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This paper presents the pulse propagation and gain saturation characteristics for different input optical pulse shapes with different energy levels in semiconductor optical amplifiers (SOAs). A finite-difference beam propagation method (FD-BPM) is used to solve the modified nonlinear Schrödinger equation (MNLSE) for the simulation of nonlinear optical pulse propagation and gain saturation characteristics in the SOAs. In this MNLSE, the gain spectrum dynamics, gain saturation are taken into account those are depend on the carrier depletion, carrier heating, spectral hole-burning, group velocity dispersion, self-phase modulation and two photon absorption. From this simulation, we obtained the output waveforms and spectra for different input pulse shapes considering different input energy levels. It has shown that the output pulse shape has changed due to the variation of input parameters, such as input pulse shape, input pulse width, and input pulse energy levels. It also shown clearly that the peak position of the output waveforms are shifted toward the leading edge which is due to the gain saturation of the SOA. We also compared the gain saturation characteristics in the SOA for different input pulse shapes. [ABSTRACT FROM AUTHOR]