Direct Solution of the Rayleigh Integral to Obtain

Direct Solution of the Rayleigh Integral to Obtain the Radiation Pattern of an Annular Array Ultrasonic Transducer Y. Qian, S. P. Beeby, N. R. Harris School of Electronics and Computer Science, University of Southampton, UK 1 Introduction l High frequency annular arrays can provide very good resolution in ultrasound bio-microscopy l Numerical simulation of their imaging patterns is essential for good design. l However, the commonly used Spatial Impulse Response (SIR) method is not ideal for calculating annular array patterns numerically. l An algorithm based on the direct solution of the Rayleigh Integral has been developed, which shows good efficiency and accuracy. l In combination with FEA (used to calculate the transducer output), a good agreement with published data is achieved, with much reduced compute time. 2 Algorithm 3 Accuracy and Results —— A 30 MHz, 5 element, 1 mm diameter annular array [2] is chosen as an example for the numerical evaluation q Principle —— The Rayleigh integral is used to evaluate the acoustic diffraction by q Accuracy calculating the pressure at point P in the focal plane [1], as illustrated § The accuracy depends on Nap and Nth , but large values increase computing time. A compromise is needed between speed and accuracy § Figure 2 shows the pattern along the focal plane with different Nap and Nth. below Figure 2 Imaging pattern by the direct method with (a) various Nap but Nth=256, and (b) Nap=128 but various Nth Figure 1 Radiating source with arbitrary shape for Rayleigh integral § Focal line is defined as a line drawn from the point on the Z-axis at the focal plane and represents the whole focal plane due to the axis-symmetric condition. —— It is found that Nap of 128 and Nth of 512 is sufficient to show accurate results within an acceptable computing time (a few minutes). §Pressure p is then expressed as follows, q Results (1) —— where rf and r are vectors representing the point in the focal and aperture planes respectively; d. S is the radiation source element with its polar coordinate (r, φ), R is the distance between d. S and P, c and ρ0 are the speed of sound and density of the fluid. §The above responses are based on an ideal radiating source, while the actual emitted pulses could be non-ideal. §To allow for practical discrepancies, FEA (Finite Element Analysis) is used to obtain practical pulse data which is then combined into the algorithm. §Pure FEA results are given below for comparison (compute time >48 hrs) q Algorithm § Instead of using Fourier or Hankel transforms as in the SIR method, Equation (1) can be solved directly in the time domain by a discretization process to prevent the calculation difficulties that emerge in annular arrays. § Polar coordinates are then introduced for the annular array geometry; three of these tiny elements are magnified to be clearly seen in Figure 1(b) (shaded region). The increment of radius and azimuth angle is set to be ∆r and ∆φ for the element, respectively. § Equation (1) is thus transferred to a discrete Rayleigh integral expressed by Equation (2) where ∆r and ∆φ are the increment of radius and azimuth angle for source d. S, respectively; Nap and Nth are the number of points along the radial and azimuth directions. § Please notice that for an annular array to focus, a time delay td is required for each element to follow the focusing rule, and this is absorbed into g(t) as described in the full paper. —— An algorithm based on Equation (2) is then developed by using Matlab (computing software). Figure 3 Imaging pattern of the 1 mm array along focal line by using different method —— The combined method not only provides reliable results compared to the practical response (or FEA), but also reduces the huge amount of computing time required if FEA is solely used for imaging evaluation 4 Conclusion The direct solution of the Rayleigh Integral successfully evaluates annular array patterns. By using a combination of FEA and the Rayleigh Integral, accurate results are achieved with a significant saving in computing time. 5 Reference [1] Szabo, T. L. , Diagnostic Ultrasound Imaging: Inside Out. 2004: Elsevier Academic Press. [2] Snook, K. A. , et al. , High-frequency ultrasound annular-array imaging. Part I: Array design and fabrication. Ieee Transactions on Ultrasonics Ferroelectrics and Frequency Control, 2006. 53(2): p. 300 -308. —— More references can be found in the full paper.