<p>We consider a homogeneous medium of atoms exhibiting a two-level system $g$ and $e$ separated by an energy $\hbar \omega$. A laser source of surface intensity $I$ is directed along the longitudinal axis of the medium. We denote by $W$ the stimulated emission rate from state $e$ to $g$, which is assumed to be equal to the absorption rate. This emission rate is related to the intensity of the incident laser source and the absorption cross-section $\sigma$,</p>
$$
W = \frac{\sigma I}{\hbar \omega}
$$<p>We note $\Gamma_e$ the spontaneous emission rate from $e$ to $g$.</p>
<p>In this elementary volume, the medium being homogeneous, there are $N_e \frac{S \mathrm{d}z}{V}$ atoms in the excited state and $N_g \frac{S \mathrm{d}z}{V}$ in the ground state.</p>
<p>By performing an energy balance</p>
$$
\underbrace{I(z+\mathrm{d}z) S \mathrm{d}t}_{\rm outgoing \: energy} = \underbrace{I(z) S \mathrm{d}t}_{\rm incoming \: energy} - \underbrace{\hbar \omega N_g \frac{S \mathrm{d}z}{V} W \mathrm{d}t}_{\rm energy \: absorbed \: by \: the \: atoms} + \underbrace{\hbar \omega N_e \frac{S \mathrm{d}z}{V} W \mathrm{d}t}_{\rm energy \: supplied \: by \: stimulated \: emission}
$$<p>which gives
$$
\frac{\mathrm{d}I}{\mathrm{d} z} = \hbar \omega \Delta N \frac{W}{V}, \quad \Delta N = N_e - N_g
$$</p>
<p>Because $W = \sigma I / (\hbar \omega)$,</p>
$$
\boxed{
\frac{\mathrm{d}I}{\mathrm{d} z} = \frac{\sigma}{V} \Delta N \times I
}
$$<p>The surface intensity is therefore $I(z) = I(0) \mathrm{exp} (\sigma \Delta N z /V)$. Beam energy/power is amplified if
$$
\boxed{\Delta N >0}
$$</p>
<ol start="2">
<li>$N_e$ is solution of the following differential equation</li>
$$<p>It is therefore impossible to have a population inversion, i.e. to amplify the power of the incident beam. It is impossible to form a laser with a two-level system.</p>
<ol start="4">
<li>For a $3$-level system, the idea is to have $\Gamma_e \gg W, \Gamma_i$ in such a way as to empty the $| e \rangle$ state relatively quickly. In addition, we need to store atoms in the $|i\rangle$ state, $\Gamma_i \ll W$. Finally,</li>
</ol>
$$
\boxed{
\Gamma_i \ll W \ll \Gamma_e
}
$$<p>The laser transition occurs between the levels $i$ and $g$.</p>
<ol start="5">
<li>With a $4$-level system, if we consider just the $|a\rangle$ and $|b\rangle$ levels as shown in the graph below (this is the laser transition), we can see that population inversion is immediate, for example if $\Gamma_a \ll \Gamma_e, \Gamma_b$.</li>
<p>$\gamma_{\rm cav}$ is proportionnal to the transmission loss rate of the mirror, and therefore corresponds to the lifetime of photons in the cavity. $\kappa$ corresponds to the characteristic time associated with the stimulated emission process.</p>
<ol start="2">
<li>The differential equation can be rewritten as follows</li>
<img alt="No description has been provided for this image" class="" 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"/>