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The impact of element size on the ultrasound field of a planar array of variable radius with a phase defined focal depth had to be considered. Since no method for addressing this has been previously reported, a geometric approach has been adopted in the present work. For elements of a finite size, it is useful to note that the path length from the outermost edge of an element to the focus is different from the path length from the element centre. Let us call this difference in path length ∆P ath. In particular, it is useful to consider the outermost element in an array which, for focussing on the central axis of symmetry, will exhibit the largest value of ∆P athof any element, which shall be called ∆P athmax.

Figure 3.4 illustrates the definition of ∆P athmax, which is the difference in

length between paths r1 and r2 for the outermost element in an array. If it is assumed that angle θ3 π₂, then angles θ1 and θ2 are equal and can be calculated

Figure 3.4: Diagram showing the definition of ∆P athmax, where ael is the element

radius, aarr is the array radius, and f is the phase-defined focal depth.

by equations 3.4 and 3.5 whereaelis the element radius andaarris the array radius.

θ1 sin1 ∆P athmax ael (3.4) θ2 sin1 aarr r2 (3.5)

Combining these two equations allows ∆P athmax to be expressed in terms

of the element radius ael, the array radius aarr, and the length of path r2. Using

Pythagoras’ theorem, r2 aa2

arr f2, where f is the phase-defined focal depth.

Hence ∆P athmax can be expressed as shown in equation 3.6.

∆P athmax ael.aarr a a2 arr f2 (3.6)

Simulations were performed to look at the impact ∆P athmax has on the

field produced by an array. This was done using an annular grid of circular elements as illustrated in figure 3.5, where the array radius, the element radius, and the phase-defined focal depth were varied in order to vary ∆P athmax.

Figure 3.6 illustrates the effect ∆P athmax had on the 1D intensity profiles

taken along the central axis of the transducer through the focal region, for an array of radiusaarrand a phase-defined focal distancef of 80mm. The relationship between

intensity and element size will be discussed shortly, but first the relationship between the focal geometry and element size will be considered.

−20 −10 0 10 20 −25 −20 −15 −10 −5 0 5 10 15 20 25 x [mm] y [mm]

Figure 3.5: Layout of the annular grid of 37 circular elements used in the simulations studying the effect of increasing the element size. ∆P athmax was variable, depending

on the chosen array radiusaarr, element radius ael and phase-defined focal distance

f.

From figure 3.6 it can be seen that the length of the focal zone was larger for arrays composed of larger elements. As ∆P athmax increased beyond approximately

0.5λ (where λ is the ultrasonic wavelength in the medium), the focal region grew significantly, with a regional minimum appearing where the focal peak was expected as ∆P athmax increased beyond about 0.7λ.

Figure 3.7 illustrates the relationship between the length of the focal region and ∆P athmax, in this case for an array radiusaarrof 25 mm and for phase-defined

focal distancesf of 60, 80 and 100 mm. Simulations were performed for other values of aarr and f and gave the same general result: the length (and hence volume) of

the focal zone increased sharply when ∆P athmax increased beyond approximately

0.5λ. Beyond this limit, waves from the edge of an outermost element in the array arrive at the focus out of phase with waves from the centre of the same element, leading to destructive interference and a local minimum within the focal region, as was seen in figure 3.6.

It should be noted that while figures 3.6 and 3.7 were both obtained from the same set of simulated fields, care must be taken if making a visual estimate of the -3 dB focal length from the curves in figure 3.6, since the peak focal intensity of the 0.4λprofile is taken as the 0 dB point for all four curves in figure 3.6. In contrast,

0 20 40 60 80 100 120 140 −9 −6 −3 0 3 6 Z−direction / mm Intensity / dB 0.7λ 0.6λ 0.5λ 0.4λ

Figure 3.6: Simulated intensity profiles along the central axis of symmetry for an annular grid of 37 circular elements, where the array radiusaarr25 mm, the wave-

length λ 1.48 mm, and the phase-defined focal distance f 80mm. The profiles shown are for arrays of four different element radii, chosen such that∆P athmax

0.4λ, 0.5λ, 0.6λ and 0.7λ. Note that the length of the focal zone increased in size significantly as∆P athmax was increased beyond 0.5λ, and a regional minimum (in-

dicated by the arrow) appeared at the focal peak as ∆P athmax was increased beyond

0.7λ. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 10 20 30 40 50 60 70 80 ∆ Pathmax / λ Focus −3dB Length / mm f=60mm f=80mm f=100mm

Figure 3.7: Graph showing the -3dB length of the focal zone from simulations of an annular grid of 37 circular elements, where the array radiusaarr25 mm, and the

wavelengthλ1.48 mm. ∆P athmax was varied by modifying the element radii and

the phase-defined focal distancef. Note the sharp increase in the length of the focal zone as∆P athmax was increased beyond 0.5λ.

in figure 3.7 the -3 dB focal lengths have been measured relative to the peak focal intensity of the profile in question. That is, for the 0.6λdata, the focal length uses the peak focal intensity of the 0.6λprofile as the 0 dB point.

A similar relationship was seen between the width of the focal region and element size. The focal width also increased with increasing ∆P athmax, although

without such a sharp increase beyond 0.5λas was seen for the focal length.

To summarise, in order to produce a tight focal region ∆P athmax should be

constrained as shown in equation 3.7. In combination with equation 3.6, this limits the size of element that can be used in a planar array.

∆P athmax  

λ

2 (3.7)

Let us now return to the relationship between intensity and element size. In the simulations described here, the power delivered to the array remained constant per unit area of PZT, meaning that an array with larger elements was capable of delivering more power. As a result, the intensity of the focus increased as the element radius (and hence ∆P athmax) increased. The relationship between element radius

and the simulated focal intensity is illustrated in figure 3.8a, which demonstrates that there is a linear relationship (when plotted on a log-log scale) for the region of the graph where the elements are small. Best fit straight lines are shown on the plots, illustrating that the intensity I increases with the element radius ael to the

fourth power as described in equation 3.8. (This relationship can be verified visually from figure 3.8a, since on a log-log scale a fourth power relationship appears as a straight line with a gradient of four.)

I9a4_el (3.8)

That the peak focal intensity increases with the element radius to the fourth power may seem unintuitive, but can be understood by considering the pressure amplitude rather than the intensity. The peak focal pressure amplitude increases linearly with the element area, since it is the integral of the pressures produced by each elemental area of the vibrating element. (This is the principle upon which Ul-

trasim is based, as described in section 3.2.1.) Intensity is proportional to the square of the pressure amplitude, and therefore the peak focal intensity is proportional to the square of the element area. This then leads to the relationship described in equation 3.8, that the peak focal intensity is proportional to the fourth power of the element radius.

As mentioned above, figure 3.8a indicates that the relationship between in- tensity and element radius is only valid where the element radius is small: As the radius is increased, the simulated data diverge from the best fit line. For different focal depths, the relationship between intensity and element radius breaks down at different values of the element radius, making it difficult to identify a convenient upper boundary for the relationship between intensity and element radius.

A clearer view of the situation can be obtained by considering the relationship between the peak focal intensity and ∆P athmax, which is shown in figure 3.8b for

the same set of simulations. Again, the straight lines in this plot represent the fits for the region of the plot where the element size was small, where the focal intensity is proportional to the element radius to the 4th power (equation 3.8). The point at which this relationship breaks down is easier to identify in this plot, and the 0.3λand 0.5λ values (labelled in figure 3.8b as A and B respectively) are used as indicators of the approximate points at which: (A) The simulated data diverges from the trend (I9a4_el); (B) The increase in intensity with ∆P athmax levels off. At

values of ∆P athmax¡ 0.5λthere is no further increase in intensity with element

size.

In summary, an analysis of the focal intensity places the same upper limit on the element size as the analysis of the focal geometry, that ∆P athmax should be

kept below λ2.

Other workers have also considered the impact of element size on the function of phased arrays. Gavrilov and Hand [2000] noted in their simulations of a 256 element array (with a diameter of 110mm and a radius of curvature of 120mm) that increasing the element diameter beyond5λresulted in a deterioration of the intensity distribution and the appearance of grating lobes. There are similarities

10−1 100 101 10−5 10−4 10−3 10−2 10−1 100 101 102 Element radius / mm

Peak focal intensity / Arb. units

f=60mm f=80mm f=100mm 10−2 10−1 100 10−5 10−4 10−3 10−2 10−1 100 101 102 ∆ Path_max / λ

Peak focal intensity / Arb. units

f=60mm f=80mm f=100mm

A B

(a) (b)

Figure 3.8: Graphs showing the peak intensity of the focal zone from simulations of an annular grid of 37 circular elements, where the array radius aarr 25 mm,

and the wavelengthλ1.48 mm. Simulations were performed for a variety element radiiael. Plot (a) shows the peak focal intensity against element radius, and plot (b)

shows the simulated peak focal intensity against∆P athmax. In each plot, the points

represent the simulation results while the straight lines illustrate the fit, whereI9a4_el. In plot (b), two vertical black lines labelled A and B are also shown, indicating the locations on the x-axis where ∆P athmax is equal to0.3λ and 0.5λ respectively. See

the main text for further comments onA and B.

between this and the finding from the present analysis that ∆P athmax must be

limited to less than λ2. Wooh and Shi [1998] also considered the impact of element size on the field produced by a phased array. Although their work did not place an upper limit on the useful element size, it did highlight that element size has an impact on the grating lobes. However, as will be demonstrated in the next section, the geometry of the element layout is the critical factor in this regard.