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Example: Three‑level laser with pump rate Rp = 1×10^9 s⁻¹, spontaneous lifetime τ = 1×10⁻⁶ s, and cross‑section σ = 1×10⁻¹⁷ cm². Step 1: Calculate spontaneous decay rate: A21 = 1/τ = 1×10⁶ s⁻¹. Step 2: Find steady‑state excited population N2 using rate equation dN2/dt = Rp – N2/τ = 0 ⇒ N2 = Rp·τ = (1×10^9)(1×10⁻⁶) = 1×10^3 atoms. Step 3: Ground state population N1 ≈ total atoms – N2. Assume total N = 1×10^4 atoms → N1 = 9×10^3. Step 4: Verify inversion: N2 > N1? No, 1×10^3 < 9×10^3. Increase pump. Step 5: Increase Rp to 1×10^10 s⁻¹ → N2 = 1×10^4, N1 ≈ 0 → inversion achieved. Step 6: Stimulated emission rate Rst = σ·Φ·N2. If photon flux Φ = 1×10^12 cm⁻² s⁻¹, then Rst = (1×10⁻¹⁷)(1×10^12)(1×10^4) = 1×10³ s⁻¹. Result: With sufficient pump, population inversion is reached and stimulated emission can dominate.