<p>We consider a homogeneous medium of atoms $-$ in a cylinder with $S$ the cross-sectional area $-$ 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>
<li>The medium contains $N_g$ in state $g$ and $N_e$ in $e$. Considering only stimulated emission and absorption, express the intensity gain per unit length $\mathrm{d} I /\mathrm{d}z$ as a function of $W$, $\hbar \omega$, $\Delta N = N_e - N_g$ and the volume $V$ of the atomic medium. Under what conditions can the beam power be amplified?</li>
</ol>
<p><em>Hint : carry out an energy balance on a position of size $\mathrm{d}z$ of the medium to find the differential equation verified by $I$.</em></p>
<p><strong>Two-level sytem</strong></p>
<ol start="2">
<li><p>Write as a function of $W$ and $\Gamma_e$ the rate equations on the two-level system (<em>i.e.</em> on $N_g$ and $N_e$ with $N = N_e + N_g$ the total atom number).</p>
</li>
<li><p>Conclude whether a laser can be produced using a two-level system.</p>
</li>
</ol>
<p><strong>Three-level sytem</strong></p>
<p>An intermediate level $i$ is added to the two-level system described above between $g$ and $e$. We note $\Gamma_e$ the spontaneous emission rate from $e$ to $i$ and $\Gamma_i$ the spontaneous emission rate from $i$ to $g$.</p>
<p>Consider a He-Ne laser whose amplifying medium is characterized by an effective absorption cross-section $\sigma = 3 \cdot 10^{-15} \rm cm^{2}$. Amplification is achieved in a four-level system. The laser transition $(\lambda = 633 ~ \rm nm)$ occurs between two levels $a$ and $b$. The $b$ level descexitizes with a rate $\Gamma = 10^7 \rm s^{-1}$ and the bottom level empties very quickly so that $N_a = 0$.</p>
<p>The amplifying medium of length $L_A$ is placed in a ring cavity of optical length $L_{\rm cav} = 1.5 \rm m$ which has selective elements making the laser single-mode. At the level of the amplifier medium, the laser is assimilated to a homogeneous cylinder with $S_A$ the cross-sectional area. The output coupling mirror has a transmission of $T = 0.01$.</p>
<p>Bar pumping is characterized by the pumping rate</p>
<p>For an A-type laser, <em>i.e.</em> for $\Gamma > \gamma_{\rm cav}$, the population inversion $\Delta N$ relaxes much faster than laser intensity. Thus, $\mathcal{N}$ can be considered constant in the population inversion evolution equation.
On the other hand, $\Delta N$ is rapidly damped at the characteristic time scale of variation of $\mathcal{N}$, so the field no longer feels the evolution of the atomic medium. We can therefore consider $\Delta N$ to be stationary.</p>
<ol start="4">
<li><p>Check that the He-Ne laser is a type A laser.</p>
</li>
<li><p>Obtain the expression of $\Delta N$.</p>
</li>
<li><p>Deduce the new equation verified by the number of photons $\mathcal{N}$.</p>
</li>
</ol>
<p><strong>Laser starting</strong></p>
<ol start="7">
<li><p>We set $r \gamma_{\rm cav} = \kappa \Delta N_0$ avec $r > 1$. Deduce that the laser starts exponentially.</p>
</li>
<li><p>Show graphically the time evolution of photon number and population inversion.</p>