Editing Stimulated emission (section)

=== Saturation intensity ===
The saturation intensity ''I''<sub>S</sub> is defined as the input intensity at which the gain of the optical amplifier drops to exactly half of the small-signal gain.  We can compute the saturation intensity as
: <math>I_S = {h \nu \over \sigma(\nu) \cdot \tau_S }</math>
where
: <math>h</math> is the [[Planck constant]], and
: <math>\tau_\text{S}</math> is the saturation time constant, which depends on the spontaneous emission lifetimes of the various transitions between the energy levels related to the amplification.
: <math>\nu</math> is the frequency in Hz
The minimum value of <math>I_\text{S}(\nu)</math> occurs on resonance,<ref>{{Cite book|last=Foot|first=C. J.|url=https://books.google.com/books?id=kXYpAQAAMAAJ|title=Atomic physics|date=2005|publisher=Oxford University Press|isbn=978-0-19-850695-9|pages=142|language=en}}</ref> where the cross section <math>\sigma(\nu)</math> is the largest. This minimum value is:
: <math>I_\text{sat} = \frac{\pi}{3}{h c \over \lambda^3 \tau_S }</math>
For a simple two-level atom with a natural linewidth <math>\Gamma</math>, the saturation time constant <math>\tau_\text{S}=\Gamma^{-1}</math>.