Effects of affine source polarization on second-harmonic generation in focused shear wave beams in tissue-like media

Recent analyses of second-harmonic generation in a focused shear wave beam, both analytically [Kaufinger et al., Wave Motion 139, 103595 (2025)] and numerically [Kaufinger et al., JASA 157 (2025); doi: 10.1121/10.0037533], have been concerned primarily with radially polarized beams for comparison with corresponding measurements in tissue-mimicking gelatin phantoms [Cormack et al., IEEE TBME 71, 621 (2024)]. A radially polarized fundamental beam yields a second harmonic that also has radial polarization. However, other source polarizations may be considered. We consider the class of affine polarizations, for which the radiated beam polarization is described as a superposition of constant polarization, e.g., linear or elliptical, and first-order stretches and shears, e.g., radial, torsional, or pure shearing motion. An analytical and numerical assessment is conducted for second-harmonic generation in focused shear wave beams in tissue-like media with arbitrary affine source polarization. In general, it is found that the polarization of the second harmonic does not match that of the fundamental beam. Particular attention is devoted to the longitudinal component of the displacement field that may be used for elastography measurements in potential medical applications. [PGK is supported by the ARL:UT Chester M. McKinney Graduate Fellowship in Acoustics.]