Sanity check: Using the Jones calculus for superposition

• boxfullofvacuumtubes
In summary: So the result should be vertical polarization rotated 45 degrees.In summary, when a light in a coherent superposition of linear horizontal polarization and linear vertical polarization passes through a linear polarizer at a 45-degree angle, its polarization and intensity remain unchanged. However, if it passes through a -45-degree linear polarizer, none of the light can pass through due to its orthogonal polarization. When passing through a quarter-wave plate with a fast axis oriented at a 45-degree angle, the resulting polarization is left circular and the intensity remains unchanged.
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)

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!

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.

boxfullofvacuumtubes

1. What is the Jones calculus and how is it used?

The Jones calculus is a mathematical tool used to analyze the polarization of light. It is based on the Jones vector, which represents the amplitude and phase of the electric field of a polarized light wave. The Jones calculus is used to calculate the effects of various optical elements on the polarization of light, such as polarizers and waveplates.

2. How does the Jones calculus work for superposition of light waves?

The Jones calculus uses matrix multiplication to combine the Jones vectors of two or more light waves that are superimposed. This allows for the calculation of the resulting polarization state of the superimposed light wave.

3. Can the Jones calculus be applied to non-linear optics?

Yes, the Jones calculus can be extended to non-linear optics by using the Jones matrix to represent the non-linear optical effect. However, this approach may not be accurate for very strong non-linear effects.

4. What are the limitations of using the Jones calculus for superposition?

The Jones calculus assumes that the light waves are monochromatic and coherent, which may not always be the case in real-world situations. It also does not take into account the effects of absorption or scattering.

5. Are there any alternative methods to the Jones calculus for analyzing superposition of light waves?

Yes, there are other mathematical tools that can be used to analyze the superposition of light waves, such as the Mueller calculus and the Stokes parameters. Each method has its own advantages and limitations, and the choice of which to use depends on the specific situation and the desired level of accuracy.

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