Power calculation in FDTD.

I'm having problems in the calculation of power using FDTD method and the poynting vector.

The Yee cell has 6 faces, so i have to calculate [(E × H) dot n] in each face, rigth?

But how i can do that? For example, the upper face. In that face, n points to z direction, and in that face we have only Ez, Hx and Hy, so the Poynting vector is in x and y direction. If Poynting vector is in x and y direction, in that face, (E × H) dot n is 0. The same happens in all other faces.

Where am I wrong?

Thanks in advance.

Added after 24 minutes:

I have to consider the other cells?

For example, in the following figure:

If I want to calcute the [(E × H) dot n] in rigth face (yellow) of cell 1, I have the result:

(EzHx - ExHz)

But in that face there are no Ez and Ex. I can use the Neighborhood cells? I can say that Ez and Ex in that face is:

Ez = 0.25 *
[Ez in the upper face in cell 1 +
Ez in the lower face in cell 1 +
Ez in the upper face in cell 2 +
Ez in the lower face in cell 2]

Ex = 0.25 *
[Ez in the front face in cell 1 +
Ez in the back face in cell 1 +
Ez in the front face in cell 2 +
Ez in the back face in cell 2]


?

Thanks

You are nowhere wrong. [added after rereading your message: Actually you are by assuming that the missing field values are zero. You need interpolation to get them].


There are two alternatives.

1) compute the missing field values for some field F from the surrounding locations using interpolation.
PRO: easy to implement
CON: inaccurate especially at low wavelength/cell size ratios

2) a) store all signals b) FT the signals c) use geometric averaging on phasors d) inverse FT e) compute Poynting
PRO: more accurate
CON: more cumbersome, potentially huge storage requirements

Added after 2 hours 54 minutes:

Let me try again. I read your message too quickly.

Basically you have to make sure that all the values you need to compute the
Poynting vector exist at a given location.

There are two basic ways to achieve it. For both you have top decide how many neighboring
points you want to use. In homogeneous media more points usually mean higher accuracy.
Nect you have to decide the interpolation method: polynomial (linear if you use 2 points),
spline etc. (Getting the best way to interpolate is not easy when you are on material interfaces.)

Finally you have to decide when to interpolate (which I tried to sketch in my previous message):
1) During the simulation,
2) After the simulation (using FT and IFT to separate the frequencies).

I think that you should calculate Poynting w.r.t to the whole cell, which contains all 6 components.
Then every cell in your recording plane will contain a vectior value for P as well.
And this only after F-transforming, since (E vec H) is a nonlinear transformation (in this I disagree with the post before..).
Then you sum up along all cells of interest, multiplying by your (Dx,Dy) increments accordingly to obtain physically meaningful units.

This should give the power flux across the recording surface, as a power density in frequency.

Hope this helps..

what do you disagree with?

I said:
No FT => inaccurate & easy,
USE FT => more accurate but more cumbersome + potential memory problems

no, I was referring to the sequence of FT and (E vec H) operations:
I think that first you should FT every component of the fields along a plane or line of interest, and with these frequency-domain fields compute the Poynting vector..

That is exactly version 2 I mentioned before as being the most accurate.