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 [1,2]. 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] and it was later shown that such core shell particles allow one to tune the plasmonic resonance over a wide range of wavelengths [4]. This was followed by the experimental demonstration of the practicality of realizing these core shell particles and reliably controlling their resonances [5]. 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], surface-enhanced Raman optical activity [7], surface enhanced Raman scattering (SERS) [8], cloaking [911] and dielectric waveguide impedance matching [12]. Plasmonic coreshells have also found use in biomedical applications [13] including targeted delivery [14] and cancer therapy [15]. 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]. 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 [1719]. 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 [1921]. 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 [2228].

Homogenization theory has also been extended to electron waves in graphene [29]. 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 . 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 [19,21]. 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. 1 Internal homogenization compared to external homogenization.

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Fig. 2 Schematic of the core-shell internal homogenization problem.

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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].

α1=4πε0(εeε0εe+2ε0)a3α2=4πε0(a3(εc+2εs)(εsε0)+b3(εcεs)(2εs+ε0))2b3(εcεs)(εsε0)+a3(εc+2εs)(εs+2ε0)a3
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]. 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)
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]. 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)
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 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]. Figure 3(c) below also shows the fit of the effective Drude model (Eq. (3)) compared to the exact effective permittivity (Eq. (2)).

 

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.

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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

In the second case we assume that the shell is a dielectric with a constant permittivity whereas the permittivity of the core is given by following equation.

εc=εωp2ω(ω+iΓ)
Following the same strategy that was used in the previous section we can represent the effective permittivity in the following form, which is Lorentzian.
εe=ε,e+feωn,e2ωn,e2ω(ω+iΓe)
The Lorentzian parameters are given by the following equations.
ε,e=εsa3(2εs+ε)+2b3(εεs)a3(2εs+ε)b3(εεs)fe=9a3b3εs2(a3b3)[a3(2εs+ε)b3(εεs)]ωn,e=ωp(a3b3)a3(2εs+ε)b3(εεs)Γe=Γ
As opposed to the previous case, the effective permittivity in this case follows the Lorentz model instead of the Drude model. This is also consistent with the dispersion of Maxwell-Garnett EMT. When the core material is Drude with a dielectric shell the free electrons inside the Drude material encounter a natural barrier in the form of the dielectric shell. This converts the free electrons to bound electrons where the electrons are bounded by geometry instead of being bound to a nucleus. The electromagnetic response of a bound electron is given by the Lorentz model and consequently the effective permittivity follows the Lorentz model. But if the Drude material occupies the shell instead of the core the free electrons in the Drude material are not bound on the outside. This is why the permittivity is described by the Drude model when the Drude material forms the shell with a dielectric material forms the core as seen in the previous section.

Figure 4 shows the variation of the Lorentz parameters of the effective permittivity when the core is set to silver and shell is set to silica (SiO2) (εs = 2.25). The permittivity of silver and silica is the same as given in the previous section. In Fig. 4 we can see that the resonance frequency in the Lorentz model (ωn) decreases from 3 eV to 0 eV as the ratio b/a is increased from 0 to 1. A resonance frequency of 0 implies that the free electrons are no longer bound by the structure. This is consistent with the fact that when the radius ratio is set to 1 the dielectric shell vanishes and the free electrons in the Drude core are no longer bounded by the structure. We can also see that the effective Lorentz model provides a very good fit to the exact effective permittivity Eq. (1) in the example case where the radius ratio is set to 0.9.

 

Fig. 4 Oscillator strength (a), resonance frequency (b) and effective ε (c) of the core-shell as a function of the radius ratio. (d) Comparing of the exact effective permittivity with the Lorentz approximation for a silica coated silver sphere with b/a = 0.9.

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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 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.

 

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.

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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

In the previous sections we looked into the theory of internal homogenization for spherical core-shell structures with several layers. The simplicity and symmetry of the spherical core-shell structure allows us to derive closed form expressions for the effective permittivity. But this approach can also be extended to arbitrary shapes with constituent materials that may even possess anisotropic permittivity tensors even though it may not be possible to derive a closed form expression and we need to rely on numerical methods for the solution. The first step is to express the polarizability of an arbitrary particle with multiple constituents. Assuming the quasi static approximation we can write down the polarizability as follows.

α¯¯E¯0=V(ε¯¯ε0I¯¯)E¯dV
Where, E¯0 represents the incident electric field vector, E¯ represents the electric field vector inside the structure, I¯¯ denotes a unit dyadic, and ε¯¯ represents the permittivity tensor which can take different values at various locations corresponding to the different constituents. The integral is carried out over the volume of the particle that is to be homogenized although the limits of the integral are automatically enforced since (ε¯¯ε0I¯¯) is zero outside the particle. We could selectively excite individual components of E¯0one at a time and obtain the corresponding three of the components of α¯¯. For the sake of brevity of notation we can instead express the interior electric field in dyadic form (E¯¯) where we retain the information about the electric field vector excited by each component of the incident electric field. In this case the incident electric field will be expressed as a diagonal dyadic (E¯¯0). Furthermore on assuming that the incident electric field has a unit magnitude along each of the directions reduces E¯¯0to a unit dyadic. This would lead to the following compact expression for the tensor polarizability.
α¯¯=V(ε¯¯ε0I¯¯)E¯¯dV
Similarly we could define the polarizability of the effective particle as follows.
αe¯¯=V(εe¯¯ε0I¯¯)Ee¯¯dV
In both of the above expressions the dot product represents the dyadic dot product. Now we can define a cost function (CF) that depends on the effective permittivity as shown below.
CF(εe¯¯)α¯¯αe¯¯2=V[(ε¯¯ε0I¯¯)E¯¯(εe¯¯ε0I¯¯)Ee¯¯]dV2
where, ·2 denotes the L2 norm of the tensor. The effective permittivity tensor for the particle is given by the permittivity tensor εe¯¯that brings the cost function to zero. In the event that such a solution does not exist the global minimum of the cost function can provide an acceptable approximation. This is essentially an optimization problem in multiple variables since the permittivity tensor consists of multiple tensor elements with real and imaginary parts. The optimization step is necessary since the problem of finding the effective permittivity tensor is essentially a non-linear inverse problem [33] that is non-invertible in the general case. If the number of parameters in this optimization problem is small we can use any fairly simple optimization algorithm like gradient descent or Nelder-Mead simplex method whereas more complicated cases might require sophisticated optimization algorithms like genetic algorithms, particle swarm optimization, etc [34]. The calculation of the cost function itself involves calculating the electric field vector inside the particle for the given incident electric field. This calculation needs to be performed only once for the real particle and multiple times for effective particle as the optimization proceeds. The electric field can be obtained by solving the following electric potential problem which reduces to the Laplace equation in the case of a homogeneous medium.
(ε¯¯ϕ)=0
where, ϕ represents the electric potential. The electric field is given by E=ϕ, which can then be used to evaluate the cost function. The electric potential problem can be solved through various numerical methods and it can be solved very efficiently and with a low simulation time since it is a static scalar problem.

3. Loss control

In the previous section we noticed that we could tune both the plasma frequency and collision frequency of the effective sphere by using Drude materials for both core and shell. This property can be exploited to tune the loss in a spherical particle. We present an example to show how effectively this could be done. Assume that we have two Drude materials with the following parameters: material 1 has the parameters, ωp = 9 eV, Γ = 0.02 eV, ε = 1, and material 2 has the parameters ωp = 1.5 eV, Γ = 0.10 eV, ε = 1. Figure 6(a) shows the extinction efficiency (Qext) of a core-shell structure formed with material 1 as the core and material 2 as the shell. The extinction efficiency is defined as the ratio of the scattering cross section σsca to the geometrical cross section of the core shell. The extinction efficiency is plotted for four values of the ratio of the radii. As expected the resonance wavelength moves to shorter wavelengths as we increase the radius ratio. Now let us look at the Qext when b/a = 0.7. The resonance occurs at a wavelength of around 1110 nm for this core-shell. This corresponds to an effective plasma frequency of around 1.95 eV, i.e. a solid sphere made up of a Drude material with ωp = 1.95 eV will have a resonance around the same wavelength. Figure 6(b) shows the extinction efficiency for a uniform sphere with ωp = 1.95 eV, ε = 1, and various values for Γ. Figure 6(b) also shows the plot from Fig. 6(a) for b/a = 0.7 for comparison. We can see that when the Γ for the uniform sphere is above 0.15 eV the core-shell provides a stronger and narrower resonance in Qext. This could be important depending on the application. If the only materials with plasma frequency around 1.95 eV happen to possess a collision frequency above 0.15 eV it would be possible to use a core-shell with above mentioned material and radius ratio. In this case the core-shell will provide a sharper resonance compared to the uniform sphere.

 

Fig. 6 (a) Extinction efficiency for a two layered core shell particle for various radius ratios. (b) Extinction efficiency for a solid sphere with different values of Γ. Dotted line shows the extinction efficiency of the core shell particle with b/a = 0.7.

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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

This work is supported in part by the Office of Naval Research (ONR) Multidisciplinary Research Program of the University Research Initiative (MURI) under grant # N00014-10-1-0942.

References and links

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References

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  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).
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    [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. Express20(14), 16104–16112 (2012).
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  13. N. Halas, “Playing with plasmons: tuning the optical resonant properties of metallic nanoshells,” MRS Bull.30(05), 362–367 (2005).
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  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).
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2012

2011

A. Alù, “First-principles homogenization theory for periodic metamaterials,” Phys. Rev. B84(7), 075153 (2011).
[CrossRef]

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]

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]

2010

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]

C. Fietz and G. Shvets, “Current-driven metamaterial homogenization,” Physica B405(14), 2930–2934 (2010).
[CrossRef]

2009

G. P. Ortiz, B. E. Martínez-Zérega, B. S. Mendoza, and L. W. Mochán, “Effective optical response of metamaterials,” Phys. Rev. B79(24), 245132 (2009).
[CrossRef]

2008

A. Alù and N. Engheta, “Plasmonic and metamaterial cloaking: physical mechanisms and potentials,” J. Opt. A10(9), 093002 (2008).
[CrossRef]

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

2007

C. R. Simovski and S. A. Tretyakov, “Local constitutive parameters of metamaterials from an effective-medium perspective,” Phys. Rev. B75(19), 195111 (2007).
[CrossRef]

2006

2005

A. B. Evlyukhin and S. I. Bozhevolnyi, “Point-dipole approximation for surface plasmon polariton scattering: Implications and limitations,” Phys. Rev. B71(13), 134304 (2005).
[CrossRef]

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

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]

2004

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

2003

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]

1998

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]

1996

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

1992

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

1989

1985

L. W. Mochán and R. G. Barrera, “Electromagnetic response of systems with spatial fluctuations. I. general formalism,” Phys. Rev. B32, 32–36 (1985).

1972

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B6(12), 4370–4379 (1972).
[CrossRef]

1951

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

1935

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

1908

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

Acevedo, R.

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]

Aden, A. L.

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

Alù, A.

A. Alù, “First-principles homogenization theory for periodic metamaterials,” Phys. Rev. B84(7), 075153 (2011).
[CrossRef]

A. Alù and N. Engheta, “Plasmonic and metamaterial cloaking: physical mechanisms and potentials,” J. Opt. A10(9), 093002 (2008).
[CrossRef]

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

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]

Averitt, R. D.

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]

Baer, R.

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

Bardhan, R.

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]

Barrera, R. G.

L. W. Mochán and R. G. Barrera, “Electromagnetic response of systems with spatial fluctuations. I. general formalism,” Phys. Rev. B32, 32–36 (1985).

Bergman, D. J.

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

Birnboim, M. H.

Bozhevolnyi, S. I.

A. B. Evlyukhin and S. I. Bozhevolnyi, “Point-dipole approximation for surface plasmon polariton scattering: Implications and limitations,” Phys. Rev. B71(13), 134304 (2005).
[CrossRef]

Brown, L. V.

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]

Bruggeman, D. A. G.

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

Chettiar, U. K.

Christy, R. W.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B6(12), 4370–4379 (1972).
[CrossRef]

Engheta, N.

M. Silveirinha and N. Engheta, “Effective medium approach to electron waves: graphene superlattices,” Phys. Rev. B85(19), 195413 (2012).
[CrossRef]

U. K. Chettiar, R. F. Garcia, S. A. Maier, and N. Engheta, “Enhancement of radiation from dielectric waveguides using resonant plasmonic coreshells,” Opt. Express20(14), 16104–16112 (2012).
[CrossRef] [PubMed]

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

A. Alù and N. Engheta, “Plasmonic and metamaterial cloaking: physical mechanisms and potentials,” J. Opt. A10(9), 093002 (2008).
[CrossRef]

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]

Evlyukhin, A. B.

A. B. Evlyukhin and S. I. Bozhevolnyi, “Point-dipole approximation for surface plasmon polariton scattering: Implications and limitations,” Phys. Rev. B71(13), 134304 (2005).
[CrossRef]

Fietz, C.

C. Fietz and G. Shvets, “Current-driven metamaterial homogenization,” Physica B405(14), 2930–2934 (2010).
[CrossRef]

Garcia, R. F.

Halas, N.

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

Halas, N. J.

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]

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]

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]

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]

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]

Hirsch, L. R.

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]

Huschka, R.

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]

Jackson, J. B.

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]

Johnson, B. R.

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]

Johnson, P. B.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B6(12), 4370–4379 (1972).
[CrossRef]

Joshi, A.

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]

Kerker, M.

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

Knight, M. W.

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]

Lal, S.

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]

Lombardini, R.

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]

Maier, S. A.

Martínez-Zérega, B. E.

G. P. Ortiz, B. E. Martínez-Zérega, B. S. Mendoza, and L. W. Mochán, “Effective optical response of metamaterials,” Phys. Rev. B79(24), 245132 (2009).
[CrossRef]

Mendoza, B. S.

G. P. Ortiz, B. E. Martínez-Zérega, B. S. Mendoza, and L. W. Mochán, “Effective optical response of metamaterials,” Phys. Rev. B79(24), 245132 (2009).
[CrossRef]

Mie, G.

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

Mochán, L. W.

G. P. Ortiz, B. E. Martínez-Zérega, B. S. Mendoza, and L. W. Mochán, “Effective optical response of metamaterials,” Phys. Rev. B79(24), 245132 (2009).
[CrossRef]

L. W. Mochán and R. G. Barrera, “Electromagnetic response of systems with spatial fluctuations. I. general formalism,” Phys. Rev. B32, 32–36 (1985).

Neeves, A. E.

Neuhauser, D.

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

Nordlander, P.

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]

Oldenburg, S. J.

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]

Ortiz, G. P.

G. P. Ortiz, B. E. Martínez-Zérega, B. S. Mendoza, and L. W. Mochán, “Effective optical response of metamaterials,” Phys. Rev. B79(24), 245132 (2009).
[CrossRef]

Pendry, J. B.

Shalaev, V. M.

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

Shvets, G.

C. Fietz and G. Shvets, “Current-driven metamaterial homogenization,” Physica B405(14), 2930–2934 (2010).
[CrossRef]

Silveirinha, M.

M. Silveirinha and N. Engheta, “Effective medium approach to electron waves: graphene superlattices,” Phys. Rev. B85(19), 195413 (2012).
[CrossRef]

Simovski, C. R.

C. R. Simovski and S. A. Tretyakov, “Local constitutive parameters of metamaterials from an effective-medium perspective,” Phys. Rev. B75(19), 195111 (2007).
[CrossRef]

Smith, D. R.

Stroud, D.

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

Tretyakov, S. A.

C. R. Simovski and S. A. Tretyakov, “Local constitutive parameters of metamaterials from an effective-medium perspective,” Phys. Rev. B75(19), 195111 (2007).
[CrossRef]

Weiss, S.

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

West, J. L.

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]

Westcott, S. L.

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]

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]

Zuloaga, J.

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]

Acc. Chem. Res.

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]

Ann. Phys.

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

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

Appl. Phys. Lett.

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]

Chem. Phys. Lett.

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]

J. Am. Chem. Soc.

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]

J. Appl. Phys.

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

J. Opt. A

A. Alù and N. Engheta, “Plasmonic and metamaterial cloaking: physical mechanisms and potentials,” J. Opt. A10(9), 093002 (2008).
[CrossRef]

J. Opt. Soc. Am. B

J. Phys. Chem. C

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]

MRS Bull.

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

Nano Lett.

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

Opt. Express

Phys. Rep.

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

Phys. Rev. B

L. W. Mochán and R. G. Barrera, “Electromagnetic response of systems with spatial fluctuations. I. general formalism,” Phys. Rev. B32, 32–36 (1985).

A. Alù, “First-principles homogenization theory for periodic metamaterials,” Phys. Rev. B84(7), 075153 (2011).
[CrossRef]

C. R. Simovski and S. A. Tretyakov, “Local constitutive parameters of metamaterials from an effective-medium perspective,” Phys. Rev. B75(19), 195111 (2007).
[CrossRef]

G. P. Ortiz, B. E. Martínez-Zérega, B. S. Mendoza, and L. W. Mochán, “Effective optical response of metamaterials,” Phys. Rev. B79(24), 245132 (2009).
[CrossRef]

M. Silveirinha and N. Engheta, “Effective medium approach to electron waves: graphene superlattices,” Phys. Rev. B85(19), 195413 (2012).
[CrossRef]

A. B. Evlyukhin and S. I. Bozhevolnyi, “Point-dipole approximation for surface plasmon polariton scattering: Implications and limitations,” Phys. Rev. B71(13), 134304 (2005).
[CrossRef]

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B6(12), 4370–4379 (1972).
[CrossRef]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys.

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]

Phys. Rev. Lett.

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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Figures (6)

Fig. 1
Fig. 1

Internal homogenization compared to external homogenization.

Fig. 2
Fig. 2

Schematic of the core-shell internal homogenization problem.

Fig. 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.

Fig. 4
Fig. 4

Oscillator strength (a), resonance frequency (b) and effective ε (c) of the core-shell as a function of the radius ratio. (d) Comparing of the exact effective permittivity with the Lorentz approximation for a silica coated silver sphere with b/a = 0.9.

Fig. 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.

Fig. 6
Fig. 6

(a) Extinction efficiency for a two layered core shell particle for various radius ratios. (b) Extinction efficiency for a solid sphere with different values of Γ. Dotted line shows the extinction efficiency of the core shell particle with b/a = 0.7.

Equations (16)

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α 1 =4π ε 0 ( ε e ε 0 ε e +2 ε 0 ) a 3 α 2 =4π ε 0 ( a 3 ( ε c +2 ε s )( ε s ε 0 )+ b 3 ( ε c ε s )( 2 ε s + ε 0 ) ) 2 b 3 ( ε c ε s )( ε s ε 0 )+ a 3 ( ε c +2 ε s )( ε s +2 ε 0 ) a 3
ε e = ε s a 3 ( ε c +2 ε s )+2 b 3 ( ε c ε s ) a 3 ( ε c +2 ε s ) b 3 ( ε c ε s )
ε e = ε a 3 ( ε c +2 ε )+2 b 3 ( ε c ε ) a 3 ( ε c +2 ε ) b 3 ( ε c ε ) 2 ω p 2 ( a 3 b 3 ) ( 2 a 3 + b 3 )ω( ω+iΓ )
ε ,e = ε a 3 ( ε c +2 ε )+2 b 3 ( ε c ε ) a 3 ( ε c +2 ε ) b 3 ( ε c ε ) ω p,e = ω p 2( a 3 b 3 ) ( 2 a 3 + b 3 ) Γ e =Γ
ε c = ε ω p 2 ω( ω+iΓ )
ε e = ε ,e + f e ω n,e 2 ω n,e 2 ω( ω+i Γ e )
ε ,e = ε s a 3 ( 2 ε s + ε )+2 b 3 ( ε ε s ) a 3 ( 2 ε s + ε ) b 3 ( ε ε s ) f e = 9 a 3 b 3 ε s 2 ( a 3 b 3 )[ a 3 ( 2 ε s + ε ) b 3 ( ε ε s ) ] ω n,e = ω p ( a 3 b 3 ) a 3 ( 2 ε s + ε ) b 3 ( ε ε s ) Γ e =Γ
ε c = ε ,c ω p,c 2 ω( ω+i Γ c ) ε s = ε ,s ω p,s 2 ω( ω+i Γ s )
ε e = ε ,e ω p,e 2 ω( ω+i Γ e ) ε ,e = ε ,s a 3 ( ε ,c +2 ε ,s )+2 b 3 ( ε ,c ε ,s ) a 3 ( ε ,c +2 ε ,s ) b 3 ( ε ,c ε ,s ) ω p,e = ω p,s a 3 ( ω p,c 2 +2 ω p,s 2 )+2 b 3 ( ω p,c 2 ω p,s 2 ) a 3 ( ω p,c 2 +2 ω p,s 2 ) b 3 ( ω p,c 2 ω p,s 2 ) Γ e = Γ s ω p,e 2 ω p,s 2 ( a 3 ( ω p,c 2 Γ s +2 ω p,s 2 Γ c ) b 3 ( ω p,c 2 Γ s ω p,s 2 Γ c ) a 3 ( ω p,c 2 Γ s +2 ω p,s 2 Γ c )+2 b 3 ( ω p,c 2 Γ s ω p,s 2 Γ c ) )
ε 1,2 = ε 2 r 2 3 ( ε 1 +2 ε 2 )+2 r 1 3 ( ε 1 ε 2 ) r 2 3 ( ε 1 +2 ε 2 ) r 1 3 ( ε 1 ε 2 )
ε e = ε 3 r 3 3 ( ε 1,2 +2 ε 3 )+2 r 2 3 ( ε 1,2 ε 3 ) r 3 3 ( ε 1,2 +2 ε 3 ) r 2 3 ( ε 1,2 ε 3 )
α ¯ ¯ E ¯ 0 = V ( ε ¯ ¯ ε 0 I ¯ ¯ ) E ¯ dV
α ¯ ¯ = V ( ε ¯ ¯ ε 0 I ¯ ¯ ) E ¯ ¯ dV
α e ¯ ¯ = V ( ε e ¯ ¯ ε 0 I ¯ ¯ ) E e ¯ ¯ dV
CF( ε e ¯ ¯ ) α ¯ ¯ α e ¯ ¯ 2 = V [ ( ε ¯ ¯ ε 0 I ¯ ¯ ) E ¯ ¯ ( ε e ¯ ¯ ε 0 I ¯ ¯ ) E e ¯ ¯ ]dV 2
( ε ¯ ¯ ϕ )=0

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