The index modulation function of edge fiber Bragg grating (EFBG) is acquired using genetic algorithms synthesizing, and a method for fabricating tunable EFBG by changing the diameter of the fiber along its axis and applying tension to the fiber grating is proposed and testified. The influence of physical parameters of the device on the edge filter's reflection spectrum is analyzed. An EFBG with a bandwidth of 0.7 nm at shorter wavelength side and a peak reflectivity of 96% is fabricated by ultraviolet laser-scanning. The edge bandwidth of the fiber Bragg grating may be adjusted by tension and the edge bandwidth at shorter wavelength side increases from 0.7 nm to 2.3 nm as the tension increased from 0 to 1.715 N.