## Abstract

The Bruggeman formalism for the homogenization of particulate composite materials is used to predict the effective permittivity dyadic of a two constituent composite material with one constituent having the ability to display the Pockels effect. Scenarios wherein the constituent particles are randomly oriented, oriented spheres and oriented spheroids are numerically explored. Thereby, homogenized composite materials (HCMs) are envisaged whose constitutive parameters may be continuously varied through the application of a low-frequency (dc) electric field. The greatest degree of control over the HCM constitutive parameters is achievable when the constituents comprise oriented and highly aspherical particles and have high electro-optic coefficients.

## 1. Introduction

In 1806, after having ascended the skies in a balloon to collect samples of air at different heights and ascertained the proportions of different gases in each sample, Jean-Baptiste Biot and François Arago published the first known homogenization formula for the refractive index of a mixture of mutually inert gases as the weighted sum of their individual refractive indexes, the weights being in ratios of their volumetric proportions in the mixture. The Arago–Biot mixture formula heralded the science and technology of particulate composite materials—particularly in optics, more generally in electromagnetics and even more generally in many other branches of physics. An intensive literature has developed over the last two centuries in optics (Lakhtakia 1996), and recent forays into the realms of metamaterials and complex mediums (Grimmeiss *et al*. 2002; Weiglhofer & Lakhtakia 2003; Mackay 2005) have reaffirmed the continued attraction of both particulate composite materials and homogenization formalisms.

Post-fabrication dynamic control of the effective properties of a mixture of two constituent materials is a technologically important capability underlying the successful deployment of a host of smart materials and structures. The dynamic control can be achieved in many ways, particularly if controllability and sensing capability are viewed as complementary attributes. One way is to infiltrate the composite material with another substance, possibly a fluid, to change, say, the effective optical response properties (Lakhtakia *et al*. 2001; Mönch *et al*. 2006). This can be adequate if rapidity of change is not a critical requirement. Another way is to tune the effective properties by the application of pressure (Finkelmann *et al*. 2001; Wang *et al*. 2003) or change of temperature (Schadt & Fünfschilling 1990). Faster ways of dynamic control could involve the use of electric fields if one constituent material is a liquid crystal (Yu *et al*. 2005) or magnetic fields if one constituent material is magnetic (Shafarman *et al*. 1986).

In this paper, our focus is the control of the effective permittivity tensor of a two-constituent composite material, wherein both the constituent materials are classified as dielectric materials in the optical regime, but only one can display the Pockels effect (Boyd 1992). Both the constituent materials can be distributed as ellipsoidal particles whose orientations can be either fixed or completely random. Section 2 contains a description of the particulate composite material of interest, as well as the Bruggeman homogenization formalism (Weiglhofer *et al*. 1997) adopted to estimate the relative permittivity dyadic of the homogenized composite material (HCM). Section 3 presents a few numerical examples to show that the Pockels effect can be exploited to dynamically control the linear optical response properties of composite materials through a low-frequency electric field. Given the vast parameter space underlying the Pockels effect, we emphasize that the examples presented are merely illustrative. Vectors are in boldface and dyadics are double underlined. A Cartesian coordinate system with unit vectors *u*_{x,y,z} is adopted. The identity dyadic is written as , and the null dyadic as . An exp(−i*ωt*) time dependence is implicit with , *ω* as angular frequency and *t* as time.

## 2. Theory

Let the two constituent materials of the particulate composite material be labelled *a* and *b*. Their respective volumetric proportions are denoted by *f*_{a} and *f*_{b}=1−*f*_{a}. They are distributed as ellipsoidal particles. The dyadic describes the shape of particles made of material *a*, with *α*_{K}>0 ∀ *K*∈[1,3] and the three unit vectors *a*_{1,2,3} being mutually orthogonal. The shape dyadic similarly describes the shape of the particles made of material *b*. A low-frequency (or dc) electric field *E*^{dc} acts on the composite material, the prediction of whose effective permittivity dyadic in the optical regime is of interest.

Material *a* does not display the Pockels effect and, for simplicity, we take it to be isotropic with relative permittivity scalar *ϵ*^{(a)} in the optical regime.

Material *b* has more complicated dielectric properties as it displays the Pockels effect. Its relative permittivity dyadic in the optical regime is written in the linear approximation as (Boyd 1992; Lakhtakia 2006*a*)(2.1)provided that(2.2)Here,(2.3)and the unit vectors(2.4)are relevant to the crystallographic structure of the material; , the Cartesian components of the dc electric field; , the principal relative permittivity scalars in the optical regime when the dc electric field is absent; and *r*_{JK}, the 18 electro-optic coefficients. Material *b* can be isotropic, uniaxial or biaxial, depending on the relative values of . Furthermore, material *b* may belong to 1 out of 20 crystallographic classes of point group symmetry, in accordance with the relative values of the electro-optic coefficients.

Let the Bruggeman estimate of the relative permittivity dyadic of HCM be denoted by . If the particles of material *a* are all identically oriented with respect to their crystallographic axes, and likewise the particles of material *b*, then is determined by solving the following equation (Weiglhofer *et al*. 1997):(2.5)In this equation, are functions of and . Thus, has to be extracted from equation (2.5) iteratively using standard numerical techniques (Michel 2000).

The Bruggeman formalism is more complicated when the relative permittivity dyadics of the particles of material *b* are randomly oriented and requires statistical averaging (Lakhtakia 1993).

## 3. Numerical results and discussion

A vast parameter space is covered by the homogenization formalism described in the previous section. The parameters include the volumetric proportions and the shape dyadics of materials *a* and *b*; the relative permittivity scalar *ϵ*^{(a)}; the three relative permittivity scalars and the up to 18 distinct electro-optic coefficients *r*_{JK} of material *b*; the angles *θ*_{b} and *ϕ*_{b} that describe the crystallographic orientation of material *b* with respect to the laboratory coordinate system; and the magnitude and direction of *E*^{dc}. To provide illustrative results here, we set *ϵ*^{(a)}=1. All the calculations were made for two choices of material *b* (Cook 1996):

zinc telluride, which belongs to the cubic crystallographic class: ,

*r*_{41}=*r*_{52}=*r*_{63}=4.04×10^{−12}m V^{−1}and all other*r*_{JK}≡0 andpotassium niobate, which belongs to the orthorhombic

*mm*2 crystallographic class: , , ,*r*_{13}=34×10^{−12}m V^{−1},*r*_{23}=6×10^{−12}m V^{−1},*r*_{33}=63.4×10^{−12}m V^{−1},*r*_{42}=450×10^{−12}m V^{−1},*r*_{51}=120×10^{−12}m V^{−1}and all other*r*_{JK}≡0.

Given the huge parameter space still left, we chose to fix *f*_{b}=0.5, the Bruggeman formalism then being maximally distinguished from the Maxwell Garnett (Weiglhofer *et al*. 1997) and the Bragg–Pippard formalisms (Sherwin & Lakhtakia 2002). Particles of material *a* and *b* were chosen to be either spherical (i.e. ) or spheroidal. Finally, two different scenarios based on the orientation of material *b* were investigated: (i) particles of material *b* are randomly oriented with respect to their crystallographic axes and (ii) all particles of material *b* are identically oriented.

The estimated permittivity dyadic of the HCM may be compactly represented as(3.1)wherein the unit vectors *u*_{M} and *u*_{N} are aligned with the optic ray axes of the HCM (Mackay & Weiglhofer 2001). For the real-symmetric relative permittivity dyadic , with three distinct (and orthonormalized) eigenvectors *e*_{1,2,3} and corresponding eigenvalues , the scalars and and the unit vectors *u*_{M,N} may be stated as(3.2)

In accordance with mineralogical literature (Klein & Hurlbut 1985; Gribble & Hall 1992), let us define the linear birefringence(3.3)the degree of biaxiality(3.4)and the angles(3.5)The linear birefringence *δ*_{n} is the difference between the largest and the smallest refractive indexes of the HCM; the degree of biaxiality *δ*_{bi} can be either positive or negative, depending on the numerical value of with respect to the mean of and ; 2*δ* is the angle between the two optic ray axes; and *θ*_{M,N} are the angles between the optic ray axes and the Cartesian *z*-axis. Thus, can be specified by six real-valued parameters: , *δ*_{n}, *δ*_{bi}, *δ* and *δ*_{M,N}, in a physically illuminating way.

### (a) Randomly oriented spherical electro-optic particles

When the particles of material *b* are randomly oriented with respect to their crystallographic axes, and the particles of both materials *a* and *b* are spherical, the HCM is an isotropic dielectric medium, i.e. . The estimate *ϵ*^{Br} is plotted in figure 1 against , with . Material *b* is zinc telluride for the dashed curve and potassium niobate for the solid curve in this figure. The range for the magnitude of *E*^{dc} in figure 1—and for all subsequent figures—was chosen in order to comply with equation (2.2). When material *b* is zinc telluride, *ϵ*^{Br} varies only slightly as changes and *ϵ*^{Br} is insensitive to the sign of *ϵ*^{Br}. A greater degree of sensitivity to is observed for the HCM which arises when material *b* is potassium niobate; then, *ϵ*^{Br} is also sensitive to the sign of , thereby underscoring the significance of crystallographic class of the electro-optic constituent material even when crystallographic orientational averaging is physically valid.

### (b) Identically oriented spherical electro-optic particles

Next, let all the particles of material *b* be identically oriented, with particles of both constituent materials being spherical. In order to identify the role of the Pockels effect clearly, let material *b* be potassium niobate, which has higher electro-optic coefficients than zinc telluride. Furthermore, unlike zinc telluride, which is isotropic in the absence of the dc electric field (Lakhtakia 2006*b*), potassium niobate is anisotropic (orthorhombic and negatively biaxial) even when the Pockels effect is not invoked.

The HCM parameters are plotted in figure 2 as functions of with . The crystallographic orientation angles of material *b* are taken as *θ*_{b}=*ϕ*_{b}=0°. Whereas the parameters , *δ*_{n}, *δ*_{bi} and *δ* are not particularly sensitive to , they do vary significantly as varies. Most notably, the HCM can be made either negatively (*δ*_{bi}<0) or positively biaxial (*δ*_{bi}>0). The two optic axes of the HCM need not be mutually orthogonal, with the included angle 2*δ* between them as low as 40°. The polar angles *θ*_{M,N} are sensitive to both and . The sign of does not influence any of the HCM parameters, but the sign of does influence the polar angles *θ*_{M,N}.

The influence of the orientation angle *θ*_{b} is explored in figure 3. The constitutive parameters of material *b* are the same as in figure 2 but with *θ*_{b}∈{45°, 90°}. The dependencies of the polar angles *θ*_{M,N} upon the components of *E*^{dc} are acutely sensitive to *θ*_{b}. The HCM parameters , *δ*_{n}, *δ*_{bi} and *δ*, which are not presented in figure 3, are insensitive to increasing *θ*_{b}; the plots for these quantities are not noticeably different from the corresponding plots presented in figure 2.

A comparison of results with those for zinc telluride (not shown) indicates that the application of *E*^{dc} is more effective when material *b* is potassium niobate rather than zinc telluride. A dc electric field that is two orders smaller in magnitude is required for changing the HCM properties with potassium niobate than with zinc telluride, and this observation is reaffirmed by comparing the upper and the lower graphs in figure 1. To a great extent, this is due to the larger electro-optic coefficients of potassium niobate; however, we cannot rule out some effect of the crystallographic structure of material *b*, which we plan to explore in the near future.

Electrical control appears to require dc electric fields of high magnitude. However, the needed dc voltages are comparable with the half-wave voltages of electro-optic materials (Yariv & Yeh 2007). We must also note that the required magnitudes of *E*^{dc} are much smaller than the characteristic atomic electric field strength (Boyd 1992). The possibility of electric breakdown exists, but it would significantly depend on the time that the dc voltage would be switched on for. Finally, the non-electro-optic constituent material may have to be a polymer that can withstand high dc electric fields.

### (c) Identically oriented spheroidal electro-optic particles

We close by considering the scenario wherein the influence of the Pockels effect is going to be highly noticeable in the HCM, i.e. when the particles of material *b* are highly aspherical and the crystallographic as well as the geometric orientations of these particles are aligned with *E*^{dc}. We chose potassium niobate—which is more sensitive to the application of *E*^{dc} than zinc telluride—for our illustrative results.

In figure 4, the HCM parameters , *δ*_{n}, *δ*_{bi}, *δ* and *θ*_{M} are plotted against , with . Both constituent materials are distributed as identical spheroids with shape parameters *α*_{1,2}=*β*_{1,2}=1 and *α*_{3}=*β*_{3}∈{3,6,9}; furthermore, *θ*_{b}=*ϕ*_{b}=0°. As *θ*_{N}=180°−*θ*_{M} for this scenario, *θ*_{N} is not plotted. All the presented HCM parameters vary considerably as increases; furthermore, all are sensitive to the sign of .

The degree of biaxiality and the linear birefringence increase in figure 4 as *α*_{3}=*β*_{3} increases. This is a significant conclusion because perovskites (such as potassium niobate) are nowadays being deposited as oriented nanopillars (Gruverman & Kholkin 2006).

To conclude, the homogenization of particulate composite materials with constituent materials that can exhibit the Pockels effect gives rise to HCMs, whose effective constitutive parameters may be continuously varied through the application of a low-frequency (dc) electric field. Observable effects can be achieved even when the constituent particles are randomly oriented. Greater control over the HCM constitutive parameters may be achieved by orienting the constituent particles. By homogenizing constituent materials which comprise oriented elongated particles rather than oriented spherical particles, the degree of electrical control over the HCM constitutive parameters is further increased. The vast panoply of complex materials presently being investigated (Grimmeiss *et al*. 2002; Weiglhofer & Lakhtakia 2003; Mackay 2005) underscores the importance of electrically controlled composite materials for a host of applications for telecommunications, sensing and actuation.

## Footnotes

- Received July 29, 2006.
- Accepted September 28, 2006.

- © 2006 The Royal Society