Apparent Poynting vector contradiction

[frustrationboltzmann]

Hello all,

Im trying to do a simulation of a poynting vector of an electromagnetic wave and I assume the following: At t=0 the E-field vector is (0,0,e^(-ikx)) and the H-field vector (0,e^(-ikx),0), hence orthogonal to it in vaccum, which is assumed, also the amplitudes are simplified both to 1 since only the direction is of interest.

When I calculate the cross product I get now: (-e^(-i2kx),0,0) and when I take the real part I get (-cos(2kx),0,0) which means the poynting vector oscillates in x-direction but this is wrong because it shouldn't oscillate.

My procedure was summarized: take the COMPLEX E-and H-vector, calculate the cross product and after get the real part of it...but this is obviously wrong.

if I take the real parts immediately of the 2 amplitudes and THEN take the cross product of the real amplitudes it works...I get a vector (cos(kx)^2,0,0) which also oscillates but not in 2 different directions but only in intensity which is correct.

between those 2 procedures is a contradiction but at the moment I am afraid I can't see why.
I appreciate any help.
Thank you in advance very much.

 

[Cryo]

You have to be careful in computing products of fields when working with complex-harmonic fields. First of all, there is nothing weird about Poynting vector oscillating, it is a bit like if you multiply two sine-functions: $\sin\left(\omega t\right)^2=\frac{1-\cos\left(2\omega t\right)}{2}$ - you get a constant + oscillating part.

Now what you were probably after is time-averaged Poynting vector. Given complex harmonic electric field ($\mathbf{E}$) and magntic field ($\mathbf{H}$), the time-averaged Poynting vector is:

$\langle \mathbf{S} \rangle = \frac{1}{2} \Re \left(\mathbf{E} \times \mathbf{H}^{\dagger}\right)$

i.e. note the complex-conjugation

 

[shrzhao333]

This is a AC problem complex Poynting theorem should be applied, in which, The Poynting vector is defined as
$$\boldsymbol{S}=\frac{1}{2}\boldsymbol{E}\times\boldsymbol{H}^{*}$$
The power is,
$$ P=\frac{1}{2} \iint_{\Gamma}\boldsymbol{E}\times\boldsymbol{H}^{*}\cdot\hat{n}d\Gamma $$

$*$ is complex conjugate that is important, that cannot be taken away.