Extension of Fresnel’s Formulas for Turbid Colloidal Suspensions: A Rigorous Treatment

Abstract

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.

Localized magnetic plasmons in all-dielectricμ<0metastructures

“…Moreover, the scattering cross section (SCS) of the individual constituents at that frequency is only about three times the geometrical one (πR 2 ). Thus, one could expect that the effective medium theory can still hold, giving accurate results without further refinements [34][35][36], as we will show below. One may wonder now, what would be the optical properties of a small particle made of such effective medium?…”

“…Since the size of the particles is extremely subwavelength and the dipolar approximation holds, we expect the usual effective medium theories to give accurate results for the effective properties of the medium, at least, away from the resonance and for low-volume filling fractions, f . Incidentally, it has been recently reported that the effective optical response of colloidal particles, in general, may exhibit a nonlocal magnetic permeability [34,35]; despite that, as a first estimate, we make use of Clausius-Mossotti relations.…”

Electromagnetic field correlations in three-dimensional speckles

“…Below this length scale, the non-locality in the effective dielectric function entering the Dyson equation cannot be avoided. The use of non-local models for the effective dielectric function of scattering media is by itself a topic of current study [76].…”

Real part of refractive index measurement approach for absorbing liquid

“…The diffuse field is generated by particle scattering of the transmitted beam into the suspension, and only a little portion of it can reach photodiode 2. So, while the concept of effective RI is inadequate to describe the property of diffuse light [24], most of the energy on the detector is coherent. To measure the effective RI of turbid liquid, there needs to be a physical model between effective RI and light intensity of the coherently reflected beam.…”

“…For instance, Goyal et al have derived an empirical model of TIR from turbid liquids [25]. Gutiérrez-Reyes and co-workers have provided extension of Fresnel formulas by rigorous derivation from Maxwell equations [24,26]. These works would be helpful to describe more general systems of suspension samples.…”