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Microcavity laser linewidth behavior close to the threshold transition region is investigated through high-resolution linewidth measurements. A local maximum is found for the linewidth of lasers operating slightly above threshold. The increase in laser linewidth close to threshold is explained by the effective contribution of the linewidth enhancement factor (alpha) to the laser linewidth. The Fokker-Planck model of laser noise is then solved using the eigenfunction expansion method to fit to the measured linewidth data. The behavior of the measured linewidth agrees with the model with an extracted linewidth enhancement factor (alpha) ranging between 3.5 and 5.0.