] .

41.

J. C. Maxwell-Garnett, “Colours in metal glasses and in metallic films,” Philos. Trans. R. Soc. London 203, 385 (1904) [CrossRef] .

42.

R. M. Hornreich and S. Shtrikman, “Theory of gyrotropic birefringence,” Phys. Rev. 171, 1065–1074 (1968) [CrossRef] .

OCIS Codes
(230.4170) Optical devices : Multilayers
(240.6680) Optics at surfaces : Surface plasmons
(260.2065) Physical optics : Effective medium theory

ToC Category:
Metamaterials

History
Original Manuscript: April 30, 2013
Revised Manuscript: June 26, 2013
Manuscript Accepted: June 26, 2013
Published: July 2, 2013

Citation
Ruey-Lin Chern, "Spatial dispersion and nonlocal effective permittivity for periodic layered metamaterials," Opt. Express 21, 16514-16527 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-14-16514


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References

  1. J. J. Hopfield and D. G. Thomas, “Theoretical and experimental effects of spatial dispersion on the optical properties of crystals,” Phys. Rev.132, 563–572 (1963). [CrossRef]
  2. L. D. Landau, E. M. Lifshitz, and L. P. Pitaevski, Electrodynamics of Continuous Media, 2nd ed. (Butterworth-Heinenan, 1984).
  3. V. M. Agranovich and V. L. Ginzburg, Crystal Optics with Spatial Dispersion, and Excitons (Springer-Verlag, 1984). [CrossRef]
  4. J. J. Hopfield, “Theory of the contribution of excitons to the complex dielectric constant of crystals,” Phys. Rev.112, 1555–1567 (1958). [CrossRef]
  5. S. I. Pekar, “Theory of electromagnetic waves in a crystal with excitons,” J. Phys. Chem. Solids5, 11–22 (1958). [CrossRef]
  6. R. Ruppin, “Reflectivity of a nonlocal dielectric with an excitonic surface potential,” Phys. Rev. B29, 2232–2237 (1984). [CrossRef]
  7. B. Chen and D. F. Nelson, “Wave propagation of exciton polaritons by a wave-vector-space method,” Phys. Rev. B48, 15372–15389 (1993). [CrossRef]
  8. G. E. H. Reuter and E. H. Sondheimer, “The theory of the anomalous skin effect in metals,” Proc. R. Soc. A-Math. Phys. Eng. Sci.195, 336–364 (1948). [CrossRef]
  9. K. L. Kliewer and R. Fuchs, “Anomalous skin effect for specular electron scattering and optical experiments at non-normal angles of incidence,” Phys. Rev.172, 607–624 (1968). [CrossRef]
  10. R. Ruppin, “Optical properties of a spatially dispersive cylinder,” J. Opt. Soc. Am. B6, 1559–1563 (1989). [CrossRef]
  11. R. Ruppin, “Extinction properties of thin metallic nanowires,” Opt. Comm.190, 205–209 (2001). [CrossRef]
  12. A. A. Maradudin and D. L. Mills, “Effect of spatial dispersion on the properties of a semi-infinite dielectric,” Phys. Rev. B7, 2787–2810 (1973). [CrossRef]
  13. A. R. Melnyk and M. J. Harrison, “Resonant excitation of plasmons in thin films by elecromagnetic waves,” Phys. Rev. Lett.21, 85–88 (1968). [CrossRef]
  14. W. E. Jones, K. L. Kliewer, and R. Fuchs, “Nonlocal theory of the optical properties of thin metallic films,” Phys. Rev.178, 1201–1203 (1969). [CrossRef]
  15. A. R. Melnyk and M. J. Harrison, “Theory of optical excitation of plasmons in metals,” Phys. Rev. B2, 835–850 (1970). [CrossRef]
  16. R. Ruppin, “Optical properties of small metal spheres,” Phys. Rev. B11, 2871–2876 (1975). [CrossRef]
  17. V. Yannopapas, “Non-local optical response of two-dimensional arrays of metallic nanoparticles,” J. Phys.-Condes. Matter20, 325211 (2008). [CrossRef]
  18. R. J. Pollard, A. Murphy, W. R. Hendren, P. R. Evans, R. Atkinson, G. A. Wurtz, A. V. Zayats, and V. A. Podolskiy, “Optical nonlocalities and additional waves in epsilon-near-zero metamaterials,” Phys. Rev. Lett.102, 127405 (2009). [CrossRef] [PubMed]
  19. J. M. McMahon, S. K. Gray, and G. C. Schatz, “Nonlocal optical response of metal nanostructures with arbitrary shape,” Phys. Rev. Lett.103, 097403 (2009). [CrossRef] [PubMed]
  20. J. M. McMahon, S. K. Gray, and G. C. Schatz, “Optical properties of nanowire dimers with a spatially nonlocal dielectric function,” Nano Lett.10, 3473–3481 (2010). [CrossRef] [PubMed]
  21. S. Raza, G. Toscano, A.-P. Jauho, M. Wubs, and N. A. Mortensen, “Unusual resonances in nanoplasmonic structures due to nonlocal response,” Phys. Rev. B84, 121412 (2011). [CrossRef]
  22. B. Gompf, J. Braun, T. Weiss, H. Giessen, M. Dressel, and U. Hubner, “Periodic nanostructures: spatial dispersion mimics chirality,” Phys. Rev. Lett.106, 185501 (2011). [CrossRef] [PubMed]
  23. S. I. Pekar and O. D. Kocherga, Crystal Optics and Additional Light Waves (Benjamin/Cummings, 1983).
  24. A. A. Orlov, P. M. Voroshilov, P. A. Belov, and Y. S. Kivshar, “Engineered optical nonlocality in nanostructured metamaterials,” Phys. Rev. B84, 045424 (2011). [CrossRef]
  25. M. F. Bishop and A. A. Maradudin, “Energy flow in a semi-infinite spatially dispersive absorbing dielectric,” Phys. Rev. B14, 3384–3393 (1976). [CrossRef]
  26. D. F. Nelson, “Generalizing the Poynting vector,” Phys. Rev. Lett.76, 4713–4716 (1996). [CrossRef] [PubMed]
  27. M. A. Vladimir and N. G. Yu, “Spatial dispersion and negative refraction of light,” Phys. Usp.49, 1029 (2006). [CrossRef]
  28. P. A. Belov, R. Marques, S. I. Maslovski, I. S. Nefedov, M. Silveirinha, C. R. Simovski, and S. A. Tretyakov, “Strong spatial dispersion in wire media in the very large wavelength limit,” Phys. Rev. B67, 113103 (2003). [CrossRef]
  29. C. R. Simovski and P. A. Belov, “Low-frequency spatial dispersion in wire media,” Phys. Rev. E70, 046616 (2004). [CrossRef]
  30. F. J. Garcia de Abajo, “Nonlocal effects in the plasmons of strongly interacting nanoparticles, dimers, and waveguides,” J. Phys. Chem. C112, 17983–17987 (2008). [CrossRef]
  31. L. Brillouin, Wave Propagation in Periodic Structures (Dover, 1953).
  32. J. Elser, V. A. Podolskiy, I. Salakhutdinov, and I. Avrutsky, “Nonlocal effects in effective-medium response of nanolayered metamaterials,” Appl. Phys. Lett.90, 191109 (2007). [CrossRef]
  33. A. V. Chebykin, A. A. Orlov, A. V. Vozianova, S. I. Maslovski, Y. S. Kivshar, and P. A. Belov, “Nonlocal effective medium model for multilayered metal-dielectric metamaterials,” Phys. Rev. B84, 115438 (2011). [CrossRef]
  34. A. V. Chebykin, A. A. Orlov, C. R. Simovski, Y. S. Kivshar, and P. A. Belov, “Nonlocal effective parameters of multilayered metal-dielectric metamaterials,” Phys. Rev. B86, 115420 (2012). [CrossRef]
  35. B. Wood, J. B. Pendry, and D. P. Tsai, “Directed subwavelength imaging using a layered metal-dielectric system,” Phys. Rev. B74, 115116 (2006). [CrossRef]
  36. A. Fang, T. Koschny, and C. M. Soukoulis, “Optical anisotropic metamaterials: Negative refraction and focusing,” Phys. Rev. B79, 245127 (2009). [CrossRef]
  37. X. Fan, G. P. Wang, J. C. W. Lee, and C. T. Chan, “All-angle broadband negative refraction of metal waveguide arrays in the visible range: theoretical analysis and numerical demonstration,” Phys. Rev. Lett.97, 073901 (2006). [CrossRef] [PubMed]
  38. E. Verhagen, R. de Waele, L. Kuipers, and A. Polman, “Three-dimensional negative index of refraction at optical frequencies by coupling plasmonic waveguides,” Phys. Rev. Lett.105, 223901 (2010). [CrossRef]
  39. E. N. Economou, “Surface plasmons in thin films,” Phys. Rev.182, 539–554 (1969). [CrossRef]
  40. P. A. Belov, “Backward waves and negative refraction in uniaxial dielectrics with negative dielectric permittivity along the anisotropy axis,” Microw. Opt. Technol. Lett.37, 259–263 (2003). [CrossRef]
  41. J. C. Maxwell-Garnett, “Colours in metal glasses and in metallic films,” Philos. Trans. R. Soc. London203, 385 (1904). [CrossRef]
  42. R. M. Hornreich and S. Shtrikman, “Theory of gyrotropic birefringence,” Phys. Rev.171, 1065–1074 (1968). [CrossRef]

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