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I was watching this video (https://www.youtube.com/watch?v=KKQJIMdEEHY) to gain a better understanding about light polarization, but I was having some trouble with the derivation of the Jones Vector.

In the video, Professor Hafner made the assumption $k_{x}=k_{y}$ and $\omega_{x}=\omega_{y}$:

$\vec{E}=real \left\langle A_{x}e^{i(k_{x}z-\omega_{x} t+\phi_{x})}, A_{y}e^{i(k_{y}z-\omega_{y} t+\phi_{y})} \right\rangle$

$\vec{E}=real( \left\langle A_{x}e^{i\phi_{x}}, A_{y}e^{i\phi_{y}} \right\rangle e^{i(kz-\omega t)} )$

I do not understand why this assumption can be made.

Jones Vector = $normalize(\left\langle A_{x}e^{i\phi_{x}}, A_{y}e^{i\phi_{y}} \right\rangle)$

where:
$normalize(\vec{X})=$ unit vector in direction of $\vec{X}$

Also in this video (https://www.youtube.com/watch?v=0USje5vTIKs), Professor Zwiebach uses a similar vector to describe a photon traveling through a beam splitter. Is this the same Jones vector as the one Professor Hafner derived or does it have a different derivation?

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