A closed-form field analysis of a broad-band annular array

A closed-form transient field response under the paraxial approximation for an impulse wave excited by a circularly symmetric ultrasonic transducer is derived. The analysis is based on linear transform techniques to obtain the impulse response of a planar or a lensed transducer such as a disk or an annulus in the whole field of view. A closed-form solution is obtained for the focal plane response of a nonuniform aperture with an apodizing function represented by a polynomial of the radial distance on the transducer surface. Moreover, a closed-form impulse response for a planar annular is derived without the paraxial approximation and taking into account the obliquity factor. For a phased annular array, the impulse responses of array elements are convolved with the delayed excitation pulse and then summed to get the resultant field disturbance of the array. These results provide an effective means of studying the focusing characteristics of a phased annular array, including its focusing delay and apodizing quantization error effects.<
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