The transmission characteristics of a Fabry-Pérot (F-P) interferometer based on a fiber Bragg grating (FBG) pair with a built-in long-period fiber grating (LPFG) are theoretically analyzed, and the shift of transmission interference fringe as a function of environmental refractive index is acquired. The influence of the lengths of F-P cavity, LPFG and FBG on the transmission characteristics of the proposed interferometer has been numerically investigated, and the simulation results indicate that the sensitivity of refractive index reaches 2.27 10-6 for an optical spectrum analyzer (OSA) with a resolution of 1 pm.