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Construct an original illustrative spectrum using ordinary frequency detuning $x={\nu }_L-{\nu }_0$. Write the longitudinal Doppler shift in frequency units as $u=v/{\lambda }_0$, where ${\lambda }_0=c/{\nu }_0$. Choose a normal distribution with standard deviation ${\sigma }_D$ and a weak-probe Lorentzian line shape with half width at half maximum ${\gamma }_{\nu }$:Thus $B(0)=1$. Choose ${\sigma }_D=120\mathrm{MHz}$ and natural linewidth $\Gamma /(2\pi )=6\mathrm{MHz}$, so ${\gamma }_{\nu }=3\mathrm{MHz}$. The Gaussian-only full width at half maximum is $2\sqrt{2\ln 2}{\sigma }_D\simeq 282.58\mathrm{MHz}$; the averaged profile also includes the finite Lorentzian line shape width.

Add a deliberately phenomenological Lamb dip:Here $A_s$ is the absorption coefficient normalized to the pump-off line-centre value. The factor $0.24$ is chosen, not inferred from populations or a fit. The dip's assigned full width at half maximum $w_s$ borrows the single-atom power broadening scale purely to illustrate contrast versus width. It is not the preceding overlap integral or a fit to historical spectra. There are no crossover features in this isolated-transition toy spectrum.

For $s=1$, the subtracted dip has depth $0.12$ and width $8.49\mathrm{MHz}$; for $s=9$, depth $0.216$ and width $18.97\mathrm{MHz}$. Its half-depth points are $x=\pm w_s/2$. Measure that width relative to the curved background $B(x)$, not a horizontal line through the total absorption minimum.

For a chosen cell optical depth ${\tau }_s(x)=0.4A_s(x)$, the line-centre pump-off transmission is $e^{-0.4}\simeq 0.6703$. At $s=1$ it is $e^{-0.352}\simeq 0.7033$. The transmitted probe therefore increases by about $4.92$ percent relative to pump off. Absorption contrast, transmission contrast and spectral width are three different quantities.

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 $\sqrt{2}\Gamma$ 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.

Theodor W. Hänsch, I. S. Shahin and Arthur Leonard Schawlow reported pulsed dye laser saturation spectroscopy of sodium D lines in 1971. They resolved hyperfine structure; delaying the probe revealed persistent changes in ground-level distributions of velocities from optical pumping. Added argon allowed observation of collisional recovery. These measurements probe population memory as well as line positions.^[ref]

The two-level saturation population model explains depletion in a closed transition; the delayed multilevel optical pumping signal needs additional populations and time dependence. A measured Doppler-free saturation spectroscopy feature can still contain natural linewidth, power broadening, pressure broadening, finite interaction time, imperfect alignment and laser spectral width. The word “Doppler-free” removes a particular thermal broadening mechanism, not every limit on resolution.

Arthur Leonard Schawlow's 1981 recognition concerned laser spectroscopy.^[ref] Theodor W. Hänsch received a distinct award in 2005 for precision laser spectroscopy.^[ref]