- #1
Wox
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The time averaged norm of the Poynting vector of this electromagnetic field (elliptically polarized light):
<br /> \begin{split}<br /> \bar{E}(t,\bar{x})=&(\bar{E}_{0x}+\bar{E}_{0y}e^{i \delta})e^{\bar{k}\cdot\bar{x}-\omega t}\\<br /> \bar{B}(t,\bar{x})=&\frac{1}{\omega}(\bar{k}\times\bar{E}(t,\bar{x}))<br /> \end{split}<br />
with \bar{E}\perp\bar{B}\perp\bar{k}, becomes (as I calculated in SI-units J/(m^{2}s))
<br /> 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})<br />
I have been trying to verify this, but I can't find a source that explicitly discusses this. For a linear polarized beam, \delta=0 so that I(\bar{x})=\frac{c\epsilon_{0}(\bar{E}_{0x}+\bar{E}_{0y})^{2}}{2}, which is correct. For general elliptical polarization I found this link which basically says that
<br /> I(\bar{x})=E_{x}E_{x}^{\ast}+E_{y}E_{y}^{\ast}= \bar{E}_{0x}^{2}+\bar{E}_{0y}^{2}<br />
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?
<br /> \begin{split}<br /> \bar{E}(t,\bar{x})=&(\bar{E}_{0x}+\bar{E}_{0y}e^{i \delta})e^{\bar{k}\cdot\bar{x}-\omega t}\\<br /> \bar{B}(t,\bar{x})=&\frac{1}{\omega}(\bar{k}\times\bar{E}(t,\bar{x}))<br /> \end{split}<br />
with \bar{E}\perp\bar{B}\perp\bar{k}, becomes (as I calculated in SI-units J/(m^{2}s))
<br /> 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})<br />
I have been trying to verify this, but I can't find a source that explicitly discusses this. For a linear polarized beam, \delta=0 so that I(\bar{x})=\frac{c\epsilon_{0}(\bar{E}_{0x}+\bar{E}_{0y})^{2}}{2}, which is correct. For general elliptical polarization I found this link which basically says that
<br /> I(\bar{x})=E_{x}E_{x}^{\ast}+E_{y}E_{y}^{\ast}= \bar{E}_{0x}^{2}+\bar{E}_{0y}^{2}<br />
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?