Jones vectors

The Jones vector is a two-component complex column vector that describes the polarization state of fully-polarized monochromatic light. It captures both the amplitude and the relative phase of the two orthogonal electric-field components, making it the natural representation for coherent polarization-dependent calculations.

Definition. Decompose the electric field of a plane wave into orthogonal $x$ and $y$ components:

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

where $E_x, E_y$ are complex amplitudes. The Jones vector is:

$$\mathbf{J} \;=\; \begin{pmatrix} E_x \\ E_y \end{pmatrix} \;=\; \begin{pmatrix} |E_x| \, e^{i\phi_x} \\ |E_y| \, e^{i\phi_y} \end{pmatrix}.$$

Normalization $|E_x|^2 + |E_y|^2 = 1$ is conventional, leaving the Jones vector to encode the polarization state and discarding the overall intensity.

Standard polarization states.

State	Jones vector
Horizontal linear	$\begin{pmatrix} 1 \\ 0 \end{pmatrix}$
Vertical linear	$\begin{pmatrix} 0 \\ 1 \end{pmatrix}$
+45° linear	$\frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ 1 \end{pmatrix}$
$-45°$ linear	$\frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ -1 \end{pmatrix}$
Right circular	$\frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ -i \end{pmatrix}$
Left circular	$\frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ i \end{pmatrix}$
Right elliptical (general)	$\begin{pmatrix} \cos\chi \\ -i \sin\chi \end{pmatrix}$ with ellipticity angle $\chi$

The sign convention for circular polarization (Right means clockwise viewed from source vs from receiver) is unfortunately not universal. Modern optics textbooks generally follow the IEEE convention: viewed from the receiver looking toward the source, the electric field rotates clockwise for right-circular polarization.

Jones matrices. Linear optical elements transform Jones vectors via 2×2 complex matrices:

$$\mathbf{J}_\text{out} \;=\; \mathbf{T} \, \mathbf{J}_\text{in}.$$

Standard Jones matrices:

Element	Jones matrix
Linear polarizer (horizontal axis)	$\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$
Linear polarizer (vertical axis)	$\begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}$
Quarter-wave plate (fast axis horizontal)	$\begin{pmatrix} 1 & 0 \\ 0 & i \end{pmatrix}$
Half-wave plate (fast axis horizontal)	$\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$
Faraday rotator by angle $\theta$	$\begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$
Rotation of element by angle $\theta$	conjugate by $R(\theta)$ rotation matrix

For an element rotated by angle $\theta$, the Jones matrix transforms as $\mathbf{T}_\theta = R(\theta) \, \mathbf{T} \, R(-\theta)$, where $R(\theta)$ is the 2×2 rotation matrix.

Cascaded systems. A sequence of optical elements is described by the product of their Jones matrices in reverse order:

$$\mathbf{T}_\text{total} \;=\; \mathbf{T}_N \cdots \mathbf{T}_2 \, \mathbf{T}_1.$$

For computational convenience, this product is calculated and the result applied to the input Jones vector once.

Limitations. Jones vectors only describe fully-polarized monochromatic light. They cannot describe:

• Partially polarized light: requires Stokes parameters (and Mueller matrices)
• Unpolarized light: same
• Polychromatic light with frequency-dependent polarization: requires either ensemble averaging (Stokes) or coherency matrix formalism
• Quantum (single-photon) polarization states: technically OK but Jones is the classical limit

For coherent optical systems with deterministic polarization control (laser systems, fiber-optic communications), Jones formalism is computationally efficient and physically transparent.

Coherency matrix. A bridge between Jones and Stokes: the coherency matrix $\mathbf{J}$ is the outer product of the Jones vector with itself:

$$\mathbf{C} \;=\; \begin{pmatrix} \langle |E_x|^2 \rangle & \langle E_x E_y^* \rangle \\ \langle E_y E_x^* \rangle & \langle |E_y|^2 \rangle \end{pmatrix}.$$

The angled brackets indicate time-averaging — essential for partially-polarized light. The trace of $\mathbf{C}$ is the total intensity; off-diagonal elements encode polarization correlations. From $\mathbf{C}$, all four Stokes parameters can be derived:

$$S_0 = C_{11} + C_{22}, \;\; S_1 = C_{11} - C_{22}, \;\; S_2 = 2\text{Re}(C_{12}), \;\; S_3 = -2\text{Im}(C_{12}).$$

Standard applications.

• Telecom polarization control: digital signal processing in coherent receivers uses Jones-matrix inversion to demultiplex polarization-multiplexed signals
• Liquid crystal displays: each LC pixel is a controllable Jones-matrix element
• Polarization-maintaining fiber: PM fiber maintains a specific Jones-vector orientation through propagation
• Polarization scramblers and analyzers: characterized by Jones matrices
• Photoelastic modulators: time-varying Jones matrices for fast polarization control
• Quantum cryptography (BB84): photon polarization states represented as Jones-vector basis states

Comparison with Stokes.

Aspect	Jones	Stokes
Components	2 complex	4 real
Phase	Explicit	Implicit (encoded in correlations)
Partial polarization	No	Yes
Most natural for	Coherent optics, simple calculations	Measurement, partial polarization, incoherent sources
Operation	2×2 Jones matrix	4×4 Mueller matrix
Physical units	Field amplitude	Intensity

Most photonics design uses both: Jones for theoretical analysis (compact, intuitive) and Stokes for experimental measurement (directly measurable).

References: Saleh & Teich, Fundamentals of Photonics (3rd ed., 2019), Ch. 6 (polarization optics); Yariv & Yeh, Photonics: Optical Electronics in Modern Communications (6th ed., 2007), Ch. 1 for the canonical electromagnetic treatment; Goldstein, Polarized Light (3rd ed., 2010) for the engineering reference.