# Laser Pumping and Population Inversion

This is a continuation from the previous tutorial - ** microbending loss**.

Let us now examine in elementary terms the kind of pumping process that can produce the population inversion needed for laser amplification.

### Four-Level Pumping Model

As a simplified but still quite realistic model of many real laser systems, we can consider the four-level atomic energy system shown in Figure 1.29.

We assume here that there is a lowest or ground energy level \(E_0\) and two higher energy levels \(E_1\) and \(E_2\), between which laser action is intended to take place, plus a still higher level, or more often a group of higher levels, into which there is effective pumping from the ground level \(E_0\).

We can for simplicity group all these higher levels into a single upper pumping level \(E_3\). At thermal equilibrium, under the Boltzmann relation, essentially all the atoms will be in the ground energy level \(E_0\).

We then assume that there is a pumping rate \(R_{p0}\) (atoms/second) from the ground level \(E_0\) into the upper pumping level or levels \(E_3\).

This pumping rate may be produced by electron impact with the ground-level atoms in a gas discharge, as in many gas lasers; or by pumping with intense incoherent light from a pulsed flashlamp or a cw arc lamp, as in many optically pumped solid-state lasers; or by several other mechanisms we have not yet discussed.

In any event, the properties of atoms do permit selective excitation from a lowest level primarily into certain selected upper levels, as assumed in this example.

It is then a realistic description of many practical lasers that a certain fraction \(\eta_p\) of the atoms excited upward will relax down, perhaps through a series of cascaded steps, from the upper pumping level \(E_3\) into the intended upper laser level \(E_2\).

We might call \(\eta_p\) the pumping efficiency for the laser system, since the effective pumping rate into the upper laser level (again in atoms/second) is \(R_p=\eta_pR_{p0}\). This pumping efficiency can be close to unity in some solid-state and organic dye lasers, and only parts per thousand or less in many gas laser systems.

We can also assume in the simplest case that atoms relax from level \(E_2\) down to level \(E_1\) with a relaxation rate \(\gamma_{21}\) and from level \(E_1\) down to level \(E_0\) with a relaxation rate \(\gamma_{10}\). The relaxation processes between these levels may be a combination of the radiative and nonradiative processes we have described in preceding sections.

In many practical lasers the fractional number of atoms lifted up out of the ground level \(E_0\) into all the upper excited levels also remains small, so that the ground-level population remains essentially unchanged whether the pumping process is on or not.

The flow of atoms between energy levels under the influence of these pumping and relaxation processes (but not laser action for the minute) can then be described by atomic rate equations which we will discuss in much more detail in later tutorial.

For example, the rate equations describing the laser-level populations in the system shown in Figure 1.29 may be written as (Figure 1.30)

\[\tag{20}\frac{\text{d}N_2}{\text{d}t}\approx{R_p}-\gamma_{21}N_2\]

and

\[\tag{21}\frac{\text{d}N_1}{\text{d}t}\approx\gamma_{21}N_2-\gamma_{10}N_1\]

These equations include the upward pumping rate and the downward relaxation rates into and out of levels \(E_1\) and \(E_2\).

If the pumping process is applied in a continuous fashion and the system comes to a steady-state equilibrium in which \(\text{d}N_1/\text{d}t=\text{d}N_2/\text{d}t\equiv0\), we can solve these equations for the steady-state populations and population difference on the laser transition, in the form

\[\tag{22}N_{2,SS}=R_p/\gamma_{21}\qquad\text{and}\qquad{N_{1,SS}}=(\gamma_{21}/\gamma_{10})N_{2,SS}\]

and hence

\[\tag{23}(N_2-N_1)_{SS}=\frac{R_p(\gamma_{10}-\gamma_{21})}{\gamma_{10}\gamma_{21}}=R_p\tau_{21}\times(1-\tau_{10}/\tau_{21})\]

where \(\tau_{21}\equiv1/\gamma_{21}\) and \(\tau_{10}\equiv1/\gamma_{10}\).

This formula shows that if the lower-level decay rate \(\gamma_{10}\) is fast compared to the upper-level decay rate \(\gamma_{21}\), so that \(\tau_{10}\lt\tau_{21}\), then there will inevitably be a population inversion on the \(2\rightarrow1\) laser transition produced by the pumping process.

Whether this inversion will be large enough to permit continuous laser amplification or oscillation on this transition is another question, obviously depending in part on the pumping efficiency and on how hard we can pump.

### Conditions for Population Inversion

The basic physical requirement to obtain continuous population inversion in this system is that atoms should relax out of the lower laser level \(E_1\) down to still lower levels faster than atoms relax into this level from the upper laser level \(E_2\).

The absolute strength of the population inversion also depends on a strong pumping rate \(R_p\) and a long upper-level lifetime \(\tau_{21}\equiv1/\gamma_{21}\); but the essential condition for population is still that the relative relaxation rates obey the condition that \(\gamma_{10}\gt\gamma_{21}\).

The rate equations for real laser systems can become considerably more complicated, and involve more energy levels and relaxation rates than this simplest example; but the essential features will still be quite similar.

The upper levels in many real lasers, for example, are more or less metastable—that is, they have comparatively long lifetimes. If we can pump efficiently into such a longer-lived upper level, and if there is a lower energy level with a short lifetime or rapid downward relaxation rate, then a population inversion is very likely to be established between these levels by the pumping process.

As we have mentioned, gas discharges and optical pumping are the two most widely used laser pumping processes. The gas discharges may be continuous (usually in lower-pressure gases) or pulsed (typically in higher-pressure gases).

Direct electron impact with atoms or ions, and transfer of energy by collisions between different atoms, are the two main mechanisms involved in gas discharge pumping.

Optical pumping techniques may also be continuous or pulsed. The sources of the pumping light may be continuous-arc lamps, pulsed flashlamps, exploding wires, another laser, or even focused sunlight.

Other more exotic pumping mechanisms include chemical reactions in gases, especially in expanding supersonic flows; high-voltage electron-beam pumping of gases or solids; and direct current injection across the junction region in a semiconductor laser.

The next tutorial discusses about ** laser oscillation and laser cavity modes**.