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But if IN-coherent light is sent into an amplifying medium, won't the
result be incoherent light which is brighter? A laser crystal
amplifies what it is given. It is transparent, so it doesn't alter
the wavefronts or change the coherence length.
No. The coherence time of the light will increase due to stimulated
emission. The amplification of this medium is highly nonlinear, the
gain being maximum near the center of the lasing frequency. Nearby
wavelengths will be amplified less.