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I understand that when there is inhomogeneous broadening in a laser cavity, the resultant gain profile is a Gaussian (as opposed to a Lorentzian profile for homogeneous broadening). Due to the wider nature of a Gaussian profile, it is natural that multiple modes fall within the gain bandwidth (FWHM/full width half maximum). This subsequently leads to the possibility of multimode oscillation.

However, I am struggling to understand why exactly this leads to multimode oscillation. Why can't there be multiple modes within the bandwidth, with only one of them oscillating?

To summarise the question: how does this specific broadening mechanism lead to multimode oscillation?

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    $\begingroup$ Broadening doesn't lead to multimode operation. It allows it. "Any thing that can happen will happen." All the conditions for oscillation are met, so oscillation occurs. $\endgroup$ – garyp May 13 '17 at 21:07
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First of all, in-homogeneous broadening doesn't have to be Gaussian, it happens to be like that only for Doppler broadening (due to Maxwell-Boltzmann distribution of velocities). Other in-homogeneous broadening don't result in a Gaussian gain profile.

In the same way, homogeneous broadening doesn't have to be a Lorentzian, this happens only for life-time broadening.

Second, the mechanism which prevents multiple wavelength from participating in the laser is called "mode competition". It is similar how at ceremonies when someone increases the microphone's volume too much we hear a single frequency oscillating. The gain curve has maximum points which with each round in the cavity are amplified the most until the amplifier gets to saturation. When this happens, because of the homogeneous nature of the broadening, the whole gain curve is lowered so some frequencies stop getting amplified, and this continues on and on until only "the last survivor" remains, a single frequency.

But, when the broadening isn't homogeneous, it means that the different frequencies don't share the amplifying atoms, some atoms amplify more one wavelength and some amplify another, and there is no mode competition. That's why with in-homogeneous broadening there are multiple wavelengths that come out of the laser.

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