Why Jones matrix of quarter-wave plate of which fast axis is vertical is $e^{i\pi/4}\left[\begin{matrix}1&0\\0&-i\\\end{matrix}\right]$? Jones matrix of quarter-wave plate of which fast axis is horizontal is $e^{i\pi/4}\left[\begin{matrix}1&0\\0&i\\\end{matrix}\right]$
If the plate is rotated by $π/2$, then Jones matrix is followed
$e^{i\pi/4}\left[\begin{matrix}\cos{\frac{\pi}{2}}&-\sin{\frac{\pi}{2}}\\\sin{\frac{\pi}{2}}&\cos{\varphi\frac{\pi}{2}}\\\end{matrix}\right]\left[\begin{matrix}1&0\\0&i\\\end{matrix}\right]\left[\begin{matrix}\cos{\frac{\pi}{2}}&\sin{\frac{\pi}{2}}\\-\sin{\varphi\frac{\pi}{2}}&\cos{\frac{\pi}{2}}\\\end{matrix}\right]=e^{i\pi/4}\left[\begin{matrix}i&0\\0&1\\\end{matrix}\right]$
Hence I think quarter-wave plate of which fast axis is vertical is $e^{i\pi/4}\left[\begin{matrix}i&0\\0&1\\\end{matrix}\right]$
But quarter-wave plate of which fast axis is vertical is known as $e^{i\pi/4}\left[\begin{matrix}1&0\\0&-i\\\end{matrix}\right]$
I don't understand why Jones matrix of quarter-wave plate of which fast axis is vertical is $e^{i\pi/4}\left[\begin{matrix}1&0\\0&-i\\\end{matrix}\right]$
 A: What source are you using in this derivation? Wikipedia, for example, uses the same notation as you, $e^{i\pi/4}\begin{pmatrix}1&0\\0&-i\end{pmatrix}$, for the fast-axis vertical case and $e^{-i\pi/4}\begin{pmatrix}1&0\\0&i\end{pmatrix}$ for the fast-axis horizontal case. Those two are compatible with your $\pi/2$ rotation condition.
Regardless, global phases are irrelevant to Jones matrix calculus, because all of the relevant physical quantities that can be measured do not depend on global phases. As such, multiplying your final Jones matrix by $-i=e^{i3\pi/2}$ does not change the physical quantity that it represents, so your final expression and your desired final expression are physically equivalent. (For example, if you would like to measure an intensity or any of the Stokes parameters, the global phase $e^{i\phi}$ and its complex conjugate $e^{-i\phi}$ will be multiplied together, yielding $1$, so the final result will never depend on the global phase.)
A: Your rotated matrix and the wiki one are indeed equal, but you dropped a minus sign on the phase term outside of the wiki matrix. Otherwise if you factor an i out of your rotated matrix they appear to be the same.
