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 Optical Society of America
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  and it was later shown that such core shell particles allow one to tune the plasmonic resonance over a wide range of wavelengths . This was followed by the experimental demonstration of the practicality of realizing these core shell particles and reliably controlling their resonances . 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 , surface-enhanced Raman optical activity , surface enhanced Raman scattering (SERS) , cloaking [9–11] and dielectric waveguide impedance matching . Plasmonic coreshells have also found use in biomedical applications  including targeted delivery  and cancer therapy . 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 . 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 [17–19]. 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 [19–21]. 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 [22–28].
Homogenization theory has also been extended to electron waves in graphene . 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 .
The polarizability of the equivalent sphere and a coated sphere with the same outer radius a is given by the following two equations under quasi static approximation .
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 . Equating the polarizability of the core-shell and the effective sphere yields the following equation for the effective permittivity.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, . 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.
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 . 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
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.
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.
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.
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.
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.
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.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 . 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.
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.
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.
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.
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