The relay optical system was widely used in optical systems such as infrared spectral imaging system, light field imaging system, optical microscopy system, polarization interference imaging system, compound eye imaging system, ring-belt panoramic optical system, multi-scale imaging system and head-mounted enhanced display system, etc, which can link up, match pupil and deflect optical systems. The structures of the existing relay optical system were studied. The design method of telecentric off-axis three-mirror optical system with front aperture and the description method of free-form surface were introduced. According to the design parameters, the telecentric relay optical system with broadband and real entrance pupil was completed. The system was an off-axis three-mirror optical system structure, and each mirror was a free-form surface described by XY polynomial. The results of CODEV software simulation show that the MTF of the system is close to the diffraction limit, the distortion is less than 1%, and the imaging quality is good with the working spectral range of 0.4-5.0 μm, f'=400 mm, F/3 and 2ω=8°.