To advance the phase by the angle ϕ, we replace ωt by ωt + ϕ (that is, we replace − ωt by − ωt − ϕ), with the result that the (complex) field is multiplied by e. So a phase advance is equivalent to multiplication by a complex constant with a negative argument. This becomes more obvious when the field (1) is factored as ‍ E k e e, where the last factor contains the time-dependence. That factor also implies that differentiation w.r.t. time corresponds to multiplication by −iω.