= Pressure-broadened lines and interferogram
Interpret the stated $\tau_1=1\,\mathrm{ns}$ as the coherence-decay time, or the mean interval between fully phase-randomizing collisions in the simple impact model of <pressure broadening>. Then each line has correlation $e^{-|\tau|/\tau_1}e^{-i\omega_j\tau}$. The <collisional phase relaxation and Lorentzian linewidth> transform gives
$$
L_j(\omega)=\frac1\pi\frac{1/\tau_1}{(\omega-\omega_j)^2+(1/\tau_1)^2},
\qquad \boxed{\Delta\omega_{\rm FWHM}=\frac2{\tau_1}=2.0\times10^9\,\mathrm{rad\,s^{-1}},\quad
\Delta\nu_{\rm FWHM}=\frac1{\pi\tau_1}=0.318\,\mathrm{GHz}.}
$$
The intrinsic spectrum is the sum of two such <Lorentzian line shapes>, still centred $5.0\,\mathrm{GHz}$ apart. With equal weights and balanced arms, its normalized visibility is
$$
\boxed{\mathcal V(\tau)=e^{-|\tau|/\tau_1}|\cos(\Delta\omega\tau/2)|.}
$$
For the intensity normalization of the first source plot, the corresponding model <interferogram> is
$$
I(\tau)=4+4e^{-|\tau|/\tau_1}\cos(\Delta\omega\tau/2)\cos(\bar\omega\tau).
$$
The constant background remains $4$; only the oscillatory contrast decays. The beating zeros and recurrence spacing remain fixed, but successive visibility revivals shrink, reaching a factor $e^{-0.2}\simeq0.819$ at $|\tau|=200\,\mathrm{ps}$ and $e^{-1}$ at $1\,\mathrm{ns}$. Long-delay fringes disappear toward the background.
\Image[/past-exam-of-the-optics-and-electrodynamics-course-of-the-university-of-cambridge/2018-b2-pressure-broadening.png]
{title=Calculated pressure-broadened interferogram, decaying visibility revivals and Lorentzian doublet for a one-nanosecond coherence time}
The sketch chooses a carrier <frequency> for legibility; the envelope and linewidth follow from the inferred spacing and specified coherence time. A finite experimental delay window additionally convolves these intrinsic Lorentzians with the instrumental line profile. If $\tau_1$ were instead a population lifetime, the lifetime-limited field decay rate would be $1/(2\tau_1)$ and the corresponding linewidth would be half the boxed value. The source does not define that time convention explicitly; pressure-collision coherence decay is the interpretation used here.
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