Arbitrary-order nonlinear contribution to self-steepening

On the basis of the recently published generalized Miller formulas, we derive the spectral dependence of the contribution of arbitrary-order nonlinear indices to the group-velocity index. We show that in the context of laser filamentation in gases, all experimentally accessible orders (up to the ninth-order nonlinear susceptibility chi((9)) in air and chi((11)) in argon) have contributions of alternative signs and similar magnitudes. Moreover, we show both analytically and numerically that the dispersion term of the nonlinear indices must be considered when computing the intensity-dependent group velocity.