Numerical studies of the nonlinear properties of composites

Using both numerical and analytical techniques, we investigate various ways to enhance the cubic nonlinear susceptibility ${\mathrm{\chi }}_{\mathit{e}}$ of a composite material. We start from the exact relation ${\mathrm{\chi }}_{\mathit{e}}$ =${\mathit{tsum}}_{\mathit{i}}$ ${\mathit{p}}_{\mathit{i}}$ ${\mathrm{\chi }}_{\mathit{i}}$〈(E⋅E ${)}^2$ ${\mathrm{〉}}_{\mathit{i},\mathrm{l}\mathrm{i}\mathrm{n}}$/ ${\mathit{E}}_0^4$, where ${\mathrm{\chi }}_{\mathit{i}}$ and ${\mathit{p}}_{\mathit{i}}$ are the cubic nonlinear susceptibility and volume fraction of the ith component, ${\mathit{E}}_0$ is the applied electric field, and 〈${\mathit{E}}^4$ ${\mathrm{〉}}_{\mathit{i},\mathrm{l}\mathrm{i}\mathrm{n}}$ is the expectation value of the electric field in the ith component, calculated in the linear limit where ${\mathrm{\chi }}_{\mathit{i}}$=0. In our numerical work, we represent the composite by a random resistor or impedance network, calculating the electric-field distributions by a generalized transfer-matrix algorithm. Under certain conditions, we find that ${\mathrm{\chi }}_{\mathit{e}}$ is greatly enhanced near the percolation threshold. We also find a large enhancement for a linear fractal in a nonlinear host. In a random Drude metal-insulator composite ${\mathrm{\chi }}_{\mathit{e}}$ is hugely enhanced especially near frequencies which correspond to the surface-plasmon resonance spectrum of the composite. At zero frequency, the random composite results are reasonably well described by a nonlinear effective-medium approximation. The finite-frequency enhancement shows very strong reproducible structure which is nearly undetectable in the linear response of the composite, and which may possibly be described by a generalized nonlinear effective-medium approximation. The fractal results agree qualitatively with a nonlinear differential effective-medium approximation. Finally, we consider a suspension of coated spheres embedded in a host. If the coating is nonlinear, we show that ${\mathrm{\chi }}_{\mathit{e}}$/${\mathrm{\chi }}_{\mathrm{coat}}$≫1 near the surface-plasmon resonance frequency of the core particle.