Saturation: when the absorbing medium runs out

Every layer so far assumed the upper-state density $n_u \ll n_l \approx N$ (the ground state holds essentially all the atoms). This is the linear regime: the absorption coefficient $k(\nu)$ is fixed by the vapour density and lineshape, independent of how hard we pump.

Push the laser hard enough and this breaks. As $n_u$ becomes comparable to $n_l$, ground-state depletion reduces the absorption coefficient itself:

\[ \kappa(\nu, \mathbf{r}, t) = k(\nu)\,\frac{n_l(\mathbf{r}, t)}{N}\left[1 - \frac{n_u}{n_l}\frac{g_l'}{g_u'}\right] \qquad\text{(M&O Eq 13.1)} \]

The single-atom saturation intensity is $I_s = A_{ul}/B_{lu} \approx 10\,\text{W/cm}^2$ for visible-range alkalis. But under trapping, the effective lifetime grows by $g_0$, so the trapped saturation intensity drops:

\[ I_s^{\text{trapped}} = \frac{I_s}{g_0} \]

For $g_0 = 100$ (typical fluorescent lamp on the 254 nm line), $I_s^{\text{trapped}} \approx 0.1\,\text{W/cm}^2$ — far below the focused-laser intensity in any modern atomic-physics lab. Saturation is the norm, not the exception, in modern experiments.

Explore

Slider the excitation flux density. Watch the upper-state density saturate. Watch the effective absorption coefficient collapse toward zero. Then turn on the three-level scheme to see where the photons actually go in a LightCell-type configuration.

Excitation intensity log₁₀(F/F_s) 0

Trapping factor g₀ 100

Scheme

τ_b / τ_c (storage ratio) 1000

Upper-state population n_u/N

Solid: actual. Dashed: linear extrapolation. Diverges past F_s^trapped = F_s/g₀.

Effective absorption κ/k₀

Vapour bleaches as ground state depletes. At F → ∞, κ → 0 (transparent).

Level populations (three-level)

Storage level b fills up; the lower-line bc transition starts trapping its own photons.

FEYNMAN's worked example. Set g₀ = 100. The trapped saturation intensity is $I_s^{\text{trapped}} = I_s/g_0 \approx 100\,\text{mW/cm}^2$ — easily exceeded by a focused diode laser. At F/F_s = 1 (above the trapped saturation but below the bare one), the upper-state population is already saturated and the absorption is bleached to half. This is the regime where every cold-atom experiment and every DPAL operates. Now switch to three-level. With τ_b/τ_c = 10³ (typical alkali metastable), the storage level fills at a flux density 1000× lower than the bare two-level prediction (M&O Sec 13.1.iii). LightCell's sodium 3p storage level fills at $F_{\text{exc}}\,\tau_b\,\sigma_{bc}\,R \approx 1$, which gates the b–c transition's effective opacity.

The Bennett hole

A narrow-band laser excites only those atoms with the matching Doppler-shifted velocity. At low intensity this doesn't matter — the absorption coefficient depends on the velocity-averaged density. At high intensity the laser burns a hole in the velocity distribution at the resonant velocity — a Bennett hole. The absorption coefficient is no longer determined by the bulk vapour density but by the rate at which velocity-changing collisions refill the hole.

For LightCell-relevant CFR conditions (collisional broadening > natural), the Bennett hole washes out fast and we can ignore it. For a low-pressure pure-Hg discharge, the hole matters and complicates the saturation kinetics. See M&O Sec 13.2.1.

What this layer does NOT do

Doesn't solve the full non-linear Holstein equation — that needs Auer-Heasley complete linearisation or operator-perturbation, both deferred to Layer 6.
Uses the single-atom $I_s/g_0$ rule of thumb for the trapping-enhanced saturation. Real spatial heterogeneity (M&O Eq 13.6) gives intermediate values; deferred to Layer 6.
Doesn't address very-strong-field effects (ac-Stark splitting, power broadening). M&O Appendix G.

FEYNMAN ✓ worked example · Victor ✓ four reactive controls · Kay ✓ reuses Layer 1 g₀ · Molisch ✓ Eqs 13.1, 13.5, 13.6 cited