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 $a_n\left(\omega \right)$, 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 $G_{•n}^{\mathrm{th}}$ and corresponding emission frequency ${\omega }_{•n}^{\mathrm{th}}$ 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.

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• Gain properties of lasers
• Plasmons
• Scattering theory
• Nano-oscillators