= Solution
Let $d=\delta t$. At the port that is bright for zero delay, the ideal 50:50 <Michelson interferometer> gives
$$
E_{\mathrm{out}}(t)=\frac12[E(t)+E(t-d)].
$$
Each arm has field <amplitude> one half after two beam-splitter interactions. For $|d|\ll\tau$, approximate both real Gaussian envelopes by $E_0e^{-at^2}$ while retaining their rapidly varying <phases>. Put $A=\omega_0t+bt^2$ and $B=\omega_0(t-d)+b(t-d)^2$. The identity $(e^{-iA}+e^{-iB})/2=e^{-i(A+B)/2}\cos[(A-B)/2]$ gives
$$
\boxed{E_{\mathrm{out}}(t)\simeq E_0e^{-at^2}\cos\!\left[bd\left(t-\frac d2+\frac{\omega_0}{2b}\right)\right]e^{-i[\omega_0t+\psi(t)]},}
$$
where
$$
\boxed{\psi(t)=bt^2-bdt+\frac{bd^2}{2}-\frac{\omega_0d}{2}.}
$$
A delay-dependent constant overall <phase> can be shifted by a port convention, but the expression above follows directly from the specified bright-port sum.
The output <intensity> is a Gaussian envelope modulated by a squared cosine:
$$
I_{\mathrm{out}}(t)\simeq I_0e^{-2at^2}\cos^2\!\left[bd(t-d/2)+\omega_0d/2\right].
$$
The modulation maxima are separated by
$$
\boxed{\Delta t\simeq\frac{\pi}{b|d|}.}
$$
This is the spacing of resolved adjacent pulse peaks when the envelope changes little over a modulation period, requiring $b|d|\tau\gg1$. The source's assumption $|d|\ll\tau$ alone does not ensure several visible peaks: a weakly chirped pulse may show only one broad maximum. At $d=0$ the output is the original pulse and the modulation-spacing formula has no finite-peak interpretation. The figure compares the exact bright-port field sum with the small-delay approximation.
\Image[/past-exam-of-the-laser-science-and-quantum-information-processing-course-of-the-university-of-oxford/2018-chirped-pulse-interference.png]
{title=Temporal <interference> of chirped pulses at small and large Michelson delays}
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