Fresnel Diffraction

In Section 13.7 of the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss the medical uses of ultrasound. One important problem we analyze is the pressure distribution produced by a piezoelectric transducer.

There are some important features of the radiation pattern from a transducer which we review next. Consider a circular transducer, the surface of which is oscillating back and forth in a fluid… Each small element of the vibrating fluid creates a wave that travels radially outward, the points of constant phase being expanding hemispheres. The amplitude of each spherical wave decreases as 1/r, the intensity falling as 1/r². We want the pressure at a point z on the axis of the transducer. It is obtained by summing up the effect of all the spherical waves emanating from the face of the transducer….

The [average intensity] is plotted in Fig. 13.13 for a fairly typical but small transducer (a = 0.5 cm, f = 2 MHz)... Close to the transducer there are large oscillations in intensity along the axis: there are corresponding oscillations perpendicular to the axis, as shown in Fig. 13.14. The maxima and minima form circular rings. This is called the near field or Fresnel zone… The depth of the Fresnel zone is approximately a²/λ [where a is the radius of the transducer, λ is the wavelength, and f is the frequency].

Fig, 13.13.

The calculated intensity along the axis, as shown in our Fig. 13.13, is interesting. In the Fresnel zone, the intensity has many points where it is zero. In Intermediate Physics for Medicine and Biology we calculate why this happens mathematically, but it is illuminating to describe what is happening physically. Basically, this is a result of wave interference. Our statement that “each small element of the vibrating fluid creates a wave that travels radially outward” is often called Huygens principle. Each point on the face of the transducer produces such a wavelet. To understand the pressure distribution, we must examine the phase relationship among these various wavelets. Very near the face of the transducer, the waves that contribute significantly to the pressure are in phase; they all interfere constructively and you get a maximum (evaluate Eq. 13.39 at z = 0 and you get a nonzero constant). However, as you move away, more distant points on the transducer face contribute to the pressure on the axis, and these points may be out of phase with the pressure produced by the point at the center. For some value of z the in-phase and out-of-phase wavelets interfere destructively, resulting in zero intensity. Increase z a little more, and not only do the in-phase points at the center and the out-of-phase points just away from the center contribute to the pressure, but so do some in-phase points even farther from the center. When you add it all up, you get a net constructive interference and a non-zero intensity. And so it goes, as you move out farther and farther along the z axis.

Fig. 13.14.

The radial distribution of the intensity is surprisingly rich and complex, given the rather simple integral that underlies the behavior. If you want to explore the radial distribution in more detail, go to the excellent website http://wyant.optics.arizona.edu/fresnelZones/fresnelZones.htm, where you can perform these calculations yourself. You can adjust the parameters as you wish and create plots such as those in Fig. 13.14, and also produce grayscale images of the full intensity distribution that provide much insight. The website was produced with optics in mind, so you have to put in strange looking parameters to model ultrasound. To reproduce the middle panel of Fig 13.14, input 770,000 for the wavelength in nm, 10,000 for the aperture diameter in microns, and 15.75 for the observation distance in mm. To my eye, the agreement between the website’s calculation and Fig. 13.14 is impressive. At small values of z the plots get very complex and beautiful. For the same wavelength and aperture, I like the richness of z = 5 mm, and for z = 4 mm you get a fairly uniform brightness except for a dramatic dark spot right at the center. It reminds me of Poisson’s spot, which I discussed in the September 17, 2010 entry in this blog, about Augustin-Jean Fresnel. Indeed, the physics behind the calculations in Fig. 13.14 and Poisson’s spot in optics are nearly identical. The circular aperture is a classic problem in Fresnel diffraction. You can find a more detailed discussion of this topic in the textbook Optics (4th edition), by Eugene Hecht. (My bookshelf contains the first edition, by Hecht and Zajac, that I used in my undergraduate optics class at the University of Kansas).

If you want to be clever, you could make the ultrasound transducer vibrate only at those radii that result in constructive interference along the axis, and have it remain stationary at radii that cause destructive interference. (Of course, this would mean you would have to design your transducer face cleverly so concentric rings vibrate, separated by rings that do not, which might make constructing the transducer more difficult.) Using such a trick eliminates the dark spots along the z axis, increasing the intensity there. This method is commonly used to focus light waves, and is called a zone plate. It has been used occasionally with ultrasound.