Second-harmonic generation in shear wave beams with different polarizations

Abstract

A parabolic equation describing the propagation of collimated shear wave beams in isotropic elastic solids was derived by Zabolotskaya [Sov. Phys. Acoust. 32, 296–299 (1986)], and was seen to contain both cubic and quadratic nonlinear terms at leading order. While second-order nonlinear effects vanish for the quasi-planar case of linearly-polarized shear wave beams, the importance of quadratic nonlinearity for more complicated polarizations is not yet well understood. The current work investigates the significance of quadratic nonlinearity by considering second-harmonic generation in shear wave beams generated by a certain class of source polarizations that includes such cases as radial and torsional polarization, among others. Corresponding to such beams with Gaussian amplitude shading, analytic solutions are derived for the propagated beam at the source frequency and the second harmonic. Diffraction characteristics are discussed, and special attention is paid to the relationship between the source polarization of the beam and the polarization of the subsequently generated second harmonic. Finally, suggestions are made for possible experiments that could be performed in tissue phantoms, exploiting the theoretical results of this work. [Work supported by the ARL:UT McKinney Fellowship in Acoustics.]