Population inversion

1. What It Is

Population inversion = a non-equilibrium condition where upper laser level population exceeds lower:

N_2 > N_1 (or N_2/g_2 > N_1/g_1 for degenerate)

This is opposite to thermal equilibrium where $N_2 = N_1 e^{-hν/kT} < N_1$ . With $N_2>N_1$ , stimulated emission dominates absorption ⇒ net amplification:

R_{net}= (N_2 - N_1) Bρ >0

Without inversion, net absorption (attenuation). Inversion is necessary (not sufficient; also need gain > cavity loss) for laser oscillation.

2. Why Not Two-Level System (impossibility)

In a strict two-level system pumped by same photons $hν$ , rates: $dN_2/dt = Bρ(N_1-N_2)-A N_2$ . At steady state with large $ρ$ , $N_2→N_1/2$ at best (equalization), never $N_2>N_1$ because absorption and stimulated emission have same $B$ — pumping equalizes. So two levels cannot give CW inversion with same $ν$ pump.

3. How to Achieve — Three- and Four-Level Schemes

Three-level (Ruby): Ground $E_1$ $E_{1}$ , pump to $E_3$ $E_{3}$ (broad), fast non-radiative decay to upper laser level $E_2$ (metastable), laser $E_2→E_1$ $E_{2} \to E_{1}$ .
- Needs >50% of ground population pumped to $E_2$ to invert $E_2$ vs $E_1$ (since lower is ground) ⇒ high threshold, pulsed often.
Four-level (Nd:YAG, He-Ne): Pump $E_0→E_3$ , fast decay to $E_2$ (upper), laser $E_2→E_1$ , fast decay $E_1→E_0$ (empty). Lower laser level $E_1$ nearly empty ⇒ easy inversion (small $N_2$ already > $N_1$ ), low threshold, CW possible.

So practical lasers use 3 or 4 levels with a metastable upper laser level (long lifetime $∼ms$ vs $ns$ spontaneous) to store population.

4. Worked Checks

Thermal at 300K for λ=600 nm: hν/kT≈ (6.6e-34·5e14)/(1.38e-23·300)≈80 ⇒ N2/N1≈e^{-80}∼10^{-35} negligible.

Three-level Ruby: need pump to lift >50% atoms to E2; pulsed flashlamp does; CW would need huge power.

Four-level He-Ne: pump excites He, transfers to Ne E2; lower Ne E1 quickly empties to ground via collisions ⇒ inversion with small pump.

5. Exam Points

Define inversion $N_2>N_1$ and why needed for net stimulated emission — 5 marks, very frequent.
Explain why two-level cannot invert (equal $B$ ) — 5 marks, very frequent.
Compare three vs four-level with diagram and threshold difference — 5-7 marks, very frequent.
Mention metastable state lifetime — 3 marks.

Population inversion