Past exam of the quantum, atomic and molecular physics course of the University of Oxford / 2016 / 3

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.

Let $A=A_{21}$ and $r=g_2/g_1$. Before stimulated emission and stimulated absorption are added, the source and decay terms giveThe upper level is fed at a specified rate of addition to the number density, its spontaneous decay feeds the lower level, and lower-level removal leads to states outside this two-level system. This is therefore an open pumped system, not a closed two-level system with conserved population.

At steady state,The degeneracy-weighted population inversion compares population per magnetic sublevel. It is positive only if $rA{\tau }_1<1$; a positive pump rate alone cannot guarantee optical gain if lower-level removal is too slow. A short lower-state lifetime is consequently helpful even when the upper-state pumping is unchanged.

The equations assume finite positive $A$ and ${\tau }_1$, approximately constant pumping and no other specified sources, sinks or branching channels. A finite steady state would not exist for continuous pumping if the lower level had no escape route. The fixed source rate also presumes access to an external reservoir; it does not imply that the two populations can grow beyond all available active atoms in a real finite medium.

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.