What is population inversion and why is it necessary for laser action?

In one line

Population inversion is the state in which more atoms are in a higher energy level than in a lower one, and it is needed so that stimulated emission outnumbers absorption and light gets amplified.

Explained simply

Imagine a classroom where students are either sitting (ground state) or standing (excited). Normally almost everyone sits. If a "stand up" signal (a photon) comes, it is mostly used up making seated students stand, so nothing builds up. Only if you first get more people standing than sitting will each signal cause more people to sit down and shout, making the noise grow.

Technically, at thermal equilibrium the populations follow the Boltzmann law, so \(N_2 < N_1\). A photon of energy \(E_2 - E_1\) is then more likely to be absorbed than to cause stimulated emission. When \(N_2 > N_1\), the system is in population inversion and stimulated emission dominates, giving optical gain.

How it works

1. At equilibrium, \(N_2/N_1 = e^{-(E_2-E_1)/kT}\), which is always less than 1.
2. Rate of absorption is proportional to \(N_1\), rate of stimulated emission to \(N_2\). For amplification we need \(N_2 > N_1\).
3. In a two-level system, strong pumping can at best make \(N_2 = N_1\), because the pump also stimulates atoms down. So inversion is not possible with two levels.
4. Lasers use three- or four-level schemes with a metastable state, a level where atoms stay for about \(10^{-3}\) s instead of \(10^{-8}\) s. Atoms pile up there, and inversion is achieved between the metastable level and a lower level.
5. Four-level lasers (like He-Ne) are easier, because the lower laser level is nearly empty to begin with.

Key formula

\(\dfrac{N_2}{N_1} = e^{-(E_2-E_1)/kT}\): \(N_2, N_1\) are numbers of atoms per unit volume in the upper and lower levels, \(k = 1.38\times10^{-23}\) J/K is Boltzmann's constant, \(T\) is absolute temperature in K.

Quick example

For the He-Ne red line, \(E_2 - E_1 = 1.96\) eV. At 300 K, \(kT \approx 0.0259\) eV, so \(N_2/N_1 = e^{-1.96/0.0259} \approx 1.4\times10^{-33}\). Practically no atoms are in the upper level naturally, which shows why artificial pumping is essential.

Common mistakes

• Saying inversion is possible in a two-level system with strong pumping.
• Confusing metastable state with the ground state; it is an excited level with a long lifetime.
• Writing the Boltzmann ratio without the minus sign in the exponent.

How MU asks it

Asked as "What is population inversion? Explain how it is achieved" (5 marks) or "Explain the role of metastable state" (3 marks). For full marks give the definition, Boltzmann relation, an energy-level diagram of a three-level or four-level system with the metastable state marked, and one line on why two-level systems fail.