# Computing Poynting Vector from phasors

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 $$\widetilde{\mathbf{H}}$$?

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.