= Solution
{c}
In <mode locking>, fixed relative <phases> between many longitudinal <normal modes> produce constructive <interference> during a short part of each <optical resonator> round trip. The round-trip time sets the repetition period. The available coherent <gain bandwidth>, the number of locked <normal modes>, the <spectral phase>, and residual <group velocity dispersion> determine the <optical pulse> duration.
For $M$ consecutive equally weighted <normal modes> of <angular frequency> spacing $\Delta\omega$, their <wave amplitude> sum is proportional to
$$
\frac{\sin(M\Delta\omega t/2)}{\sin(\Delta\omega t/2)}.
$$
Its central peak has width of order $1/(M\Delta\omega)$, while its repetition period is $2\pi/\Delta\omega$. A <transform-limited pulse> therefore becomes shorter as more coherent <gain bandwidth> is used. This mechanism can give picosecond or femtosecond <optical pulses>: it does not require depletion of the whole stored <population inversion> during each peak, as <Q-switching> does. <Mode locking> generally accesses a much shorter <time-bandwidth product> scale than the gain-discharge time of <Q-switching>.
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