Obtaining limited diffraction beams with the wavelet transform

Generating limited diffraction (or nondiffracting theoretically) beams involves deriving special solutions to a homogeneous wave equation. Previous results have been derived by the Fourier and Laplace transforms. In this paper, we use the wavelet transform to obtain a novel nondiffracting solution. It can be shown that this new solution is equivalent to the second derivative of Lu-Greenleaf's zero-th order X wave or the first derivative of Donnelly's Localized wave. The advantage of the wavelet beams is their localization property, that is, they have smaller sidelobes compared with the previous results. The magnitude decays as 1/r³ along the lateral (r) direction. Although the slowest decay is still 1/√r asymptotically, the sidelobes are reduced to about half those of the broadband X wave. We also show that this new nondiffracting beam can be realized as a limited diffraction beam with finite energy and finite aperture ultrasound transducers.