Connection between X-Waves, Fourier-Bessel series and optimal modelling aperture for circular symmetric arrays

Fox, P.D., Lu, J-Y., Holm, S. and Tranquart, F. (2005) Connection between X-Waves, Fourier-Bessel series and optimal modelling aperture for circular symmetric arrays. Proceedings of the IEEE International Ultrasonics Symposium, Rotterdam. 17 - 20 Sep 2005. pp. 1644-1647 .

Abstract

This paper addresses various unresolved issues raised in publications [1], [2], and [3], in connection with the study and application of limited-diffraction and non-diffracting beams. Nondiffracting beams have the property of a constant radial profile with propagation distance, subject to an infinite aperture source, and the related theories advanced in the context of medical imaging have resulted in the possibility of extremely high fame rates. However, all the fundamental theory assumes an infinite-aperture source being available. In reality this is not possible, and when nondiffracting beams are implemented on finite source apertures they become limited diffraction beams. [1], [2], [3] have studied the use of and numerical differences between nondiffracting theory and limited diffraction implementation of such beams for field computation and tuning. In [1], [2] it was shown that an iterative technique could be applied to extend the effective modelling aperture from the physical limits of the transducer (limited diffraction basis functions) towards an infinite modelling aperture (Bessel beams / X-waves), to both tune and compute the emitted field from any given circular symmetric flat array with linear propagation conditions. This technique involved the concept of a modelling aperture spanning the gap between the physical limit of the transducer and infinity, which, when increased iteratively resulted in convergence of the corresponding computed field and source driving function. However, the technique relied on a combination of iterative combinations and 137 numerical field convergence to within a pre-selected limit in order to terminate the computations at an appropriate point. In this paper, a formal mathematical connection between the limited-aperture (limited diffraction) basis functions and the nondiffracting infinite aperture theory (Bessel beams / X-Waves) is established as a function of the increasing modelling aperture. The result is that a specific optimal modelling aperture may then be specified as a function of frequency spectrum, spatial field extent to be investigated, and pulse repetition frequency. Consequently the previous iterative technique may be replaced by a single one-shot computation to achieve the same result. As a result, the new technique is significantly more efficient than the previous technique and the specific saving in computation depends on the particular transducer considered, but typically computational reductions are in the order of 50%. The global contribution of the paper is twofold : firstly a formal mathematical connection between limited diffraction beams and nondiffracting beams as function of increasing modelling aperture, and secondly the derivation of the optimal modelling aperture required for computation and tuning of circular symmetric fields with minimal computational demands.

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