In this paper., we present the results of the calculation and design for waveguide and the dependence on the performance of mid-infrared InAlAs/InGaAs/InP quantum cascade (QC) lasers grown by gas source molecular beam epitaxy. The dispersion of refractive indices are calculated by taking into account the contribution of the free-carrier concentration absorption. We also report the first GSMBE grown InAlAs/InGaAs/InP QC lasers emitting at 5.1 mum, operated in pulse mode up to 130 K with T-0 of 208 K and J(0) of 2.6 kA/cm(2). For the QC laser designed with lower waveguide cladding InP concentration of 1 x 10(18) cm(-3), it does not lase. Presented theoretical and experimental results indicate that the free-carrier absorption in both upper InGaAs contact layer and lower InP waveguide cladding does play an important role in the dispersion of the index of refraction of semiconductors at mid-far-infrared wavelengths and influence the performance of QC lasers.