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Polarization & Jones Calculus

Linear, circular, and elliptical polarization — encoded as 2-component Jones vectors.

For a plane EM wave propagating along $\hat z$, the transverse electric field is described by

$$\mathbf E(z, t) = \mathrm{Re}\!\left[ \begin{pmatrix} E_x \\ E_y \end{pmatrix} e^{i(kz - \omega t)} \right].$$

The complex 2-vector $(E_x, E_y)^T$ — a Jones vector — fully specifies the polarization state (up to overall phase + amplitude):

  • $(1, 0)^T$: linearly polarized along $\hat x$.
  • $\tfrac{1}{\sqrt 2}(1, 1)^T$: linear at 45°.
  • $\tfrac{1}{\sqrt 2}(1, i)^T$: right circular ($\hat y$ lags $\hat x$ by 90°).
  • $\tfrac{1}{\sqrt 2}(1, -i)^T$: left circular.

Polarizers, wave plates, etc., are $2\times 2$ Jones matrices acting on the Jones vector. Composition is matrix product. Malus's law for an ideal linear polarizer at angle $\theta$ to incident linear polarization: transmitted intensity $I = I_0 \cos^2\theta$.

Interactive: animated polarization ellipse

Quiz

1. Jones vector $\tfrac{1}{\sqrt 2}(1, i)^T$ describes:
2. Malus's law: intensity after a linear polarizer at angle $\theta$:
3. A quarter-wave plate, oriented at 45° to incoming linear, produces:
4. Linear at 45°, in Jones notation, is:
5. Unpolarized light cannot be represented by a single Jones vector because:
6. Brewster's angle is the incidence angle at which: