= Solution
Write the optical system's spectral response as $A(\omega)e^{i\phi(\omega)}$. Expanding its <spectral phase> around the pulse's central <angular frequency> gives
$$
\phi(\omega)=\phi(\omega_0)+\phi^{(1)}(\omega_0)(\omega-\omega_0)+\frac12\phi^{(2)}(\omega_0)(\omega-\omega_0)^2+\cdots.
$$
The <group delay> is $\phi^{(1)}=d\phi/d\omega$, with units of time. The linear <spectral phase> term delays the pulse envelope by this amount. The <group delay dispersion> is $\phi^{(2)}=d^2\phi/d\omega^2$, with units of time squared. It makes the <group delay> <frequency> dependent: $t_g(\omega)\simeq\phi^{(1)}+\phi^{(2)}(\omega-\omega_0)$. Positive <group delay dispersion> delays higher <frequencies> more and gives positive <chirp>. A nonzero quadratic <spectral phase> broadens an initially transform-limited pulse; applied to an already chirped pulse, a suitable opposite sign can compress it.
In <chirped pulse amplification>, an ultrashort seed pulse first passes through a dispersive stretcher, often a grating-based system. This correlates <frequency> with arrival time and increases pulse duration while approximately preserving pulse <energy>, thereby lowering peak <intensity>. An optical amplifier can then increase pulse <energy> without exposing the gain medium to the original short pulse's enormous peak <intensity>, reducing optical damage and unwanted nonlinear <phase> accumulation. A final dispersive compressor, typically using gratings, cancels the accumulated <spectral phase> to recover a short, high-energy pulse. The essential components are therefore a short-pulse source, stretcher, amplifier and matched compressor; spectral gain bandwidth and higher-order <dispersion> limit the achievable compression.
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