4.1 Polarization states and their representations

Light waves can wiggle in different ways as they travel. This wiggling is called polarization. There are three main types: linear, circular, and elliptical. Each type has unique properties that affect how light interacts with materials.

We use special math tools to describe polarization. Jones vectors and Stokes parameters help us represent polarization states precisely. The Poincaré sphere gives us a visual way to understand how polarization changes as light moves through different materials.

Polarization States

Linear vs circular vs elliptical polarization

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• Linear polarization occurs when the electric field oscillates in a single plane, which can be horizontal, vertical, or at any angle between
• Circular polarization happens when the electric field rotates in a circular path, either right-handed (RHCP) or left-handed (LHCP), with equal amplitudes in two orthogonal components and a 90° phase difference
• Elliptical polarization is characterized by the electric field tracing an elliptical path due to unequal amplitudes in two orthogonal components with a phase difference other than 0° or 90°
  • Linear and circular polarizations are special cases of elliptical polarization (phase difference of 0° or 90° and equal or unequal amplitudes)

Jones vectors and Stokes parameters

• Jones vectors are complex 2D vectors representing the amplitude and phase of the electric field, expressed as $\vec{E} = \begin{pmatrix} E_x \\ E_y \end{pmatrix} = \begin{pmatrix} a_x e^{i\delta_x} \\ a_y e^{i\delta_y} \end{pmatrix}$
  • Horizontal linear polarization: $\begin{pmatrix} 1 \\ 0 \end{pmatrix}$, vertical linear polarization: $\begin{pmatrix} 0 \\ 1 \end{pmatrix}$, RHCP: $\frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ -i \end{pmatrix}$, LHCP: $\frac{1}{\sqrt{2}} \begin{pmatrix} 1 \\ i \end{pmatrix}$
• Stokes parameters are real-valued 4D vectors representing the polarization state, given by $S = \begin{pmatrix} S_0 \\ S_1 \\ S_2 \\ S_3 \end{pmatrix} = \begin{pmatrix} I \\ Q \\ U \\ V \end{pmatrix}$
  • $S_0$ represents the total intensity, while $S_1$, $S_2$, and $S_3$ represent the polarization state
  • Horizontal linear polarization: $\begin{pmatrix} 1 \\ 1 \\ 0 \\ 0 \end{pmatrix}$, vertical linear polarization: $\begin{pmatrix} 1 \\ -1 \\ 0 \\ 0 \end{pmatrix}$, RHCP: $\begin{pmatrix} 1 \\ 0 \\ 0 \\ 1 \end{pmatrix}$, LHCP: $\begin{pmatrix} 1 \\ 0 \\ 0 \\ -1 \end{pmatrix}$

Poincaré sphere for polarization

• The Poincaré sphere is a 3D representation of polarization states where Stokes parameters $S_1$, $S_2$, and $S_3$ form the Cartesian coordinates
  • Linear polarization states lie on the equator, circular polarization states are at the poles, and elliptical polarization states are on the surface of the sphere
• Polarization state changes are represented by rotations on the Poincaré sphere
  • Birefringent elements cause rotations about an axis in the $S_1$-$S_2$ plane, while polarization rotators cause rotations about the $S_3$ axis

Optical elements and polarization effects

• Linear polarizers transmit light with electric field parallel to the polarizer's axis, following Malus' law: $I = I_0 \cos^2 \theta$, where $\theta$ is the angle between the polarizer and the incident polarization
• Wave plates (retarders) introduce a phase difference between orthogonal components
  • Quarter-wave plate (QWP) introduces a 90° phase difference, converting linear polarization to circular polarization and vice versa
  • Half-wave plate (HWP) introduces a 180° phase difference, rotating linear polarization by twice the angle between the fast axis and the incident polarization
• Polarization rotators change the polarization state
  • Faraday rotator uses the Faraday effect to rotate the polarization state, with the rotation angle depending on the magnetic field strength and the material's Verdet constant
  • Optical activity in chiral materials rotates the polarization state, with the rotation angle depending on the material's specific rotation and the path length