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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.

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]

In the ideal three-level laser, pumping excites atoms out of the ground state to a higher auxiliary level, rapid relaxation fills a relatively long-lived upper laser level, and the laser transition returns to the ground state. A ruby laser is a standard example of this classification. Since the lower laser state initially holds most atoms, substantial pumping is needed to reverse the population per sublevel. With upper and lower degeneracies $g_2,g_1$ and $N_1+N_2\simeq N_T$, the positive-gain condition isFor equal degeneracies more than half the active atoms must be excited before optical gain is even positive.

In a four-level laser the lower laser level is above the ground reservoir and empties rapidly after laser emission. Pumping again feeds a higher auxiliary level which relaxes into the upper laser state, but the lower laser-state population can remain small. The helium-neon laser provides a multilevel physical realization often represented by this scheme for a four-level laser. Positive population inversion then requires much less depletion of the ground reservoir, giving a much lower laser pump threshold for comparable lifetimes, emission cross sections and cavity losses.

The threshold of a four-level laser is not literally zero. Oscillation requires round-trip optical gain to overcome output coupling and other losses, and competing decay or incomplete lower-level removal raises the required pump. Real laser media contain more atomic states than ideal diagrams for a three-level laser or a four-level laser; the labels describe the relevant pumping and laser-transition cycle rather than the total number of states in the material.

Define $\sigma (\omega )$ as the emission cross section for an upper-level atom, averaged over the participating sublevels, and let $W=\sigma I/(\hslash \omega )$. Detailed balance gives the upward stimulated rate per lower-level atom as $rW$, where $r=g_2/g_1$. With $D=N_2-r{N}_1$ the complete laser rate equations areAdding them eliminates the internal transitions. Therefore steady state has $N_1=R_2{\tau }_1$ even when the radiation is strong. Substitution into the upper equation givesOne net stimulated transition changes the beam energy by $\hslash \omega$, so the coefficient of optical gain for intensity is $\alpha (\omega ,I)=\sigma (\omega )D$. At line center, with ${\sigma }_0=\sigma ({\omega }_{21})$,Here ${\alpha }_0$ is the unsaturated intensity optical gain per length, which may be negative, and $I_s$ is the intensity at which its magnitude is halved in this model with homogeneous broadening. The cancellation of ${\tau }_1$ from $I_s$ follows from exclusive upper-to-lower decay and constant upper-state pumping. Keeping $N_1=A{\tau }_1{N}_2$ in the stimulated problem would discard the stimulated feeding term and give an incorrect saturation scale. Other decay branches or different pump constraints lead to different expressions.

For a traveling beam with uniform pump, no additional loss and local steady populations,If ${\alpha }_0>0$ and $I\gg I_s$, the increment becomes $dI/dz\simeq {\alpha }_0{I}_s=\hslash {\omega }_{21}{R}_2(1-rA{\tau }_1)$, approximately constant. Intensity then grows linearly, rather than exponentially, with distance while that strong-saturation regime persists. Negative ${\alpha }_0$ gives saturable absorption and a decreasing beam.

Keep the given emission cross section in square centimetres so that the density remains per cubic centimetre. For the first transition, $r=5/3$ andThusThe separate populations are $N_2^{(0)}=1.667\times 10^{13}$ cm${}^{-3}$ and $N_1^{(0)}=1.00\times 10^{12}$ cm${}^{-3}$. Subtracting $5N_1/3$, rather than $N_1$ alone, reproduces the population inversion used above.

Here $rA{\tau }_1=0.10<1$, so upper-state pumping and sufficiently rapid lower-level removal produce positive small-signal optical gain. This coefficient is an intensity optical gain; a weak field amplitude would grow with half this exponent. A positive single-pass coefficient also does not by itself establish laser oscillation without a cavity-loss budget. No transition wavelength was supplied for this numerical example, so a numerical saturation intensity is not needed or invented to calculate ${\alpha }_0$.

For the second transition, $r=3$ and the much longer lower-level lifetime changes the sign of the available population inversion:If its emission cross section equals the first transition's value, the estimate isThe word “comparable” does not specify exact equality, so the more general answer is ${\alpha }_0=-(0.1133\mathrm{c}{\mathrm{m}}^{-1})[{\sigma }_2/(4\times 10^{-15}\mathrm{c}{\mathrm{m}}^2)]$. The quoted numerical value is an explicitly equal-cross-section estimate, while the negative sign is robust for any positive comparable emission cross section.

In this case $rA{\tau }_1=2.7>1$. At fixed pump, $N_1=1.50\times 10^{13}$ cm${}^{-3}$, so its degeneracy-weighted contribution exceeds the upper population. The medium therefore absorbs a weak beam at this transition. Increasing the radiation saturates the photon absorption magnitude toward zero according to c; it does not convert this fixed-pump model into an optical amplifier with positive gain. Simply inserting the magnitude $|{\alpha }_0|$ into the propagation law for positive optical gain would discard the physically important population balance.