Radiation field calculations of pulsed ultrasonic transducers Part 2: spherical disc- and ring-shaped transducers A. WEYNS The exact equation to describe the propagation of ultrasonic pressure waves in a lossless, homogeneous medium was evaluated assuming pulse-excited radiators. Acoustical field patterns were studied using the dimensions and shape of the transducer, the ultrasonic frequency and number of cycles within the pulse as parameters. The results emphasize the influence of the piston shape on interference phenomena within the near- and far-field and on beam-narrowing effects in the intermediate range between near- and far-field. The following transducer shapes are studied: spherical disc and spherical ring. Introduction As the information about pulse-excited, spherical radiators is very scanty, it is the purpose of this paper to report the results of numerical calculations describing the beam pattern of these transducers, using a method of presentation similar to that used for plane radiators in Part 1 of this paper.’ The effects on the radiation field caused by transducer shape and size, frequency, pulse duration and number of cycles within the pulse, will be studied. As in the previous paper’, the exact equation describing the propagation of ultrasonic pressure waves in a homogeneous non-absorptive medium of infinite extent is evaluated assuming thickness mode vibration and sine-modulated Gaussian pulse excitation. Numerical calculations show that the malized beam pattern of the number -10 dB pulse width, n,., as well as the ducer radius, a, and the frequency, f, that for plane circular radiators.
influence of cycles influence is almost
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With respect to the lateral resolution of a transducer, two parameters are used: wf and & For each beam pattern a different point of maximal constriction exists for each dB-contour. For this contour the relative value, wr, is defined as the smallest distance between the two corresponding dBcontours divided by the radius a, and dr as the axial distance of this point of minimum beam width divided by the normalization factor (a”/X). The following transducer shapes are studied: spherical disc and spherical ring. Spherical transducer For the spherical radiators considered, secondary diffraction, that is, the fact that waves radiated from any part of the
The author is with Siemens AG, Medical Engineering Group, Sonography Branch, Erlangen, FRG,and with the University of Louvein, Department of Electrical Engineering, Belgium. Paper received 23 July 1979. Revised 2 January 1980.
surface are diffracted by other parts of this surface, will be neglected because the surface S is taken to be much larger than the wavelength. Here, the coordinate system of O’Neil is used’. For possibility of comparison with the acoustic fields of plane circular transducers’, the same transducer radius a and the same normalization factor a’/h, are assumed. Concerning the influence of the normalized radius of curvature A, that is, (radius of curvature)/(a2/Q, Figs la and lb illustrate the dependence of the relative width, wf, and the normalized distance, df, of the -3, -10 and -20 dB contours, as results from numerical calculations. The form of the graphs for wf and df shows that the smaller A is, the greater the reduction rate of both parameters will be; this means that a decrease of the radius of curvature causes an increase of the beam-narrowing effect, which is more important the smaller the radius of curvature. This especially holds for the -20 dB contours. As Fig. lb illustrates, reducing A from m to 2.0 results in a decrease of the -3 dB df distance of about 30%, whereas a supplementary reduction to 1 .O or 0.5 augments this percentage to 47% or 64% respectively. Fig, la shows that the same holds for the -3 dB wf value width, but here the corresponding reductions are about 2% higher. According to the slope of the graphs in both figures, the increase of the constrictive effect for the -3 dB contour occurs especially for A < 1.7. Dealing with the -10 dB graphs, the corresponding reduction rates of dr are 37%, 55% or 72% respectively, whereas for wf these values are about 8% higher. So, comparing these percentages to those of the -3 dB graphs, the decrease in df of the -10 dB contour as a function of A is about 8% higher than that of the -3 dB contour, whereas for wf this even amounts to 14%. Considering the -20 dB graphs, at least for A < 1 .O the relation between df and A as well as wf and A is almost linear. Here the same influence of A holds : the gradient of wf is higher than that of dr. Another interesting feature : for A < 1 .O the gradient of df of the -20 dB graph equals unity, which means that the
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for the beam width of the -3 dB and -10 dB contours respectively. In Figs 2a and 2b, a comparison with the values of wf and by the Kossoff formula for curved radiators (1) is made. For df as well as wf, the values obtained by Kossoff are always larger than the numerically calculated values, but the difference diminishes for a decreasing A. Dealing with the different pressure contours, Kossoff s approach results in the same df value for all of them.
df of the -3 dB contour values obtained
On the other hand, A does influence the angle of aperture of the beam pattern as much. As the pressure contours beyond the range of minimum beam width are not always straight lines, as can be seen in Figs 4,.5 and 6, it is impossible to define an angle of aperture that describes the whole beam pattern beyond the range of maximal constriction. Nevertheless, since for diagnostic ultrasonic equipment this range is the most important part of the radiation field, the angle of aperture is here ascertained as the angle between the radiator axis and the straight line through the point of maximal constriction and the point at which twice this minimum beam width, wf, is attained. Thus the formula of 8 is: 8 = arctan [wf/(d,f - df)]
Numerical approach * Kossoff’s apprwch
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o Numerical approach x Kossoff’s approach
where Z is the normalized axial distance. (The Kossoff formula was modified for normalized radii of curvature.) Equation (1) holds for harmonic excitation of the transducer. So, according to the directivity factor of the pressure function, the db values must be multiplied by a factor of 0.42 or 0.70
In his paper3 , Kossoff presents a useful formula for the calculation of the theoretical beam width, defined as the diameter between the first off axis zeros: (1)
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) db =A sin [(n/2) (A-Z) (ZA)]
I I5 A
Fig. 1 : a - Influence of normalized radius of curvature,A,on relative -3, -10 and -20 dB minimal beam width, wf, for a spherical discshaped radiator of radius a = 7.5 mm, frequency 2 MHz, nc = 2, modulated Gaussian pulse; b - influence of normalized radius of curvature,A, on normalized -3, -10 and -20 dB focal distance, df, of the point of maximal constriction, for spherical disc-shaped radiator of radius a = 7.5 mm, frequency 2 MHz,nc = 2, modulated Gaussian pulse
focal distance of the -20 dB contour equals the radius of curvature. This is also true for the -10 dB contour, but only ifA <0.3.
A a - Influence of normalized radius of curvature,A, on relaFig. 2: tive -3 dl3 width, wf, for a spherical disc-shaped radiator of radius a = 7.5 mm, frequency 2 MHz, IT= = 2, comparison numerical-Kossoff’! approach; b - influence of normalized radius of curvature, A, on normalized -3 dB distance,df, for a spherical discshaped radiator of radius a = 7.5 mm, frequency 2 MHz,“, = 2, comparison numericalKossoff’s approach
with d,r the normalized distance on the piston- or Z-axis where twice the minimum beam width is attained. The influence of A on B is shown in Fig. 3. Firstly, it is important to notice that the angle of aperture not only depends on A but also on the pressure contour considered, for instance -3 dB or -10 dB. However, the same holds for both contours: a reduction inA increases the angle of aperture, especially for A < 0.5. The considerable increase in 8 for the -3 dB pressure contour in this range is caused by the non-linear beam pattern between the distances df and dzf. Thus for spherical radiators, the optimum value of A, combining a good beam-narrowing effect and a small angle of aperture, lies within the range 0.4
Fig. 5 Ultrasonic pressure pattern for a spherical disc-shaped transducer, A = 0.53, radius a = 7.5 mm, frequency 2 MHz, nc = 2, modulated Gaussian pulse
Considering -3 dB, Kossoff’s formula delivers higher values especially for 0.3 1 S. As the angle 6’ does not give an exact picture of the whole pressure contours beyond the point of minimum beam width, a complete comparison with the Kossoff formula can only be made by comparing corresponding far-field patterns (see Fig. 4). Looking at Figs 5,6 and 7, two surprising characteristics of ultrasonic fields of spherical radiators come to the fore: firstly, as still reported the increase in the beam width beyond the point of maximal constriction is not a linear function of Z, as it is with the plane transducers; secondly, the existence SOT
Fig. 6 Ultrasonic pressure pattern for a spherical disc-shaped transducer, A = 0.32, radius a = 7.5 mm, frequency 2 MHz, nc = 2, modulated Gaussian pulse
Fig. 3 Influence of normalized radius of curvature, A, on -3 dB and -10 dB angle of aperture, 0, for a spherical disc-shaped radiator, radius a = 7.5 mm, frequency 2 MHz, nc = 2, modulated Gaussian pulse Kossoff’s approach 2aFig. 7 Ultrasonic pressure pattern for a spherical disc-shaped transducer,A = 0.27, radiusa = 7.5 mm, frequency 2 MHz,nc= modulated Gaussian pulse :o . I
of interference minima in that range. As Figs 6 and 7 illustrate the location of the zone of the non-linear beam pattern as well as the position of the unusual interference phenomena depend on the value of A. When A has large values, these two characteristics occur for large values of Z.
Fig. 4 Comparison of numerical-Kossoff’s tours for A = 0.80, A = 0.53 and A = 0.27
of -3 dB con-
Thus, in the region of interest, these radiators have a linear beam pattern without the minimum on the Z-axis. A decrease in A causes a reduction in the size of this far-field interference zone as well as a decrease in the distance from the radiator at which it happens. The latter also holds for the range of inhomogeneity.
~__F----A---b A=O.32 Fig. 8 Ultrasonic pressure pattern for a spherical disc-shaped transducer, A = 0.32, radiusa = 7.5 mm, frequency 2 MHz, continuous wave mode (nc = -)
Beam pattern measurements on strongly-focused transducers verify this unusual behaviour. In the case of impulse excitation, Pentinnen und Luukkala4 also found a minimum on the Z-axis beyond the focal range at Z = 0 53 for A = 0.28. This agrees with the location of the minimum in Fig. 7 for A = 0.27. The interference structure beyond the focal range also exists for ‘continuous’ excitation, as shown in Fig. 8. Thus the existence of higher frequencies within the modulated Gaussian pulse, for which the near-field structure extends over greater distances from the transducer, cannot be an explanation for these phenomena. Finally, in contrast to the ultrasonic fields of annular radiators, an increase in the constrictive effect by reduction of A is not accompanied with a complication of the near-field structure or an increase in the number and size of the sidelobes. Spherical annular transducer A combination of a spherical and annular transducer shape, that is, a geometrically focused ring, causes a further increase in the constrictive effect, as shown in Figs 9 and 10. A comparison of the spherical with the plane ring proves that the increase of the beam-confining effect (decrease of wf and &I becomes the more important the smaller A is. Assuming ai = ae/2, the decrease of 35% for df at A = 1 .O rises to 65% at A = 0.32. The same holds for the reduction
a, /a, Fig. 9 Influence of inner radius,ai, on relative -3 dB width, wf, for a spherical annular transducer, ag = 7.5 mm, frequency 2 MHz, nc = 2, modulated Gaussian pulse
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Fig. 10 Influence of inner radius,ai,on normalized -3 dB distance, df, for a spherical annular transducer, ag = 7.5 mm, frequency 2 MHz, nc = 2, modulated Gaussian pulse
of wf, but now the percentage decreases are about 5% greater than the corresponding values for df. Thus the spherical ring shape has a little more influence on wf than on df, which is important for ultrasonic imaging systems. Nevertheless, a comparison of the influence of the ring shape on plane transducers to that on spherical transducers, (that is, comparing Figs 9 and 10 with Fig. 10 of Part 1 of this paper’) demonstrates that the ring structure does not increase the constrictive effect for the spherical transducer as strongly as it does for the plane radiator. The smaller A, the more pronounced this effect is. For instance, at A = 0.53 a decrease from ae/ai = m to ae/Ui = 2 results in a reduction in wf by about 30% in contrast to 50% for the plane annular transducer (Fig. 9) and in a decrease in df by about 15% as opposed to 35% (Fig. 10). For A = 2.0, however, the same reduction in a,,/4 results in a decrease in wf by about 40% and a reduction in df by about 25%. A much more important feature of the geometrical focused ring is the fact that this radiator shape reduces the negative characteristics of the beam patterns of spherical as well as of annular transducers, which were inherent in an increase in the beam-narrowing effect. For instance, for the plane annular radiator a reduction in wf by more than 40% as compared to the plane circular, disc-shaped radiator requires an inner radius of more than one half the outer radius and results in a considerable increase in the side lobes. For spherical discshaped transducers, a reduction in wf by more than 65% as compared to the plane circular disc-shaped radiator requires a value of A smaller than 0.5 and results in an important increase in the angle of aperture, 0, together with an approaching of the far-field interference structure. In the case of spherical ring-shaped radiators, a decrease in 0 can be obtained by increasing the inner radius as shown in Table 1 and in Fig. 11. As the ultrasonic field of the spherical ring has larger side-lobes than that of the spherical disc-shaped radiator, there exists an optimum for the magnitude of the inner radius which depends on the characteristics of the imaging system. Furthermore, Fig. 11 also demonstrates the considerable reduction of the side-lobes using a spherical annular transducer in comparison with those of a plane ring shape. In consequence, the use of an annular radiator can best be combined with a geometrically focused shape. For spherical
reduction reduction reduction
of the near-field structure of the side-lobes in size and number of the constrictive effect
The transducer shape itself also has important consequences on the beam pattern (Fig. 11). Compared to the radiation field of a circular, planar, disc-shaped radiator, the following characteristics apply: The annular shape increases the focusing effect without an augmentation of the angle of aperture, but at the cost of an amplification of the side lobes.
The spherical disc shape improves the beam-narrowing effect more significantly without an amplification of the side lobes, but at the expense of an augmentation of the angle of aperture and of an approach of the far-field interference structure and inhomogeneity in the beam pattern.
shape: -3 dB pressure contour for plane circular, plane annular (et = a,-,/2), spherical (A = 0.53) and spherical annular (A = 0.53,ai =ao/2) transducer,ag= 7.5 mm, frequency 2 MHz,nc = 2, modulated Gaussian pulse. Fig. 11
Table 1. Influence of inner radius, ai, on -3 dB angle of aperture 0, for spherical annular transducer;a, = 7.5 mm, frequency 2 MHz, nc = 2, modulated Gaussian pulse
transducers, the combination with a ring shape should be used if wr or df values less than 0.20 or 0.37 respectively are demanded.
The spherical annular shape reduces the negative features of both previous radiators as well as amplifying their positive effects. Concerning the influence of radiator size and frequency on the normalized beam pattern the same holds as with plane transducers’, that is, a decrease in ap causes a simplification of the near-field structure and a reduction of the side-lobes. For ‘pulse’ excitation, the smaller n, is the less significant both effects are. In the beamconfming range, df and wf are always independent of the radiator size. The same applies to the frequencyf, assuming a ‘continuous’ excitation, whereas for ‘pulse’ excitation, a decrease in f causes a small increase in df and small decrease in wr.
The ultrasonic field of pulse-excited spherical radiators strongly depends on the number of cycles within the pulse. As for plane transducers the transition from ‘continuous’ to ‘pulse’ excitation occurs from n, = 6 onwards and shows the following features:
Weyns, A., Radiation Field Calculations~of Pulsed Ultrasonic Transducers: Part 1 -Planar Circular, Square and Annular ‘I’ransducers, Ultrasonics 18 (4) (1980) 183-188 O’Neil, H.T. Theory of Focussing Radiators,J. Acoust. Sot. Am. 21 (1949) 516-524 Kossoff, G., Robinson, D.E., Garnett, WJ. Ultrasonic TwoDimensional Visualization for Medical Diagnosis, .I. Acoust. Sot. Am. 44 (1968) 1310-1318 Pentinnen, A., Luukkala, M. The impulse response and pressure nearfield of a curved ultrasonic radiator, J. Phys. D. Appl. Phys. 9 (1976) 1547-1557