Mode-dependent emission thresholds and frequencies in metal nanoparticles within gain-enhanced media

We present a mode-resolved analysis of emission thresholds in metal nanoparticles embedded within an infinite gain medium, using Mie scattering theory as a rigorous framework. Focusing on the first three resonant modes, we identify the onset of emission by locating the complex zeros of the electric scattering coefficient a_n(ω), which serve as indicators of lasing conditions. Our hybrid numerical scheme—combining coarse bracketing and two-dimensional Newton refinement—enables precise determination of both the threshold gain G^th _{• n} and corresponding emission frequency ω_•^th_n across varying particle radii. The results uncover strong size-dependent behavior, with higher-order modes in larger particles exhibiting significantly reduced threshold gain. These insights elucidate the modal dynamics underlying plasmonic nanolasing and provide design guidelines for active photonic systems leveraging gain-enhanced nanoparticles.