We have developed terahertz frequency quantum cascade lasers that exploit a double-periodicity distributed feedback grating to control the emission frequency and the output beam direction independently. The spatial refractive index modulation of the gratings necessary to provide optical feedback at a fixed frequency, and simultaneously, a far-field emission pattern centered at controlled angles, was designed through use of an appropriate wavevector scattering model. Single mode terahertz (THz) emission at angles tuned by design between 0 and 50 was realized, leading to an original phase-matching approach for highly collimated THz quantum cascade lasers. Terahertz (THz) frequency quantum cascade lasers (QCLs) have undergone rapid development in performance since their first demonstration, 1 finding potential application in a number of fields including astronomy, security screening, biomedicine, and cultural heritage, inter alia. 2,3 Operating across the 1.2-5 THz range, QCLs can provide high peak output powers (1 W), 4 a high spectral purity, 5,6 frequency, phase and amplitude stability, 7-9 and an ultrabroadband gain spanning an octave in frequency 10,11 at temperatures 199 K. 12 Most of the mentioned applications require sources with a low divergent spatial profile in the far-field as well as a fine spectral control of the emitted radiation. However, the double-metal waveguides conventionally employed to maximize the THz QCL operating temperature 12 suffer from a lack of efficient extraction and a poor collimation of the output radiation, 13 owing to the sub-wavelength dimensions of the resonant cavities. Also, the strong longitudinal confinement provided by such microstrip waveguide configuration typically induces the laser to operate in a multimode regime.Distributed feedback (DFB) is commonly employed in semiconductor lasers to tailor the shape, symmetry, and frequency of the optical resonator eigenmodes, while simultaneously allowing stable single-mode operation. A DFB operates by introducing a periodic refractive index modulation along the propagation direction, which provides scattering between two guided states (feedback) or between a guided state and a radiative one (extraction), so that distributed optical feedback is achieved by coupling two counterpropagating modes through one of the spatial harmonics of a refractive index modulation. In fact, the latter is never purely sinusoidal and a grating with periodicity L contains spatial harmonics with wavevector k n ¼ n/L. Depending on whether the first, second, or third harmonic is used for feedback (n ¼ 1, 2, 3), the grating is described to be of the first, second or third order, respectively. 14 In the case of second 15,16 and third order 17 gratings, although a high spatial harmonic is used for optical feedback (k fb ¼ k 2 , k 3… ), the fundamental frequency is employed for radiation extraction (k e ¼ k 1 ). The ratio between the feedback wavevector k fb and the extraction wavevector k e determines the behavior of the laser: second order gratings emit...