Theoretical description of a focused Gaussian beam

Abstract

The nonlinear equation in nondimensional form used by S. I. Aanonsen et al. [J. Acoust. Soc. Am. 75, 749 (1984)] is solved to obtain an expression for the sound‐pressure distribution produced by a plane Gaussian radiator with a concave solid lens in an absorbing fluid medium. Analysis of the solutions shows that the sound field of a focusing lens still maintains a Gaussian distribution across the beam without maxima and minima in the nearfield as well as without sidelobes in the farfield. The theory also shows that the focal distance is not equal to that predicted by geometrical acoustics, but is always shorter because of diffraction. The second harmonic pressure also exhibits Gaussian behavior throughout the beam, even in the focal plane, where the second harmonic is observed to be more narrow than the fundamental. The growth of total energy in the second harmonic also is evaluated. The results indicate that the second harmonic initially increases quite rapidly up to the focus. Beyond the focus, the rate of growth of the second harmonic is found to decrease, and, at a certain distance, the second harmonic is found to have an even smaller magnitude than would be the case without focusing. A set of curves is presented which helps to describe the features of a focused Gaussian beam. [Research supported by the Office of Naval Research.]