[MATLAB] [\(\LaTeX\)]
Vortex beam radiated by Gaussian vortex source: dimensionless code for Eq. (14) of
C. A. Gokani, M. R. Haberman, M. F. Hamilton. "Paraxial and ray approximations of acoustic vortex beams," J. Acoust. Soc. Am., 155, 2707-2723 (2024).The dimensionless form of Eq. (14) is in terms of the parameters \(Q = q/p_0\), \(R = r/a\), \(Z =z/z_R = 2z/ka^2\), and \(G = ka^2/2d\), where \(r\) and \(z\) are the cylindrical coordinates, \(a\) is the characteristic radius of the source, \(k\) is the wavenumber, and \(d\) is the focal length: \begin{align} {|Q|(R,Z)} &= \sqrt{2\pi} \frac{Z}{R^2} \big|\chi^{3/2} e^{-\chi} \big[I_{(\ell-1)/2}(\chi) - I_{(\ell+1)/2}(\chi)\big]\big|\,,\label{eq:gauss}\\ \chi(R,Z) &= \frac{iR^2/2Z}{1+ Z(i-G)} \,. \label{eq:chi} \end{align} Unfocused Gaussian vortex beams are obtained by setting \(G = 0\) in Eq. \eqref{eq:chi}. Both focused and unfocused cases are handled in the code.
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