Second harmonic generation due to quadrupole interaction in a photonic crystal slab: Angle dependence and symmetry of the unit cell

We investigate second harmonic generation (SHG) from a photonic crystal slab consisting of centrosymmetric materials. The SHG signal is observed in the transmission direction when the incident laser excites the quasiwaveguide mode. As the SHG frequency approaches the exciton level, the SHG intensity increases resonantly. When the incident angle is exactly 0, the SHG signal vanishes even if the transmission dip is excited. This fact is readily explained by a quadrupole theory based on the Lorentz oscillator model, where the source of the nonlinearity is the Lorentz force. When the unit cell in the photonic crystal lacks inversion symmetry, the SHG signal is expected even for the normal incidence. It is experimentally demonstrated for a square array of triangular semiconductor slabs.