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

1 Answer

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.

• Thanks for the answer, but that result was known for me and it is not what I want. I have broadened the question to contemplate possible expressions different from what I proposed Commented Jan 30, 2019 at 0:32
• @MPA95 Yeah I figured this isn't what you were looking for. Otherwise the answer is (as far as I can tell) no: there is no such phasor form. I'll leave the answer so others know this isn't what you want. Commented Jan 30, 2019 at 0:34
• The Poynting vector is usually defined as the current which appears in the local energy conservation law (defined up to a divergence-free contribution), and if you go by that definition, then ZeroTheHero's expression is the Poynting vector, not the “complex Poynting vector” that you define. So I think your question makes no sense. If you start with the complex Poynting vector, then there are situations where its imaginary part manifests itself physically (see e. g. Section 6.9 in Jackson's book on electromagnetism). Commented Jan 30, 2019 at 4:25