A derivation of an analytical expression for the propagation of a parabolic pulse in an optical fiber amplifier with a finite Lorentzian gain bandwidth is presented. Through a separation of variables combined with the method of stationary phase, we derive equations in both time and frequency spaces to obtain the analytical solution. This results in a compact analytical form that has a number of physical meanings. It shows that the finite gain bandwidth seriously limits the performance of parabolic amplification by distorting both the chirp and the frequency envelope, thus preventing efficient pulse compression required for high-power femtosecond pulse generation. The validity of the analytical derivation is verified through numerical simulations using the split-step Fourier method, showing an excellent agreement with the derived analytical solution, in pulse shape, chirp, and the optical spectrum.
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