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A new fractional-order derivative operator is presented and applied to nonlinear Sasa-Satsuma equation having Kerr nonlinearity law with third-order dispersion. Two novel families of singular, dark and bright soliton solutions for the fractional-order nonlinear Sasa-Satsuma equation were obtained by considering the modified simple equation method and the F-expansion method. Behavior of these soliton solutions are illustrated in 3d graphs, contour plots and 2d plots. Furthermore, periodic singular solutions of the equation are obtained by both analytical techniques. Optical type soliton solutions of the nonlinear Schrödinger equation is generated as well. To show out the behavior of the modulation gain spectra with the effects of the fractional derivative order, the linear stability technic was used. The effects of the fractional derivative order have been investigated in normal and anomalous dispersion regime associated to the Kerr nonlinearity effects. MI gain spectra were depicted by choosing adequate parameters of the fractional nonlinear Sasa-Satsuma equation. [ABSTRACT FROM AUTHOR]