Theory of Acoustically Induced Optical Harmonic Generation

Our recent theory of nonlinear electrodynamics of elastic anisotropic dielectrics is applied to acoustically induced optical harmonic generation (AIOHG) in which two input optical waves and an input acoustic wave are mixed to produce an output optical wave at a frequency displaced from the optical harmonic by the much lower acoustic frequency. The susceptibility governing AIOHG is derived from a fundamental point of view for acentric dielectrics of arbitrary symmetry. It consists of (a) a direct effect represented by a fifth-rank material tensor whose symmetry, frequency dispersion, and relation to other nonlinearities are derived, and (b) five indirect contributions, three being two-step processes and two being three-step processes. The indirect contributions are expressible in terms of lower-order directly measurable material tensors and various wave vectors of the interacting waves. Because of the latter dependence these contributions possess symmetry different from the direct effect and from each other. They can be comparable in magnitude to the direct effect. Rotations present in shear waves are shown to contribute to AIOHG to an extent comparable to that from strains in materials whose second-order optical mixing tensor is large. This shows that the displacement gradient, not the strain, is the measure of elastic deformation relevant to AIOHG. The form of the phase-matched output AIOH wave is derived for waves having an arbitrary orientation in an anisotropic medium. The concept of double phase matching is introduced, whereby not only the output wave is phase matched, but also the intermediate step in one of two types of two-step indirect contributions. Under this condition the output wave grows as the fourth power of the crystal length if pump depletion is negligible. Double phase matching can give an output-power enhancement, compared to single phase matching, of ∼ ${10}^9$ under reasonable conditions.