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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 21 — Oct. 10, 2011
  • pp: 20035–20047
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General properties of two-dimensional conformal transformations in electrostatics

Yong Zeng, Jinjie Liu, and Douglas H. Werner  »View Author Affiliations


Optics Express, Vol. 19, Issue 21, pp. 20035-20047 (2011)
http://dx.doi.org/10.1364/OE.19.020035


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Abstract

Electrostatic properties of two-dimensional nanosystems can be completely described by their non-trivial geometry modes. In this paper we prove that these modes as well as the corresponding eigenvalues are invariant under any conformal transformation. This invariance suggests a new way to study electrostatic conformal transformations, while also providing an in-depth interpretation of the behavior exhibited by singular plasmonic nanoparticles.

© 2011 OSA

An intriguing property of Maxwell’s equations is their form invariance under arbitrary coordinate transformations, assuming the field quantities and the material properties are transformed accordingly [1

1. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006). [CrossRef] [PubMed]

]. This invariance leads to a powerful design tool, referred to as transformation optics or transformation electromagnetics [1

1. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006). [CrossRef] [PubMed]

7

7. D.-H. Kwon and D. H. Werner, “Transformation optical designs for wave collimators, flat lenses and right-angle bends,” N. J. Phys. 10, 115023 (2008). [CrossRef]

]. Since the introduction of transformation optics in 2006, a wealth of novel and unique devices have been theorized and demonstrated [7

7. D.-H. Kwon and D. H. Werner, “Transformation optical designs for wave collimators, flat lenses and right-angle bends,” N. J. Phys. 10, 115023 (2008). [CrossRef]

,8

8. H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9, 387–396 (2010). [CrossRef] [PubMed]

]. Some examples include: electromagnetic cloaks [9

9. D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006). [CrossRef] [PubMed]

11

11. T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science 328, 337–339 (2010). [CrossRef] [PubMed]

], event cloaks [12

12. M. W. McCall, A. Favaro, P. Kinsler, and A. Boardman, “A spacetime cloak, or a history editor,” J. Opt. 13, 024003 (2011). [CrossRef]

], optical black holes [13

13. E. E. Narimanov and A. V. Kildishev, “Optical black hole: broadband omnidirectional light absorber,” Appl. Phys. Lett. 95, 041106 (2009). [CrossRef]

] and field splitters [14

14. M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, and D. R. Smith, “Optical design of reflectionless complex media by finite embedded coordinate transformations,” Phys. Rev. Lett. 100, 063903 (2008). [CrossRef] [PubMed]

, 15

15. D.-H. Kwon and D. H. Werner, “Polarization splitter and polarization rotator designs based on transformation optics,” Opt. Express 16, 18731–18738 (2008). [CrossRef]

].

When the electromagnetic wavelength is far longer than the characteristic size of the structure, quasi-static approximations can be employed. As a result, the interactions between light and a medium can be approximately described by Laplace’s equation [16

16. J. D. Jackson, Classical Electrodynamics, 3rd ed. (John Wiley & Sons, 2001).

]. As with the full-wave Maxwell equations, the two-dimensional Laplacian equation is also invariant under conformal transformations. Additionally, the electrostatic transformation does not alter the properties of the constituent material, in direct contrast with the full-wave coordinate transformations which generally result in a physical medium with complex or exotic properties. This unique property of the Laplacian equation has been used to design near-field perfect lenses and reflectors [17

17. J. B. Pendry and S. A. Ramakrishna, “Near field lenses in two dimensions,” J. Phys.: Condens. Matter 14, 8463–8479 (2002). [CrossRef]

21

21. B. Gralak and S. Guenneau, “Transfer matrix method for point sources radiating in classes of negative refractive index materials with 2n-fold antisymmetry,” Waves Random Complex Media 17, 581–614 (2007). [CrossRef]

]. Recently, it has been employed to study two-dimensional plasmonic particles. Wherein, the transformed geometry can be studied analytically by mapping a complicated two-dimensional structure into a simple one-dimensional configuration, such as a metallic slab of finite-thickness [22

22. A. Aubry, D. Y. Lei, S. A. Maier, and J. B. Pendry, “Broadband plasmonic device concentrating the energy at the nanoscale: the crescent-shaped cylinder,” Phys. Rev. B 82, 125430 (2010). [CrossRef]

27

27. D. Y. Lei, A. Aubry, S. A. Maier, and J. B. Pendry, “Broadband nano-focusing of light using kissing nanowires,” N. J. Phys. 12, 093030 (2010). [CrossRef]

]. It was further found that these singular plasmonic nanoparticles, including crescent and touching or non-touching cylindrical dimers, can exhibit broadband properties and significantly enhance the local electric fields.

In this paper, we will study electrostatic conformal transformations by taking advantage of the invariance of geometry modes and their associated eigenvalues. We will show that transformation of a geometry can be equivalently explained by transforming the excitation source and modifying the amplitudes of the associated geometry modes. A new approach is developed that exclusively depends on the structural geometry. This method is then used to interpret two important features of singular plasmonic structures, i.e., broadband response and divergent electric fields around the singularities.

To begin, we will invoke the spectral Bergman-Milton theory [28

28. D. J. Bergman and D. Stroud, in Solid State Physics , H. Ehrenreich and D. Turnbull, eds. (Academic, 1992), Vol. 46, pp. 148–270. [CrossRef]

, 29

29. G. W. Milton, The Theory of Composites (Cambridge University Press, 2002). [CrossRef]

], which states that the electrostatic behavior of a nanosystem consisting of two different mediums, with permittivities of ɛ1(ω) and ɛ2(ω) respectively, can be described by a set of non-trivial eigenmodes φn of the following generalized eigenproblem
[θ(r)φn(r)]=sn2φn(r),
(1)
where sn represents the corresponding eigenvalues (see Appendix A). The function θ(r) characterizes the geometry of the composite: θ(r) = 1 when r is inside the medium with dielectric constant ɛ1 and equals 0 elsewhere. To simplify the following discussion, ɛ2(ω) is assumed to be 1. Since this equation depends exclusively on the geometry, but not on the material composition, the resultant eigenmodes are therefore referred to as geometry modes [28

28. D. J. Bergman and D. Stroud, in Solid State Physics , H. Ehrenreich and D. Turnbull, eds. (Academic, 1992), Vol. 46, pp. 148–270. [CrossRef]

]. Moreover, by defining a scalar product as
(ϕ|ψ)=θϕ*ψdv,
(2)
we find that sn of a non-trivial mode must be real and limited to the range 0 < sn < 1, and the normalized eigenmodes φn can be used to expand an arbitrary function for points r that are inside the ɛ1 material [30

30. D. J. Bergman and M. I. Stockman, “Surface plasmon amplification by stimulated emission of radiation: quantum generation of coherent surface plasmons in nanosystems,” Phys. Rev. Lett. 90, 027402 (2003). [CrossRef] [PubMed]

33

33. D. J. Bergman, “The dielectric constant of a simple cubic array of identical spheres,” J. Phys. C: Solid State Phys. 12, 4947–4960 (1979). [CrossRef]

].

The invariance of the non-trivial eigenmodes φn and their corresponding eigenvalue sn immediately suggest a new way to interpret and study the electrostatic response of the transformed structure. This new approach depends exclusively on the geometry, as described by the function θ(r), and does not need any information regarding the material parameters ɛ(ω) or the external potential. For instance, we can expand the total potential φt in terms of the associated eigenmodes,
ns(ω)(φn|φ0)φns(ω)snnβn(φn|φ0)φn,
(5)
where k = ω/c, s(ω) = 1/(1 – ɛ1) being the Bergman’s spectral parameter and φ0 being the external potential. Notice that we do not include the radiation damping here. Consequently, once these eigenmodes of the original geometry are known, we can obtain the potential φt of any transformed or derivative geometry by simply calculating the expansion coefficients (φn|φ0). In other words, transforming geometry is equivalent to transforming the external source, or more precisely, changing the expansion coefficient of each eigenmode. Furthermore, the time-averaged power Pa absorbed by the structure can be written as (See Appendix B)
Pa=ωɛ0Im(ɛ1)Ie/2,
(6)
where Ie = Σn|βn(φn|φ0)|2 represents the integration of the electric field intensity |E|2 inside the particle. In a similar way, the extinction may be expressed as
Pex=ωɛ02n|(φn|φ0)|2Im[(ɛ11)βn].
(7)
When we transform the external field φ0 accordingly, i.e., φ0(x, y) = φ0 (u, v) with u and v being the new coordinates, all the energy quantities, Pa, Ie and Pex, are evidently invariant. On the other hand, if the external potential is fixed with respect to the w coordinate where w = u + iv, it follows that these energy quantities are proportional to a2 when the conformal mapping has a form a × w(z) with z = x + iy and a is real. In other words, the larger the particle area, the stronger the absorption and extinction.

It should be mentioned that the geometry modes actually are the surface modes of the particle [34

34. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley & Sons, 1998). [CrossRef]

, 35

35. Y. Zeng, Q. Wu, and D. H. Werner, “Electrostatic theory for designing lossless negative permittivity metamaterials,” Opt. Lett. 35, 1431–1433 (2010). [CrossRef] [PubMed]

], and the corresponding resonance condition for the nth mode is strictly s(ω) = sn, or equivalently ɛ1 = 1 – 1/sn, when we do not include the radiation loss. Notice that the corresponding permittivity ɛ1 of the particle should be real and negative since 0 < sn < 1 [34

34. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley & Sons, 1998). [CrossRef]

]. The most striking property of the surface mode is that the resultant total electrostatic energy of the entire system, including the free space and the particle, is exactly zero [36

36. F. Wang and Y. R. Shen, “General properties of local plasmons in metal nanostructures,” Phys. Rev. Lett. 97, 206806 (2006). [CrossRef] [PubMed]

]. Furthermore, the complex frequency of the nth surface plasmonic resonance (SPR) is given by s(ωnn) = sn with ωn being real resonant frequency and γn being the relaxation rate [28

28. D. J. Bergman and D. Stroud, in Solid State Physics , H. Ehrenreich and D. Turnbull, eds. (Academic, 1992), Vol. 46, pp. 148–270. [CrossRef]

,37

37. D. R. Fredkin and I. D. Mayergoyz, “Resonant behavior of dielectric objects (electrostatic resonances),” Phys. Rev. Lett. 91, 253902 (2003). [CrossRef]

]. Since the eigenvalues sn are conserved under any conformal mapping, the transformed structure will have the same SPRs as the original structure at the identical resonant frequencies. For instance, since a metallic cylinder can be transformed from a metal-dielectric interface by w = ez, its SPR hence can be determined by the nonretarded surface-plasmon condition of the metal-dielectric interface [38

38. A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. 408, 131–314 (2005). [CrossRef]

].

As a proof of principle, we consider a one-dimensional dielectric slab with finite thickness (Figure 1(a)) and its derivative systems obtained through different conformal transformations. The eigenvalues of the slab are given by (See Appendix C)
sk,±=[1±e|k|(d2d1)]/2,
(8)
where k is a nonzero real valued parameter, and the corresponding eigenmodes αkφk,±eiky are given by
{[e2|k|d1e|k|(d1+d2)]e|k|x,xd1[e|k|xe|k|(d1+d2)]e|k|x,d1<x<d2(1e|k|(d2d1))e|k|x,xd2
(9)
where the normalization coefficient αk2=4π|k|(e2|k|d1e2|k|d2). Evidently, in terms of the symmetry of the electric field, these eigenmodes can be cataloged into two groups. The subscript +(−) corresponds to an even (odd) mode respectively, while their associated eigenvalues are larger (smaller) than 0.5. Since the electric fields of the odd modes penetrate the metal weakly, they can propagate longer along the metallic surface than the even modes [38

38. A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. 408, 131–314 (2005). [CrossRef]

]. Furthermore, the resonant condition, approximately given by Re[s(ωn)] = sn when the relaxation rate is weak, suggests that only the odd modes will be excited resonantly when Re(ɛ1) is smaller than −1. Moreover, these eigenmodes and eigenvalues can be directly applied to the complementary structure of the finite slab, i.e. a free-space gap sandwiched by two semi-infinite metallic spaces. Since these two structures have identical geometry, they therefore share the same eigenvalues and eigenmodes. To study the complementary structure, all we need to do is interchange ɛ1 and ɛ2. As a direct consequence, when Re(ɛ1) is smaller than −1, only the even modes will be resonantly excited in the semi-infinite structure [27

27. D. Y. Lei, A. Aubry, S. A. Maier, and J. B. Pendry, “Broadband nano-focusing of light using kissing nanowires,” N. J. Phys. 12, 093030 (2010). [CrossRef]

].

Fig. 1 A one-dimensional finite-thickness slab in xy coordinates (a) is transformed to an annulus (b) with a conformal transformation of w = ez, a crescent (c) and two kissing cylinders (d) with a conformal transformation of w = 1/z, in the new uv coordinates.

We now calculate the resultant expansion coefficients of a structure transformed from the finite slab through a conformal mapping of w = w(z). It is further assumed that the external potential φ0(u, v) = puu + pvv, which corresponds to a uniform electric field −pueupvev. The expansion coefficients in this case can be written as
(φk,±|φ0)=2kαk[ρ2pFkρ1p*Fk*],
(10)
where
Fk=d1d2dxdygekz*,p12(puipv),gdwdz.
(11)
In addition, ρ1 = 1 and ρ2 = ∓e–|k|(d1+d2) for positive k, and they should be interchanged when k < 0. Here we use the Cauchy-Riemann equations of conformal mappings
ux=vy=g+g*2,uy=vx=g*g2i,
(12)
as well as
φ0x=gp+g*p*,φ0y=i(gpg*p*).
(13)

We first transform the slab to two coaxial cylinders (Figure 1(b)) with w = ez, where the radii of the two cylinders are r1 = ed1 and r2 = ed2 respectively [39

39. P. B. Catrysse and S. Fan, “Understanding the dispersion of coaxial plasmonic structures through a connection with the planar metal-insulator-metal geometry,” Appl. Phys. Lett. 94, 231111 (2009). [CrossRef]

]. The expansion coefficients are found to be (See Appendix D)
(φk,±|φ0)=αkpued1+d2δ(|k|1)/4.
(14)
Here pv is assumed to be zero because of the structural symmetry. Evidently, only eigenmodes with k = ±1 are excited. We further use these coefficients to calculate the induced potential φi. It is found that σφi/[φ0(1 – ɛ1)], with σ=r22(1+ɛ1)2r12(1ɛ1)2, equals
{(r12r22)(1ɛ1),rr1r12(1ɛ1)+r22(1+ɛ1)2r12r22/r2,r1<r<r2r22(r22r12)(1+ɛ1)/r2,rr2
(15)
Note that our results are identical to those obtained by expanding the potential in terms of Bessel functions [34

34. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley & Sons, 1998). [CrossRef]

].

Since the eigenvalues of the finite slab cover a wide region (0,1) (note that the eigenvalues of Eq. (1) are limited to between 0 and 1), any negative Re(ɛ1) will excite a surface mode as long as its expansion coefficient is not zero. Consequently the transformed system will present broadband response in principle. One example is the crescent studied in [22

22. A. Aubry, D. Y. Lei, S. A. Maier, and J. B. Pendry, “Broadband plasmonic device concentrating the energy at the nanoscale: the crescent-shaped cylinder,” Phys. Rev. B 82, 125430 (2010). [CrossRef]

], which can be obtained by using w = 1/z and d1 > 0 (Figure 1(c)). The expansion coefficients are found to be (See Appendix D)
(φk,±|φ0)=αk[pu+isgn(k)pv]/4,
(16)
which can be further used to achieve identical electric fields as those presented in Ref. [22

22. A. Aubry, D. Y. Lei, S. A. Maier, and J. B. Pendry, “Broadband plasmonic device concentrating the energy at the nanoscale: the crescent-shaped cylinder,” Phys. Rev. B 82, 125430 (2010). [CrossRef]

] (See Appendix E). Evidently, each eigenmode is excited by the external potential. Furthermore, |(φk,±|φ0)| depends exclusively on the amplitude of the external electric field. The energy quantities such as the absorption hence do not depend on the direction of the incident field [22

22. A. Aubry, D. Y. Lei, S. A. Maier, and J. B. Pendry, “Broadband plasmonic device concentrating the energy at the nanoscale: the crescent-shaped cylinder,” Phys. Rev. B 82, 125430 (2010). [CrossRef]

], a property which is not so obvious. Figure 2 shows the dependence of |(φk,±|φ0)|2 on k achieved by setting pu = pv = 1, d1 = d and d1 = 2d with d being an arbitral real quantity. As mentioned, each eigenmode of the crescent is excited, consequently |(φk,±|φ0)|2 is always positive. Furthermore, in this case these expansion coefficients are strongly localized in the regime 0.3 ≤ |k|d ≤ 1.6 with a maximum around |k|d = 0.72.

Fig. 2 The dependence of the expansion coefficient on the mode index k, when it is assumed that pu = pv = 1. Notice that two eigenmodes with identical k while different symmetry have identical amplitudes of the expansion coefficients, mode symmetry therefore is not specified. Here d is an arbitrary real quantity.

Another example consists of the kissing cylinders suggested in [27

27. D. Y. Lei, A. Aubry, S. A. Maier, and J. B. Pendry, “Broadband nano-focusing of light using kissing nanowires,” N. J. Phys. 12, 093030 (2010). [CrossRef]

], which is obtained by transforming two semi-infinite slabs, with d1 < 0 and d2 > 0, using the conformal mapping given by w = 1/z (Figure 1(d)). The corresponding expansion coefficients are calculated as (See Appendix D)
(φk,±|φ0)=αk4[pue|k|(d2+d1)±1e|k|(d2d1)±1+isgn(k)pve|k|(d2+d1)1e|k|(d2d1)1].
(17)
Again all the eigenmodes are excited by the external source. Note that when d1 = −d2 the cylinders have the same radius. The kissing cylinders then possess both u and v mirror symmetry. Hence, the even mode φk,+ or the odd mode φk, can be excited by a u- or v- polarized electric field respectively. An example dependence of |(φk,±|φ0)|2 on k is plotted in Figure 2, with pu = pv = 1 as well as d1 = −d and d1 = 2d. Similar to the curve associated with the crescent, the eigenmodes with 0.2 ≤ |k|d ≤ 1.6 contribute significantly to the absorption cross section, which guarantees that the kissing cylinders will exhibit broadband behavior.

Considerable field enhancement and confinement has been observed around the singularity of the metallic crescent or cylinder dimers under a uniform illumination [22

22. A. Aubry, D. Y. Lei, S. A. Maier, and J. B. Pendry, “Broadband plasmonic device concentrating the energy at the nanoscale: the crescent-shaped cylinder,” Phys. Rev. B 82, 125430 (2010). [CrossRef]

27

27. D. Y. Lei, A. Aubry, S. A. Maier, and J. B. Pendry, “Broadband nano-focusing of light using kissing nanowires,” N. J. Phys. 12, 093030 (2010). [CrossRef]

]. To interpret this phenomenon, we assume the frequency of the external excitation coincides with ωn, the resonant frequency of the n-th surface mode. The total potential of Eq. (5) can therefore be approximated as βn(φn|φ0)φn. Since only φn is position dependent, the amplitude of the corresponding electric field is proportional to |∇wφn| = |dz/dw||∇zφn|. Taking the crescent as an example, |dz/dw| = x2 + y2 since w = 1/z. Consequently, when we map the point (a,∞) in xy coordinates to the singular point of the crescent (the origin of the uv coordinates), the corresponding electric field tends to infinity. On the other hand, |dz/dw| = 1/ex for the two coaxial cylinders we studied above, and the resultant electric field for this case only depends on r and does not exhibit singular behavior. Note that |dz/dw|2 is actually the stretching factor for area when we transform the xy coordinates to the uv coordinates. The field enhancements because of |dz/dw| hence are purely induced by the coordinate transformations. Furthermore, we want to point out that, although the electric field at a few specific points can be infinite, Ie, the integration of the electric field intensity over the system, is finite and invariant under any conformal transformation.

In conclusion, we proved that the electrostatic eigenmodes and eigenvalues of two-dimensional nanosystems with proper boundary conditions are invariant under any conformal transformation. Based on this property, we suggested a new approach to studying the electrostatic responses of the transformed structures. Namely, transforming a geometry is equivalent to transforming the external potential or the expansion coefficients of the invariant eigenmodes. This method is then used to interpret two important features of singular plasmonic structures, i.e., broadband response and divergent electric fields in the neighborhood of the singularities.

A. Properties of Eq. (1)

It is useful to reformulate the generalized eigenproblem, Eq. (1), as an integral equation [33

33. D. J. Bergman, “The dielectric constant of a simple cubic array of identical spheres,” J. Phys. C: Solid State Phys. 12, 4947–4960 (1979). [CrossRef]

]
snφn(r)=G^φn(r)dvθ(r)G(r,r)φn(r),
(A.1)
with the help of the Green’s function G(r,r′) which satisfies the following Laplace’s equation
2G(r,r)=δ(rr).
(A.2)
Notice that both φn and G should satisfy the homogeneous Dirichlet-Neumann boundary conditions. If a scalar product of two functions further is defined as
(ϕ|ψ)=θ(r)ϕ*ψdv,
(A.3)
the operator Ĝ is then Hermitian. To prove this fact, we can show that
(ϕ|G^ψ)=θ(r)ϕ*(G^ψ)dv=dvθ(r)ϕ*[dvθ(r)G(r,r)ψ]=dvdvθ(r)θ(r)ϕ*[G(r,r)ψ],
(A.4)
as well as
(G^ϕ|ψ)=θ(r)(G^ϕ*)ψdv=dvθ(r)ψ[dvθ(r)G(r,r)ϕ*]=dvdvθ(r)θ(r)ψ[G(r,r)ϕ*]=dvdvθ(r)θ(r)ψ[G(r,r)ϕ*]=dvdvθ(r)θ(r)ψ[G(r,r)ϕ*],
(A.5)
where the symmetry of the Green’s function has been employed. Further notice that
ϕ*[G(r,r)ψ]=ψ[G(r,r)ϕ*],
(A.6)
and hence the operator Ĝ is Hermitian, such that its eigenvalues sn are all real. Moreover,
|φm|2dv=1smθ(r)|φm|2dv,
(A.7)
which suggests that the eigenvalues lie between 0 and 1. As pointed out in Reference [33

33. D. J. Bergman, “The dielectric constant of a simple cubic array of identical spheres,” J. Phys. C: Solid State Phys. 12, 4947–4960 (1979). [CrossRef]

], the eigenmodes corresponding to the eigenvalues of 0 and 1 are unimportant because their scalar product with any function vanishes. We therefore only consider the non-trivial modes and normalize them in the following way
(φm|φn)=δmn.
(A.8)

B. Absorption and extinction

The normalized non-trivial eigenmodes φn are a complete orthogonal set [33

33. D. J. Bergman, “The dielectric constant of a simple cubic array of identical spheres,” J. Phys. C: Solid State Phys. 12, 4947–4960 (1979). [CrossRef]

], and the external potential, inside the ɛ1 material, can be expanded in terms of these modes
φ0=n(φn|φ0)φn.
(B.1)
Using the fact that
(θφ0)=s(ω)2φi(θφi),
(B.2)
where φi being the induced potential, we achieve the following
φi=nsnφns(ω)sn(φn|φ0).
(B.3)
The total potential hence can be given as
φt=φi+φ0=ns(ω)φns(ω)sn(φn|φ0)nβn(φn|φ0)φn.
(B.4)
Moreover, the corresponding electric field is given by E = −∇φ. Here the radiation loss is not included.

In electromagnetics, the extinction of a particle illuminated by an external field is given by [16

16. J. D. Jackson, Classical Electrodynamics, 3rd ed. (John Wiley & Sons, 2001).

]
Pex=ωɛ02Im(ɛ11)E0*Etdv.
(B.5)
Here the integration is performed over the particle, and the surrounding material is assumed to be air. Substituting Eq. (B.1) and (B.4) into the above equation yields
Pex=ωɛ02n|(φn|φ0)|2Im[βn(ɛ11)].
(B.6)
Furthermore, the electrostatic energy inside the metallic region is expressed as
Ie=vθEt*Etdv=n|βn(φn|φ0)|2.
(B.7)
In a similar way, the absorption may be represented as
pa=ωɛ02Im(ɛ1)θEt*Etdv=ωɛ02Im(ɛ1)Ie.
(B.8)

C. Geometry modes of one-dimensional finite slab

We consider a one-dimensional dielectric slab with finite thickness (See Figure (1a)). The slab sits between [d1, d2]. The possible solution has a form ae|k|x eiky for x < d1, ce–|k|x eiky for x > d2, and b1 e|k|x eiky + b2e−|k|x eiky in the middle. The boundary conditions at x = d1 and x = d2 yield
(ab1)m2=b2,am2=(s1)s(b1m2b2)cb2=b1n2,c=(1s)s(b1n2b2),
(C.1)
where m = e|k|d1 and n = e|k|d2. Two sets of eigenmodes can be obtained. The eigenvalues of the odd modes are given by
sk,+=1+e|k|(d2d1)2,
(C.2)
and the associated eigenmodes are
αkφk,+eiky={[e2|k|d1e|k|(d1+d2)]e|k|x,xd1e|k|(d1+d2)e|k|x+e|k|x,d1<x<d2(1e|k|(d2d1))e|k|x,xd2.
(C.3)
The eigenvalues of the even modes are
sk,=1e|k|(d2d1)2,
(C.4)
and the corresponding eigenmodes are
αkφk,eiky={[e2|k|d1+e|k|(d1+d2)]e|k|x,xd1e|k|(d1+d2)e|k|x+e|k|x,d1<x<d2(1+e|k|(d2d1))e|k|x,xd2.
(C.5)
The normalization coefficient αk2=4π|k|(e2|k|d1e2|k|d2).

D. Expansion coefficients of different structures

Two coaxial cylinders: The conformal mapping used here is w = ez. To obtain the expansion coefficients, we use Eqs. (10) and (11). It is found that
Fk=d1d2dxdyezekz*=π(e2d2e2d1)δ(k1),
(D.1)
as well as
Fk*=d1d2dxdyez*ekz=π(e2d2e2d1)δ(k+1),
(D.2)
where use has been made of the fact that
eikydy=2πδ(k).
(D.3)
Because of the structural symmetry we assume that pv = 0, and the resultant expansion coefficients are
(φk,±|φ0)=αk4pued1+d2δ(|k|1).
(D.4)
Evidently, only eigenmodes with k = ±1 are excited. Making use of these coefficients, the induced potential φi = Σn(βn – 1)(φn|φ0)φn can be expressed as
φi(u,v)=pur1r24[(β1,1)(α1φ1,+α1φ1,)(β1,+1)(α1φ1,++α1φ1,+],
(D.5)
with r1 = ed1, r2 = ed2, and
β1,±1=(1ɛ1)(r2±r1)(1+ɛ1)r2(1ɛ1)r1.
(D.6)
Finally we achieve, by using the fact that φ0 = puu = puex cos y = pur cosθ, the following result:
σφiφ0={(r12r22)(1ɛ1)2,rr1r12(1ɛ1)2+r22(1ɛ12)2(1ɛ1)r12r22/r2,r1<r<r2r22(r22r12)(1ɛ12)/r2,rr2
(D.7)
with σ=r22(1+ɛ1)2r12(1ɛ1)2.

Crescent: It is known that a slab, with d1 > 0, can be transformed to a crescent by a conformal mapping of w = 1/z. Consequently g = dw/dz = −1/z2. By using Eqs. (10) and (11), it is found that
Fk=d1d2dxdy1(x+iy)2ek(xiy)=d1d2ekxdxdy1(x+iy)2eiky.
(D.8)
Since xd1 > 0, the second integration has a pole at y = ix in the upper half complex plane. Consequently for negative k, we have
Fk<0=2πkd1d2e2kxdx=π(e2kd2e2kd1).
(D.9)
Similar procedures can be applied to calculate Fk* for positive k, which yield
Fk*=d1d2dxdy1(xiy)2ek(x+iy)=π(e2kd2e2kd1),
(D.10)
because of the fact that
dy1(xiy)2eiky=2πkekx.
(D.11)
Here the integration has a pole at y = −ix in the lower half complex plane. Hence the resultant expansion coefficients are given by
(φk,±|φ0)=αk4[pu+isgn(k)pv].
(D.12)
Notice that for same k, the even mode and the odd mode result in identical expansion coefficients.

Kissing cylinder dimer: A kissing dimer can be transformed from a slab with d1 < 0 and d2 > 0, by using a conformal mapping of w = 1/z.

We first calculate Fk,
Fk=d1d2dxdy1(x+iy)2ek(xiy)=d1d2ekxdxdy1(x+iy)2eiky.
(D.13)
To evaluate the second integration, we notice that its integrand has a pole at y = ix, which is located in the upper half plane for positive x while in the lower half plane for negative x. By carefully choosing the integral contour, it is found that
Fk=π(e2kd21),Fk*=π(e2kd11),
(D.14)
for negative k, and
Fk=π(e2kd11),Fk*=π(e2kd21),
(D.15)
for positive k. The corresponding expansion coefficients are then given by
(φk,±|φ0)=αk4[pue|k|(d2+d1)±1e|k|(d2d1)±1+isgn(k)pve|k|(d2+d1)1e|k|(d2d1)1].
(D.16)

E. Electric field of the crescent

The total electric field, including both incident and scattered fields, can be expressed as
E=wφt=nβn(φn|φ0)wφn,
(E.1)
which can be further manipulated into the form
Eu=nβn(φn|φ0)[φnxxu+φnyyu],Ev=nβn(φn|φ0)[φnxxv+φnyyv],
(E.2)
with
xu=yv=zw+zw*2,xv=yu=izwzw*2,
(E.3)
as well as zw = dz/dw. By representing the geometrical mode φk,± as
φk,±(x,y)=1αkeiky(ak,±e|k|x+bk,±e|k|x),
(E.4)
we finally arrive at
Eu=k,±|k|eikyαkβk,±(φk,±|φ0)[(ak,±e|k|xzwbk,±e|k|xzw*)ϑ(k)+(ak,±e|k|xzw*bk,±e|k|xzw)ϑ(k)],Ev=ik,±keikyαkβk,±(φk,±|φ0)[(ak,±e|k|xzw+bk,±e|k|xzw*)ϑ(k)+(ak,±e|k|xzw*+bk,±e|k|xzw)ϑ(k)],
where ϑ(x) = 1 for positive x while ϑ(x) = 0 when x < 0. Furthermore, using Eq. (8), βk,± can be reformulated as
βk,±=s(ω)s(ω)sk,±=2ɛ1+111±eγ|k|t,
(E.5)
where t = d2 – d1 being the slab thickness, and = (ɛ1 1)/(ɛ1 + 1). Moreover, since k is continuous, the electric field, taking the u component as an example, may be expressed as
Eu=pu+ipv2(ɛ1+1)0dkkeiky(ak,+e|k|xzwbk,+e|k|xzw*1+eγkt+ak,e|k|xzwbk,e|k|xzw*1eγkt)puipv2(ɛ1+1)0dkkeiky(ak,+e|k|xzw*bk,+e|k|xzw1+eγ+kt+ak,e|k|xzw*bk,e|k|xzw1eγ+kt).
We now assume that, in a certain frequency region, Re(ɛ1) < −1 as well as the Im(ɛ1) is positive and close to zero. Consequently eγ is located in the first quadrant of the complex plane. Further assuming that only the poles of the integrand, given by k = ±γ/t, contribute to the integration, we finally obtain
Eu=iπγ(pu+ipv)(ɛ1+1)t2(aeγx/teiγy/tzwbeγx/teiγy/tzw*)
(E.6)
for positive y, and
Eu=iπγ(puipv)(ɛ1+1)t2(aeγx/teiγy/tzw*beγx/teiγy/tzw)
(E.7)
for negative y. The electric field in the x < d1 region, where a = (1+eγ)e−2γd1/t and b = 0, can then be expressed as
Eu=πγ(ipusgn(y)pv)(ɛ1+1)t2(1+eγ)(x+i|y|)2eγ(x+i|y|2d1)/t.
(E.8)
By using the same notation as Ref. [22

22. A. Aubry, D. Y. Lei, S. A. Maier, and J. B. Pendry, “Broadband plasmonic device concentrating the energy at the nanoscale: the crescent-shaped cylinder,” Phys. Rev. B 82, 125430 (2010). [CrossRef]

] as well as setting g = 1, the above equation is found to be identical to Eq. (28) of Ref. [22

22. A. Aubry, D. Y. Lei, S. A. Maier, and J. B. Pendry, “Broadband plasmonic device concentrating the energy at the nanoscale: the crescent-shaped cylinder,” Phys. Rev. B 82, 125430 (2010). [CrossRef]

].

Acknowledgments

This work was supported in part by the Penn State MRSEC under NSF grant no. DMR 0213623.

References and links

1.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006). [CrossRef] [PubMed]

2.

U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006). [CrossRef] [PubMed]

3.

U. Leonhardt and T. G. Philbin, “Transformation optics and the geometry of light,” Prog. Opt. 53, 69–152 (2009). [CrossRef]

4.

D.-H. Kwon and D. H. Werner, “Transformation electromagnetics: an overview of the theory and its application,” IEEE Antennas Propag. Mag. 52, 24–45 (2010). [CrossRef]

5.

M. Kadic, S. Guenneau, and S. Enoch, “Transformational plasmonics: cloak, concentrator and rotator for SPPs,” Opt. Express 18, 12027–12032 (2010). [CrossRef] [PubMed]

6.

J. Renger, M. Kadic, G. Dupont, S. S. Acimovic, S. Guenneau, R. Quidant, and S. Enoch, “Hidden progress: broadband plasmonic invisibility,” Opt. Express 18, 15757–15768 (2010). [CrossRef] [PubMed]

7.

D.-H. Kwon and D. H. Werner, “Transformation optical designs for wave collimators, flat lenses and right-angle bends,” N. J. Phys. 10, 115023 (2008). [CrossRef]

8.

H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9, 387–396 (2010). [CrossRef] [PubMed]

9.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006). [CrossRef] [PubMed]

10.

R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science 323, 366–369 (2009). [CrossRef] [PubMed]

11.

T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science 328, 337–339 (2010). [CrossRef] [PubMed]

12.

M. W. McCall, A. Favaro, P. Kinsler, and A. Boardman, “A spacetime cloak, or a history editor,” J. Opt. 13, 024003 (2011). [CrossRef]

13.

E. E. Narimanov and A. V. Kildishev, “Optical black hole: broadband omnidirectional light absorber,” Appl. Phys. Lett. 95, 041106 (2009). [CrossRef]

14.

M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, and D. R. Smith, “Optical design of reflectionless complex media by finite embedded coordinate transformations,” Phys. Rev. Lett. 100, 063903 (2008). [CrossRef] [PubMed]

15.

D.-H. Kwon and D. H. Werner, “Polarization splitter and polarization rotator designs based on transformation optics,” Opt. Express 16, 18731–18738 (2008). [CrossRef]

16.

J. D. Jackson, Classical Electrodynamics, 3rd ed. (John Wiley & Sons, 2001).

17.

J. B. Pendry and S. A. Ramakrishna, “Near field lenses in two dimensions,” J. Phys.: Condens. Matter 14, 8463–8479 (2002). [CrossRef]

18.

J. B. Pendry and S. A. Ramakrishna, “Focusing light with negative refractive index,” J. Phys.: Condens. Matter 15, 6345–6364 (2003). [CrossRef]

19.

S. Guenneau, B. Gralak, and J. B. Pendry, “Perfect corner reflector,” Opt. Lett. 30, 1204–1206 (2005). [CrossRef] [PubMed]

20.

S. Guenneau, A. C. Vutha, and S. A. Ramakrishna, “Negative refraction in 2-D checkerboards by mirror antisymmetry and 3-D corner lenses,” N. J. Phys. 7, 164 (2005). [CrossRef]

21.

B. Gralak and S. Guenneau, “Transfer matrix method for point sources radiating in classes of negative refractive index materials with 2n-fold antisymmetry,” Waves Random Complex Media 17, 581–614 (2007). [CrossRef]

22.

A. Aubry, D. Y. Lei, S. A. Maier, and J. B. Pendry, “Broadband plasmonic device concentrating the energy at the nanoscale: the crescent-shaped cylinder,” Phys. Rev. B 82, 125430 (2010). [CrossRef]

23.

A. Aubry, D. Y. Lei, A. I. Fernández-Domínguez, Y. Sonnefraud, S. A. Maier, and J. B. Pendry, “Plasmonic light-harvesting devices over the whole visible spectrum,” Nano Lett. 10, 2574–2579 (2010). [CrossRef] [PubMed]

24.

Y. Luo, J. B. Pendry, and A. Aubry, “Surface plasmons and singularities,” Nano Lett. 10, 4186–4191 (2010). [CrossRef] [PubMed]

25.

A. Aubry, D. Y. Lei, S. A. Maier, and J. B. Pendry, “Conformal transformation applied to plasmonics beyond the quasistatic limit,” Phys. Rev. B 82, 205109 (2010). [CrossRef]

26.

A. Aubry, D. Y. Lei, S. A. Maier, and J. B. Pendry, “Interaction between plasmonic nanoparticles revisited with transformation optics,” Phys. Rev. Lett. 105, 233901 (2010). [CrossRef]

27.

D. Y. Lei, A. Aubry, S. A. Maier, and J. B. Pendry, “Broadband nano-focusing of light using kissing nanowires,” N. J. Phys. 12, 093030 (2010). [CrossRef]

28.

D. J. Bergman and D. Stroud, in Solid State Physics , H. Ehrenreich and D. Turnbull, eds. (Academic, 1992), Vol. 46, pp. 148–270. [CrossRef]

29.

G. W. Milton, The Theory of Composites (Cambridge University Press, 2002). [CrossRef]

30.

D. J. Bergman and M. I. Stockman, “Surface plasmon amplification by stimulated emission of radiation: quantum generation of coherent surface plasmons in nanosystems,” Phys. Rev. Lett. 90, 027402 (2003). [CrossRef] [PubMed]

31.

M. I. Stockman, S. V. Faleev, and D. J. Bergman, “Localization versus delocalization of surface plasmons in nanosystems: can one state have both characteristics?” Phys. Rev. Lett. 87, 167401 (2001). [CrossRef] [PubMed]

32.

M. I. Stockman, D. J. Bergman, and T. Kobayashi, “Coherent control of nanoscale localization of ultrafast optical excitation in nanosystems,” Phys. Rev. B 69, 054202 (2004). [CrossRef]

33.

D. J. Bergman, “The dielectric constant of a simple cubic array of identical spheres,” J. Phys. C: Solid State Phys. 12, 4947–4960 (1979). [CrossRef]

34.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley & Sons, 1998). [CrossRef]

35.

Y. Zeng, Q. Wu, and D. H. Werner, “Electrostatic theory for designing lossless negative permittivity metamaterials,” Opt. Lett. 35, 1431–1433 (2010). [CrossRef] [PubMed]

36.

F. Wang and Y. R. Shen, “General properties of local plasmons in metal nanostructures,” Phys. Rev. Lett. 97, 206806 (2006). [CrossRef] [PubMed]

37.

D. R. Fredkin and I. D. Mayergoyz, “Resonant behavior of dielectric objects (electrostatic resonances),” Phys. Rev. Lett. 91, 253902 (2003). [CrossRef]

38.

A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep. 408, 131–314 (2005). [CrossRef]

39.

P. B. Catrysse and S. Fan, “Understanding the dispersion of coaxial plasmonic structures through a connection with the planar metal-insulator-metal geometry,” Appl. Phys. Lett. 94, 231111 (2009). [CrossRef]

OCIS Codes
(000.3860) General : Mathematical methods in physics
(250.5403) Optoelectronics : Plasmonics

ToC Category:
Physical Optics

History
Original Manuscript: May 27, 2011
Revised Manuscript: July 13, 2011
Manuscript Accepted: September 17, 2011
Published: September 29, 2011

Citation
Yong Zeng, Jinjie Liu, and Douglas H. Werner, "General properties of two-dimensional conformal transformations in electrostatics," Opt. Express 19, 20035-20047 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-21-20035


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References

  1. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science312, 1780–1782 (2006). [CrossRef] [PubMed]
  2. U. Leonhardt, “Optical conformal mapping,” Science312, 1777–1780 (2006). [CrossRef] [PubMed]
  3. U. Leonhardt and T. G. Philbin, “Transformation optics and the geometry of light,” Prog. Opt.53, 69–152 (2009). [CrossRef]
  4. D.-H. Kwon and D. H. Werner, “Transformation electromagnetics: an overview of the theory and its application,” IEEE Antennas Propag. Mag.52, 24–45 (2010). [CrossRef]
  5. M. Kadic, S. Guenneau, and S. Enoch, “Transformational plasmonics: cloak, concentrator and rotator for SPPs,” Opt. Express18, 12027–12032 (2010). [CrossRef] [PubMed]
  6. J. Renger, M. Kadic, G. Dupont, S. S. Acimovic, S. Guenneau, R. Quidant, and S. Enoch, “Hidden progress: broadband plasmonic invisibility,” Opt. Express18, 15757–15768 (2010). [CrossRef] [PubMed]
  7. D.-H. Kwon and D. H. Werner, “Transformation optical designs for wave collimators, flat lenses and right-angle bends,” N. J. Phys.10, 115023 (2008). [CrossRef]
  8. H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater.9, 387–396 (2010). [CrossRef] [PubMed]
  9. D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science314, 977–980 (2006). [CrossRef] [PubMed]
  10. R. Liu, C. Ji, J. J. Mock, J. Y. Chin, T. J. Cui, and D. R. Smith, “Broadband ground-plane cloak,” Science323, 366–369 (2009). [CrossRef] [PubMed]
  11. T. Ergin, N. Stenger, P. Brenner, J. B. Pendry, and M. Wegener, “Three-dimensional invisibility cloak at optical wavelengths,” Science328, 337–339 (2010). [CrossRef] [PubMed]
  12. M. W. McCall, A. Favaro, P. Kinsler, and A. Boardman, “A spacetime cloak, or a history editor,” J. Opt.13, 024003 (2011). [CrossRef]
  13. E. E. Narimanov and A. V. Kildishev, “Optical black hole: broadband omnidirectional light absorber,” Appl. Phys. Lett.95, 041106 (2009). [CrossRef]
  14. M. Rahm, S. A. Cummer, D. Schurig, J. B. Pendry, and D. R. Smith, “Optical design of reflectionless complex media by finite embedded coordinate transformations,” Phys. Rev. Lett.100, 063903 (2008). [CrossRef] [PubMed]
  15. D.-H. Kwon and D. H. Werner, “Polarization splitter and polarization rotator designs based on transformation optics,” Opt. Express16, 18731–18738 (2008). [CrossRef]
  16. J. D. Jackson, Classical Electrodynamics, 3rd ed. (John Wiley & Sons, 2001).
  17. J. B. Pendry and S. A. Ramakrishna, “Near field lenses in two dimensions,” J. Phys.: Condens. Matter14, 8463–8479 (2002). [CrossRef]
  18. J. B. Pendry and S. A. Ramakrishna, “Focusing light with negative refractive index,” J. Phys.: Condens. Matter15, 6345–6364 (2003). [CrossRef]
  19. S. Guenneau, B. Gralak, and J. B. Pendry, “Perfect corner reflector,” Opt. Lett.30, 1204–1206 (2005). [CrossRef] [PubMed]
  20. S. Guenneau, A. C. Vutha, and S. A. Ramakrishna, “Negative refraction in 2-D checkerboards by mirror antisymmetry and 3-D corner lenses,” N. J. Phys.7, 164 (2005). [CrossRef]
  21. B. Gralak and S. Guenneau, “Transfer matrix method for point sources radiating in classes of negative refractive index materials with 2n-fold antisymmetry,” Waves Random Complex Media17, 581–614 (2007). [CrossRef]
  22. A. Aubry, D. Y. Lei, S. A. Maier, and J. B. Pendry, “Broadband plasmonic device concentrating the energy at the nanoscale: the crescent-shaped cylinder,” Phys. Rev. B82, 125430 (2010). [CrossRef]
  23. A. Aubry, D. Y. Lei, A. I. Fernández-Domínguez, Y. Sonnefraud, S. A. Maier, and J. B. Pendry, “Plasmonic light-harvesting devices over the whole visible spectrum,” Nano Lett.10, 2574–2579 (2010). [CrossRef] [PubMed]
  24. Y. Luo, J. B. Pendry, and A. Aubry, “Surface plasmons and singularities,” Nano Lett.10, 4186–4191 (2010). [CrossRef] [PubMed]
  25. A. Aubry, D. Y. Lei, S. A. Maier, and J. B. Pendry, “Conformal transformation applied to plasmonics beyond the quasistatic limit,” Phys. Rev. B82, 205109 (2010). [CrossRef]
  26. A. Aubry, D. Y. Lei, S. A. Maier, and J. B. Pendry, “Interaction between plasmonic nanoparticles revisited with transformation optics,” Phys. Rev. Lett.105, 233901 (2010). [CrossRef]
  27. D. Y. Lei, A. Aubry, S. A. Maier, and J. B. Pendry, “Broadband nano-focusing of light using kissing nanowires,” N. J. Phys.12, 093030 (2010). [CrossRef]
  28. D. J. Bergman and D. Stroud, in Solid State Physics, H. Ehrenreich and D. Turnbull, eds. (Academic, 1992), Vol. 46, pp. 148–270. [CrossRef]
  29. G. W. Milton, The Theory of Composites (Cambridge University Press, 2002). [CrossRef]
  30. D. J. Bergman and M. I. Stockman, “Surface plasmon amplification by stimulated emission of radiation: quantum generation of coherent surface plasmons in nanosystems,” Phys. Rev. Lett.90, 027402 (2003). [CrossRef] [PubMed]
  31. M. I. Stockman, S. V. Faleev, and D. J. Bergman, “Localization versus delocalization of surface plasmons in nanosystems: can one state have both characteristics?” Phys. Rev. Lett.87, 167401 (2001). [CrossRef] [PubMed]
  32. M. I. Stockman, D. J. Bergman, and T. Kobayashi, “Coherent control of nanoscale localization of ultrafast optical excitation in nanosystems,” Phys. Rev. B69, 054202 (2004). [CrossRef]
  33. D. J. Bergman, “The dielectric constant of a simple cubic array of identical spheres,” J. Phys. C: Solid State Phys.12, 4947–4960 (1979). [CrossRef]
  34. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley & Sons, 1998). [CrossRef]
  35. Y. Zeng, Q. Wu, and D. H. Werner, “Electrostatic theory for designing lossless negative permittivity metamaterials,” Opt. Lett.35, 1431–1433 (2010). [CrossRef] [PubMed]
  36. F. Wang and Y. R. Shen, “General properties of local plasmons in metal nanostructures,” Phys. Rev. Lett.97, 206806 (2006). [CrossRef] [PubMed]
  37. D. R. Fredkin and I. D. Mayergoyz, “Resonant behavior of dielectric objects (electrostatic resonances),” Phys. Rev. Lett.91, 253902 (2003). [CrossRef]
  38. A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “Nano-optics of surface plasmon polaritons,” Phys. Rep.408, 131–314 (2005). [CrossRef]
  39. P. B. Catrysse and S. Fan, “Understanding the dispersion of coaxial plasmonic structures through a connection with the planar metal-insulator-metal geometry,” Appl. Phys. Lett.94, 231111 (2009). [CrossRef]

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