Full dynamic corrections to the Maxwell Garnett mixing formula and corresponding extensions beyond the dipolar approximation

Abstract

Here we derive a full dynamic extension to the Maxwell Garnett mixing formula for the effective refractive index of a random system of particles that takes proper account of scattering by the particles even for moderate densities. The new formula exhibits the so-called dependent-scattering effects, which are strongest, in relative terms, in colloidal systems with small non-absorbing particles. Then we extend our result to systems of large particles having a multipolar electromagnetic response and offer a new approximation to the effective refractive index for these systems. The resulting formula can also be seen as an extension of the well-known Foldy-Lax approximation through dynamic local-field corrections. This formula reduces to all previously established approximations in the appropriate limits. We compare predictions of this formula for the imaginary part of the effective refractive index with experimental data of colloidal suspensions of non-absorbing particles with sizes comparable to and larger than the wavelength of the incident radiation and find excellent agreement. We provide some new insights into the origin of dependent scattering for systems of very small particles, such as nanofluids in the optical regime, and finally show predictions of the here-derived formulas for the effective refractive index of a system of plasmonic particles, showing redshifts of the resonance which differ from the ones predicted by other approximations.