Nonlinear susceptibilities of granular matter

Femtosecond nearly resonant self-focusing in gold nanorod colloids

We evaluate the threshold power for self-focusing in gold nanorod colloids of varying concentration by a power limiting method in the femtosecond filamentation regime. The pulses are tuned near the longitudinal plasmon peak of the nanorods, leading to saturation of linear absorption and reshaping of the particles. We evaluated the last two effects by optical transmission measurements and spectroscopic analysis and estimated that considerable particle deformation does not occur before the collapse of the beam. We performed numerical simulations based on the experimental results, and evaluated only a subtle, monotonically increasing enhancement of the nonlinear refractive index of the host material (water) as the nanoparticles concentration increases. The role of higher-order contributions is discussed. Our work provides an alternative characterization approach of ultrafast nonlinearities in absorbing media. It further emphasizes that self-focusing of intense femtosecond pulses in gold nanocomposites is hampered by the ultrafast modulation of the susceptibility of the metal.

Retrieving nonlinear refractive index of nanocomposites using finite-difference time-domain simulations

In this Letter, a method is proposed that utilizes three-dimensional finite-difference time-domain simulations of light propagation for restoring the effective Kerr nonlinearity of nanocomposite media. In this approach, a dependence of the phase shift of the transmitted light on the input irradiance is exploited. The reconstructed values of the real parts of the nonlinear refractive index of a structure of randomly arranged spheres are in good agreement with the predictions of the effective medium approximations.

Engineered nonlinear materials using gold nanoantenna array

Gold dipole nanoantennas embedded in an organic molecular film provide strong local electromagnetic fields to enhance both the nonlinear refractive index (n₂) and two-photon absorption (2PA) of the molecules. An enhancement of 53× for 2PA and 140× for nonlinear refraction is observed for BDPAS (4,4′-bis(diphenylamino)stilbene) at 600 nm with only 3.7% of gold volume fraction. The complex value of the third-order susceptibility enhancement results in a sign change of n₂ for the effective composite material relative to the pure BDPAS film. This complex nature of the enhancement and the tunability of the nanoantenna resonance allow for engineering the effective nonlinear response of the composite film.

Effective nonlinear optical properties and optical bistability in composite media containing spherical particles with different sizes

We study the effective nonlinear optical properties of composite media in which identical nonlinear nanospheres are randomly embedded in the linear host medium. In the weakly-nonlinear case, we aim at the effective linear permittivity and effective third-order nonlinear susceptibility with effective medium theory combined with the linear Mie theory. We show that large enhancement of optical nonlinear susceptibility can be achieved at the surface plasmon resonant wavelength, which can be tuned by changing the size of nanoparticles. Our numerical results are compared with those in the quasistatic limit or/and from Comsol simulations, good agreement is found. In the strong-nonlinear case, based on nonlinear Mie theory and self-consistent mean-field method, we study the optical bistability of the composite media. The optical bistability and tristability are found, and the bistable threshold fields are found to be strongly dependent on the sizes of nanoparticles and the incident wavelength. Such nonlinear nanocomposites with large optical nonlinearity and tunable bistable behavior are envisioned for use as nonlinear optical nanodevices such as optical nanoswitches, optical nanomemories and so on.

Linear and nonlinear optical characteristics of composites containing metal nanoparticles with different sizes and shapes

We study the effective linear and nonlinear optical parameters of composites containing noble metal nanoparticles and their dependence on the shape and size of the particles. Our numerical approach is based on the effective medium approximation combined with discrete dipole approximation, which results in a fast and accurate numerical method. The results demonstrate the possibility to achieve large enhancements of the linear and nonlinear optical parameters by tuning the plasmon resonance to a desired frequency by changing the size and the shape of the nanoparticles.

Structural control of nonlinear optical absorption and refraction in dense metal nanoparticle arrays

The linear and nonlinear optical properties of a composite containing interacting spherical silver nanoparticles embedded in a dielectric host are studied as a function of interparticle separation using three dimensional frequency domain simulations. It is shown that for a fixed amount of metal, the effective third-order nonlinear susceptibility of the composite chi((3))(omega) can be significantly enhanced with respect to the linear optical properties, due to a combination of resonant surface plasmon excitation and local field redistribution. It is shown that this geometry-dependent susceptibility enhancement can lead to an improved figure of merit for nonlinear absorption. Enhancement factors for the nonlinear susceptibility of the composite are calculated, and the complex nature of the enhancement factors is discussed.

Response of Ferrogels Subjected to an AC Magnetic Field

When a ferrogel, which is chemically cross-linked polymer networks swollen with a ferrofluid, consisting of magnetic particles having nonlinear characteristics is subjected to an alternating current (ac) magnetic field, the magnetic response will generally consist of ac fields at frequencies of the higher-order harmonics. By using a perturbation approach, we investigate nonlinear ac responses of ferrogels, under an ac magnetic field either coupled with a dc magnetic field or not. It is shown that it is possible to detect the volume fraction and shape of particles in ferrogels by measuring such ac responses. Our results are very well understood in spectral representation and are favorably compared with the experimental observations of suspensions being beyond ferrogels.

Spectral representation of the effective dielectric constant of graded composites

We generalize the Bergman-Milton spectral representation, originally derived for a two-component composite, to extract the spectral density function for the effective dielectric constant of a graded composite. This work has been motivated by a recent study of the optical absorption spectrum of a graded metallic film [Huang and Yu, Appl. Phys. Lett. 85, 94 (2004)] in which a broad surface-plasmon absorption band was shown to be responsible for enhanced nonlinear optical response and an attractive figure of merit. It turns out that, unlike in the case of homogeneous constituent components, the characteristic function of a graded composite is a continuous function because of the continuous variation of the dielectric function within the constituent components. Analytical generalization to three-dimensional graded composites is discussed, and numerical calculations for multilayer composites are given as a simple application.

Magnetization of polydisperse colloidal ferrofluids: effect of magnetostriction

We exploit magnetostriction in polydisperse ferrofluids in order to generate nonlinear responses and apply a thermodynamical method to derive the desired nonlinear magnetic susceptibility. For an ideal gas, this method has been demonstrated to be in excellent agreement with a statistical method. In the presence of a sinusoidal ac magnetic field, the magnetization of the polydisperse ferrofluid contains higher-order harmonics, which can be extracted analytically by using a perturbation approach. We find that the harmonics are sensitive to the particle distribution and the degree of field-induced anisotropy of the system. In addition, we find that the magnetization is higher in the polydisperse system than in the monodisperse one, as also found by a recent Monte Carlo simulation. Thus, it seems possible to detect the size distribution in a polydisperse ferrofluid by measuring the harmonics of the magnetization under the influence of magnetostriction.

Nonlinear ac responses of electro-magnetorheological fluids

We apply a Langevin model to investigate the nonlinear ac responses of electro-magnetorheological (ERMR) fluids under the application of two crossed dc magnetic (z axis) and electric (x axis) fields and a probing ac sinusoidal magnetic field. We focus on the influence of the magnetic fields which can yield nonlinear behaviors inside the system due to the particles with a permanent magnetic dipole moment. Based on a perturbation approach, we extract the harmonics of the magnetic field and orientational magnetization analytically. To this end, we find that the harmonics are sensitive to the degree of anisotropy of the structure as well as the field frequency. Thus, it is possible to real-time-monitor the structure transformation of ERMR fluids by detecting the nonlinear ac responses.

New nonlinear dielectric materials: linear electrorheological fluids under the influence of electrostriction

The usual approach to the development of new nonlinear dielectric materials focuses on the search for materials in which the components possess an inherently large nonlinear dielectric response. In contrast, based on thermodynamics, we have presented a first-principles approach to obtain the electrostriction-induced effective third-order nonlinear susceptibility for electrorheological (ER) fluids in which the components have inherent linear, rather than nonlinear, responses. In detail, this kind of nonlinear susceptibility is in general of about the same order of magnitude as the compressibility of the linear ER fluid at constant pressure. Moreover, our approach has been demonstrated to be in excellent agreement with a different statistical method. Thus, such linear ER fluids can serve as a new nonlinear dielectric material.

Nonlinear alternating current responses of dipolar fluids

The frequency-dependent nonlinear dielectric increment of dipolar fluids in nonpolar fluids is often measured by using a stationary relaxation method in which two electric fields are used: The static direct current (dc) field of high strength causing the dielectric nonlinearity, and the probing alternating current (ac) field of low strength and high frequency. When a nonlinear composite is subjected to a sinusoidal electric field, the electric response in the composite will, in general, consist of ac fields at frequencies of higher-order harmonics. Based on the Fröhlich model, we present a theory to investigate the nonlinear ac responses of dipolar fluids containing both polarizable monomers and dimers. In the case of monomers only, our theory reproduces the known results. We obtain the fundamental, second-, and third-order harmonics of the Fröhlich field by performing a perturbation expansion. The even-order harmonics are induced by the coupling between the ac and dc fields, although the system under consideration has a cubic nonlinearity only. The harmonics of the Fröhlich field can be affected by the field frequency, temperature, dispersion strength, and the characteristic frequency of the dipolar fluid, as well as the dielectric constant of the nonpolar fluid. The results are found to be in agreement with recent experimental observations.

Nonlinear alternating current responses of graded materials

When a composite of nonlinear particles suspended in a host medium is subjected to a sinusoidal electric field, the electrical response in the composite will generally consist of alternating current (ac) fields at frequencies of higher-order harmonics. The situation becomes more interesting when the suspended particles are graded, with a spatial variation in the dielectric properties. The local electric field inside the graded particles can be calculated by the differential effective dipole approximation, which agrees very well with a first-principles approach. In this work, a nonlinear differential effective dipole approximation and a perturbation expansion method have been employed to investigate the effect of gradation on the nonlinear ac responses of these composites. The results showed that the fundamental and third-harmonic ac responses are sensitive to the dielectric-constant and/or nonlinear-susceptibility gradation profiles within the particles. Thus, by measuring the ac responses of the graded composites, it is possible to perform a real-time monitoring of the fabrication process of the gradation profiles within the graded particles.

Force between two spherical inclusions in a nonlinear host medium

In an attempt to investigate the effect of nonlinear characteristics on particle interactions, a self-consistent mean-field theory in combination with a multiple image method has been employed to compute the interparticle force. Taking the nonlinearity of the host medium into account, the interparticle force exhibits a nonmonotonic behavior as the applied electric field is increased. We show that the interparticle force increases initially at low fields, goes through a maximum at an optimal electric field, then decreases with increasing field, and vanishes at a critical electric field at which the effective dielectric contrast between the host and the inclusions becomes zero. The influence of a larger volume fraction of inclusions on the interparticle force is also investigated.

Bounds and estimates for the properties of nonlinear heterogeneous systems

A powerful and versatile variational principle, allowing the estimation of the effective properties of nonlinear heterogeneous systems, has been introduced recently by Ponte Castañeda (1992). The central idea is to express the effective energy-density function of a given nonlinear composite in terms of an optimization problem involving the effective energy-density functions of linear comparison composites with similar microstructure. This permits the computation of bounds and estimates for the effective properties of given classes of nonlinear heterogeneous systems directly from well-known bounds and estimates for the effective properties of corresponding classes of linear comparison composites. In this paper, we review the variational principle and apply it to determine bounds and estimates for the effective properties of certain classes of nonlinear composite dielectrics with homogeneous, isotropic phases. Thus, nonlinear bounds of the Hashin-Shtrikman and Beran types are obtained for composites with overall isotropy and prescribed volume fractions (of the phases). While nonlinear (second-order) bounds of the Hashin-Shtrikman type have been obtained previously, in different form, by other methods, the nonlinear (higher-order) Beran bounds are the first of their type. Finally, exact estimates are also obtained for nonlinear composites with ‘sequentially layered’ microstructures. These special composites, which have proved to be extremely useful in assessing the optimality of bounds for linear systems, are also useful, although to a lesser extent, in assessing the sharpness of the nonlinear bounds.