Extinction, absorption, and scattering of light by plasmonic spheres embedded in an absorbing host medium

Although the general Lorenz–Mie formalism for spheres in an absorbing host has been developed, no correct analytical expressions in the small-particle limit have been published so far. Here, we derive two sets of analytical expressions for the extinction, absorption, and far- and near-field scattering cross sections of small particles embedded in an absorbing host. One set is a modification of the electrostatic approximation (EA) for an absorbing host, whereas the other represents an improved electrostatic approximation (IEA) based on the generalized Lorenz–Mie theory and a new form of Mie coefficients for the internal field expansion. To illustrate the accuracy of the derived approximations, we consider Au and Ag nanospheres embedded in model hosts (real part of the refractive index, 1.33; imaginary part, 0–0.3), in a lossless poly(methyl methacrylate) (PMMA), and a lossy poly(3-hexylthiophene) (P3HT) matrix. In general, the IEA cross sections agree with those calculated using Lorenz–Mie theory if the particle diameter is not greater than 50 nm. Two small-particle limits are found for the near-field scattering cross sections. When host absorption is negligible, the scattering efficiency scales as the fourth power of the size parameter. In contrast, for nonzero absorption, the scattering efficiency scales as the first power of the size parameter. For a spectrally independent host, an increase in host absorption broadens and suppresses plasmonic peaks. We found an exception to this general tendency for near-field scattering by small (10–50 nm) particles; for these, an increase in host absorption increases the scattering peak. This surprising behavior is explained analytically. For 10–30 nm Au particles in the PMMA and P3HT matrixes, the EA and IEA data perfectly agree with the exact Lorenz–Mie simulations, in contrast to the previously reported conclusions. In particular, replacing PMMA with P3HT shifts the plasmonic peaks of the 10 nm particles from 540 nm to 650 nm and strongly enhances near- and far-field scattering. However, far-field scattering does not contribute to the extinction derived from the generalized optical theorem.