Past exam of the atomic and laser physics course of the University of Oxford / 2022 / 4

Define $r=g_2/g_1$ and let $\sigma ={\sigma }_{21}(\omega )$ be the stimulated-emission cross section, averaged over upper magnetic sublevels. The Einstein coefficients relation $g_1{B}_{12}=g_2{B}_{21}$ gives the absorption cross section ${\sigma }_{12}=r\sigma$. Assume equally populated magnetic sublevels within each energy level, a steady pump, and a monochromatic beam. The photon flux is $I/(\hslash \omega )$; define $W=\sigma I/(\hslash \omega )$ and the effective population inversion $D=N_2-r{N}_1$. Then the rate equations areThe $WD$ term includes both stimulated emission and absorption. In steady state, adding the two rate equations gives $N_1=R_2{\tau }_1$. The first equation then yieldsThe directed beam gains energy at rate $\hslash \omega WD$ per volume, giving $dI/dz=\sigma DI$ and the optical gain coefficient $\alpha =\sigma D$. At line centre, write ${\sigma }_0={\sigma }_{21}({\omega }_{21})$. ThereforeHere ${\alpha }_0$ is the small-signal optical gain coefficient and $I_s$ is the saturation intensity, at which the local optical gain is half its small-signal value. Positive optical gain requires $r{A}_{21}{\tau }_1<1$: the lower energy level must empty sufficiently quickly.

The cancellation of ${\tau }_1$ from $I_s$ is specific to this model's steady fixed pump and the fact that every upper spontaneous emission feeds the lower energy level. Adding an extra lower-state accumulation factor to $I_s$ would contradict these rate equations. Alternative conventions using the absorption cross section must replace ${\sigma }_0$ by ${\sigma }_{12}/r$ throughout. Homogeneous spectral broadening permits the whole selected ensemble to share this local saturation response.

In a uniform steady amplifier, with no extra propagation losses, the local optical gain law givesIntegrate between the two faces of length $L$ to obtainFor $I\ll I_s$, this reduces to exponential amplification. In deep saturation, the intensity increment approaches ${\alpha }_0{I}_{s}L$: finite pumping limits the extracted power.

Let $x=I_0/I_s$ and $G={\alpha }_{0}L$. Applying the integrated amplifier law to each section givesEliminating $G$ givesAs a consistency check, both amplifiers have $G\simeq 26.1175$. The second section receives a larger intensity and adds $80I_0$, compared with $69I_0$ in the first, while its multiplicative optical gain is much smaller. This is the expected approach to constant extracted power in saturation.

For a closed absorbing two-level system, write $N=N_1+N_2$, let $A=1/\tau$, and use ${\sigma }_a={\sigma }_{12}$ for the absorption cross section. This differs from the stimulated-emission cross section convention in part (a). With $r=g_2/g_1$, the rate equations areAt steady state, $N_2=W_{a}N/[A+(1+1/r)W_a]$ for $W_a={\sigma }_{a}I/(\hslash \omega )$. Hence the positive absorption coefficient isFor the given sodium fine-structure transition, the statistical weights are $g_1=2$, $g_2=4$, so $r=2$ and $1+1/r=3/2$. The small-signal transmission gives the optical depthTo account for the finite decrease of intensity through the sample, integrate $dI/dz=-\kappa I$ rather than treating the stronger beam as exactly uniform. If $T=I_{\mathrm{out}}/I_{\mathrm{in}}$, this givesTaking $270\mathrm{W}{\mathrm{m}}^{-2}$ as the incident intensity and $T=0.967$,Using the photon energy $hc/\lambda$ then givesIgnoring the small variation of the stronger intensity would instead give about $1.11\times 10^{-13}{\mathrm{m}}^2$; the integrated value uses the stated transmissions more consistently. The length is not needed to eliminate $N$, but gives the check $N\simeq 9.3\times 10^{14}{\mathrm{m}}^{-3}$.

The resonant absorption cross section is of optical-wavelength-squared size, much larger than a typical atomic geometric collision area. Pressure broadening cross sections involve phase-changing interatomic encounters and need not equal hard-sphere collision areas; their numerical value depends on collision partners and speed. No precise ratio follows from the supplied data. A useful distinction is that resonant absorption involves coherent response on the optical scale, so its cross section need not resemble an chemical atom's geometric area.

For homogeneous spectral broadening, all participating chemical atoms must share the same resonance and single-atom response width. In a dilute sample dominated by natural linewidth, the weak-beam requirement isUsing the sodium atomic mass, equality occurs near $T=0.017\mathrm{K}$, so $T\ll 17\mathrm{mK}$ is a sufficient thermal condition. Spatial Zeeman energy shifts, light shifts and transit effects should also be small, the laser should address the same selected component, and the interaction should last long enough for steady populations. Uniform impact pressure broadening can also be homogeneous, but would change the width and effective cross section; it is not included in the lifetime-only numerical model.

Real sodium also has hyperfine structure and optical pumping among magnetic sublevels. The inferred ${\sigma }_a$ is the effective cross section of the stated $g_J=2J+1$ two-level system model, assuming statistical sublevel populations and no loss into unaddressed states. A measurement on a resolved, polarized closed hyperfine structure component requires its actual transition dipole moment and selection rules; a universal polarization-independent sodium cross section cannot be inferred from this simplified model. The dependence of resonant cross sections and saturation intensities on sublevel preparation is tabulated explicitly for the real sodium D lines.^[ref]