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 ${a}_{n}(\ensuremath{\omega})$, 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 ${G}_{\ifmmode\bullet\else\textbullet\fi{}\phantom{\rule{0.16em}{0ex}}n}^{\mathrm{th}}$ and corresponding emission frequency ${\ensuremath{\omega}}_{\ifmmode\bullet\else\textbullet\fi{}\phantom{\rule{0.16em}{0ex}}n}^{\mathrm{th}}$ 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.