The electric field in phasor/complex notation

Question

The electric field in phasor notation is often written \begin{align} \mathbf{E}(x,y,z,t)&=\Re\{\mathbf{E}_0\mathrm{e}^{j\phi}\mathrm{e}^{j\omega t}\}\\ &=\Re\{\tilde{\mathbf{E}}\mathrm{e}^{j\omega t}\}. \end{align} Does it mean $\mathbf E_0$ is a constant vector, i.e. $$\mathbf E_0=E_{0x}\mathbf{\hat{x}}+E_{0y}\mathbf{\hat{y}}+E_{0z}\mathbf{\hat{z}}?$$

Or a vector field, $\mathbf E_0:\mathbb R^3\rightarrow \mathbb R^3$, i.e. $$ \mathbf{E}_0(x,y,z)=E_{0x}(x,y,z)\mathbf{\hat x}+E_{0y}(x,y,z)\mathbf{\hat y}+E_{0z}(x,y,z)\mathbf{\hat z}? $$

Answer 1

It means a combination of the two: the complex amplitude can depend on space, i.e. it is a complex-valued vector field $\tilde{\mathbf{E}}: \mathbb R^3\to\mathbb C^3$, which gives you a space-dependent polarization, \begin{align} \mathbf{E}(x,y,z,t) &=\mathrm{Re}\mathopen{}\bigg[\tilde{\mathbf{E}}(x,y,z)\mathrm{e}^{-i\omega t}\bigg]\mathclose{}, \end{align} including spatial dependence of everything from intensity and ellipticity through the orientation of the polarization plane and of the polarization ellipse within that plane.

For examples of this in action, see e.g. the references in this answer or, say, this paper or this one or this one.