The single lobe far-field patterns produced from terahertz quantum cascade lasers (QCLs) are greatly demanded for various applications, such as imaging, data transmission, etc. However, for a ridge waveguide terahertz QCL, the far-field beam divergence is large due to the fact that the waveguide aperture is far smaller than the terahertz wavelength. This is the case typically for double-metal waveguide terahertz QCL which emits terahertz photons in almost every direction in the space. Even for a single plasmon waveguide terahertz QCL, the divergence angle is as large as 30 in both horizontal and vertical direction. Here, in this work we design and fabricate a double metal third-order distributed feedback terahertz QCL emitting around 4.3 THz, and investigate the characteristics of the longitudinal and transverse modes. This work aims to achieve high beam quality for terahertz QCL by exploiting the third-order distributed feedback geometry, and in the meantime to achieve single longitudinal mode operation. The electromagnetic field distribution in the waveguide is modelled by employing a finite element method. The mode selection mechanism is studied by using the eigen frequency analysis, and the far-field beam is simulated by applying the near-field to far-field Fourier transform technique. The QCL active region used in this work is based on the resonant-phonon design, which is grown by a molecular beam epitaxy (MBE) system on a semi-insulating GaAs (100) substrate. The wafer bonding and traditional semiconductor device fabrication technology, i.e., optical lithography, electron beam evaporation, lift-off, wet and dry etching, are used to process the MBE-growth wafer into the third-order distributed feedback geometry with double-metal waveguides. By carefully designing the grating structures and optimizing the fabrication process, we achieve third-order distributed feedback terahertz QCL with quasi-single-longitudinal mode operation and single lobe far-field beam pattern with low beam divergence in both vertical and horizontal directions. The effect of grating duty cycle on the far-field beam divergence is systematically studied theoretically and experimentally. By the simulation, we finally achieve the divergence angle of 1213 for a third-order distributed feedback laser with a grating duty cycle of 12% that results in an effective refractive index close to 3. The experimental results show good agreement with the simulation. There is still room to further reduce the beam divergence of third-order distributed feedback terahertz QCL by improve the accuracy of the simulation and the fabrication.