Bessel beams of finite amplitude in absorbing media

The profile of a Bessel beam is J₀(ρ), where J₀ is the zeroth‐order Bessel function and ρ is a dimensionless distance from the axis. The beam propagates without diffraction, a property that has stimulated interest in connection with medical ultrasound imaging. Previous analyses of second‐harmonic generation in Bessel beams are limited to lossless media. Du, Zhang, and Zhu [Proc. 14th Intl. Symp. Nonlin. Acoust., edited by R. J. Wei (Nanjing U.P., Nanjing, 1996), pp. 189–194] showed that the beam profile of the second harmonic in the near field is given approximately by J²₀(ρ). Ding and Lu [Appl. Phys. Lett. 68, 608 (1996)] obtained J₀(2ρ) for the far field. We provide a more general analysis of second‐harmonic generation, first by investigating solutions for lossless propagation in greater detail, and second by including absorption. It is shown that the far‐field beam profile is J₀(2ρ) only for very small values of a characteristic absorption parameter. As the absorption parameter increases, the beam profile evolves toward a distribution given approximately (i.e., away from minima) by J²₀(ρ). Numerical results are presented for higher harmonics and for waveform distortion in a Bessel beam that forms a shock. [Work supported by ONR.]