Sanity check: Using the Jones calculus for superposition

[boxfullofvacuumtubes]

Homework Statement

Suppose light is prepared in a coherent superposition of linear horizontal polarization and linear vertical polarization. What is the resulting polarization according to Jones calculus if it passes through:

• a linear polarizer at a 45-degree angle (0 degrees would be vertical)
• a linear polarizer at a -45-degree angle(0 degrees would be vertical)
• a quarter-wave plate with a fast axis oriented at a 45-degree angle(0 degrees would be vertical)

Homework Equations

Jones calculus.

The Attempt at a Solution

Superposition of LHP and LVP appears to be the same as 45-degree polarization (L+45):
$$\begin{pmatrix}1\\0\end{pmatrix} + \begin{pmatrix}0\\1\end{pmatrix} = \begin{pmatrix}1\\1\end{pmatrix}$$

If this light passes through a 45-degree polarizer, its polarization and intensity should remain unchanged:
$$\begin{pmatrix}0.5 & 0.5\\0.5 & 0.5\end{pmatrix} × \begin{pmatrix}1\\1\end{pmatrix} = \begin{pmatrix}1\\1\end{pmatrix}$$

None of this light can pass through a minus-45-degree polarizer because of its orthogonal polarization:
$$\begin{pmatrix}0.5 & -0.5\\-0.5 & 0.5\end{pmatrix} × \begin{pmatrix}1\\1\end{pmatrix} = \begin{pmatrix}0\\0\end{pmatrix}$$

A quarter-wave plate at 45 degrees turns the polarization to left circular and does not affect intensity:
$$\begin{pmatrix}1 & 0\\0 & -i\end{pmatrix} × \begin{pmatrix}1\\1\end{pmatrix} = \begin{pmatrix}1\\-i\end{pmatrix}$$

Did I make an error somewhere? Thanks for a sanity check!

 

[MisterX]

I think you have in the last portion applied the quarter wave plate at 0 degrees instead of 45 degrees. Wikipedia gives an expression for the quarter wave plate matrix rotated at any angle here.
$$e^{-\frac{i\pi}{4}}\begin{pmatrix}
\cos^2\theta + i\sin^2\theta & (1 - i)\sin\theta \cos\theta \\
(1 - i)\sin\theta \cos\theta & \sin^2\theta + i\cos^2\theta
\end{pmatrix}$$
Some people may leave off the prefactor $e^{-\frac{i\pi}{4}}$.
I believe this can also be expressed as a rotated version of the vertical quarter wave plate
$$R\begin{pmatrix}1 & 0\\0 & -i\end{pmatrix}
R^{-1} $$
Applying the 45 degree quarter wave plate to 45 degree polarization should be like applying a 0 degree plate to vertical polarization, except everything is rotated.