= Power broadening
{title2=$W_\omega=\Gamma\sqrt{1+s}$}
In the <two-level saturation population model>, $p_e$ and the population-difference reduction $q$ reach half their resonant values at $|\Delta|=(\Gamma/2)\sqrt{1+s}$. Their half width at half maximum is therefore $(\Gamma/2)\sqrt{1+s}$ and their <full width at half maximum> is $W_\omega=\Gamma\sqrt{1+s}$ in <angular frequency>, or $W_\nu=W_\omega/(2\pi)$ in <hertz>. This is <power broadening>: stronger excitation addresses a wider range of detunings even though the lifetime has not changed.
Increasing $s$ improves the local population-difference reduction $q(0)=s/(1+s)$ but broadens the response. From $s=1$ to $s=9$, this reduction increases from $0.5$ to $0.9$, while the width increases from $\sqrt2\,\Gamma$ to $\sqrt{10}\,\Gamma$. The contrast approaches a limit and the width keeps growing. This single-atom width is not automatically the width of a velocity-averaged pump–probe feature.
For example, if the distribution of <velocities> is approximately constant across the narrow resonant classes, the population-only overlap integral contains two <Lorentzian line shapes>: a probe of half width $\gamma=\Gamma/2$ and a pump hole of half width $a=\gamma\sqrt{1+s}$. Their centres in $kv$ are separated by $2\delta$. The <convolution> of these profiles has half width $a+\gamma$ in that separation, so its <full width at half maximum> as a function of $\delta$ is $a+\gamma=(\Gamma/2)(1+\sqrt{1+s})$. Even this simplified pump–probe width differs from the single-atom <power broadening> formula. Finite <Doppler broadening> and coherent or multilevel effects require further analysis.
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