= Solution
A <spectral line> has finite width when the emitted or absorbed field loses coherence, or when different emitters have different transition frequencies. It is useful to distinguish <homogeneous spectral broadening> of each emitter from <inhomogeneous spectral broadening> across the ensemble.
\b[Finite lifetime and <dephasing>.] For an upper-state population lifetime $\tau$ and a stable lower state, the radiating <amplitude> decays as $e^{-t/(2\tau)}$. Its Fourier spectrum is Lorentzian, with <full width at half maximum>
$$
\Delta\omega_{\rm nat}=\frac1\tau,\qquad
\Delta\nu_{\rm nat}=\frac1{2\pi\tau}.
$$
More generally, an <optical coherence time> $T_2$ gives $\Delta\omega_{\rm hom}=2/T_2$. Both level lifetimes and pure <dephasing> can contribute to $1/T_2$. Distinguishing population lifetime from <amplitude> coherence time prevents the usual factor-of-two error.
\b[Thermal motion.] To first order in velocity, $\delta\nu=\nu_0v_\parallel/c$. A <Maxwell-Boltzmann velocity distribution> therefore produces a Gaussian profile, with
$$
\sigma_\nu=\frac{\nu_0}{c}\sqrt{\frac{k_BT}{M}},\qquad
\boxed{\Delta\nu_D=\frac{\nu_0}{c}\sqrt{\frac{8k_BT\ln2}{M}}}.
$$
This is inhomogeneous <doppler broadening>: different velocity classes have different <resonance> frequencies. A distribution of bulk velocities can add further broadening, while a common bulk velocity shifts the line.
\b[Interactions with the surroundings.] Collisions can interrupt the dipole <wave phase> and perturb the <energy> levels, producing <pressure broadening> and shifts. In the impact approximation, approximately exponential <wave phase> relaxation produces a Lorentzian contribution; its rate increases with perturber density times an appropriate velocity-averaged collision cross section. Slowly varying electric microfields can instead produce non-Lorentzian Stark profiles. Neutral perturbers, charged perturbers and <resonance> interactions need not produce the same line shape. These mechanisms and the Gaussian thermal limit are distinguished in NIST's atomic-spectroscopy compendium.<Atomic Spectroscopy - A Compendium of Basic Ideas, Notation, Data, and Formulas by William C. Martin and Wolfgang Wiese>[section 20, Spectral Line Shapes, Widths, and Shifts: Doppler and pressure broadening]{ref}
Finite observation time limits coherence; <transit-time broadening> also occurs for particles crossing a narrow optical beam. Strong resonant excitation causes <power broadening>: for an ideal lifetime-limited two-level transition, $\Delta\omega=\Gamma\sqrt{1+s}$ with population decay rate $\Gamma$ and <saturation parameter> $s$. Unresolved <fine structure>, <hyperfine structure>, isotope components, or spatially varying Zeeman and Stark shifts may look like a single broadened feature even though several frequencies are present.
The observed spectrum is additionally convolved with the instrument response. A Gaussian Doppler contribution convolved with Lorentzian homogeneous broadening gives a <Voigt profile>; their full widths do not generally add in quadrature. An informative experiment varies temperature, perturber pressure and excitation intensity separately, calibrates the spectrometer response with a narrower reference, and checks optical thickness or self-absorption. Extrapolation to weak excitation and low pressure helps isolate the lifetime contribution, while a Doppler-free technique is needed if velocity spread still dominates.
\Image[/past-exam-of-the-optics-and-electrodynamics-course-of-the-university-of-cambridge/2015/paper-1-line-profiles.png]
{title=Original normalized Gaussian, Lorentzian and Voigt line-shape comparison}
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