Waveguide having a phase-matching region

A waveguide comprising:

• a slab of transparent material having first and second opposed sides;
• first and second diffraction gratings supported by the slab at the first side thereof;
• a third diffraction grating supported by the slab at the second side thereof and disposed opposite the second diffraction grating;
• and
• a first phase-matching region conterminous with the first and second diffraction gratings and disposed in an optical path extending between the first and third diffraction gratings and including the second diffraction grating, wherein, for an optical beam propagating along the optical path:
• the first diffraction grating adds a first phase shift ϕ1 to the optical beam reflecting therefrom;
• the second diffraction grating adds a second phase shift ϕ2 to the optical beam reflecting therefrom;
• and
• the first phase-matching region adds a first matching phase shift ϕM1 to the optical beam reflecting therefrom, such that
• min(ϕ1,ϕ2)≤ϕM1≤max(ϕ1,ϕ2).