Effective Medium Theory for Heat Generation Using Plasmonics: A Parabolic Transmission Problem Driven by the Maxwell System

Abstract: The excitation of plasmonic nanoparticles by incident electromagnetic waves at frequencies near their subwavelength resonances induces localized heat generation in the surrounding medium. We develop a mathematical framework to rigorously quantify this heat generation in systems of arbitrarily distributed nanoparticles. 1. For an arbitrary discrete distribution of M nanoparticles within a bounded domain, the effective heat distribution is described by a coupled system: Volterra-type integral equations for the heat conduction and a Foldy-Lax-type system governing the self consistent electric field intensities. These equations are parameterized by the particle geometries and the local electromagnetic field interactions. The effective heat generation is computed by solving these coupled systems, with the computational complexity scaling as M^2. 2. In the case M >> 1, under natural scaling regimes, the discrete system converges to a continuum model, yielding an effective parabolic equation for the heat distribution. The source term in this homogenized parabolic model is characterized by the solution of the homogenized Maxwells equations, incorporating an effective permittivity distribution derived from the Drude model under resonance conditions. Our analysis utilizes advanced tools in potential theory, asymptotic analysis and homogenization. By leveraging layer potential representations, we derive point-wise field approximations. The coupling between the Maxwell and heat equations is resolved by analyzing the spectral properties of the nanoparticles and their scaling limits. This framework reduces the problem to two mathematical challenges: a control problem for the effective parabolic system and an internal phase-less inverse problem for the Maxwell system, thus providing a unified approach to modeling heat generation in nanoparticle clusters.