Optical Fiber Communications 4th Edition by Gerd Keiser

Exercise 15

Since the gain constant a depends on the frequency, the amplifier gain is also frequency- dependent. The 3-dB bandwidth (full-width at half-maximum, FWHM) is defined as the frequency for which the power gain G ( v ) is reduced by a factor of 2. Assume the gain parameter g has a gaussian profile
where v is the optical bandwidth (the spectral width of the gain profile) and v 0 is the maximum-gain frequency. Starting with Eq. (11.9), show that the ratio of the 3-dB bandwidth 2( v - v 0 ) to the optical bandwidth v is
where log 2 X is the base-2 logarithm of X. What does this equation show concerning the relationship between the amplifier gain and the optical bandwidth