- #1
Wox
- 70
- 0
The time averaged norm of the Poynting vector of this electromagnetic field (elliptically polarized light):
[tex]
\begin{split}
\bar{E}(t,\bar{x})=&(\bar{E}_{0x}+\bar{E}_{0y}e^{i \delta})e^{\bar{k}\cdot\bar{x}-\omega t}\\
\bar{B}(t,\bar{x})=&\frac{1}{\omega}(\bar{k}\times\bar{E}(t,\bar{x}))
\end{split}
[/tex]
with [itex]\bar{E}\perp\bar{B}\perp\bar{k}[/itex], becomes (as I calculated in SI-units [itex]J/(m^{2}s)[/itex])
[tex]
I(\bar{x})=\left<\left\|\bar{P}(t,\bar{x})\right\|\right>=\frac{c\epsilon_{0}}{2}(\bar{E}_{0x}^{2}+2\bar{E}_{0x}\cdot\bar{E}_{0y}\cos\delta+\bar{E}_{0y}^{2})
[/tex]
I have been trying to verify this, but I can't find a source that explicitly discusses this. For a linear polarized beam, [itex]\delta=0[/itex] so that [itex]I(\bar{x})=\frac{c\epsilon_{0}(\bar{E}_{0x}+\bar{E}_{0y})^{2}}{2}[/itex], which is correct. For general elliptical polarization I found this link which basically says that
[tex]
I(\bar{x})=E_{x}E_{x}^{\ast}+E_{y}E_{y}^{\ast}= \bar{E}_{0x}^{2}+\bar{E}_{0y}^{2}
[/tex]
which can't be right (as it doesn't work for linear polarized light). Does anyone know of a proper reference for this? Or even better, can someone verify my solution?
[tex]
\begin{split}
\bar{E}(t,\bar{x})=&(\bar{E}_{0x}+\bar{E}_{0y}e^{i \delta})e^{\bar{k}\cdot\bar{x}-\omega t}\\
\bar{B}(t,\bar{x})=&\frac{1}{\omega}(\bar{k}\times\bar{E}(t,\bar{x}))
\end{split}
[/tex]
with [itex]\bar{E}\perp\bar{B}\perp\bar{k}[/itex], becomes (as I calculated in SI-units [itex]J/(m^{2}s)[/itex])
[tex]
I(\bar{x})=\left<\left\|\bar{P}(t,\bar{x})\right\|\right>=\frac{c\epsilon_{0}}{2}(\bar{E}_{0x}^{2}+2\bar{E}_{0x}\cdot\bar{E}_{0y}\cos\delta+\bar{E}_{0y}^{2})
[/tex]
I have been trying to verify this, but I can't find a source that explicitly discusses this. For a linear polarized beam, [itex]\delta=0[/itex] so that [itex]I(\bar{x})=\frac{c\epsilon_{0}(\bar{E}_{0x}+\bar{E}_{0y})^{2}}{2}[/itex], which is correct. For general elliptical polarization I found this link which basically says that
[tex]
I(\bar{x})=E_{x}E_{x}^{\ast}+E_{y}E_{y}^{\ast}= \bar{E}_{0x}^{2}+\bar{E}_{0y}^{2}
[/tex]
which can't be right (as it doesn't work for linear polarized light). Does anyone know of a proper reference for this? Or even better, can someone verify my solution?