Mixing Rule for Calculating the Effective Refractive Index Beyond the Limit of Small Particles

TL;DR

This paper develops an improved mixing rule for calculating the effective refractive index in disordered media with large particles, surpassing traditional models like Maxwell-Garnett that assume small particles.

Contribution

It introduces a quadratic polynomial mixing rule with empirically derived coefficients, extending effective medium theory to larger particles beyond the small-particle limit.

Findings

Maxwell-Garnett theory fails for large particles.

The new quadratic mixing rule outperforms existing models.

The rule is validated through FDTD simulations.

Abstract

Considering light transport in disordered media, the medium is often treated as an effective medium requiring accurate evaluation of an effective refractive index. Because of its simplicity, the Maxwell-Garnett (MG) mixing rule is widely used, although its restriction to particles much smaller than the wavelength is rarely satisfied. Using 3D finite-difference time-domain simulations, we show that the MG theory indeed fails for large particles. Systematic investigation of size effects reveals that the effective refractive index can be instead approximated by a quadratic polynomial whose coefficients are given by an empirical formula. Hence, a simple mixing rule is derived which clearly outperforms established mixing rules for composite media containing large particles, a common condition in natural disordered media.