Effective medium approximation for strongly nonlinear media

The effective medium approximation is one of the most popular approximations used for calculating the effective coefficients of linear composite media. When the same approach is applied to the case of power-law nonlinear composite media the obtained expression contains a function whose values are unknown. In order to determine the form of this function and to calculate some coefficients related to it, we calculate the electric field for the case of a single inclusion. The numerical solution is based on the relaxation method for solving differential equations, but involves some modifications due to the nonlinearity. After the solution of the differential equation the function and the coefficients are calculated and examined. The results differ considerably from those obtained earlier by simple approximations. An erratum to this article is available at .     

The Effective AC Response of Nonlinear Composites

A perturbative approach is used to study the AC response of nonlinear compostie media,which obey a current-field relation of the form J=σE χ|E|^2E with comp[onents having nonlinear response at finite frequencies.For a sinusoidal applied field,we extend the local potential in terms of sinusoidal components at fundamental frequency and high-order harmonic frequencies to treat the nonlinear comosites.For nonlinear composite media with a low concentrations of spherical inclusions,we give the formulae of the nonlinear effective AC susceptibility χ3ω at the third harmonic frequency.     

Dielectric responses of graded composites having generalized gradation profiles

Under an external uniform electric field, the dielectric response of graded cylindrical composites having generalized dielectric profile inclusions is investigated. The generalized dielectric profile of graded cylindrical inclusion is expressed in the form, $\varepsilon _i (r)=c(b+ r)^k\e^{\beta r}$ where $r$ is the radial variable of the cylindrical inclusions and $c$, $b$, $k$ and $\beta $ are parameters. The local potential solution of generalized dielectric profile graded composites is derived by means of the power series method and the effective dielectric response is predicted in the dilute limit. Moreover, from the result of generalized profile, the analytical solutions of local potentials and the effective responses of graded composites having three cases of dielectric profiles, i.e., the exponential profile $\varepsilon_\i (r)=c\e^{\beta r}$, the general power law profile $\varepsilon_\i (r)=c(b+r)^k$ and the profile $\varepsilon_\i (r)=cr^k\e^{\beta r}$, are sorted out, respectively. In the dilute limit, our exact results are used to test the validity of differential effective dipole approximation (DEDA) for estimating the effective response of graded cylindrical composites, and it is shown that the DEDA is in excellent agreement with the exact result.     

磁性颗粒膜法拉第转角的研究

在外磁场作用下,复合介质的法拉第磁光效应依赖于颗粒膜电介质张量。而复合介质的电介质张量的计算相当复杂。运用了有效介质近似理论,利用非均匀复合介质的有效电场等于单个颗粒中局域场的平均值的自恰条件,由电介质张量εe方程及自洽条件导出了计算磁性颗粒膜系统磁光法拉第转角的解析公式。并应用导出的关系,以Cu金属颗粒为例,讨论了颗粒膜中金属颗粒含量及对应的基质、离子浓度、颗粒形状对法拉第转角的影响,结果表明,利用有效介质近似理论计算的结果与实验结果一致。     

具有各向异性的梯度颗粒复合材料的有效介电响应

本文研究了具有梯度壳层椭球颗粒复合体系的有效介电响应，在稀释条件下用准静态近似方法, 推导了颗粒壳层具有任意介电梯度形式的椭球颗粒复合体系的有效介电常数和部分共振条件的一般表达式。并以壳层的介电常数为主轴的幂函数形式为例，得出了通过调节壳层的介电梯度形式、颗粒的结构和形状,可以提高该体系的有效介电常数和实现部分共振的结论。     

Effective AC and DC Responses of Nonlinear Composites at Higher Concentration

For higher concentration of inclusions, an effective medium approximation （EMA） method is used to investigate the bulk effective response of weakly nonlinear composites, which are subject to the constitutive relation of electric displacement and electric field, Dα = εαE ＋Xα｜E｜^2 E. As an example of three dimensions, under the external AC and DC electric fields Eapp = Eα（1 ＋ sinωt）, we have derived the general effective nonh＇near response of composites by the EMA method for higher concentration of spherical inclusions. Furthermore, the effective nonlinear responses at harmonics are predicted.     

梯度界面层对复合材料有效光学性质的影响

 研究了球形颗粒被无规地浸没在基质中,具有梯度分界面和光滑分界面的球形颗粒复合物界面层对介电响应和光学性质的影响;利用第一原理近似,计算了在稀释极限下具有梯度分布界面层复合物的有效介电常数,讨论了其实部和吸收光谱随外加场频率的变化关系。对结果进行比较,发现梯度分布的界面层使复合材料在等离子体共振频率处的有效介电常数的涨落明显减小,吸收谱线明显宽化并伴有“蓝移”现象,梯度界面层能很好地控制等离子体共振和减小有效介电响应的涨落。     

Effective responses of nonlinear elliptic cylindrical composites

The higher-order dielectric responses of composites composed of weakly nonlinear elliptic cylindrical inclusions randomly embedded in linear media are investigated. The general formulae for effective DC coefficients are newly derived by the average field method up to the ninth order and then applied to analyze the effects of inclusion shapes on the effective response. The results are directly transformed to the quasi-static AC response of the composites under the sinusoidal external electric field with fundamental and third harmonic frequencies. The more general relationships between the effective DC and AC coefficients are established up to the seventh order and third harmonic, which are applicable to all weakly nonlinear isotropic composites.     

Electric field distribution and effective nonlinear AC and DC responses of graded cylindrical composites

The perturbation method is used to study the localization of electric field distribution and the effective nonlinear response of graded composites under an external alternating-current (AC) and direct-current (DC) electric field E_app = E₀ (1+sin ωt). The dielectric profile of the cylindrical inclusions is modeled by function ε_i (r) = C_k r^k (r ≤ a), where r is the radius of the cylindrical inclusion, and C_k, k, a are parameters. In the dilute limit, the local potentials and the effective nonlinear responses at all harmonics are derived. Meanwhile, the general effective nonlinear responses are also derived and compared with the effective nonlinear responses at harmonics under the AC and DC external field. It is found that the effective nonlinear AC and DC responses at harmonics can be calculated by those of the general effective nonlinear of the graded composites under the external DC electric field. Moreover, the obtained local electrical fields show that the electrical field distribution in the cylindrical inclusions is controllable, and the maximum of the electric field inside the cylinder is at its center.     

弱非线性复合体中的高阶非线性响应

利用谱表示理论和微扰展开法，从理论上给出了适合于一般微结构复合体系的有效非线性响 应的一般表示式，并结合有效媒质近似（EMA），在弱非线性条件下研究了由三阶非线性组 分（体积分数为p）和线性组分构成的非线性复合体系的有效非线性响应，讨论了复合 体系的有效介电常数ε～e=εe+χe|E0|2+ηe|E0| 4中的有效三次非线性响应χe和有效高次非线性响应ηe与体积分数p和 退极化因子L之间的关系，分析了非线性组分的介电常数为复数情形时体系的有效高阶非线 性响应，从理论上说明了组分的高次非线性响应对整个复合体系的有效介电常数的影响.  关键词： 非线性复合介质 有效非线性响应 谱表示理论     

Giant enhancement of optical nonlinearity in mixtures of graded particles with dielectric anisotropy

To investigate the effective linear dielectric constant and third-order nonlinear susceptibility of composite media, in which graded inclusions with radial dielectric anisotropy are randomly embedded in a linear isotropic matrix, we develop an nonlinear anisotropic differential effective dipole approximation (NADEDA). Alternatively, based on a first-principles approach, the exact expressions for and are also derived for the linear dielectric profiles with small slopes. Then, excellent agreement between the two methods is numerically demonstrated. As an application, we further apply the NADEDA to a nonlinear metal-dielectric composite, in which the metal particles possess spatially varying radial dielectric anisotropy, in an attempt to study the nonlinearity enhancement and the figure of merit of the composite. To this end, it is shown that the presence of gradation in the radial dielectric constant plays a crucial role in enhancing the optical nonlinearity as well as the figure of merit.Received: 16 August 2003, Published online: 30 January 2004PACS: 77.22.Ej Polarization and depolarization - 42.65.-k Nonlinear optics - 42.79.Ry Gradient-index (GRIN) devices - 77.84.Lf Composite materials     

Study of the relationship between porous ratio and effective index in nanoporous film

In this paper, a theory model called 'composite media in parallel' is proposed to explain the connection between the porous ratio and effective index of the nanoporous film. This model comes from theory of composite media's effective dielectric constant in solid material field. From this model, a function relationship between the porous ratio and effective index of the nanoporous film is obtained. And the Finite Difference Time Domain (FDTD) method is used to construct the model of nanoporous film, simulate the propagation of lightwave in the film and calculate some effective indices according to different ratios of pore. Compared with Maxwell Garnett theory model and Bruggeman theory model, it is found that the function curve based on the composite media in parallel model is most consistent with the results simulated by FDTD method.     

A theory of nonlinear AC response in coated composites

A method for determining effective dielectric responses of Kerr-like coated nonlinear composites under the alternating current (AC) electric field is proposed by using perturbation approach. As an example, we have investigated the composite with coated cylindrical inclusions randomly embedded in a host under an external sinusoidal field with finite frequency ω. The local field and potential of composites in general consists of components with all harmonic frequencies. The effective nonlinear AC responses at all harmonics are induced by the coated nonlinear composites because of the nonlinear constitutive relation. Moreover, we have derived the formulae of effective nonlinear AC responses at the fundamental frequency and the third harmonic in the dilute limit.     

Modeling of the electric-field distribution of plasma photonic crystals having inhomogeneity in the materials

We study the electric-field distributions in binary 1D-PPCs in the presence of inhomogeneity in the materials of unit cells. Here we consider four 1D-PPC structures: the first two structures have linear and exponential graded dielectric materials with a homogeneous plasma, and the other two structures have homogeneous dielectric material with linear and exponential density profiles of the plasma. Matching the electric fields and their derivatives on the dielectric-plasma interface, we obtain the required field-distribution relations. We observe that the amplitude of electric field changes in the presence of inhomogeneity in the materials, and the angle of incidence of electromagnetic waves highly affects the electric-field distributions.     

Effective Response of Nonlinear Composite under External AC and DC Electric Field

A perturbation method is used to study effective response of nonlinear Kerr composites, which are subject to the constitutive relation of electric displacement and electric field, Dα=εαE＋xα｜E｜^2E. Under the external AG and DC electric field Eapp = Eα（1 ＋ sinωt）, the effective nonlinear responses and local potentials are induced by the cubic nonlinearity of Kerr materials at all harmonics. As an example in three dimensions, we have investigated this kind of nonlinear composites with spherical inclusions embedded in a host. At all harmonic frequencies, the potentials in inclusion and host regions are derived. Furthermore, the formulae of the effective linear and nonlinear responses are given in the dilute Iimit.     

颗粒复合介质在高温下的光学双稳特性

利用Maxwell-Garnett近似，结合谱表示方法，理论研究了金属/绝缘颗粒复合介质的光学双稳特性随环境温度的依赖关系.数值结果表明，体系的光学双稳区域随环境温度的升高而逐渐变窄，甚至消失.还进一步研究了复合体系的折射系数和消光系数随温度的变化关系. 关键词： 颗粒复合介质 光学双稳     

Tuneability amplification factor and loss of high-contrast composites

The problem of tuneability of nonlinear ferroelectric–dielectric composite materials is addressed. Attention is concentrated on the analysis of the tuneability amplification factor, K, of a composite material, which is introduced as the ratio of the effective tuneability of a composite material to the tuneability of its ferroelectric (tuneable) component. Previously, ferroelectric–dielectric composite materials have been designed with an effective tuneability amplification factor slightly greater than 1 (i.e. 1.1–1.4) [A.K. Tagantsev et al., J. Electroceramics 11 (2003) p.5; A.G. Kolpakov et al., J. Electroceramics 18 (2007) p.129]. It is demonstrated that the tuneability amplification factor can take values significantly greater than 1. Numerical prototypes (structural designs integrated with finite element method tools) of microstructures are presented with K in the range 3 to 30. The effective tuneability of nonlinear composite materials strongly depends on the microgeometry and microtopology of the material and, in general, cannot be described in terms of volume fraction of components of composite material. In the designs presented, the increased tuneability is due to concentration of the high electric field in narrow regions with carefully selected geometry. The problem of loss in ferroelectric–dielectric composite materials is discussed. In the general case, the loss tangent stays between that of the components of the composite. For high-contrast ferroelectric–dielectric composite materials, the effective loss tangent is practically equal to the loss tangent of the ferroelectric.     

绝缘颗粒液体主体基质复合介质的非线性光学性质

利用Maxwell-Garnett近似，并结合谱表示方法，理论研究了具有一般非线性的绝缘颗粒，无规则浸入液体主体基质复合体系的非线性光学性质-在弱非线性条件下，数值研究了体系的有效线性介电函数和光学非线性随体积分数的变化，并发现体系的线性介电函数比光学非线性更依赖于绝缘颗粒组分的体积分数-在一般非线性条件下，研究体系的有效介电函数随入射光强度的依赖关系，还研究了s方向和p方向极化的总反射率的行为- 关键词： 复合介质 谱表示 非线性光学性质     

Calculation of Dielectric Function of a Quark-Gluon Plasma

With the hard thermal loop （HTL） resummation technique, we calculate the dielectric function excited by hard gluon in quark-gluon plasma （QCP）. We find that in a section of the space-like region po/p ∈ [0.78, 0.95], there are two extremum structures on the dielectric function curve, while the dielectric function in the HTL approximation decreases monotonously without these properties. Through the analyses of the imaginary part of the dielectric function, we conclude that the character of the dielectric function in this region reflects effects of the Landau damping.     

Effect of concentration gradient on the first-, second- and third-order polarizabilities

The effect of concentration gradients on the polarizabilities of chemical systems of mixed composition is examined theoretically. It is found that the first-, second- and third-order polarizabilities all exhibit linear dependences on concentration gradient but nonlinear dependences on dielectric gradient. Furthermore, sample calculations imply that under selected conditions, concentration and/or dielectric gradient effects should be readily observable experimentally and indeed, under some conditions could dominate the experimental response. Examples of systems that might display the predicted effects include compositionally graded polymer, semiconductor, composite films, as well as thermally graded liquid mixtures and colloidal suspensions.