  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 THz emission at angles tuned by design between
0{\deg} and 50{\deg} was realized, leading to an original phase-matching
approach, lithographically independent, for highly collimated THz QCLs.
