Substrate including photonic crystal and method for manufacturing the same, and surface emitting quantum cascade laser

A method for manufacturing a substrate on which an active layer of a quantum cascade laser is grown, the method comprising:

• on a surface of a compound semiconductor substrate, forming a dielectric film including a nitride film or an oxide film and having a refractive index lower than the refractive index of the compound semiconductor substrate;
• patterning the dielectric film and forming dielectric layers constituting grating points of a two-dimensional diffraction grating, such that each of the dielectric layers has an asymmetric shape in relation to at least one edge of the two-dimensional diffraction grating;
• on the surface of the compound semiconductor substrate and on the dielectric layers, growing a first semiconductor layer having a thickness not greater than 2 μm, and
• flattening the first semiconductor layer by chemical and mechanical polishing process, such that a deviation of a lattice constant of the first semiconductor layer is within a range of minus 1% to plus 1% (±1%) to a lattice constant of a material of the compound semiconductor substrate.