Second-harmonic generation and unique focusing effects in the propagation of shear wave beams with higher-order polarization
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This dissertation is a continuation of the work by Zabolotskaya (Sov. Phys. Acoust. 32, 296-299 (1986)) and Wochner et al. (J. Acoust. Soc. Am. 125, 2488-2495 (2008)) on the nonlinear propagation of shear wave beams in an isotropic solid. In those works, a coupled pair of nonlinear parabolic equations was derived for the transverse components of the particle motion in a collimated shear wave beam, accounting consistently for the effects of diffraction, viscosity and nonlinearity. The nonlinearity includes a cubic nonlinear term that is equivalent to the nonlinearity present in plane shear waves, as well as a quadratic nonlinear term that is unique to diffracting beams. The purpose of this work is to investigate the quadratic nonlinear term by considering second-harmonic generation in Gaussian beams as a second-order nonlinear effect using standard perturbation theory. Since shear wave beams with translational polarizations (linear, elliptical, and circular) do not exhibit any second-order nonlinear effects, we broaden the class of source polarizations considered by including higher-order polarizations that account for stretching, shearing and rotation of the transverse plane. We find that the polarization of the second harmonic generated by the quadratic nonlinearity is not necessarily the same as the polarization of the source-frequency beam, and we are able to derive a general analytic solution for second-harmonic generation that gives explicitly the relationship between the polarization of the source-frequency beam and the polarization of the second harmonic. Additionally, we consider the focusing of shear wave beams with this broader class of source polarizations, and find that a tightly-focused, radially-polarized shear wave beam contains a highly-localized region of longitudinal motion at the focal spot. When the focal distance of the beam becomes sufficiently short, the amplitude of the longitudinal motion becomes equal to the amplitude of the transverse motion. This phenomenon has a direct analogy in the focusing properties of radially-polarized optical beams, which was investigated experimentally by Dorn et al. (Phys. Rev. Lett. 91, 233901 (2003)).