Why laser needs population inversion

Population inversion

Population inversion is the condition in which the number of atoms N₂ in a higher energy level exceeds the number N₁ in a lower level (N₂ > N₁), which is the opposite of the thermal equilibrium situation.

Why laser action needs it

Three radiative processes occur between levels 1 and 2 (E₂ − E₁ = hν):

• Absorption: rate = B₁₂ N₁ ρ(ν)
• Stimulated emission: rate = B₂₁ N₂ ρ(ν)
• Spontaneous emission: rate = A₂₁ N₂ (random phase and direction)

where ρ(ν) is the energy density of the radiation, and Einstein showed B₁₂ = B₂₁.

1. Net gain from an incident beam: (stimulated emission) − (absorption) = B ρ (N₂ − N₁).
2. Amplification of light requires stimulated emission to beat absorption, so N₂ > N₁.
3. At thermal equilibrium, Boltzmann's law gives N₂/N₁ = exp(−hν/kT), which is always less than 1. For red light (632.8 nm) at 300 K, hν = 1.96 eV and kT = 0.0259 eV, so N₂/N₁ = e⁻⁷⁵·⁸ ≈ 10⁻³³. Almost every atom is in the lower level.
4. With N₂ < N₁ a beam of light is absorbed more than it is amplified; the intensity falls along the medium. At best, with N₂ = N₁ the medium is transparent (absorption equals stimulated emission), and no net gain is possible.
5. Spontaneous emission alone produces incoherent light, not a laser beam.
6. To get N₂ > N₁, atoms are pumped to a metastable upper level (lifetime about 10⁻³ s instead of 10⁻⁸ s), where they stay long enough to build up a larger population than the lower level. This needs at least three levels, since a two-level system cannot be inverted (pumping stops once N₁ = N₂).
7. With inversion the medium has gain: the intensity grows as I = I₀ e^(γx), and with mirrors feeding the beam back, the gain exceeds the losses and laser oscillation starts.

Hence laser action (Light Amplification by Stimulated Emission of Radiation) cannot occur without population inversion.