Application of the simplified bond-hyperpolarizability model to fourth-harmonic generation

Using the simplified bond-hyperpolarizability model, we obtain analytic expressions for the first-forbidden ͑spatial dispersion, magnetic dipole/electric quadrupole͒ bulk contributions to second-harmonic generation for centrosymmetric materials. Applying these to oxidized Si, we show theoretically and by comparison to experiment that the relative bulk contribution near 800 nm is minor, less than half that of the interface, but that the coherent superposition of bulk and interface contributions is important and cannot be neglected.