What Is Gain Narrowing in Lasers?

[unscientific]

A laser is amplified exponentially, with spectral intensity

$$I (\omega,z) = I (\omega,0) e^{\alpha z}$$

The small-signal gain coefficient is given by
$$\alpha_{21}(\omega-\omega_0) = N^* \sigma_{21}(\omega-\omega_0) = N^* \frac{\hbar \omega_0}{c}B_{21} g_B(\omega-\omega_0) = N^* \frac{\hbar \omega_0}{c}B_{21} \frac{1}{\pi} \frac{(\frac{\Delta \omega_L}{2})}{(\omega-\omega_0)^2+(\frac{\Delta \omega_L}{2})^2}$$

So obviously, the gain will be the strongest when $\omega=\omega_0$. But I don't understand the concept of 'Gain narrowing' as described:

 

[blue_leaf77]

An alternative way to understand power density amplification is to think the gain medium as a frequency filter. If the width of the filter, or the gain bandwidth, is smaller than the input beam bandwidth, the output will obviously be cut off by some fraction resulting in narrower bandwidth, which termed as gain narrowing.

 

[unscientific]

An alternative way to understand power density amplification is to think the gain medium as a frequency filter. If the width of the filter, or the gain bandwidth, is smaller than the input beam bandwidth, the output will obviously be cut off by some fraction resulting in narrower bandwidth, which termed as gain narrowing.

So you're saying: Input wide beam, only part of beam that is close to $\omega_0$ gets amplified exponentially. Then output beam: narrow close to $\omega_0$.

 

[blue_leaf77]

So you're saying: Input wide beam, only part of beam that is close to ω0\omega_0 gets amplified exponentially. Then output beam: narrow close to ω0\omega_0.

For small signal case, every part in the beam's spectrum will get amplified exponentially. It's just that the amount of amplification is nonuniform such that frequencies near $\omega_0$ experience much stronger gain than those farther from $\omega_0$.
Actually the main reason of why people arouse this issue of gain narrowing is due to the fact that the laser pulse duration is insensitive to the absolute magnitude of the spectrum, it's only sensitive to the normalized profile of the field spectrum (imagine two Gaussian profile spectra with equal FWHM but different max values, then their time domain fields will have the same FWHM). Therefore even if all parts in the spectrum get amplified, but the output pulse will be longer in time compared to the input due to the narrowing the output.bandwidth.