Polarization & Jones/Mueller Calculus
Polarization & Jones/Mueller Calculus
Polarization describes the orientation of the electric-field vector of a light wave. For a monochromatic plane wave along $z$, the field traces an ellipse in the $xy$-plane; linear and circular polarizations are special cases.
The Jones calculus uses $2\times1$ complex vectors for fully polarized states and $2\times2$ matrices for optical elements. The Mueller calculus uses $4\times1$ real Stokes vectors, suitable for partially polarized and incoherent light.
Jones Vector & Stokes Parameters
Jones: $\mathbf{E}=(E_x,E_y)^T$. Stokes: $S_0=|E_x|^2+|E_y|^2$, $S_1=|E_x|^2-|E_y|^2$, $S_2=2\operatorname{Re}(E_xE_y^*)$, $S_3=2\operatorname{Im}(E_xE_y^*)$.
Cascade Rule
For $N$ elements: $\mathbf{E}_\text{out}=M_N\cdots M_1\mathbf{E}_\text{in}$ (Jones). Mueller: $\mathbf{S}_\text{out}=\mathcal{M}_N\cdots\mathcal{M}_1\mathbf{S}_\text{in}$.
Example 1
Horizontal polarizer followed by quarter-wave plate (fast axis 45°) acts on horizontal light. Find output polarization.
Solution: Polarizer passes $\mathbf{E}=(1,0)^T$; QWP at 45° converts it to right circular polarization $(1,-i)^T/\sqrt{2}$.
Example 2
Malus\'s law: find intensity through a polarizer at angle $\theta$ to the polarization axis.
Solution: $I=I_0\cos^2\theta$ — directly from the Jones matrix projection.
Practice
- Write the Jones vector for left circular polarization.
- What is the Mueller matrix of an ideal linear horizontal polarizer?
- How does a half-wave plate rotate linear polarization?
- When must you use Mueller rather than Jones calculus?
Show Answer Key
1. Left circular polarization: $\mathbf{J} = \frac{1}{\sqrt{2}}\begin{pmatrix}1\\-i\end{pmatrix}$ (convention: looking toward the source, the field rotates counterclockwise). Right circular: $\frac{1}{\sqrt{2}}\begin{pmatrix}1\\i\end{pmatrix}$.
2. Ideal horizontal linear polarizer Mueller matrix: $\frac{1}{2}\begin{pmatrix}1&1&0&0\\1&1&0&0\\0&0&0&0\\0&0&0&0\end{pmatrix}$. It transmits the horizontal component and blocks the vertical, reducing the Stokes vector accordingly.
3. A half-wave plate ($\delta = \pi$) reverses the sign of the component perpendicular to its fast axis. If linearly polarized light enters at angle $\theta$ to the fast axis, it exits at angle $-\theta$ (reflected about the fast axis). Net rotation: $2\theta$ from the original direction. Used to rotate polarization by placing the HWP at half the desired rotation angle.
4. Jones calculus works only for fully polarized light (coherent, deterministic polarization state). Mueller calculus handles partially polarized and unpolarized light using Stokes vectors and Mueller matrices. Use Mueller when dealing with depolarizing elements, incoherent superposition, or partially polarized beams.