= Solution
A time delay $d$ multiplies the spectral field by $e^{i\omega d}$. The bright-port <transfer function> is therefore
$$
H(\omega)=\frac{1+e^{i\omega d}}2=e^{i\omega d/2}\cos\frac{\omega d}{2},\qquad
|H(\omega)|^2=\cos^2\frac{\omega d}{2}.
$$
The spectral transmission maxima have <angular frequency> spacing $\Delta\omega=2\pi/|d|$. For a strongly chirped pulse the local frequency-time mapping is $\omega(t)=\omega_0+2bt$. Thus the filter's adjacent transmitted <frequency> bands emerge at times separated by
$$
\boxed{\Delta t=\frac{\Delta\omega}{d\omega/dt}=\frac{\pi}{b|d|},}
$$
as obtained directly from the temporal fields. The spectral <transfer function> itself is exact; mapping its features directly onto time requires a sufficiently strong chirp and overlapping envelopes.
As $|d|$ approaches the pulse duration, the two envelopes overlap only partly and the small-delay formula fails. For $|d|\gg\tau$, the selected port contains two separate replicas of field <amplitude> $E_0/2$, each with one quarter of the input pulse <energy>. Their combined <energy> is half the input; the other output port carries the remainder. There is no temporal overlap or corresponding time-resolved <interference>. A sufficiently resolving spectrometer can still reveal fine spectral fringes, whose spacing decreases as $1/|d|$.
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