I see that the harmonic field is sometimes written in exponential form. But sometimes the complex amplitude of this form is just a constant and in others (like when talking about modes) is dependent on the position. Why sometimes is dependent and what does it represent in both cases?
First case $E(\vec{r},t)=he^{-i\omega t}$
Second case $E(\vec{r},t)=h(\vec{r})e^{-i\omega t}$
Thanks
As explained in detail in What is the physical significance of the imaginary part when plane waves are represented as $e^{i(kx-\omega t)}$?, when complex field amplitudes like $E(\vec r,t) = h(\vec r)e^{-i\omega t}$ are presented, there is a broad convention that the physical field is obtained as its real part, $E_\mathrm{phys}(\vec r,t) = \mathrm{Re}\left[h(\vec r)e^{-i\omega t}\right]$.
The amplitude will sometimes be space-dependent and sometimes be spatially uniform because sometimes we are interested in configurations where the field is uniform, and sometimes we do care about the spatial dependence.
