Tensor of the second-order nonlinear susceptibility in asymmetrically strained silicon waveguides: analysis and experimental validation.

We theoretically consider the existence of multiple nonzero components of the second-order nonlinear susceptibility tensor, χ(2), generated via strain-induced symmetry breaking in crystalline silicon. We determine that, in addition to the previously reported χ(xxy)(2) component, the χ(yyy)(2) component also becomes nonzero based on the remaining symmetry present in the strained material. In order to characterize these two nonlinearities, we fabricate Fabry-Perot waveguide resonators on 250 nm thick silicon-on-insulator wafers clad with 180 nm of compressively stressed (-1.275  GPa) silicon nitride. We measure the shifts in these devices' modal effective indices in response to several bias electric fields and calculate the χ(eff,xxy)(2) and χ(eff,yyy)(2) nonlinear susceptibility tensor elements induced by the breaking of the guiding material's inversion symmetry. Through the incorporation of finite element simulations encompassing the theoretical distribution of strain, the applied bias field, and the optical modes supported by the waveguide geometry, we extract two phenomenological scaling coefficients which relate the induced optical nonlinearities to the local strain gradient.