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At the lab I work at, we have a laser capable of emitting light at a spectrum of wavelength. As far as I know, this spectrum is continuous, in that it can emit from 800 nm to 1200 nm for example.

Conventional lasers emit a single wavelength of light due to the emission signature of specific atoms, which is clearly not the case here. My question is how does a laser like this work? How can it emit variable wavelengths?

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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!

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My guess is that the laser is what's known as a frequency comb. It uses a technique that causes the laser to emit light at many frequencies. The frequencies are separated by a constant small amount. The output in frequency space is something like a series of delta functions. The mechanism for accomplishing this is beyond me though. You should read about a frequency comb.

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