Short-pulse propagation in semiconductor optical amplifiers (SOAs) has been widely studied for applications in optical signal processing and optical communications areas. Even though it is possible to integrate differential equations numerically, such implementations may not provide adequate insight into the device operation. We propose a systematic way to construct analytical solutions for the gain-recovery dynamics of SOAs and show excellent agreement with numerically integrated results. Our approach makes use of the multiple-scale technique. The main contribution of this work is to put earlier heuristic approaches into a firm theoretical base so that approximate analytical solutions for carrier-recovery dynamics can be systematically constructed for different variants of SOA models. We derive analytical solutions for the signal gain and pulse-energy gain at an arbitrary point with the SOA waveguide. Surpassing previous work in this area, we also show that it is possible to obtain analytical solutions when waveguide attenuation is not negligible.
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