Biomedical Engineering Reference
decreasing exponentially with distance. The directivity function
in the far field is given by
where J 1 is the first order Bessel function, θ is the angle that the
line joining the point of interest to the center of the transducer
makes with the axis, and λ is the acoustic wavelength in the
propagation medium. For most therapeutic ultrasound appli-
cations involving plane transducers, the tissues being targeted
lie within the near field, and so uniform ultrasound exposure is
most easily achieved by moving the transducer.
FIGURE 5.2 A spherical bowl, multi-element focused ultrasound trans-
ducer. The central aperture allows for the insertion of an imaging device.
5.2.5 Focused Fields
5.2.4 plane transducers
The majority of plane transducers are formed from circular
discs. It is usually assumed that the plane face of these sources
responds linearly to an applied alternating voltage and vibrates
as a whole with uniform amplitude and phase. The field from
such a transducer is commonly described as being composed
of two regions: the near field (Fresnel zone) and the far field
(Fraunhofer zone). Interference effects in the near field result
in complex pressure distributions both on axis and off axis.
Theoretical prediction of the near field is complex. The intensity
on axis can be approximated to
22.214.171.124 Focused Bowl transducers
The simplest way to achieve a focused field is to use a single-
element spherical curved shell of piezoelectric or piezoceramic
material. It is possible to calculate the field from such a trans-
ducer geometry (O'Neill 1949, Penttinen and Luukala 1976).
Figure 5.3 illustrates the pressure distribution within a focused
ultrasound field, both along the sound axis (a) and in a trans-
verse plane through the focal peak (b).
The ultrasound focus lies on the central axis, near the cen-
ter of curvature of the bowl. As 2 π h/λ increases (where h is the
depth of the bowl, as shown in Figure 5.4), the focal point moves
closer to the geometric center of curvature. The gain factor ( G ,
the ratio of the intensity at the center of curvature to the average
intensity at the transducer surface, I C /I 0 ) is given by
where z is the axial distance and I z is the intensity at point z . I z
oscillates in amplitude between I 0 and zero. The intensity var-
ies smoothly on axis in the far field, with the intensity on axis
Axial (z) position from focal peak (mm)
Horizontal (x) position from focal peak (mm)
Pressure distribution close to the focus of a typical focused ultrasound field in the axial (a) and transverse (b) planes.