# The units of gain and number of atoms in population inversion in a laser

I am following my university course notes on amplification in laser media, and have come across expressions for the gain of a medium, but the notes are not exactly rigorous... The expression given for the gain $G$ in units of dB m$^{-1}$ is:

$$G = 10 \log_{10}(e^\gamma) \mathrm{dB} \mathrm{m}^{-1}$$ where $\gamma$ is also the gain coefficient for the media with units m$^{-1}$. However later in the notes it gives the expression $$\gamma = \Delta N \sigma$$ where $\Delta N$ is the "number of atoms in population inversion" and $\sigma = [\mathrm{m}]^{-2}$ is the "cross-section for stimulated emission". This clearly shows that $\Delta N = [\mathrm{m}]$, which doesn't make much sense to me. Also the first expression implies that $\gamma$ is unitless, so I am mighty confused.

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