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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 21 — Oct. 8, 2012
  • pp: 22976–22986
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Internal homogenization: Effective permittivity of a coated sphere

Uday K. Chettiar and Nader Engheta  »View Author Affiliations


Optics Express, Vol. 20, Issue 21, pp. 22976-22986 (2012)
http://dx.doi.org/10.1364/OE.20.022976


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Abstract

The concept of internal homogenization is introduced as a complementary approach to the conventional homogenization schemes, which could be termed as external homogenization. The theory for the internal homogenization of the permittivity of subwavelength coated spheres is presented. The effective permittivity derived from the internal homogenization of coreshells is discussed for plasmonic and dielectric constituent materials. The effective model provided by the homogenization is a useful design tool in constructing coated particles with desired resonant properties.

© 2012 OSA

1. Introduction

Spherical solid metal particles possess plasmonic resonances which are well understood using the solutions provided by the Mie theory [1

1. G. Mie, “Beitrage zur optik trüber medien, speziell kolloidaler metallösungen,” Ann. Phys. 330(3), 377–445 (1908). [CrossRef]

,2

2. U. Kreibig and M. Vollmer, Optical Properties of Metal Clusters (Springer, 2011).

]. Mie theory was originally used to explain the colors in metal colloids, which are now known to have their origins in the plasmonic resonance of the metal particles. Mie theory has also been extended for layered spheres [3

3. A. L. Aden and M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22(10), 1242–1246 (1951). [CrossRef]

] and it was later shown that such core shell particles allow one to tune the plasmonic resonance over a wide range of wavelengths [4

4. A. E. Neeves and M. H. Birnboim, “Composite structures for the enhancement of nonlinear-optical susceptibility,” J. Opt. Soc. Am. B 6(4), 787–796 (1989). [CrossRef]

]. This was followed by the experimental demonstration of the practicality of realizing these core shell particles and reliably controlling their resonances [5

5. S. J. Oldenburg, R. D. Averitt, S. L. Westcott, and N. J. Halas, “Nanoengineering of optical resonances,” Chem. Phys. Lett. 288(2-4), 243–247 (1998). [CrossRef]

]. The core shells were fabricated through the use of colloid reduction chemistry over dielectric spheres. This has led to a significant amount of research into these versatile structures. Coated spheres (core-shells) have been studied widely in the recent years due to their tunable resonant response with applications ranging from enhanced absorption [6

6. R. Baer, D. Neuhauser, and S. Weiss, “Enhanced absorption induced by a metallic nanoshell,” Nano Lett. 4(1), 85–88 (2004). [CrossRef]

], surface-enhanced Raman optical activity [7

7. R. Lombardini, R. Acevedo, N. J. Halas, and B. R. Johnson, “Plasmonic enhancement of Raman optical activity in molecules near metal nanoshells: theoretical comparison of circular polarization methods,” J. Phys. Chem. C 114(16), 7390–7400 (2010). [CrossRef]

], surface enhanced Raman scattering (SERS) [8

8. J. B. Jackson, S. L. Westcott, L. R. Hirsch, J. L. West, and N. J. Halas, “Controlling the surface enhanced Raman effect via the nanoshell geometry,” Appl. Phys. Lett. 82(2), 257–259 (2003). [CrossRef]

], cloaking [9

9. A. Alù and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(1), 016623 (2005). [CrossRef] [PubMed]

11

11. A. Alù and N. Engheta, “Multifrequency optical invisibility cloak with layered plasmonic shells,” Phys. Rev. Lett. 100(11), 113901 (2008). [CrossRef] [PubMed]

] and dielectric waveguide impedance matching [12

12. U. K. Chettiar, R. F. Garcia, S. A. Maier, and N. Engheta, “Enhancement of radiation from dielectric waveguides using resonant plasmonic coreshells,” Opt. Express 20(14), 16104–16112 (2012). [CrossRef] [PubMed]

]. Plasmonic coreshells have also found use in biomedical applications [13

13. N. Halas, “Playing with plasmons: tuning the optical resonant properties of metallic nanoshells,” MRS Bull. 30(05), 362–367 (2005). [CrossRef]

] including targeted delivery [14

14. R. Huschka, J. Zuloaga, M. W. Knight, L. V. Brown, P. Nordlander, and N. J. Halas, “Light-induced release of DNA from gold nanoparticles: nanoshells and nanorods,” J. Am. Chem. Soc. 133(31), 12247–12255 (2011). [CrossRef] [PubMed]

] and cancer therapy [15

15. R. Bardhan, S. Lal, A. Joshi, and N. J. Halas, “Theranostic nanoshells: from probe design to imaging and treatment of cancer,” Acc. Chem. Res. 44(10), 936–946 (2011). [CrossRef] [PubMed]

]. The resonance can be tuned to different wavelengths by changing the ratio of the radius of the core and the shell.

Such core-shells could also be used as inclusions in a substrate making it possible to engineer the effective permittivity of the inclusion-substrate composite. The effective permittivity in presence of inhomogeneity is provided by some form of homogenization. The theory of homogenization has been developed for a range of physical properties including elasticity, conductivity, thermodynamics, electromagnetism, etc [16

16. G. W. Milton, The Theory of Composites (Cambridge University Press, 2004).

]. Even a superficially homogeneous material is inhomogeneous at the molecular scale. In the case of electromagnetic homogenization the permittivity is given by the Clausius-Mossotti relation which relates the macroscopic permittivity to the microscopic molecular polarizability of the constituent molecules [17

17. D. J. Bergman and D. Stroud, “Physical properties of macroscopically inhomogeneous media,” Solid State Phys. 46, 147–269 (1992). [CrossRef]

19

19. V. M. Shalaev, Nonlinear Optics of Random Media: Fractal Composites and Metal-dielectric Films (Springer, 2000).

]. In the presence of more macroscopic inhomogeneity the Clausius-Mossoti relation could be used in conjunction with mean-field theory to yield various homogenization schemes. Two such schemes are the Maxwell-Garnett effective medium theory (EMT) and Bruggeman EMT [19

19. V. M. Shalaev, Nonlinear Optics of Random Media: Fractal Composites and Metal-dielectric Films (Springer, 2000).

21

21. A. H. Sihvola, Electromagnetic Mixing Formulas and Applications (Institution of Electrical Engineers, 2008).

]. Maxwell-Garnett EMT is applicable in case of a dilute inclusion whereas Bruggeman EMT is valid over a wider range of concentration for the inclusion. In all these effective medium theories the material response is averaged over a macroscopic scale to yield the effective permittivity. With the advent of metamaterials the homogenization theories have been extended to more complex systems. Such systems can includes effects like anisotropy, chirality, bianisotropy, spatial dispersion, etc. which require very careful first principle considerations for deriving the proper homogenization theory [22

22. L. W. Mochán and R. G. Barrera, “Electromagnetic response of systems with spatial fluctuations. I. general formalism,” Phys. Rev. B 32, 32–36 (1985).

28

28. G. P. Ortiz, B. E. Martínez-Zérega, B. S. Mendoza, and L. W. Mochán, “Effective optical response of metamaterials,” Phys. Rev. B 79(24), 245132 (2009). [CrossRef]

].

Homogenization theory has also been extended to electron waves in graphene [29

29. M. Silveirinha and N. Engheta, “Effective medium approach to electron waves: graphene superlattices,” Phys. Rev. B 85(19), 195413 (2012). [CrossRef]

]. We term all such approaches as external homogenization. In the present work, we develop and examine the concept of internal homogenization, in which we explore the possibility of assigning a single effective permittivity to coated spheres made of several material layers, thus representing the entire multilayered spherical particle as a particle of the same size (or outer radius) but with a single effective permittivity filling the entire sphere. While the conventional homogenization techniques provide the bulk material properties based on the polarizability of constituent inclusions, i.e. they effectively start from the polarizability of inclusions and look “outward” towards effective bulk properties, here the notion of internal homogenization looks “inward”, and attempts to assign a single permittivity to an inclusion with internal layers of different materials, such as coated spheres.

2. Internal homogenization theory and results

The concept of internal homogenization is illustrated in Fig. 1
Fig. 1 Internal homogenization compared to external homogenization.
. The top panel shows the classical external homogenization scheme where a distribution of inclusions in a matrix is assigned an effective material property. There are several methods of assigning such an effective property depending of various parameters and the region of operation such as Maxwell-Garnett effective medium theory (EMT) and Bruggeman EMT [19

19. V. M. Shalaev, Nonlinear Optics of Random Media: Fractal Composites and Metal-dielectric Films (Springer, 2000).

,21

21. A. H. Sihvola, Electromagnetic Mixing Formulas and Applications (Institution of Electrical Engineers, 2008).

]. As opposed to the external homogenization, in the case of internal homogenization we assign an effective permittivity for a single inclusion which may have a complicated internal structure. This is shown in the bottom panel of Fig. 1. For simplicity we will be using the core-shell geometry in this manuscript; however, this concept can be applied to particles with any geometry. Also in order for EMT to be applicable we assume that the core-shells are subwavelength and consequently we can use the quasi-static small-radii analysis in order to derive the effective properties. The core-shell and the effective sphere schematic are shown in Fig. 2
Fig. 2 Schematic of the core-shell internal homogenization problem.
.

The polarizability of the equivalent sphere α1 and a coated sphere α2 with the same outer radius a is given by the following two equations under quasi static approximation [30

30. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles, 2nd ed. (Wiley-VCH, 2008).

].
α1=4πε0(εeε0εe+2ε0)a3α2=4πε0(a3(εc+2εs)(εsε0)+b3(εcεs)(2εs+ε0))2b3(εcεs)(εsε0)+a3(εc+2εs)(εs+2ε0)a3
(1)
whereεc and εs are the permittivity of the core and the shell, respectively, b is the core radius, and εe is the effective permittivity of the equivalent sphere to be determined. The polarizability equations can be derived by evaluating the scattered field when the core-shell is under a plane wave excitation. The scattered field can then be compared to the field radiated by an electric dipole. This yields the effective dipole moment of the core-shell under a plane wave excitation and on taking its ratio with the incident electric field we get the polarizability.

If the effective sphere is to behave just like the core-shell both the effective sphere and the core-shell should possess the same polarizability. Under the quasi-static approximation, only the dipolar terms are important and so as long as the two particles possess the same polarizability they will interact with any excitation in an identical manner. This is true even when we have a collection of such particles, as long as the particles are not too close to excite the multipolar terms. The dipole approximation has been used for a diverse range of problems in plasmonics, see for example [31

31. A. B. Evlyukhin and S. I. Bozhevolnyi, “Point-dipole approximation for surface plasmon polariton scattering: Implications and limitations,” Phys. Rev. B 71(13), 134304 (2005). [CrossRef]

]. Equating the polarizability of the core-shell and the effective sphere yields the following equation for the effective permittivity.
εe=εsa3(εc+2εs)+2b3(εcεs)a3(εc+2εs)b3(εcεs)
(2)
The effective permittivity thus obtained is similar in form to the effective permittivity given by Maxwell-Garnett EMT if we assume that the core material is embedded in a medium of the shell material, and the filling fraction of the core material is given by b3/a3, the same as the filling fraction of the core material in the core-shell structure. Hence, we would expect the frequency dispersion of effective permittivity given by internal homogenization to have similar properties as the effective permittivity given by Maxwell-Garnett EMT [21

21. A. H. Sihvola, Electromagnetic Mixing Formulas and Applications (Institution of Electrical Engineers, 2008).

]. This is confirmed in the following sections.

2.1 Shell is a Drude material and core is a dielectric

Assuming that the shell is a Drude material we may replace the permittivity of shell material with the Drude model for permittivity, εs=εωp2ω(ω+iΓ). Let us assume the core is a dielectric with a constant permittivity. Substituting the Drude model for εs in Eq. (2) would give us the exact effective permittivity. But after some algebraic simplification and ignoring the terms corresponding to the resonance at the core-shell interface we can mold the effective permittivity into an effective Drude model as shown in the following equation.
εe=εa3(εc+2ε)+2b3(εcε)a3(εc+2ε)b3(εcε)2ωp2(a3b3)(2a3+b3)ω(ω+iΓ)
(3)
The Drude parameters of the effective permittivity can be listed as shown below.

ε,e=εa3(εc+2ε)+2b3(εcε)a3(εc+2ε)b3(εcε)ωp,e=ωp2(a3b3)(2a3+b3)Γe=Γ

The effective plasma frequency scales between ωp and 0 as b/a is increased from 0 to 1. On the other hand the effective collision frequency remains the same. This can be justified on physical grounds since the collision frequency classically represents the rate at which the free electrons in the Drude material undergo collisions. If we ignore the size effect consideration the collision frequency is a property of the material. Hence even if we have the Drude material in the form of a shell around a dielectric core the frequency of collisions of the free electrons inside the Drude shell should effectively remain unchanged. Furthermore the plasma frequency is proportional to the square root of the free electron density in the material. As the shell becomes thinner the effective density of the free electrons over the volume of the whole core-shell reduces and consequently the effective plasma frequency also reduces. Figure 3
Fig. 3 Effective plasma frequency (a) and effective ε (b) of the core-shell as a function of the radius. (c) Comparing of the exact effective permittivity with the Drude approximation for a silver coated silica sphere with b/a = 0.9.
shows the variation of the effective plasma frequency (ωp) and effective ε as a function of b/a when the core is set to silica (εc = 2.25) and the shell is set to silver (ωp = 9.2 eV, Γ = 0.0212 eV, ε = 5.0), where 1 eV = 241.8 THz [32

32. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972). [CrossRef]

]. Figure 3(c) below also shows the fit of the effective Drude model (Eq. (3)) compared to the exact effective permittivity (Eq. (2)).

As expected, the Drude model for the effective permittivity does not capture the resonance around 300 nm which occurs at the interface between the core and the shell, but it describes the permittivity away from the resonance accurately, especially the region where the real part the effective permittivity is close to zero. This region is of importance in tuning the resonance of the core-shell particles. Figures 3(a) and 3(b) shows the dependence of the effective plasma frequency (ωp) and effective ε as we change the radius ratio (b/a). As seen from the figure we can tune the Drude parameters of the effective permittivity over a wide range by simply changing the value of b/a. This provides us with a simple design strategy to obtain a spherical particle with a desired plasma frequency. For any desired plasma frequency between 0 and ωp we can simply read the required radius ratio (b/a) from Fig. 3(a). Such an effective particle will behave like a spherical particle with the desired plasma frequency. The effective particle could further be used as an inclusion to design more complicated materials.

2.2 Core is a Drude material and shell is a dielectric

2.3 Both core and shell are Drude materials

In this case we assume that both the core and shell possess a Drude permittivity as shown below.
εc=ε,cωp,c2ω(ω+iΓc)εs=ε,sωp,s2ω(ω+iΓs)
As in the case of a Drude shell we can express the effective permittivity in terms of a Drude model. The effective Drude model is shown in the following equation.

εe=ε,eωp,e2ω(ω+iΓe)ε,e=ε,sa3(ε,c+2ε,s)+2b3(ε,cε,s)a3(ε,c+2ε,s)b3(ε,cε,s)ωp,e=ωp,sa3(ωp,c2+2ωp,s2)+2b3(ωp,c2ωp,s2)a3(ωp,c2+2ωp,s2)b3(ωp,c2ωp,s2)Γe=Γsωp,e2ωp,s2(a3(ωp,c2Γs+2ωp,s2Γc)b3(ωp,c2Γsωp,s2Γc)a3(ωp,c2Γs+2ωp,s2Γc)+2b3(ωp,c2Γsωp,s2Γc))

Figure 5
Fig. 5 Effective plasma frequency (a), effective ε (b) and collision frequency (c) of the core-shell as a function of the radius ratio. (d) Comparison of the exact effective permittivity with the Drude approximation when b/a = 0.6.
shows the variation of the effective plasma frequency, effective ε and the effective Γ as a function of b/a when the core is set to a plasmonic material with the properties given by (ωp = 1 eV, Γ = 0.05 eV, ε = 10), where 1 eV = 241.8 THz and the shell is set to a plasmonic material with parameters given by (ωp = 9 eV, Γ = 0.1 eV, ε = 1). The figure also shows the fit between the exact effective permittivity and the Drude approximation when the radius ratio is set to 0.6. Similar to the case with a Drude shell and dielectric core, the Drude model for the effective permittivity does not capture the resonance around 300 nm (Fig. 5(d)), but it describes the permittivity away from the resonance accurately, especially the region where the real part the effective permittivity is close to zero, similar to the case where the shell material is a Drude material.

2.4 Core shells with more than two layers

The concept of internal homogenization is not limited to just core-shell structures with two layers, and it can be easily extended to multi-layered structured. For example, assume that we have a three-layered spherical particle with radii given by r1, r2 and r3 (r1 < r2 < r3) and the corresponding permittivity given by ε1, ε2, and ε3. In order to arrive at the effective permittivity of the whole sphere we can use the concept of internal homogenization in an iterative manner. The two innermost layers can be replaced by a single sphere with a radius of r2 and with an effective permittivity given by the following equation.
ε1,2=ε2r23(ε1+2ε2)+2r13(ε1ε2)r23(ε1+2ε2)r13(ε1ε2)
After replacing the two inner layers with the effective sphere the problem reduces to a two layered core-shell structure and the overall effective permittivity of the effective sphere can be written as follows.
εe=ε3r33(ε1,2+2ε3)+2r23(ε1,2ε3)r33(ε1,2+2ε3)r23(ε1,2ε3)
The effective permittivity of such multi-layered core-shell structures could also be approximated by an effective Drude or Lorentz model under various conditions. But this discussion is not included in the current manuscript.

2.5 Internal homogenization for arbitrary shapes

3. Loss control

4. Conclusions

We have introduced the concept of internal homogenization wherein a confined composite structure is homogenized to provide an effective structure with the same dimensions, but an effective permittivity such that it has the same response to an electromagnetic excitation as the original composite structure irrespective of the ambient medium. The homogenization equations for multi-layered spherical core-shell structures were also presented. We have also provided simplified forms for two-layered core-shell structures under various conditions. A particular application was demonstrated where you could provide a relatively low-loss spherical particle by using a core-shell implementation instead of a uniform material sphere. This approach could also find use in numerical computation where core-shell elements could be replaced by their effective uniform sphere counterparts. This will be especially useful in the presence of thin shells which could result in a high overhead in terms of the number of mesh cells that would be required to simulate such a structure. Replacing the core-shells with the effective sphere would reduce the number of mesh cell by removing the need to represent the thin shell as a separate material.

Acknowledgments

References and links

1.

G. Mie, “Beitrage zur optik trüber medien, speziell kolloidaler metallösungen,” Ann. Phys. 330(3), 377–445 (1908). [CrossRef]

2.

U. Kreibig and M. Vollmer, Optical Properties of Metal Clusters (Springer, 2011).

3.

A. L. Aden and M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22(10), 1242–1246 (1951). [CrossRef]

4.

A. E. Neeves and M. H. Birnboim, “Composite structures for the enhancement of nonlinear-optical susceptibility,” J. Opt. Soc. Am. B 6(4), 787–796 (1989). [CrossRef]

5.

S. J. Oldenburg, R. D. Averitt, S. L. Westcott, and N. J. Halas, “Nanoengineering of optical resonances,” Chem. Phys. Lett. 288(2-4), 243–247 (1998). [CrossRef]

6.

R. Baer, D. Neuhauser, and S. Weiss, “Enhanced absorption induced by a metallic nanoshell,” Nano Lett. 4(1), 85–88 (2004). [CrossRef]

7.

R. Lombardini, R. Acevedo, N. J. Halas, and B. R. Johnson, “Plasmonic enhancement of Raman optical activity in molecules near metal nanoshells: theoretical comparison of circular polarization methods,” J. Phys. Chem. C 114(16), 7390–7400 (2010). [CrossRef]

8.

J. B. Jackson, S. L. Westcott, L. R. Hirsch, J. L. West, and N. J. Halas, “Controlling the surface enhanced Raman effect via the nanoshell geometry,” Appl. Phys. Lett. 82(2), 257–259 (2003). [CrossRef]

9.

A. Alù and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(1), 016623 (2005). [CrossRef] [PubMed]

10.

A. Alù and N. Engheta, “Plasmonic and metamaterial cloaking: physical mechanisms and potentials,” J. Opt. A 10(9), 093002 (2008). [CrossRef]

11.

A. Alù and N. Engheta, “Multifrequency optical invisibility cloak with layered plasmonic shells,” Phys. Rev. Lett. 100(11), 113901 (2008). [CrossRef] [PubMed]

12.

U. K. Chettiar, R. F. Garcia, S. A. Maier, and N. Engheta, “Enhancement of radiation from dielectric waveguides using resonant plasmonic coreshells,” Opt. Express 20(14), 16104–16112 (2012). [CrossRef] [PubMed]

13.

N. Halas, “Playing with plasmons: tuning the optical resonant properties of metallic nanoshells,” MRS Bull. 30(05), 362–367 (2005). [CrossRef]

14.

R. Huschka, J. Zuloaga, M. W. Knight, L. V. Brown, P. Nordlander, and N. J. Halas, “Light-induced release of DNA from gold nanoparticles: nanoshells and nanorods,” J. Am. Chem. Soc. 133(31), 12247–12255 (2011). [CrossRef] [PubMed]

15.

R. Bardhan, S. Lal, A. Joshi, and N. J. Halas, “Theranostic nanoshells: from probe design to imaging and treatment of cancer,” Acc. Chem. Res. 44(10), 936–946 (2011). [CrossRef] [PubMed]

16.

G. W. Milton, The Theory of Composites (Cambridge University Press, 2004).

17.

D. J. Bergman and D. Stroud, “Physical properties of macroscopically inhomogeneous media,” Solid State Phys. 46, 147–269 (1992). [CrossRef]

18.

V. M. Shalaev, “Electromagnetic properties of small-particle composites,” Phys. Rep. 272(2-3), 61–137 (1996). [CrossRef]

19.

V. M. Shalaev, Nonlinear Optics of Random Media: Fractal Composites and Metal-dielectric Films (Springer, 2000).

20.

D. A. G. Bruggeman, “Berechnung verschiedener physikalischer konstanten von heterogenen substanzen,” Ann. Phys. 416(7), 636–664 (1935). [CrossRef]

21.

A. H. Sihvola, Electromagnetic Mixing Formulas and Applications (Institution of Electrical Engineers, 2008).

22.

L. W. Mochán and R. G. Barrera, “Electromagnetic response of systems with spatial fluctuations. I. general formalism,” Phys. Rev. B 32, 32–36 (1985).

23.

M. G. Silveirinha, “Nonlocal homogenization theory of structured materials,” in Theory and Phenomena of Metamaterials, F. Capolino ed. (CRC Press, 2009).

24.

A. Alù, “First-principles homogenization theory for periodic metamaterials,” Phys. Rev. B 84(7), 075153 (2011). [CrossRef]

25.

D. R. Smith and J. B. Pendry, “Homogenization of metamaterials by field averaging,” J. Opt. Soc. Am. B 23(3), 391–403 (2006). [CrossRef]

26.

C. Fietz and G. Shvets, “Current-driven metamaterial homogenization,” Physica B 405(14), 2930–2934 (2010). [CrossRef]

27.

C. R. Simovski and S. A. Tretyakov, “Local constitutive parameters of metamaterials from an effective-medium perspective,” Phys. Rev. B 75(19), 195111 (2007). [CrossRef]

28.

G. P. Ortiz, B. E. Martínez-Zérega, B. S. Mendoza, and L. W. Mochán, “Effective optical response of metamaterials,” Phys. Rev. B 79(24), 245132 (2009). [CrossRef]

29.

M. Silveirinha and N. Engheta, “Effective medium approach to electron waves: graphene superlattices,” Phys. Rev. B 85(19), 195413 (2012). [CrossRef]

30.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles, 2nd ed. (Wiley-VCH, 2008).

31.

A. B. Evlyukhin and S. I. Bozhevolnyi, “Point-dipole approximation for surface plasmon polariton scattering: Implications and limitations,” Phys. Rev. B 71(13), 134304 (2005). [CrossRef]

32.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972). [CrossRef]

33.

R. C. Aster, C. H. Thurber, and B. Borchers, Parameter Estimation and Inverse Problems (Elsevier Academic Press, 2005).

34.

J. A. Snyman, Practical Mathematical Optimization (Springer, 2005).

OCIS Codes
(260.3910) Physical optics : Metal optics
(160.3918) Materials : Metamaterials

ToC Category:
Metamaterials

History
Original Manuscript: August 14, 2012
Revised Manuscript: September 13, 2012
Manuscript Accepted: September 14, 2012
Published: September 24, 2012

Citation
Uday K. Chettiar and Nader Engheta, "Internal homogenization: Effective permittivity of a coated sphere," Opt. Express 20, 22976-22986 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-21-22976


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References

  1. G. Mie, “Beitrage zur optik trüber medien, speziell kolloidaler metallösungen,” Ann. Phys.330(3), 377–445 (1908). [CrossRef]
  2. U. Kreibig and M. Vollmer, Optical Properties of Metal Clusters (Springer, 2011).
  3. A. L. Aden and M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys.22(10), 1242–1246 (1951). [CrossRef]
  4. A. E. Neeves and M. H. Birnboim, “Composite structures for the enhancement of nonlinear-optical susceptibility,” J. Opt. Soc. Am. B6(4), 787–796 (1989). [CrossRef]
  5. S. J. Oldenburg, R. D. Averitt, S. L. Westcott, and N. J. Halas, “Nanoengineering of optical resonances,” Chem. Phys. Lett.288(2-4), 243–247 (1998). [CrossRef]
  6. R. Baer, D. Neuhauser, and S. Weiss, “Enhanced absorption induced by a metallic nanoshell,” Nano Lett.4(1), 85–88 (2004). [CrossRef]
  7. R. Lombardini, R. Acevedo, N. J. Halas, and B. R. Johnson, “Plasmonic enhancement of Raman optical activity in molecules near metal nanoshells: theoretical comparison of circular polarization methods,” J. Phys. Chem. C114(16), 7390–7400 (2010). [CrossRef]
  8. J. B. Jackson, S. L. Westcott, L. R. Hirsch, J. L. West, and N. J. Halas, “Controlling the surface enhanced Raman effect via the nanoshell geometry,” Appl. Phys. Lett.82(2), 257–259 (2003). [CrossRef]
  9. A. Alù and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.72(1), 016623 (2005). [CrossRef] [PubMed]
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  11. A. Alù and N. Engheta, “Multifrequency optical invisibility cloak with layered plasmonic shells,” Phys. Rev. Lett.100(11), 113901 (2008). [CrossRef] [PubMed]
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