= Solution
A line with <lifetime broadening> has a <Lorentzian line shape>. Write its half-width at half-maximum in angular-frequency units as $\kappa>0$. The <Wiener-Khinchin theorem>, or the <Fourier transform> of the Lorentzian, gives
$$
\Gamma(\tau)=I_0e^{-\kappa|\tau|}e^{-i\omega\tau}.
$$
Therefore in <vacuum>
$$
\boxed{I(y)=I_0\left[1+e^{-\sqrt2\kappa|y|/c}\cos(\sqrt2ky)\right].}
$$
The sketch is a cosine fringe pattern about baseline $I_0$, bounded by $I_0[1\pm e^{-\sqrt2\kappa|y|/c}]$. Its central value is $2I_0$, its spatial <interference fringe spacing> is $\lambda/\sqrt2$, and its visibility decays on length $c/(\sqrt2\kappa)$. For an isolated transition whose population lifetime is $T_1$ and which has no extra dephasing, $\kappa=1/(2T_1)$, so the field <coherence time> is $2T_1$, not $T_1$. Introducing $\kappa$ avoids confusing population, <wave amplitude> and spectral-width conventions when no numerical linewidth is given.
\Image[/past-exam-of-the-optics-and-electrodynamics-course-of-the-university-of-cambridge/2021-b5-coherence-fringes.png]
{title=Original lifetime-broadened interference sketch and the direction of the mean <Poynting vector>}
Back to article page
