Extension of Fresnel's formulas for turbid colloidal suspensions: a rigorous treatment

We provide new expressions for the reflection amplitudes of a half space of randomly located identical spherical particles that can be regarded as an extension of Fresnel's formulas when scattering is prominent. We derive them rigorously from Maxwell's equations by solving an integral equation for the electric field within the effective-field approximation. The integral equation is given in terms of the nonlocal conductivity tensor of an isolated sphere. Approximate expressions for the reflection amplitudes are also proposed and their accuracy is analyzed, first for the case of a self-sustained suspension of silver particles, and then for the more realistic situation of silver particles in water. In this latter case the integral equation is modified by introducing the half-space Green's function dyadic instead of the one in free-space, but the method of solution is analogous in both. This extension of Fresnel's formulas, together with the numerical comparison of the different approximations proposed here, is necessary for an accurate interpretation of reflection-spectroscopy measurements in dilute colloidal suspensions of practical interest. The connection between the nonlocal conductivity tensor and the T-matrix operator of scattering theory is also made manifest.