ИСТИНА

One-way parabolic equations are often used for numerical simulations of diffracting acoustic beams in optics, ocean and atmospheric acoustics, geophysics, and in ultrasound physics and its applications [1]. Most frequently the standard parabolic equation (SPE) based on the paraxial approximation is used, which is relatively easy to solve by numerical methods in a general case of inhomogeneous propagation medium. However, in many applications, for example, in therapeutic ultrasound, strongly focused beams are considered and accuracy of the SPE is not sufficient. Accuracy of the propagation model can be improved by using wide-angle parabolic equations, which are derived from factorized Helmholtz equation with the help of operator formalism. Commonly, wide-angle equations are constructed by using Padé approximations of the propagator, which is a formal solution to the one-way equation [2]. This method perfectly works when applied to two dimensional or effectively two dimensional propagation problems. In a three dimensional case, computations become very intensive since standard numerical approaches lead to large block sparse matrices, which are difficult to invert directly [3]. A method of alternating directions (ADI), which is effective for the standard parabolic equation, is not applicable here, since it generates a zero-order error term [4]. To work around this problem, either iterative linear algebra methods are proposed [5] or approximate treatment of cross terms arising from a combination of two components of Laplacian operator in two transverse directions [6]. Here it is proposed to use an alternative method for constructing wide-angle propagation models, which use the idea of finite Fourier series approximation of the one-way propagation operator [7]. The main advantage of such approach appears from the fact that individual terms of these series are exponentials of the same kind as in the formal propagator solution form of the SPE. As a consequence, all standard numerical methods for solving SPE, including finite differences, are applicable here. As a disadvantage we can note, that for a proper representation of the propagator, a relatively large number of terms should be retained in Fourier series to be a subject of optimization. In this study we demonstrate an ability of the proposed model to accurately simulate pressure fields of strongly focused ultrasound transducers of biomedical devices. This work was supported by the Russian Science Foundation grant # 23-22-00220, the student stipend from the BASIS Foundation, and performed within the framework of the MSU Scientific School "Photonic and Quantum Technologies. Digital Medicine". References [1] D. Lee, A.D. Pierce, J. Comput. Acoust., 8, 2000, 527–637. [2] M.D. Collins, J. Acoust. Soc. Am., 93, 1993, 1736–1742. [3] K. Lee et al., J. Acoust. Soc. Am., 146, 2019, 2050–2057. [4] E.V. Bekker et al, J. Light. Technol., 27, 2009, 2595–2604. [5] G. R. Hadley, J. Comput. Phys., 203, 2005, 358–370. [6] K. Lee et al, J. Acoust. Soc. Am., 146, 2019, 2041–2049. [7] A. Balkenohl, D. Schulz, J. Light. Technol., 23, 2014, 3917–3925.