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We study, theoretically and experimentally, interband four-wave mixing in semiconductor optical amplifiers whose gain recovery is accelerated by amplified spontaneous emission (ASE). Across a broad range of wavelength shifts, we observe a considerable increase (over 20 dB) in the conversion efficiency, and a corresponding increase in the optical SNR (over 12 dB), as the device current is increased from 100 to 500 mA. For input pump powers below 1 mW, gain recovery in our device is dominated by internal ASE. Higher pump power levels reduce the conversion efficiency because of the pump-induced gain saturation near the output end. We show that wavelength shifts of up to 25 nm are possible, while maintaining a >;10% conversion efficiency and a high optical SNR (>;25 dB). A major advantage of our scheme is that the use of relatively low pump powers (<; 1 mW) reduces the electrical power consumption for such wavelength converters by more than a factor of 10. We discuss in detail the issue of optimum pump and signal powers. Our study is useful for realizing energy-efficient, modulation-format transparent, wavelength converters for optical networks.