# Laser Pumping and Population Inversion

This is a continuation from the previous tutorial - ** polarization maintaining fibers**.

The atomic rate equations introduced in the two previous tutorials are of great value in analyzing laser pumping, population inversion, and gain saturation in laser systems. The primary objective of this tutorial is to illustrate this point by solving the atomic rate equations and examining these solutions for some simple but important atomic systems.

## 1. Steady-State Laser Pumping and Population Inversion

One of the most common applications for rate equations is in analyzing laser pumping. In this section, therefore, we will develop and solve the rate equations to analyze steady-state laser pumping in simplified four-level and three-level laser systems.

### Elementary Four-Level Laser System

Figure 6.1 shows a complicated multienergy-level system typical of many real laser systems; this one is for a solid-state laser system using the Nd^{3+} ion in Nd:YAG or Nd:glass.

The upward arrows indicate upward pumping rates to various higher levels produced by flashlamp pumping on strong absorption lines from the ground state; the downward arrow indicates the widely used 1.064 μm laser transition from the \(^4F_{3/2}\) level to the \(^4I_{11/2}\) level.

(There are actually at least eight different laser transitions of widely varying strengths between these two clusters of closely spaced levels, with wavelengths extending from 1.0520 to 1.1226 μm. In addition, generally weaker laser action is possible from the same \(^4F_{3/2}\) levels to the cluster of \(^4I_{9/2}\) levels at four wavelengths between 0.89 and 0.9462 μm; to the \(^4I_{13/2}\) levels at four wavelengths between 1.319 and 1.358 μm; and—very weakly—at 1.833 μm to one of the \(^4I_{15/2}\) levels slightly above the lowest or ground level in this cluster.)

Figure 6.1 does not show the numerous downward radiative and nonradiative relaxation paths among all these levels. As in many other rare-earth and other solid-state laser systems, however, atoms excited into the higher excited levels in this material will nearly all relax, primarily by fast nonradiative relaxation, into the sharp and long-lived \(^4F_{3/2}\) metastable level that provides the upper laser level in this system. A sizable population density can thus build up in this upper laser level.

For purposes of analysis this complicated set of levels can then be simplified into the idealized four-level laser system shown in Figure 6.2.

This four-level model will in fact provide a simple but surprisingly accurate analytical model for many real laser systems. In this model level 4 represents the combination of all the levels lying above the upper laser level in the real atomic system. It is desirable that many of these levels be in fact broad absorption bands, so that the optical pumping into these levels by a broadband pump lamp can be very efficient.

Level 3 represents the upper laser level, usually a fairly sharp and long-lived level, with a large gap below it. Level 2 then represents the lower laser level, and level 1 the lowest or ground level. Other low-lying levels that may be present in the material, both above and below the lower laser level, are ignored in the model because they play no real role in the laser action. They act only as temporary way stations through which atoms may pass in relaxing from the other levels to the ground level \(E_1\).

### Four-Level Pumping Analysis

To analyze this system we will write down the relevant rate equations one after another, using the model shown in Figure 6.3, and then solve for their steady-state solutions, making reasonable approximations as we go along. Since the condition \(\hbar\omega/kT\gg1\) is usually very well satisfied for all transitions in a visible laser system, we will write all of the following rate equations using the "optical approximation."

We begin by assuming that the laser pumping process, whatever its physical cause, produces a stimulated pump transition probability \(W_{14}=W_{41}=W_p\) between levels 1 and 4. The rate equation for level 4 in the optical approximation is then

\[\tag{1}\begin{align}\frac{dN_4}{dt}&=W_p(N_1-N_4)-(\gamma_{43}+\gamma_{42}+\gamma_{41})N_4\\&=W_p(N_1-N_4)-N_4/\tau_4\end{align}\]

where the lifetime \(\tau_4\) given by

\[\tag{2}\frac{1}{\tau_4}\equiv\gamma_4=\gamma_{43}+\gamma_{42}+\gamma_{41}\]

is the total lifetime for decay of level 4 to all lower levels. The steady-state population of level 4, when \(dN_4/dt=0\), is then given by

\[\tag{3}N_4=\frac{W_p\tau_4}{1+W_p\tau_4}N_1\approx{W_p}\tau_4N_1\quad\text{if}\quad{W_p\tau_4}\ll1\]

The normalized pumping rate \(W_p\tau_4\), which will appear in many of the following expressions, will in fact have a value much less than unity in many (though not all) practical laser systems.

Direct pumping up from the ground level into the upper laser level 3 in this model can very often be assumed negligible, either because the \(1\rightarrow3\) transition will have a weaker absorption cross section than the collected \(1\rightarrow4\) transitions, or because this transition will be much narrower than the strong absorption bands from the ground level to the groups of levels that make up level 4. The rate equations for the two levels \(N_3\) and \(N_2\) are then

\[\tag{4}\frac{dN_3}{dt}=\gamma_{43}N_4-(\gamma_{32}+\gamma_{31})N_3=\frac{N_4}{\tau_{43}}-\frac{N_3}{\tau_{3}}\]

and

\[\tag{5}\frac{dN_2}{dt}=\gamma_{42}N_4+\gamma_{32}N_3-\gamma_{21}N_2=\frac{N_4}{\tau_{42}}+\frac{N_3}{\tau_{32}}-\frac{N_2}{\tau_{21}}\]

The first of these then gives at steady state (\(d/dt=0\))

\[\tag{6}N_3=\frac{\tau_3}{\tau_{43}}N_4\]

In a good laser system the \(4\rightarrow3\) relaxation rate will be very fast, but the upper laser level 3 will have a long lifetime by comparison, so that \(\tau_3\gg\tau_{43}\) and hence \(N_3\gg{N_4}\).

Combining Equation 6.5 and 6.6 then gives the result

\[\tag{7}N_2=\left(\frac{\tau_{21}}{\tau_{32}}+\frac{\tau_{43}\tau_{21}}{\tau_{42}\tau_3}\right)N_3=\beta{N_3}\]

where the parameter \(\beta\) is defined to be

\[\tag{8}\beta\equiv\frac{\tau_{21}}{\tau_{32}}+\frac{\tau_{43}\tau_{21}}{\tau_{42}\tau_3}\]

This parameter \(\beta\) thus depends only on relaxation-time ratios, not absolute values. If this quantity is less than unity, the steady-state result will be \(N_2\lt{N_3}\), which means there will be the desired population inversion on the \(3\rightarrow2\) transition.

In a good laser system the upper levels \(E_4\) will relax primarily into the upper laser level \(E_3\), so that \(\gamma_{42}\approx0\) or \(\tau_{42}\approx\infty\). In this case \(\beta\approx\tau_{21}/\tau_{32}\), and the condition for population inversion becomes simply

\[\tag{9}\beta\equiv\frac{N_2}{N_3}\approx\frac{\tau_{21}}{\tau_{32}}\ll1\]

In other words, to have good inversion on the \(3\rightarrow2\) transition, atoms should relax out of the lower laser level \(E_2\) down into lower levels much faster than atoms relax into \(E_2\) from above. Even if level 4 does not relax only into level 3, if the upper laser level has a long lifetime (both \(\tau_{32}\) and \(\tau_3\) long) and the lower laser level has a short lifetime (\(\tau_{21}\) short), then population inversion on the \(3\rightarrow2\) transition is virtually certain.

Whether this population inversion will be large enough to give sufficient gain to achieve laser action in a practical cavity is another matter. Nonetheless, these conditions are met and laser action can be produced on many transitions in many real atomic systems.

### Fluorescent Quantum Efficiency

Another dimensionless parameter often used in evaluating laser materials is the ** fluorescent quantum efficiency** \(\eta\), defined as the number of fluorescent photons spontaneously emitted on the laser transition divided by the number of pump photons absorbed on the pump transition(s) when the laser material is below threshold. For the four-level system this quantum efficiency is given by

\[\tag{10}\eta=\frac{\gamma_{43}}{\gamma_4}\times\frac{\gamma_\text{rad}}{\gamma_3}=\frac{\tau_4}{\tau_{43}}\times\frac{\tau_3}{\tau_\text{rad}}\]

where \(\gamma_\text{rad}\equiv\gamma_\text{rad}(3\rightarrow2)\) is the radiative decay rate on the \(3\rightarrow2\) transition. The first ratio in this expression tells what fraction of the total atoms excited to level 4 relax directly into the upper laser level 3, rather than bypassing 3 and dropping to lower levels, and the second ratio tells what fraction of the total decay out of level 3 is purely radiative decay to level 2.

### Four-Level Population Inversion

With the aid of the parameters \(\beta\) and \(\eta\), plus the conservation of atoms condition that \(N_1+N_2+N_3+N_4=N\), we can solve for the population inversion \(N_3-N_2\) versus pumping strength on the four-level system. After some algebra we can obtain

\[\tag{11}\frac{N_3-N_2}{N}=\frac{(1-\beta)\eta{W_p}\tau_\text{rad}}{1+[1+\beta+2\tau_{43}/\tau_\text{rad}]\eta{W_p}\tau_\text{rad}}\]

where \(\tau_\text{rad}\equiv\tau_\text{rad}(3\rightarrow2)\) is the radiative decay rate on the laser transition itself. In a good laser material the lifetime \(\tau_{43}\) from the upper pump level into the upper laser level will be short compared to this radiative decay time, and this expression can then be simplified into

\[\tag{12}\frac{N_3-N_2}{N}\approx\frac{(1-\beta)\eta{W_p}\tau_\text{rad}}{1+(1+\beta)\eta{W_p}\tau_\text{rad}}\approx\frac{W_p\tau_\text{rad}}{1+W_p\tau_\text{rad}}\quad\text{if }\beta\rightarrow0\]

The optimum situation is obviously \(\beta\approx\tau_{21}/\tau_{32}\rightarrow0\).

Figure 6.4 shows a plot of the inversion \(N_3-N_2\) on the four-level laser transition versus the normalized pumping rate \(W_p\tau\), assuming \(\beta=0\). For a four-level system, the population inversion on the \(3\rightarrow2\) transition increases linearly with the pumping intensity \(W_p\) at lower pump levels, but then approaches a limiting value for \(W_p\tau\gg1\) as the ground state \(E_1\) is depleted and a large fraction of the atoms are lifted into the upper laser level.

This four-level pumping model provides a surprisingly good analytical model for understanding the behavior of a large number of real laser systems, as we will show in later sections.

### Three-Level Laser System

Figure 6.5 illustrates how a three-level laser system can be similarly employed as a model for the real energy levels of the familiar ruby laser, just as the four-level system provided a model for the Nd:YAG and many similar solid-state and dye lasers.

A three-level laser differs from the four-level system in that the lower laser level is the ground level \(E_1\). This is a serious disadvantage, since more than half the atoms initially in the ground state must be pumped through the upper pumping level \(E_3\) into the upper laser level \(E_2\) before any inversion at all is obtained on the \(2\rightarrow1\) transition.

Three-level lasers are, therefore, usually not as efficient as four-level lasers. One reason for analyzing the three-level system, nonetheless, in addition to general background knowledge, is that the 694 nm ruby laser—the first laser ever to be operated and still a useful solid-state laser—is a nearly ideal three-level laser system.

Suppose the pumping process in a three-level system produces a stimulated transition probability \(W_{13}=W_{31}=W_p\). Then the rate equations for the two upper levels are

\[\tag{13}\frac{dN_3}{dt}=W_p(N_1-N_3)-\frac{N_3}{\tau_3}\]

and

\[\tag{14}\frac{dN_2}{dt}=\frac{N_3}{\tau_{32}}-\frac{N_2}{\tau_{21}}\]

There is also the usual conservation equation \(N_1+N_2+N_3=N\), and it is again useful to define a fluorescence quantum efficiency given by

\[\tag{15}\eta=\frac{\tau_3}{\tau_{32}}\times\frac{\tau_{21}}{\tau_\text{rad}(2\rightarrow1)}\]

The important relaxation-time ratio in this model is given by

\[\tag{16}\beta\equiv\frac{N_3}{N_2}=\frac{\tau_{32}}{\tau_{21}}\]

The steady-state population difference on the \(2\rightarrow1\) transition can then be found to be

\[\tag{17}\frac{N_2-N_1}{N}=\frac{(1-\beta)\eta{W_p}\tau_\text{rad}-1}{(1+2\beta)\eta{W_p}\tau_\text{rad}+1}\]

Inversion in the three-level system can be obtained only if \(\beta\lt1\), and even then inversion can occur only when the pumping rate exceeds a threshold value given by

\[\tag{18}W_p\tau_\text{rad}\ge\frac{1}{\eta(1-\beta)}\]

The optimum situation obviously occurs when the relaxation from the upper pumping level 3 into the upper laser level 2 is very fast, so that \(\beta\rightarrow0\), and when the relaxation from the upper laser level 2 down to the ground level 1 is purely radiative, so that \(\eta\rightarrow1\). The inversion versus normalized pumping strength then reduces to

\[\tag{19}\frac{N_2-N_1}{N}\approx\frac{W_p\tau_\text{rad}-1}{W_p\tau_\text{rad}+1}\quad\text{if }\eta\rightarrow1\text{ and }\beta\rightarrow0\]

The significant differences in inversion versus pumping for a three-level and a four-level system are illustrated in Figure 6.4. Other things being equal, a four-level laser system should have a much lower pumping threshold than a three-level laser system.

### More on the Ruby Laser

The well-known ruby laser is a special case that overcomes the inherent disadvantages of the three-level laser, since the ruby laser can in fact have a moderately low pulsed laser threshold, and can even (with some difficulty) be operated as a cw laser. The ruby laser works as well as it does because of an unusually favorable combination of other factors, including:

- Unusually broad and well-located pump absorption bands that make very efficient use of the broadband radiation from standard flashlamps.
- A fluorescent quantum efficiency \(\eta\) very close to unity.
- An unusually narrow atomic linewidth for the laser transition.
- An unusually long and almost purely radiative lifetime \(\tau_{21}\) ~ 4.3 ms for the upper laser level.
- The availability of ruby synthetic crystals with very good optical quality, good thermal conductivity, and high optical power handling capability.

The ruby laser has been particularly useful because it has an oscillation wavelength located in (or at least on the edge of) the visible region, where more sensitive photodetectors are generally available, in contrast to the infrared wavelengths of most of the solid-state rare-earth lasers. On the other hand, modern flashlamp-pumped dye lasers also give good laser action all across the visible region.

## 2. Laser Gain Saturation

In many real laser systems laser action takes place between two excited levels that are located high above the ground level, and the population density in these excited laser levels always remains small compared to the total density of atoms in the lowest energy level \(E_0\) (as we will label the ground level in this section).

This is particularly true in gas lasers, where linewidths are narrow, transitions are relatively strong, and only small inversion densities are necessary to give significant gain. It may be less true in solid-state lasers, such as the ruby example of the preceding section, where large fractions of the total atomic density may sometimes be pumped into the upper laser levels.

In any event, in this section we will use this as a simplified model to develop some further rate equation analyses, showing in particular how the laser gain itself saturates with increasing signal power in typical laser systems.

### Laser Gain Saturation Analysis

Figure 6.6 gives a simplified but yet realistic model for many laser systems of this type. Atoms are pumped by some pumping mechanism from the ground level \(E_0\) into some upper level \(E_3\). They then relax down (perhaps by cascade processes) into the upper laser level \(E_2\), from where they relax or make stimulated laser transitions down to the lower laser level \(E_1\), and thence back to ground.

Note that we have specifically included a laser signal, corresponding to laser amplification or laser oscillation, and represented by the stimulated transition probability \(W_\text{sig}\) in this diagram.

Suppose the upper-level populations all remain small compared to the initial ground-state population. Then the pumping rate from the ground level \(E_0\) into the upper atomic level \(E_3\) caused by a pumping transition probability \(W_{03}=W_{30}=W_p\) may be written as

\[\tag{20}\left.\frac{dN_3}{dt}\right|_\text{pump}=W_p(N_0-N_3)\approx{W_p}N_0\]

where \(N_0\approx{N}\) is very nearly the total density of laser atoms in the system.

In this situation there is essentially no "back-pumping" from \(E_3\) to \(E_0\), since very few atoms accumulate in the upper levels and hence \(N_3\ll{N_0\). It is then common and convenient practice to speak not of a pumping transition probability \(W_p\) (probability per atom per second), but of a net pumping rate (atoms per second, per unit volume) being lifted up out of the ground level, as given by \(W_pN_0\approx{W_p}N\).

This pumping rate \(W_pN_0\) in a real laser system will be more or less directly proportional to the pump light intensity (in an optically pumped laser), or to the discharge current density (in a discharge-pumped gas laser), or to a chemical reaction rate (in a chemically pumped laser).

Moreover, in many real lasers some fixed fraction \(\eta_p\) of the atoms pumped into an upper energy level will decay, often through some cascade process, down into the longer-lived upper laser level \(E_2\).

The number of atoms per second reaching the upper laser level is then given by an effective pumping rate \(R_p=\eta_pW_pN_0\), where \(\eta_p\) represents the quantum efficiency for pump excitation into this upper laser level. (This pumping efficiency may be quite high, even approaching unity, for many solid-state and organic dye lasers, and may be very small for many typical discharge-pumped gas lasers.)

With these generally valid assumptions, the rate equations for the excited laser levels \(E_2\) and \(E_1\), including a laser signal with stimulated transition probability \(W_{12}=W_{21}=W_\text{sig}\) on the laser transition, may be written as

\[\tag{21}\frac{dN_2}{dt}=R_p-W_\text{sig}(N_2-N_1)-\gamma_2N_2\]

and

\[\tag{22}\frac{dN_1}{dt}=W_\text{sig}(N_2-N_1)+\gamma_{21}N_2-\gamma_1N_1\]

where \(\gamma_2=\gamma_{21}+\gamma_{20}\) is the total decay rate downward from the upper laser level \(E_2\) to all lower levels.

The steady-state solutions to these equations are then given by

\[\tag{23}\begin{align}N_1&=\frac{W_\text{sig}+\gamma_{21}}{W_\text{sig}(\gamma_1+\gamma_{20})+\gamma_1\gamma_2}R_p\\N_2&=\frac{W_\text{sig}+\gamma_1}{W_\text{sig}(\gamma_1+\gamma_{20})+\gamma_1\gamma_2}R_p\end{align}\]

Note particularly that only the two rate equations 6.21 and 6.22 were used in obtaining these results. No "conservation of atoms" condition stating that \(N_1+N_2\) remains constant was necessary or even possible in this case, since in fact \(N_1+N_2\) does not remain constant in this system when either the pump rate \(R_p\) or the signal strength \(W_\text{sig}\) changes. Two rate equations for the populations at the two levels are thus both necessary and also sufficient in this particular type of rate-equation calculation.

### Gain Saturation Behavior

The steady-state population difference \(\Delta{N}_{21}=N_2-N_1\) on the laser transition is then given by

\[\tag{24}\Delta{N}_{21}\equiv{N_2}-N_1=\left(\frac{\gamma_1-\gamma_{21}}{\gamma_1\gamma_2}\right)\times\frac{R_p}{1+[(\gamma_1+\gamma_{20})/\gamma_1\gamma_2]W_\text{sig}}\]

The inverted population difference in this simple example varies with both pumping rate and signal intensity in the simple form

\[\tag{25}\Delta{N}_{21}=\Delta{N_0}\frac{1}{1+W_\text{sig}\tau_\text{eff}}\]

where \(\Delta{N_0}\) is a small-signal or unsaturated population inversion given by

\[\tag{26}\Delta{N}_0=\frac{\gamma_1-\gamma_{21}}{\gamma_1\gamma_2}R_p=(1-\tau_1/\tau_{21})\times{R_p}\tau_2\]

and \(\tau_\text{eff}\) is an effective recovery time or lifetime for the signal gain given by

\[\tag{27}\frac{1}{\tau_\text{eff}}=\frac{\gamma_1\gamma_2}{\gamma_1+\gamma_{20}}\quad\text{or}\quad\tau_\text{eff}=\tau_2(1+\tau_1/\tau_{20})\]

If the upper laser level \(E_2\) relaxes primarily into the lower laser level \(E_1\) and not directly down to any lower levels \(E_0\), then the expression for the population inversion reduces to simply

\[\tag{28}\Delta{N_{21}}\approx{R_p}(\tau_2-\tau_1)\times\frac{1}{1+W_\text{sig}\tau_2}\]

where we have used the approximation that \(\gamma_2\approx\gamma_{21}\) and \(\gamma_{20}\approx0\).

#### Discussion

The analytical results in Equations 6.24 through 6.28 illustrate several typical aspects of laser behavior, including:

- The crucial requirement for obtaining inversion on this transition is that the time-constant ratio \(\tau_{21}/\tau_1\) should be > 1, or that the condition \(\gamma_1\gt\gamma_{21}\) be satisfied. In physical terms, this means that inversion is obtained only if atoms relax downward out of the lower level \(E_1\) at rate \(\gamma_1\) faster than they relax in at rate \(\gamma_{21}\) from the upper level.
- If this condition is met, the small-signal or unsaturated population difference \(\Delta{N_0}\) between the two laser levels is then directly proportional to the pump rate \(R_p\) times an effective "population integration time," which is basically the upper-level lifetime \(\tau_2\) reduced by the factor (\(1-\tau_1/\tau_{21}\)).
- The effective recovery time \(\tau_\text{eff}\), which determines the signal saturation behavior in Equation 6.25, is in general a combination of the various inter-level relaxation rates or lifetimes in the system. If the lower-level lifetime becomes short enough, \(\tau_1\rightarrow0\), so that little or no population can accumulate in the lower level \(E_1\), then \(\tau_\text{eff}\) becomes just the upper-level lifetime, \(\tau_\text{eff}\approx\tau_2\).
- Finally, in this system as in many real lasers, the saturation intensity of the inverted population depends only on the relaxation lifetimes between the atomic levels, and does not depend directly on the pumping intensity \(R_p\). That is, the signal intensity \(W_\text{sig}\) needed to reduce the population inversion or laser gain to half its initial value does not depend at all (at least in this example) on how hard the atoms are being pumped.

The final point implies in particular that turning up the pump intensity will not increase the saturation intensity of the material, or the signal level required to reduce the gain to half its small-signal value.

Turning up the pump intensity does increase the unsaturated population inversion \(\Delta{N_0}\) and hence the unsaturated gain, so that the laser must oscillate harder to bring the saturated gain down to match the losses; but the value of \(W_\text{sat}\) or \(I_\text{sat}\) is basically independent of how hard the system is pumped. This same behavior is characteristic of most real laser systems.

### The Factor of 2*

We showed earlier that in a simple two-level system with a constant total population, or \(N_1+N_2=N\), the rate equation for the absorbing population difference \(\Delta{N}\equiv{N_1}-N_2\) takes the general form

\[\tag{29}\frac{d}{dt}\Delta{N}=-2W_{12}\Delta{N}-\frac{\Delta{N}-\Delta{N_0}}{T_1}\]

and so the saturation behavior takes the form

\[\tag{30}\Delta{N}=\Delta{N_0}\frac{1}{1+2W_{12}T_1}\]

where \(T_1\) is the recovery time for the population difference.

Suppose instead that we have a pumped and possibly inverted two-level system like that just discussed, in which the total population is no longer necessarily constant; and let us suppose in addition that the relaxation rate \(\gamma_{10}\) out of the lower level is extremely rapid, so that essentially no atoms ever collect in level 1, and that \(N_1\approx0\) under all circumstances. According to the preceding results, the rate equation for the inverted population difference \(\Delta{N}\equiv{N_2-N_1}\approx{N_2}\) then becomes

\[\tag{31}\frac{d}{dt}\Delta{N}\approx-W_\text{sig}\Delta{N}-\frac{\Delta{N}-\Delta{N_0}}{\tau_2}\]

and so the saturation behavior is

\[\tag{32}\Delta{N}\approx\Delta{N_0}\frac{1}{1+W_\text{sig}\tau_2}\]

where \(\tau_2\) is the upper-level lifetime; \(W_\text{sig}\equiv{W}_{12}\); and \(\Delta{N_0}\equiv{R_p}\tau_2\).

These two situations thus lead to exactly the same basic equations except that there is an additional factor of 2 in the stimulated-transition term in one case and not the other. This factor occurs in the simple two-level absorbing system because the transition of one atom between levels reduces the population difference \(\Delta{N}\) by two.

Now, there are also many inverted laser systems in which an atom that is stimulated to make a downward transition from level 2 to level 1 remains for some considerable lifetime in level 1 before relaxing down to still lower levels. If the lifetime in level 1 is particularly long, we sometimes say that the atoms are more or less "bottlenecked" in level 1.

The stimulated transition of one atom from \(N_2\) to \(N_1\) thus again reduces the population difference \(\Delta{N}\equiv{N_2}-N_1\) by two, and we must write the stimulated-transition term as \((d/dt)\Delta{N}\approx-2W_\text{sig}\Delta{N}\) here also. If level 1 is not bottlenecked, however, and its population empties out very rapidly, so that \(N_1\approx0\) at all times, we can write the stimulated term as \((d/dt)\Delta{N}\approx-W_\text{sig}\Delta{N}\), where \(\Delta{N}\approx{N_2}\) as above.

To handle both of these situations in a single notation, and also to take account of the fact that the population difference most often recovers to some small-signal or unsaturated value \(\Delta{N_0}\) with an effective time constant \(\tau_\text{eff}\), we can write the saturation equation for the inverted population in many different practical laser systems (and also the absorbing population difference in many absorbing systems) in the general form

\[\tag{33}\frac{d}{dt}\Delta{N(t)}=-2^*W_\text{sig}\Delta{N(t)}-\frac{\Delta{N(t)}-\Delta{N_0}}{\tau_\text{eff}}\]

where \(2^*\) is a numerical factor with a value somewhere between \(2^*=2\) (for strongly bottlenecked systems) and \(2^*=1\) (for systems with no bottlenecking).

We will use this simple form several times later on to analyze the gain saturation and atomic dynamics in many laser problems, although a more complex rate-equation analysis may be needed to evaluate the actual values of \(2^*\) and \(\tau_\text{eff}\). The effective value of \(2^*\) will turn out to make a significant difference in the energy and power outputs of laser devices. (In general, the absence of bottlenecking, or \(2^*\approx1\), is good; and the presence of bottlenecking, or \(2^*\approx2\), is not so good.)

## 3. Transient Laser Pumping

The full transient solution to a set of multilevel rate equations can become very complicated since there will be in general \(M-1\) transient decay terms for an \(M\)-level atomic system.

The transient solution for the build-up of inversion in a multilevel pulse-pumped laser can, therefore, also become a complicated problem. We will illustrate one or two such transient situations in this section, however, using very simplified models, in order to give some idea of the kind of behavior to be expected.

### Transient Rate-Equation Example: Upper-Level Laser

As a first example, let us consider the "upper-level" laser model shown in Figure 6.6 and described in Equations 6.20 through 6.23. To simplify this still further, assume that no signal is present on the \(E_2\) to \(E_1\) transition, and that the relaxation rate out of the lower \(E_1\) level is sufficiently fast that \(N_1\approx0\) under all conditions. The transient pumping equation for the upper laser level population \(N_2(t)\) is then

\[\tag{34}\frac{dN_2(t)}{dt}=R_p(t)-\gamma_2N_2(t)\]

where \(R_p(t)\) is the (possibly) time-varying pump rate (in atoms lifted up per second) applied to the atomic system. A formal solution to this equation is

\[\tag{35}N_2(t)=\int_{-\infty}^tR_p(t')e^{-\gamma_2(t-t')}dt'\]

This equation says, of course, that of the number of atoms \(R_p(t')dt'\) lifted up during a little time interval \(dt'\), only a fraction \(e^{-\gamma_2(t-t')}\) will remain in the upper level at a time \(t-t'\) later.

Suppose we put in a square pump pulse with constant amplitude \(R_{p0}\) and duration \(T_p\), i.e., \(R_p(t)=R_{p0}\), \(0\le{t}\le{T_p}\). The maximum upper-level population, reached just at the end of the pumping pulse, is then given by

\[\tag{36}N_2(T_p)=R_{p0}\tau_2\left[1-e^{-T_p/\tau_2}\right]\]

where \(\tau_2\equiv1/\gamma_2\) is the lifetime of the upper laser level.

This tells us that in a pulse-pumped laser of the type in which one first pumps up the upper-level population, and then "dumps" this population by \(Q\)-switching, it is of very little use to continue the pump pulse for longer than about two upper-level lifetimes or so, since beyond that point the upper-level population no longer increases much with further pumping.

Alternatively, we might define a pumping efficiency \(\eta_p\) for this case as the ratio of the maximum number of atoms stored in the upper level, just at the end of the pumping pulse, to the total number of pump photons sent in or atoms lifted up during the pump pulse. Since the total number of atoms lifted up during the pump pulse is \(R_{p0}T_p\), this pumping efficiency is given by

\[\tag{37}\eta_p=\frac{N_2(t=T_p)}{R_{p0}T_p}=\frac{1-e^{-T_p/\tau_2}}{T_p/\tau_2}\]

In other words, this efficiency depends only on the ratio of pump pulse-width \(T_p\) to upper-level time constant, \(\tau_2\). A little work with your pocket calculator will show that if these time constants are equal, i.e., \(T_p=\tau_2\), then the pumping efficiency is only about \(\eta_p\approx63\%\). For the pumping efficiency to reach 90% requires \(T_p/\tau_2\approx0.2\), i.e., the total pump energy must be delivered in a pulse whose width \(T_p\) is only about 1/5 of the upper-level time constant \(\tau_2\).

### Transient Rate-Equation Example: Pulsed Ruby Laser

A transient solution for the simplified three-level ruby laser model given earlier in this tutorial can also rather easily be obtained and used to demonstrate both the techniques of rate-equation analysis and the good agreement with experiment that can be provided by even a simple rate-equation description.

In ruby the \(\gamma_{32}\) relaxation rate is so fast (\(\gt10^{10}\text{ sec}^{-1}\)) that atoms pumped into level 3 may be assumed to relax instantaneously into level 2. Hence we may assume that \(N_3\approx0\) at all times, even with the strongest practical pump powers that we can apply. The three-level rate equations for a ruby laser, including pumping but not signal terms, can then be reduced to the single rate equation

\[\tag{38}\frac{dN_1}{dt}=-\frac{dN_2}{dt}\approx-W_p(t)N_1(t)+\frac{N_2(t)}{\tau}\]

combined with the conservation of atoms condition that \(N_1(t)+N_2(t)\approx{N}\). Here the lifetime \(\tau\) is the total (and, in ruby, mostly radiative) decay time of approximately 4.3 ms for relaxation downward from level 2 to level 1.

These two equations can then be combined into a single rate equation for the inverted population difference \(\Delta{N}(t)=N_2(t)-N_1(t)\) in the form

\[\tag{39}\frac{d}{dt}\Delta{N(t)}=-[W_p(t)+1/\tau]\Delta{N(t)}+[W_p(t)-1/\tau]N\]

For the special case of constant pump intensity, this can be written in the even simpler form

\[\tag{40}\frac{d}{dt}\Delta{N(t)}=-(W_p+1/\tau)\times[\Delta{N(t)}-\Delta{N_\text{ss}}]\]

This equation has exactly the same form as the relaxation of an elementary two-level system toward thermal equilibrium, except that the population difference \(\Delta{N(t)}\) here relaxes toward a nonthermal steady-state equilibrium value \(\Delta{N_\text{ss}}\) given by

\[\tag{41}\Delta{N_\text{ss}}\equiv\frac{W_p\tau-1}{W_p\tau+1}N\]

and the relaxation rate toward this value is \((W_p+1/\tau)\) rather than simply \(1/\tau\). If the pumping rate is above threshold, or \(W_p\tau\gt1\), then \(\Delta{N(t)}\) of course actually relaxes toward an inverted value of \(\Delta{N_\text{ss}}\).

### Pulsed Inversion

Suppose a square pump pulse with constant pump intensity \(W_p\) is turned on at \(t=0\) in this particular system. (Some sort of pulse-forming network rather than just a single charged capacitor will be required to produce such a square pulse with a standard flashlamp.) The population inversion as a function of time during the pump flash is then given by the transient solution to the preceding equations with the initial condition that \(\Delta{N}(t=0)=-N\). This solution is

\[\tag{42}\frac{\Delta{N(t)}}{N}=\frac{(W_p\tau-1)-2W_p\tau\exp[-(W_p\tau+1)t/\tau]}{W_p\tau+1}\]

Suppose that this pumping rate is left on for a pump pulse time \(T_p\) which is short compared to the atomic decay time \(\tau\); and that the pumping rate \(W_p\tau\) is \(\gg1\), which says that if the pump rate were left on for a full atomic lifetime \(\tau\), it would create a very strong inversion. The inversion just at the end of the pump pulse, or \(t=T_p\), is then given to a good approximation by

\[\tag{43}\frac{\Delta{N(T_p)}}{N}\approx1-2e^{-W_pT_p}\qquad(T_p\ll\tau\quad\text{and}\quad{W_p}\tau\gg1)\]

This then says that (a) the inversion at the end of the pump pulse depends only on the total energy \(W_pT_p\) in the pulse, and not on its duration (or even shape); and (b) the pulsed pump can produce complete inversion of the system, if the pumping energy is large enough (\(W_pT_p\gg1\)).

Figure 6.7 illustrates a ruby-laser amplifier experiment in which a square pump pulse of length \(T_p\) short compared to the atomic decay time \(\tau\) was used to pump a ruby amplifier rod without end mirrors. (Practical values for the pumping circuit might be a flashlamp pulse length \(T_p\approx200\) μs, compared to a ruby fluorescent decay time of \(\tau=4.3\) ms.) A separate probe ruby laser was then used to measure the single-pass gain through the ruby rod just at the end of this pump pulse.

We will learn later that the gain or loss through a laser amplifier measured in dB is directly proportional to the laser population difference \(\Delta{N}\). For \(T_p\ll\tau\) as in these experiments, the ratio of gain \(G_\text{dB}\) just after the pump pulse to initial loss \(L_\text{dB}\) just before the pump pulse is predicted from the preceding equation to be

\[\tag{44}\frac{G_\text{dB}(t=T_p)}{L_\text{dB}(t=0)}=\frac{\Delta{N}(t=T_p)}{\Delta{N}(t=0)}\approx1-2e^{-W_pT_p}\]

Figure 6.8 shows experimental data for this ratio for different values of the total flashlamp pump pulse energy, which is in turn directly proportional to the normalized pump quantity \(W_pT_p\).

The experimental results are in excellent agreement with the simple theoretical formula. Note in particular that a sufficiently powerful and rapid pump pulse can come very close to complete inversion of the ruby transition; i.e., it can pump essentially all of the Cr^{3+} atoms into the upper laser level.

### Laser Oscillation Time Delay

Figure 6.9 shows another simple experimental examination of the transient behavior of populations in a pumped laser system. The small insert in this figure shows the oscillation output from a typical flash-pumped ruby laser, including the time delay \(t_d\) between the start of the pump flash and the onset of laser oscillation.

The pumping pulse \(W_p(t)\) requires a certain amount of time, or a certain amount of integrated pumping energy, before it can pump enough atoms up to the upper laser level to create a population inversion, especially in a three-level system such as ruby.

Once a net population inversion is created, however, the laser oscillation then builds up extremely rapidly, as illustrated by the sharp leading edge of the laser oscillation. (Note also the strong characteristic spiking behavior in the oscillation output.) The smoothly rising curve before the oscillation starts represents pump light leakage and preoscillation fluorescence from the laser rod.

This figure also shows how the reciprocal of the oscillation time delay varies with the total energy in a long flashlamp pulse. For a long square-topped pump pulse, the time delay \(t_d\) to the onset of oscillation can be estimated from the transient solution for \(\Delta{N}(t)\) by finding the time \(t=t_d\) at which \(\Delta{N}(t)\) passes through zero (assuming that the oscillation signal will build up very rapidly once inversion is obtained). This time delay to inversion is given for constant pumping by

\[\tag{45}t_d=\frac{\tau\ln[2W_p\tau/(W_p\tau-1)]}{W_p\tau+1}\]

There is a minimum value of pumping intensity \(W_p\) below which the laser will not reach oscillation threshold at all. If the pumping pulse has a pulse length \(T_p\) several times longer than the upper laser level lifetime \(\tau\), then this threshold pump intensity is given by \(W_{p,\text{th}}\approx1/\tau\) for \(T_p\gg\tau\). For pump intensities below this value, inversion is never reached no matter how long the pump pulse continues.

The reciprocal oscillation time delay \(t_d\) normalized to the upper-level lifetime \(\tau\) can then be written as

\[\tag{46}\frac{\tau}{t_d}=\frac{r+1}{\ln[2r/(r-1)]}\]

where the parameter \(r\) represents the normalized pumping energy above threshold, i.e., \(r\equiv{W_p}/W_{p,\text{th}}\). This simple expression gives a moderately good fit to the experimental data in Figure 6.9.

The next tutorial introduces ** LDPC-coded differential modulation decoding algorithms**.