Jones calculus and Mueller matrices are powerful tools for understanding polarized light. They help us describe how light's polarization changes as it passes through different optical elements.

These mathematical methods allow us to predict and analyze complex optical systems. By using Jones vectors and matrices or Stokes vectors and Mueller matrices, we can calculate how light behaves in various setups.

Jones Calculus and Mueller Matrices

Jones calculus for polarization states

• Represents the polarization state of light using Jones vectors, which are 2x1 complex vectors
  • Horizontal linear polarization: $\begin{bmatrix} 1 \ 0 \end{bmatrix}$
  • Vertical linear polarization: $\begin{bmatrix} 0 \ 1 \end{bmatrix}$
  • +45° linear polarization: $\frac{1}{\sqrt{2}}\begin{bmatrix} 1 \ 1 \end{bmatrix}$
  • Right circular polarization: $\frac{1}{\sqrt{2}}\begin{bmatrix} 1 \ -i \end{bmatrix}$
• Enables calculation of the output polarization state when light passes through polarization devices represented by Jones matrices
• Suitable for completely polarized light and non-depolarizing devices

Jones and Mueller matrices for polarization

• Jones matrices represent the effect of polarization devices on the electric field of light
  • 2x2 complex matrices that operate on Jones vectors
  • Cannot account for depolarization or partially polarized light
  • Examples: linear polarizers ($\begin{bmatrix} 1 & 0 \ 0 & 0 \end{bmatrix}$ for horizontal, $\begin{bmatrix} 0 & 0 \ 0 & 1 \end{bmatrix}$ for vertical), wave plates ($\frac{1}{\sqrt{2}}\begin{bmatrix} 1 & -i \ -i & 1 \end{bmatrix}$ for quarter-wave plate with fast axis at 45°, $\begin{bmatrix} \cos 2\theta & \sin 2\theta \ \sin 2\theta & -\cos 2\theta \end{bmatrix}$ for half-wave plate with fast axis at angle $\theta$)
• Mueller matrices describe the effect of polarization devices on the Stokes parameters of light
  • 4x4 real matrices that operate on Stokes vectors
  • Can handle depolarization and partially polarized light
  • Examples: linear polarizer ($\frac{1}{2}\begin{bmatrix} 1 & \cos 2\theta & \sin 2\theta & 0 \ \cos 2\theta & \cos^2 2\theta & \sin 2\theta \cos 2\theta & 0 \ \sin 2\theta & \sin 2\theta \cos 2\theta & \sin^2 2\theta & 0 \ 0 & 0 & 0 & 0 \end{bmatrix}$ with transmission axis at angle $\theta$), quarter-wave plate ($\begin{bmatrix} 1 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 \ 0 & 0 & 0 & -1 \ 0 & 0 & 1 & 0 \end{bmatrix}$ with fast axis at 0° or 90°), half-wave plate ($\begin{bmatrix} 1 & 0 & 0 & 0 \ 0 & \cos 4\theta & \sin 4\theta & 0 \ 0 & \sin 4\theta & -\cos 4\theta & 0 \ 0 & 0 & 0 & -1 \end{bmatrix}$ with fast axis at angle $\theta$)

Mueller matrices of optical elements

• Represent the polarization properties of optical elements using Stokes parameters
  • Stokes vector: $S = \begin{bmatrix} I \ Q \ U \ V \end{bmatrix}$, where $I$ is total intensity, $Q$ is the difference between horizontal and vertical polarization intensities, $U$ is the difference between +45° and -45° polarization intensities, and $V$ is the difference between right and left circular polarization intensities
• Mueller matrices can describe depolarization, unlike Jones matrices
• The first row and column of a Mueller matrix relate to the total intensity, while the remaining 3x3 submatrix describes the polarization properties of the element

Polarized light in complex systems

• Calculate the output polarization state and intensity of light passing through a series of polarization devices using Jones calculus
  1. Multiply the Jones matrices of the devices in reverse order to obtain the total Jones matrix: $J_\text{total} = J_n \cdot J_{n-1} \cdot ... \cdot J_2 \cdot J_1$
  2. Multiply the input Jones vector by the total Jones matrix to get the output Jones vector: $E_\text{out} = J_\text{total} \cdot E_\text{in}$
  3. Calculate the output intensity, which is proportional to the square of the absolute value of the output electric field: $I_\text{out} \propto |E_\text{out}|^2 = |J_\text{total} \cdot E_\text{in}|^2$
• Solve problems involving the propagation of polarized light through complex optical systems using Mueller calculus
  1. Multiply the Mueller matrices of the optical elements in reverse order to obtain the total Mueller matrix: $M_\text{total} = M_n \cdot M_{n-1} \cdot ... \cdot M_2 \cdot M_1$
  2. Multiply the input Stokes vector by the total Mueller matrix to get the output Stokes vector: $S_\text{out} = M_\text{total} \cdot S_\text{in}$
  3. Analyze the polarization properties using the output Stokes vector, such as calculating the degree of polarization (DOP): $\text{DOP} = \frac{\sqrt{Q^2 + U^2 + V^2}}{I}$
• Combine Jones and Mueller calculus when necessary by converting between Jones and Mueller matrices or vectors
  • Jones to Mueller matrix conversion: $M = A(\mathbf{J} \otimes \mathbf{J}^)A^{-1}$, where $A = \begin{bmatrix} 1 & 0 & 0 & 1 \ 1 & 0 & 0 & -1 \ 0 & 1 & 1 & 0 \ 0 & i & -i & 0 \end{bmatrix}$ and $\otimes$ denotes the Kronecker product
  • Stokes to Jones vector conversion: $\mathbf{E} = \begin{bmatrix} \sqrt{I + Q} \ \sqrt{I - Q} e^{i\arctan(V/U)} \end{bmatrix}$