Electromagnetic fields can be composed of a spectrum, not just a single frequency/wavenumber relation. If there is a spectrum for the electric field in question (and yes, spectra can include the quasi-static or DC parts as well), then there is a frequency dependence in the amplitude. Even in the case of the plane wave mentioned above, there is technically a frequency dependence. Mathematically, $A$ is not explicitly defined as a function of $\omega$ or $k$. However, if one calculated the FFT of such a signal, it would show a single peak at the corresponding $\omega$ or $k$.
In more complicated situations (i.e., nonlinear waves), then $A$ can equal $A(\omega, k)$. Meaning, a generalized nonlinear signal can have both temporal and spatial dependencies in the amplitude and both $\omega$ and $k$ need not be purely real functions.
So if this is an experiment, then either produce the FFT or hook up the signal to a oscilloscope to determine the spectrum of the electric field. The gradient operator will only affect/act-on $k$, but if $k$ is coupled to $\omega$ (e.g., through a dispersion relation), then the frequency dependence could still play a role. If this is theory, then it is an issue of whether you want to approximate with linear theory or try nonlinear generalizations.