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The approximation done, by which the results are accurate only for directions close to k_0, is called paraxial approximation. The assumption that the higher order derivatives are considerably smaller that the first order derivative and can be therefore neglected is not a consequence of the SVEA, but it is the paraxial approximation itself. It is indeed an assumption that usually accompanies the SVEA, but it is not necessary. See for example: Hadley (1992), Optics Letters, Wide-Angle beam propagation using Pade approximant operators.
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