## 1. Introduction

Metamaterials with extraordinary optical properties can find a variety of applications, in particular for manufacturing superlenses, prisms, nonreflecting (absorptive) materials, as well as controlling the optical beam intensity and propagation direction, etc. (Cai & Shalaev, 2010; Hutter & Fendler, 2004). In recent years, a large number of structures have been proposed and theoretically and experimentally investigated, which possess negative, high or small refractive indices (Moiseev et al., 2007; Oraevskii & Protsenko, 2000; Shalaev et al., 2005; Shen et al., 2005; Sukhov, 2005; Yuan et al., 2007), selective absorption of or transparency to optical light (Kachan et al., 2006; Kravets et al., 2008; Protsenko et al.,2007).

The promising candidates for a design of structures with extraordinary optical properties are plasmonic materials. Collective electronic excitations called plasmons lead to strong dispersion and absorption of light in such artificial media. For practical applications, it is important that the optical properties of composite medium incorporating metallic nanoparticles can be controlled by careful selection of geometric parameters of nanoparticles (Moiseev, 2004, 2009; Oraevskii & Protsenko, 2000). By selecting distributions of the form of metallic nanoparticles, it is possible to achieve absorption of electromagnetic radiation in specified spectral regions of visible or near IR radiation (Protsenko et al., 2007). An additional effective way of the plasmon resonance control is by spatial arrangement of metal-dielectric nanocomposites, for example, forming the multilayer systems consisting of metallic nanoparticles separated by dielectric layers (Kachan et al., 2006). It was proposed that such composite materials with controlled absorption of light can be used as cut-of filters and high-absorbing coatings.

In this work, a more detailed investigation of optical properties of a matrix metal-dielectric medium with spheroidal silver inclusions is performed, and the possibility to realizeplasmonicstructures with beneficial effectsin the visible region is considered. In so doing, it is assumed that the characteristic size of inclusions and their volume concentration satisfy the conditions of applicability of the effective medium Maxwell–Garnett model.

## 2.Mainformulae

### 2.1. Size-dependent dielectric function of silver nanoparticles

From the optics of metal nanosized particles it is well known that their optical properties significantly depend not only on their chemical composition but also on their size, so the permittivity of metal nanoparticles differs from the bulk permittivity of the medium (Khlebtsov, 2008). Throughout this communication, silver nanoparticles are considered (they are chosen becauseof their lowest absorption in the optical range). In order to estimate variation in the dielectric permittivity *ε*of a metal nanoparticle in comparison with that of a bulk sample, let us use a classical model taking into account the limitation in the electron free path length due to its collision with the particle boundary (Kreibig & Vollmer, 2004). According to this model, the finite size of a metal particle leads to a change in the relaxation rate of conduction band electrons:

where *r* is the characteristic particle radiusalong the electric field direction of external electromagnetic wave. Then, the size-dependent permittivity of the nanoparticle has the form (Yannopapas et al., 2002):

where*
* is the frequency of the incident field, _{p} is the plasma frequency, ε_{b} is the experimentally determined permittivity for a bulk sample. The tabular data for bulk silver are

Calculations show that the correction taking into account the finite size of the particle leads mainly to a variation of the imaginary part of the permittivity ε(see Fig. 1),while the real

part of ε changes slightly for the particle with radius as large as 5 nm. For instance, for a wavelength of external radiation

### 2.2. Maxwell-Garnett approach

Let the light wavelength significantly exceed the size of particles suspended in a dielectric matrix.If the interference effects on the nanoparticles are ruled out (the spatial positions of the nanoparticles don’t have to be periodic and can be random) and their volume fraction is as small as 1/3, it is considered that Maxwell-Garnett approach can be used successfully to analyse the optical properties of such composite medium (Golovanet al., 2007; Moiseev et al., 2007; Moiseev, 2009, 2010). The Maxwell – Garnett relation assumes a generalization for the case of shape anisotropy of the inclusion (particles), thus providing a possibility of analyzing optical characteristics of a composite medium with non-spherical inclusions based on analytical expressions.

For composite medium with uniformly oriented metallic spheroids in a dielectric matrix, the effective permittivity ε_{eff} according to Maxwell-Garnett approach is determined by the following expression (Golovan et al., 2007; Maxwell-Garnett,1904, 1906):

whereε_{m} is the matrix permittivity,ε is the permittivity of inclusions (Eq. (2)), η is the volume concentration of the nanoparticles (filling factor), Lis the geometrical factor (factor of depolarization) that accounts for the shape of a particle (Bohren&Huffman, 1998).In the long-wavelength limit, factor L is a real value, that depends on the ratio ξ of the length of polar semi-axis a and equatorial semi-axis b of spheroid,

for the field directed along the axis of revolution of spheroid and as

for the field directed perpendicular to this axis. The case where

By a comparison with exact electrodynamical calculation, it wasshown in (Moiseev et al., 2007; Moiseev, 2009, 2010)that for matrices with a moderatecontent of inclusions (0.01<η<0.3) resultsobtained withinthe Maxwell-Garnettmodel are in fairlygood agreement with the results of exact electrodynamiccalculation.

## 3.Spectral characteristics of Maxwell-Garnett composite with plasmon resonances

Eq. (3) is written for a composite medium with uniformly oriented spheroids, where the field vector is directed either along or perpendicular to the ellipsoid axis. For non-spherical particles, depolarization factors (4), (5) and dielectric permittivity (2) depend on the electric field direction.Therefore in the general case we dealwith an anisotropic composite,which possesses properties of uniaxial crystalhaving an optical axis collinear with the polar axis aofspheroids. A light ray in this medium is ordinary if itspropagation direction is collinear with the polar axis of spheroids; in this case, the light polarization vectorlies in the equatorial plane of spheroids.

In the case of a spheroid and an applied electric field oriented along a spheroid axis, the static polarizability of nanoparticle is

whereVis the particle volume. As follows from Eq. (6), the plasmon resonance frequency depends on the form of the particle. If a transparent dielectric is used as surrounding host medium, the resonance wavelength λ_{res} of metal spheroid can be estimated by using simple algebraic equation

In the case of spheroidal (

Plasmon resonances of nanoparticles give rise to‘extraordinary’values oftheeffectivepermittivity

and the effective extinction coefficient

obtained from the Maxwell–Garnett relation (3). As a matrix, we chose a non-absorbing medium with the refractive index

## 4. Applications of plasmonic heterogeneous materials

### 4.1. Heterogeneous composite with the unite refractive index

Let us have a closer look at the conditions where a unit effective refractive index (

The conditions where an effective refractive index is observed could be derived analytically directly from the Maxwell – Garnett relation (3). For a fixed refractive index

The dependences plotted in Figs. 3 and 4 imply that the conditions

The situation is different when inclusions are nonspherical (

The dependences presented in Fig. 4 also demonstrate that

### 4.2. TransparentMaxwell-Garnett composite with the unite refractive index

It follows from the Maxwell-Garnett formula (3) that because the permittivity of the metal particles is complex, condition of optical transparency

cannot be satisfied exactly for the matrix with real ε_{m}. Let us try to compensate the effect of the imaginary part of the metal permittivity on the optical properties of the entire composite medium by choosing an amplifying medium as the matrix. For this purpose, we will simulate the optical parameters of the amplifying medium by adding an imaginary part to the permittivity (Moiseev et. al., 2007):

where g>0 is the gain (extinction coefficient), which is equal to the gain coefficient multiplied by

Consider the dependence of the gain g, which is required for compensating the absorption of external electromagnetic radiation, on geometric and material parameters of composite medium. When the gain of the matrix is much smaller than its refractive index, from equation (3) one can obtain

Unlike other metals, the permittivity of silver has comparatively small imaginary part. Thus, having taken silver inclusions we result in decreasing in the required gain. Decrease in the volume concentration of the nanoparticles contributes to decrease in the gain g, although this way is not worth considering because

From equation (12) it follows that g depends on the factor of depolarization L. The variation of L leads to the shift of the plasmon resonance, and correspondingly, the wavelength λ, on which the unique index of refraction is observed. Thus, analysis of the transparency conditions is to be taken basing on predefined value of n_{eff}. Here we consider the case

The conditions of invisibility of Maxwell-Garnett composite with inclusions of different shapes are presented in Fig. 5. Horizontal axis shows the region of wavelengths, on which the unit

index of refraction can be obtained. The dependence obtained allows defining the filling factor, whereby the gain required for the transparency of composite medium is minimal. Contrasting parts (a) and (b) of Fig. 5, the following conclusions can be made. Firstly,composite media with nonspherical nanoparticles require less value of gain of the active component. It can be explained in the following way. In spite of volume of particle being equal the characteristic diameter of the needle-like particle in the direction of the field is bigger, and consequently, the imaginary part of the permittivity ε is less than that of spherical nanoparticles. Therefore in the case (b) the values of the gain necessary for compensation of absorption are smaller. Secondly, the condition

### 4.3. Heterogeneous medium for anti-reflection coating application

Optical properties of heterogeneous metal-dielectric composites can be efficiently tailored by nanoparticle sizes, shapes, and concentration.As it is shown in Section 4.1, for nonspherical inclusions theeffective coefficient of composite extinction is severaltimes smaller than for spherical ones. Let us determinethe values of the structural parameters at whicha composite slab with nonspherical inclusions can be usedas an interference antireflection coating in the visiblespectral range.

The use of the effective-medium model significantlysimplifies the study of the dispersion characteristicsof composite coating: disregarding the discrete-continuousstructure of the composite, one can calculatethe reflectance and transmittance using theAiry formulas (Born& Wolf, 1999). According to these formulas, fornormal incidence of light the reflection from dielectricis completely suppressed by a layer of material witha complex refractive index

where

The anisotropy of optical properties of orderedcomposite imposes certain limitations on the antireflectioncoating design. To make the compositecoating reflectance and transmittance independent ofthe light vector orientation for normal incidence oflight, the optical axis of the composite must be perpendicularto the interfaces, i.e., the equatorial planeof spheroids must be parallel to the plane of compositeplate. With allowance for this circumstance, we willconsider the optical properties of the composite onlyfor an ordinary ray.For definiteness, we consider a transparent dielectricwith a refractive index

Figure 6 shows the spectral dependencies of theeffective refractive index and extinction coefficient ofthe composite, calculated from (3). The peaks inthe curve

Thus, the antireflection coating design is determinedby the features of composite optical properties.According to the results obtained, the compositecoating for natural light must be formed by nanoparticlesin the form of oblate spheroids, whose polar axisis oriented perpendicular to the surface of underlyingmedium.

The calculation showsthat a composite with a moderate volume fraction ofinclusions(η~0.01−0.1) satisfies all necessary conditionsfor the visible spectral range. Figure 7 showsthe reflectance and transmittance of the antireflectioncomposite layer with the following parameters:ξ =0.1 and η =0.05. The layer thickness, calculatedfrom formula (14), is h = 93 nm.

The dependencies in Fig. 7 show that coating of dielectric surface by a composite layer gives a positive effect. In a wide (>100 nm) spectral range the total intensity of reflected light decreases by a factor of more than two, and the minimum reflectance of the composite coating is lower than that of the dielectric by a factor of 20. Unfortunately, the refracted wave intensity increases only slightly in this case, and even decreases in comparison with the initial dielectric at λ > 500 nm. The latter circumstance can be explained as follows: some part of the light wave energy spent on excitation of free-electron oscillations in composite nanoparticles is transformed into heat. Thus, the light wave energy is partially absorbed by the dispersed subsystem of the composite coating, as a result of which the surface of transparent material cannot be made totally antireflective.

### 4.4. Thin-film composite polarizing splitter

As it is shown in Section 3, the absorption of the composite medium incorporating metallic nanoparticles can be controlled by careful selection of geometric parameters of nanoparticles. Here we investigate the possibility to realizea thin-film plasmonicpolarizing splitter in the visible region. For definiteness, we consider a transparent dielectricwith a refractive index n_{m} = 1.5 as amatrix. In our design, the high polarization contrast of composite film is obtained by using uniformly oriented silver nanoparticles of ellipsoidal shape.

The shape of nanoparticles can be selected so as to observe only one plasmon resonance in the visible region of light. According to Fig. 2, this condition complies with

Let us assume that a layer of the composite material with uniformly oriented prolate silver spheroids has been deposited on surface of transparent medium. Let light be incident normally on this layer from vacuum. To provide maximum optical anisotropy, nanoparticles have to be oriented parallel to the plane interface between media.Using the relation (3) and Airy equation(Born& Wolf, 1999), the reflectance and transmittance can be calculated for arbitrary values of filling factor η and layer thickness h.

Results of calculation show that for 600 nm < λ< 650 nm, values of η ~ 0.1 and

## 5. Conclusion

We explore the utility of effective medium representation to simplify the electromagnetic analysis of composite system, and demonstrate the use of this simplification in solving of the boundary problem under consideration. This approach allows us to easily control the parameters of a system and predictably change its optical properties, expressing the necessary conditions in an analytical form. The data obtained with the help of formula (3) show that a heterogeneous medium with plasmonic impurities, such as silver nanoparticles with a concentration of 10^{-2}–10^{-1} per unit volume, is new and interesting object of research with many perspectives for applications. Such plasmonicmediumcan be used as a transparent anti-reflection coating, weakly reflecting light-absorbing filter,or polarizing beam splitter with high performance in transmission and reflection.It is shown also that an extraordinary refractive index