Bessel and conical beams and approximation with annular arrays

September 25, 1997, TO BE PUBLISHED IN IEEE TRANS. UFFC 1 Bessel and conical beams and approximation with annular arrays Sverre Holm Department of Informatics, University of Oslo P. O. Box 18, N-316 Oslo, Norway Abstract The Bessel beam is one of the relatively new limiteddiffraction beams that have been discovered. It is compared with the conical transducer which also gives an approximate limited-diffraction solution to the wave equation. It is shown that the conical transducer s field deviates from the predicted field in the nearfield, where it is wider. Therefore the Bessel beam is better for use in a hybrid system where a limited-diffraction beam is used for transmission and a dynamically focused beam for reception. It is also shown that the limited-diffraction Bessel beam of order zero can be excited on an annular transducer with equal-area division of elements and with a fixed prefocus, i.e. conventional transducers used in commercial medical imaging equipment. The element division implies that the scaling parameter must be chosen to contain the first lobe of the Bessel function in the first element. In addition, the prefocus must be such that the array is steerable to infinite depth with minor loss. Even when the Bessel beam yields a larger depth of field than that of an unfocused transducer, we show that it has the advantage of a narrower beam. Simulated examples are shown where the approximate Bessel beam compares favorably with a spherically focused beam with a fixed focus, an unfocused beam, and a conical transducer. I. INTRODUCTION The problem of obtaining good lateral resolution and a large depth of field is a much studied problem in medical ultrasound. The nature of the problem is such that it is impossible to satisfy both requirements at the same time, a characteristic it shares with many problems in engineering and science. The baseline for comparison is the spherically focused beam. Other alternatives trade off the two properties in different ways. The line focus produced by an axicon is one such alternative [1], [2], [3]. An axicon can be realized as a thin annulus or a conical transducer, but because of the larger radiating area, the cone has the better sensitivity. Since the conical transducer has a relatively large sidelobe level it cannot be used in an imaging system on both reception and transmission. It has however, been proposed for use in a hybrid system in combination with a spherical focus on reception [1]. In recent years solutions to the wave equation called Bessel beams have been studied [4]. These beams were originally proposed by Stratton in 1941 [5]. The beams have characteristics that are similar to the axicons in that they give a line focus with depth-independent beamwidth out to a certain depth. Lu and Greenleaf have proposed extensions and generalizations called X-waves [6]. Lu and Greenleaf have also made a great effort in studying both Bessel beams and X-waves, and they have been realized with circular transducers [7], they have been used succesfully for medical imaging [8], and they have been studied for realization with two-dimensional arrays [9]. In this paper the properties of the conical and Bessel beams are compared and the similarities are discussed. Due to the improved nearfield of the Bessel beam, it is of special interest to study these beams for use in a hybrid system where a transducer designed for dynamic focusing is used to generate an approximate Bessel beam. In this way a Bessel beam for transmission and a dynamically focused beam for reception can be generated by the same transducer. II. CONICAL TRANSDUCERS AND BESSEL BEAMS In this section, the main characteristics of conical transducers and Bessel beams are defined. The similarities and differences are found and discussed. Most of the discussion is based on a continuous wave model. A. Conical transducers The conical transducer is defined by the angle θ between the cone and the line perpendicular to the acoustical axis, and where θ = is equivalent to a flat radiator. The acoustic field cannot be found exactly, but an approximate analysis based on similarity with a thin annulus is given in [1]. The result is that the sound pressure is proportional to U(ρ) J (kρ sin θ) (1) where k = 2π/λ is the wavenumber, λ is the wavelength, and ρ = x 2 + y 2 is the distance from the center axis of the transducer. The conical transducer gives a line focus that extends almost from the tip of the cone to the depth where the line perpendicular to the cone edge meets the transducer s axis of symmetry [1]. The depth of focus (DOF) can be shown from geometry to be DOF cone R/ tan θ (2) where R is the transducer s radius. The beam is characterized by a relatively large first sidelobe of -8 db. Patterson and Foster [1] concluded that a conical focus for transmission in combination with a spherical transducer for reception, gave a system with a large depth of field and a good lateral point response. Dietz [2] also recommended a conical focus for transmission and dynamic spherical focusing for reception. B. Bessel beams Since the main work on axicons was done, a new solution to the wave equation called the Bessel beam has been studied [4]. In contrast to the conical transducer it has exact limiteddiffraction properties. For an infinite aperture, the sound pressure for a wave of frequency ω, at the observation point (ρ, z)

September 25, 1997, TO BE PUBLISHED IN IEEE TRANS. UFFC 2 where z is the depth, and at time t is proportional to [7]: U(ρ, z, t) = J n (αρ)e j(βz ωt) (3) where J n is the Bessel function of order n. The real parameters α and β are related to the wavenumber by: α 2 + β 2 = k 2 (4) Note that α = gives the plane wave solution. Eq. 3 describes a family of solutions of the wave equation which all have a relatively poor sidelobe to mainlobe ratio. A solution to this problem, which consists of multiple transmissions and subtraction of beams with different orders, has been proposed [1]. Since the Bessel beam of order n = has the best mainlobe to sidelobe ratio, we are only concerned with that beam here. The wavenumber is decomposed into β, which is the component along the z-axis and α, which is the component in the radial direction (ρ-direction). Therefore the Bessel beam is composed of plane waves travelling at an angle to the z-axis given by: sin θ = α/k (5) At a certain depth and with a finite aperture, the plane waves will no longer overlap. This results in a conical shadow zone which defines the depth of field [4]. The shadow zone starts at a depth: DOF Bessel R/ tan θ = R (k/α) 2 1 (6) The Bessel beam is characterized by a -6 db beamwidth of: ρ F W HM = 3.4/α (7) The distance between the first zeros of the beam is: ρ = 4.81/α (8) C. Comparison of conical beams and Bessel beams It is evident that there are many similarities between the conical beam and the Bessel beam. The key to finding the similarities is (5). It makes the lateral fields equivalent (compare (1) and (3)), and the expressions for the depth of field (2 and 6). The differences are: 1. The conical beam is generated by a uniform excited conically shaped transducer (constant θ) while the Bessel beam is excited by a flat circular transducer with Bessel amplitude weighting (n =, constant α and z = in (3)). 2. The cone gives a field which is dispersive in the lateral direction (insert 5 in 7), while the Bessel beam has a lateral beamwidth which is independent of the frequency. 3. Since the equivalent cone angle varies with frequency, a broad-band pulse will be distorted in the radial direction. The dispersion relation is given by (4). The dispersion is minimized for α k [11]. The conical beam does not have this dispersion in the radial direction. These properties are also indirectly evident from (2) and (6), since the depth of field for the Bessel beam will vary with frequency. Properties 2 and 3 are important for broadband pulse excitation, and the result is a slight pulse elongation for Bessel beams [11]. Property 1 gives the important difference between how the beams are generated. There are also other differences between Bessel beams and axicon beams that are not evident from the equations alone. Some simulations will show that. The simulations were done by directly summing a discrete version of the Rayleigh integral as described in [12]. A conical transducer and a Bessel transducer with equivalent parameters were compared. The frequency was 3.5 MHz, the velocity of sound was 154 m/s, and the transducer had a diameter of 15 mm. The scaling parameter for the Bessel beam was α = 1154 m 1. This value was chosen to give a zero at the transducer edge, and the Bessel amplitude weighting was implemented exactly. The field is shown in Fig. 1. The field from the equivalent conical transducer with a cone angle of θ = 4.635 is shown in Fig. 2. It is evident that there is a major difference in the field closest to the transducer, where the conical transducer has a field of the same size as the radiating aperture. This was, to our knowledge, not shown in past papers. In [1] they were concerned with a frustum-shaped transducer and thus the nearfield was not excited, and in [2], [3] one did not explicitly investigate the nearfield. It seems justified to conclude that (1) is indeed only an approximate expression for the field, especially close to the transducer. In contrast, the Bessel beam s properties are accurately predicted by (3). There is also a minor difference in that the Bessel beam seems to degrade faster beyond the predicted value for the depth of field of 92.5 mm (6). In conclusion the Bessel transducer has an advantage in that the nearfield also is determined by the Bessel beam parameters. III. APPROXIMATION OF BESSEL BEAMS The axicons can be implemented by specially designed conically shaped transducers or hybrid combinations of conical and spherical transducers [1]. Axicon beams may also be generated with a single annular transducer by varying the delay profile [3]. Our interest here is how to approximate Bessel beams by more or less standard transducers as in [13]. In [7] and [11] it is shown that the Bessel beam can be well approximated by a flat, finite circular aperture when annular rings are made at the locations of the zeroes of the Bessel excitation function. An example from [7] is a 2.5 MHz, 5 mm diameter, 1 ring Bessel transducer made according to this principle, and where the amplitude on each ring was adjusted to be equal to the peak of the respective Bessel lobe. Lu and Greenleaf also demonstrated experimentally that limited-diffraction beams could be generated. The first ring had a radius of 2 mm and thus from (8), α = 4.81/.4 = 122.5 m 1 and the depth of field (6) is 216 mm. Characteristics of this transducer are: The Bessel function contains many peaks (ten) over the surface of the transducer, thus ensuring that the beam is narrow compared to the radius of the transducer. The depth of field is much smaller than the depth of field of a flat, spherical transducer given by the nearfield-farfield transition z near = R 2 /λ (9) For this example, the transition point is 114 mm. Due to the high sidelobe level, the Bessel beam of order cannot be used for both the transmit and receive beams in an

September 25, 1997, TO BE PUBLISHED IN IEEE TRANS. UFFC 3 imaging system. The receive beam should be a conventional dynamically focused beam. Lu and Greenleaf s Bessel transducer could in principle be used on both transmit and receive, since dynamic focus could be applied by dynamically varying the delay on each ring with time, but it would be even better if one could use an annular array transducer of the type that is in common use. The preferred transducer for use with dynamic focusing is the equal-area transducer. When each element has the same area, the division between elements n and n 1 is found at: r n = R n/n, n [1, N] (1) where N is the number of elements. The annular array should be prefocused to a depth, F, in order to minimize the number of elements and the length of the delay lines. There are two advantages of the equal-area division. First, the phase error over each element will be the same as the beam moves away from the prefocus, i.e. the elements will defocus equally much as one moves away from the prefocus (Fresnel condition). Second the elements will have the same electrical impedance. The phase error over each element for a beam coming from infinite depth can be found from the geometry to be: φ π NS, S = F R 2 /λ (11) where S is the Fresnel number, i.e. the ratio of the fixed focus and the nearfield-to-farfield transition depth. To focus to infinite depth, this delay should be less than about π/2. This array is the starting point of our study: approximation of Bessel beams on equal-area, mechanically focused annular transducers. The equal-area assumption gives a relatively large radius for the inner ring, forcing the first zero of the Bessel excitation to be at this location. The scaling parameter is therefore found by combining (8) and (1): The solution is: ρ = 2r 1 (12) α = 2.45 N (13) R For small and moderate number of rings, this approximation to the Bessel transducer may very well differ much from the main characteristics of Lu and Greenleaf s transducer: The number of peaks in the Bessel function over the surface of the transducer will be small, and the depth-of-field may be comparable to and even larger than the nearfield-to-farfield transition point. The significance of these points therefore needs to be studied. The combination of (6) and (9) gives the condition α > 2π/R in order for the Bessel beam to have a smaller depth of field than a flat piston transducer. Durnin [4] gives this value as the lower limit for α. One would have thought that there would be no point in generating beams with a larger depth of field than that. In combination with (13) this gives a minimum number of equal-area rings of N = 7. On closer examination, this is not necessarily the case since the flat piston gives a beam with a width approximately equal to the size of the piston, while the approximate Bessel transducer gives a beam with a width given by the inner ring. Thus it is in fact possible to generate a beam with a width which is a fraction of the size of the transducer and which extends beyond the nearfield-farfield transition point. The following example will demonstrate this. IV. EXAMPLE Fig. 3 shows the element division and excitation (envelope of (3) for z = ) for a 3.5 MHz equal-area transducer with four rings and diameter 15 mm. The scaling parameter from (13) is α = 641.3 m 1. The contour plot of the acoustic field for an ideally excited transducer, i.e. one where the excitation follows the continuous curve in Fig. 3, is shown in Fig. 4. Note the existence of the extra peaks from depth 175 mm and outwards as predicted by (6) for the depth-of-field (167 mm). This transducer has only two peaks in the Bessel excitation, but still gives a good Bessel beam (compare with Fig. 1). A flat transducer of this size has a nearfield-farfield transition point at 128 mm (9). The field in this case is shown in Fig. 5. Compared to Fig. 4 it is evident that the Bessel beam achieves both a narrower beam and a larger depth of field than the flat piston transducer. It also has a better nearfield than the equivalent conical beam in Fig. 6 (cone angle θ = 2.574 ). They are however, quite similar beyond a range of about 5 mm. The effect of the step-wise approximation to the Bessel function is shown in Fig. 7. It results in some nearfield artifacts. The effect of the mechanical curvature must then be found. With a typical value for fixed focus of F = 75 mm, the result is shown in Fig. 8. The delay for each element has been set to compensate for the mechanical curvature. The Fresnel parameter is S =.59 and the phase error for operation at infinity is.42π. Thus the effect of the curvature can be successfully compensated for by delays. There are some differences from Fig. 4, with the main one being an increased level of near-field artifacts and a slight reduction of the depth-of-field. The beam profile should be compared to the standard spherically focused transducer in Fig. 9. Further comparison can be made when the one-way profiles at specific depths are compared. The profiles at depths 12.5, 25, 75, 125, and 175 mm have been plotted with a 2 db increment in Figs. 1 and 11. The nearfield of the approximate Bessel beam is sharper, and there is a more uniform beamwidth. The spherically focused transducer has a sharper beam at its focus and it does not have the increase in sidelobes at the end of the depth-of-field. V. CONCLUSION The acoustic fields from a conical transducer and from a Bessel transducer have been compared. They have comparable expressions for beamwidth and for depth of field for a single frequency. For broadband operation, the Bessel beam is seen to be equivalent to a conical beam with frequency-dependent cone angle. It is found that the Bessel transducer follows the predicted field much better than the conical transducer does. In particular in the nearfield, the Bessel transducer gives an improved nearfield compared to the conical transducer. The Bessel and cone transducers can be applied in a hybrid imaging system

September 25, 1997, TO BE PUBLISHED IN IEEE TRANS. UFFC 4 where the limited-diffraction beam is used for transmission and dynamic spherical focusing for reception. Of the two, the Bessel transducer is best suited for such use. In the second part of the study, it is found that the Bessel beam can be generated by an equal-area annular transducer designed for dynamic focusing. Thus, no special transducer design is required in order to implement the hybrid imaging system based on the Bessel beam. The scaling parameter of the Bessel beam of order must be chosen to fit the first lobe of the Bessel function on the first element of the array, and the mechanical focus must be such that it can be compensated for by electronic delays, i.e. the rings must be small enough that the array can be focused to infinity. When these two conditions are met, good approximations to Bessel beams can be generated by equal-area, mechanically focused annular transducers. For a small number of rings (less than seven), the Bessel beam yields a larger depth of field than that of an unfocused transducer. Even in this case we show that it has the advantage of a narrower beam compared to that of a flat transducer. In comparison with a spherically focused beam, the Bessel beam will usually give better beams in the near- and farfields at the expense of a wider lateral beamwidth and increased sidelobe level. It should be mentioned that the X-waves [6] can be viewed as a transducer that combines the best properties of both Bessel beams and conical transducers. Even for broadband excitation it has a constant cone angle and it generates a beam which has a predictable beamwidth even at close range. These beams are generated by a complex frequency dependent amplitude excitation of the surface. Recently Lu has shown that X-waves also can be designed for implementation on two-dimensional arrays [9] and on one-dimensional linear arrays [14]. The X-wave solution might therefore be a prospective candidate for elevation focusing (1.5-dimensional transducer). The use of these beams for high frame rate 3D imaging has also been proposed [15]. [12] L. Ødegaard, S. Holm, F. Teigen, and K. T., Acoustic field simulation for arbitrarily shaped transducers in an aberrating medium, in Proc. IEEE Symp. Ultrasonics, (Cannes), pp. 1535 1538, Nov. 1994. [13] S. Holm and G. Hossein Jamshidi, Approximation of Bessel beams with annular arrays, in Proc. IEEE Symp. Ultrasonics, (San Antonio, TX), pp. 881 884, Nov. 1996. [14] J.-Y. Lu, Designing limited diffraction beams, IEEE Trans. Ultrason., Ferroelect., Freq. Contr., vol. 44, pp. 181 193, Jan. 1997. [15] J.-Y. Lu, 2d and 3d high frame rate imaging with limited diffraction beams, IEEE Trans. Ultrason., Ferroelect., Freq. Contr., vol. 44, pp. 839 856, July 1997. REFERENCES [1] M. S. Patterson and F. S. Foster, Acoustic fields of conical radiators, IEEE Trans. Sonics and Ultrason., vol. SU-29, pp. 83 92, Mar. 1982. [2] D. R. Dietz, Apodized conical focusing for ultrasound imaging, IEEE Trans. Sonics and Ultrason., vol. SU-29, pp. 128 138, May 1982. [3] M. O Donnell, A proposed annular array imaging system for contact B-scan applications, IEEE Trans. Sonics and Ultrason., vol. SU-29, pp. 331 338, Nov. 1982. [4] J. Durnin, Exact solutions for nondiffracting beams. I. The scalar theory, J. Opt. Soc. Am., vol. 4, pp. 651 654, Apr. 1987. [5] J. A. Stratton, Electromagnetic Theory, p. 356. New York and London: McGraw-Hill Book Company, 1941. [6] J.-Y. Lu and J. F. Greenleaf, Nondiffracting X waves - exact solutions to free-space scalar wave equation and their finite aperture realizations, IEEE Trans. Ultrason., Ferroelect., Freq. Contr., vol. 39, pp. 19 31, Jan. 1992. [7] J.-Y. Lu and J. F. Greenleaf, Ultrasonic nondiffracting transducer for medical imaging, IEEE Trans. Ultrason., Ferroelect., Freq. Contr., vol. 37, pp. 438 447, Sept. 199. [8] J.-Y. Lu, T. K. Song, R. R. Kinnick, and J. F. Greenleaf, In vitro and in vivo real-time imaging with ultrasonic limited diffraction beams, IEEE Trans. Medical Imaging, vol. 4, pp. 819 829, Dec. 1993. [9] J.-Y. Lu and J. F. Greenleaf, A study of two-dimensional array transducers for limited diffraction beams, IEEE Trans. Ultrason., Ferroelect., Freq. Contr., vol. 41, pp. 724 739, Sept. 1994. [1] J.-Y. Lu and J. F. Greenleaf, Sidelobe reduction for limited diffraction pulse-echo systems, IEEE Trans. Ultrason., Ferroelect., Freq. Contr., vol. 4, pp. 735 746, Nov. 1993. [11] J. A. Campbell and S. Soloway, Generation of nondiffracting beam with frequency-independent beamwidth, J. Acoust. Soc. Amer, vol. 88, pp. 2467 2477, Nov. 199.

September 25, 1997, TO BE PUBLISHED IN IEEE TRANS. UFFC 5 Exact Bessel() beam, alpha = 1154 m^ 1 3 Exact Bessel() beam, alpha = 641.3 m^ 1 3 2 1 1 2 2 1 1 2 3 5 1 15 2 25 3 5 1 15 2 25 Fig. 1. Normalized contour plot showing -6, -12, and -18 db contours for exact, finite aperture Bessel beam with α = 1154 m 1. Fig. 4. Normalized contour plot showing -6, -12, and -18 db contours for exact, finite aperture Bessel beam. Conical transducer, angle 4.635 deg 3 Annular array 3 2 1 1 2 2 1 1 2 3 5 1 15 2 25 3 5 1 15 2 25 Fig. 2. Normalized contour plot showing -6, -12, and -18 db contours for exact, conical beam with θ = 4.635. Fig. 5. Normalized contour plot showing -6, -12, and -18 db contours for field from a flat piston transducer. 1.8.6 Bessel excitation and approximations Conical transdcuer, angle 2.574 deg f = 3.5 [MHz], osc = Inf 3 Relative amplitude.4.2.2.4.6.8 2 1 1 2 1 1 2 3 4 5 6 7 Radius [mm] 3 5 1 15 2 25 Fig. 3. Equal-area, exact Bessel excitation, and approximate excitation (piecewise constant curve). Fig. 6. Normalized contour plot showing -6, -12, and -18 db contours for exact, finite aperture conical beam.

September 25, 1997, TO BE PUBLISHED IN IEEE TRANS. UFFC 6 Equal area Bessel() beam, alpha = 641.3 m^ 1 3 2 1 1 2 [db] 8 6 4 2 175 mm 125 mm 75 mm Approximate Bessel beam 3 5 1 15 2 25 2 25 mm Fig. 7. Normalized contour plot showing -6, -12, and -18 db contours for equalarea approximated Bessel beam. 4 12.5 mm Equal area Bessel() beam, alpha = 641.3 m^ 1 FF = 75 [mm], f = 3.5 [MHz], osc = Inf 3 6 3 2 1 1 2 3 [mm] Fig. 1. One-way CW response at selected depths for approximate Bessel beam (cuts through Fig. 8). 2 1 1 2 3 5 1 15 2 25 Fig. 8. Normalized contour plot showing -6, -12, and -18 db contours for equalarea approximated Bessel beam with fixed focus. 8 6 Fixed focus 4 175 mm Annular array FF = 75 [mm], f = 3.5 [MHz], osc = Inf 3 [db] 2 125 mm 75 mm 2 1 1 2 2 4 25 mm 12.5 mm 6 3 2 1 1 2 3 [mm] Fig. 11. One-way response at selected depths for beam with fixed focused at 75 mm (cuts through Fig. 9). 3 5 1 15 2 25 Range in [mm], Azimuth focus=75 [mm] Envelope Fig. 9. Contour plot with -6, -12, and -18 db contours for spherically focused transducer with fixed focus.

September 25, 1997, TO BE PUBLISHED IN IEEE TRANS. UFFC 7 LIST OF FIGURES 1 Normalized contour plot showing -6, -12, and -18 db contours for exact, finite aperture Bessel beam with α = 1154 m 1................. 5 2 Normalized contour plot showing -6, -12, and - 18 db contours for exact, conical beam with θ = 4.635........................ 5 3 Equal-area, exact Bessel excitation, and approximate excitation (piece-wise constant curve).... 5 4 Normalized contour plot showing -6, -12, and -18 db contours for exact, finite aperture Bessel beam. 5 5 Normalized contour plot showing -6, -12, and -18 db contours for field from a flat piston transducer. 5 6 Normalized contour plot showing -6, -12, and -18 db contours for exact, finite aperture conical beam. 5 7 Normalized contour plot showing -6, -12, and -18 db contours for equal-area approximated Bessel beam......................... 6 8 Normalized contour plot showing -6, -12, and -18 db contours for equal- area approximated Bessel beam with fixed focus................ 6 9 Contour plot with -6, -12, and -18 db contours for spherically focused transducer with fixed focus.. 6 1 One-way CW response at selected depths for approximate Bessel beam (cuts through Fig. 8).... 6 11 One-way response at selected depths for beam with fixed focused at 75 mm (cuts through Fig. 9). 6