We demonstrate how the gradient of the tapering in a tapered fiber can significantly affect the trapping and blueshift of dispersive waves (DWs) by a soliton. By modeling the propagation of a fundamental soliton through tapered fibers with varying gradients, it is shown that the soliton traps and blueshifts an increased fraction of the energy in its DW when the gradient is decreased. This is quantified by the group-acceleration mismatch between the soliton and DW at the entrance of the taper. These findings have direct implications for the achievable power in the blue edge of a supercontinuum generated in a tapered fiber and explain observations of a lack of power in the blue edge.
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