First, the spectrum isn't really continuous because you need a standing wave in the laser cavity. But the difference between them is $\frac{c}{2L}$ where L is the cavity's length which makes the spectrum look continuous in a regular spectrum analyzer, for example a $30cm$ long resonator will have a difference of $500MHz$ which is negligible compared to the optical frequency.
Second, every atom has its own "gain spectrum" - the frequencies which it can easily amplify, which is wide when compared to this difference between frequencies, so what actually makes a laser so monochromatic is the competition between frequencies on the amplifying atoms, the standing wave frequency which is amplified the most eventually wins the competition.
Mode locked lasers, like the kind you have, have a gain medium with a very wide gain spectrum. Here you just need to find such a material, e.g. Ti:Sap. But how do you cancel the mode competition and let all of the frequencies to participate? You force them cooperating by entering something into the cavity which will let pulses laze and doesn't let CW. This laser will have a "time mode competition" which the pulses will win. Pulses are a superposition of a lot of frequencies so you made a laser with a wide spectrum!