Minimum radiation force target size for power measurements in focused ultrasonic fields with circular symmetry

The time-averaged ultrasonic power emitted by medical ultrasonic equipment is mostly measured using a radiation force balance, and the question of the necessary target size is of practical importance. The question is answered here by calculations based on a Rayleigh integral algorithm for fields from circular, focusing transducers. This case occurs particularly in the field of high-intensity therapeutic ultrasound. The calculation yields the necessary size of an absorbing target so that the radiation force is 98% of that exerted on an absorber of infinite lateral size, and this as a function of the transducer-to-target distance, of the transducer radius in comparison with the wavelength and of the focus (half-)angle. Several distributions of the transducer vibration amplitude are considered. The Rayleigh integral strictly applies only to planar transducers, but among the amplitude distributions there is also one that allows the simulation of the spherically curved transducer type often found in practice.