Computing Poynting Vector from phasors

Question

The Poynting Vector is defined as:

$$\mathbf{N} = \mathbf{E} \times \mathbf{H}$$

With $\mathbf{N}$, $\mathbf{E}$ and $\mathbf{H}$ being vector fields.
I'm working with monochromatic plane waves, so I'm using the phasorial expressions $\widetilde{\mathbf{E}}$ and $\widetilde{\mathbf{H}}$. As the computation of the Poynting Vector is not a linear operation, taking the vector product of the phasors would not result in a phasor whose real part equals $\mathbf{N}$.
However, is it valid to say that the following (with $*$ being the complex conjugate operator)

$$\widetilde{\mathbf{N}} =\widetilde{\mathbf{E}}\times\widetilde{\mathbf{H}}^*$$

is a valid phasorial representation for $\mathbf{N}$? If not, is there a valid expression for $\mathbf{N}$ in terms of the phasors $\widetilde{\mathbf{E}}$ and $\widetilde{\mathbf{H}}$?

Answer 1

Well: $\tilde{\textbf E}\times \tilde {\textbf H}^*$ will result in an expression that does not depend on time so this will not be the instantaneous Poynting vector. Of course $$ \frac{1}{2}\text{Re}\left(\tilde{\textbf E}\times \tilde {\textbf H}^*\right) $$ is the time-averaged power density, and it may not be what you want.