= Solution
{c}
A <chirp> is variation of the instantaneous <frequency> through a signal. Here the <group delay> relative to the pulse centre at the <optical lens> is
$$
t_{\rm in}(\omega)=\frac{d\psi}{d\omega}=2\eta(\omega-\omega_0).
$$
Consequently $\eta>0$ orders lower <frequencies> before higher <frequencies>; $\eta<0$ reverses that ordering. A representative strongly chirped <optical pulse> is
$$
E(t)=A(t)\cos\!\left(\omega_0t+\frac{t^2}{4\eta}+\phi_0\right),
\qquad \omega_{\rm inst}(t)=\omega_0+\frac{t}{2\eta},
$$
with a smooth localized envelope $A(t)$. The carrier oscillations become closer together towards later times for positive $\eta$ and farther apart for negative $\eta$.
The relation between the <spectral phase> and a locally assigned instantaneous <frequency> is a <stationary phase method>, not an exact reconstruction from <spectral phase> alone: the spectral <wave amplitude> also matters. For example, if the spectral envelope is $e^{-(\omega-\omega_0)^2/(2\sigma^2)}$, a <Fourier transform of a Gaussian> gives the exact real-field <phase> $\omega_0t+\eta t^2/(4|a|^2)$ up to a constant, where $a=1/(2\sigma^2)-i\eta$. The slope retains the sign of $\eta$ and approaches $1/(2\eta)$ for $|\eta|\sigma^2\gg1$.
\Image[/past-exam-of-the-laser-science-and-quantum-information-processing-course-of-the-university-of-oxford/2021-chirp-and-flying-focus.png]
{title=Opposite <chirps> and the velocity branches of a <flying focus>}
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