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We derive an analytical expression to describe the exact solution of the Helmholtz equation in cylindrical coordinates as a product of two Kummer functions. The solution is presented as a sum of two terms that describe the nonparaxial hypergeometric light beams propagated along the optical axis in the positive and negative directions. With the distance from the initial plane becoming much larger than the wavelength of light, the expression derived for the nonparaxial hypergeometric beam coincides with that for a paraxial hypergeometric mode.
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Kotlyar et al. (2009) studied this question.
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