Quantum Mechanics and Laser Gain

Stub. Full prose lives in STUDY_PLAN.md §Week 2. The densest week — give it 7 hrs.

Goals

• Know what a photon is and the relation $E_\text{photon} = h\nu$.
• Understand spontaneous emission, absorption, and stimulated emission as Einstein's three processes.
• Be able to write and physically interpret the Frantz–Nodvik equation for amplifier gain in both small-signal and saturated regimes.

Master equations

Small-signal gain (Frantz–Nodvik):

$E_{out} = E_{in}\, e^{g_0 L}$

Saturated regime:

$E_{out} = E_{sat} \ln\!\left[ 1 + (e^{E_{in}/E_{sat}} - 1)\,e^{g_0 L} \right]$

with $E_{sat} = h\nu / \sigma_{em} \approx 4\,\text{J/cm}^2$ for Nd:phosphate glass.

Einstein relation:

$\frac{A_{21}}{B_{21}} = \frac{8\pi h \nu^3}{c^3}$

NIF tie-in

• The PAM runs in the small-signal regime: 1 nJ seed → 6 J at the output. Almost the full $e^{g_0L}$ multiplier ($6\times 10^9$).
• The main amplifier runs deeply saturated: each 4-pass extraction draws stored energy. This is the most efficient operating point but also the most sensitive to slab darkening — darkened glass doesn't just reduce $g_0$, it reduces extractable stored energy, hitting $E_\text{out}$ directly.

Self-check