Glossary entry

Jones vectors

A 2×1 complex column vector representation of the polarization state of fully-polarized light. Compact mathematical framework for analyzing polarization optics in the coherent regime.

Optics fundamentalsUpdated September 2026

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:

E (z, t) = Re ⁣ {(\begin{array}{c} E_{x} \\ E_{y} \end{array}) e^{i (k z - ω t)}},

where $E_{x}, E_{y}$ are complex amplitudes. The Jones vector is:

J = (\begin{array}{c} E_{x} \\ E_{y} \end{array}) = (\begin{array}{c} ∣ E_{x} ∣ e^{i ϕ_{x}} \\ ∣ E_{y} ∣ e^{i ϕ_{y}} \end{array}) .

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	$(\begin{array}{c} 1 \\ 0 \end{array})$
Vertical linear	$(\begin{array}{c} 0 \\ 1 \end{array})$
+45° linear	$\frac{1}{\sqrt{2}} (\begin{array}{c} 1 \\ 1 \end{array})$
$- 45 °$ linear	$\frac{1}{\sqrt{2}} (\begin{array}{c} 1 \\ - 1 \end{array})$
Right circular	$\frac{1}{\sqrt{2}} (\begin{array}{c} 1 \\ - i \end{array})$
Left circular	$\frac{1}{\sqrt{2}} (\begin{array}{c} 1 \\ i \end{array})$
Right elliptical (general)	$(\begin{array}{c} \cos χ \\ - i \sin χ \end{array})$ with ellipticity angle $χ$

The sign convention for circular polarization (Right means clockwise viewed from source vs from receiver) is unfortunately not universal. The table follows the convention common in optics textbooks (Hecht, Born and Wolf): viewed from the receiver looking toward the source, the electric field rotates clockwise for right-circular polarization. The IEEE convention used in radio engineering is the opposite, so check which one a source uses before comparing signs.

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

J_{out} = T J_{in} .

Standard Jones matrices:

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

For an element rotated by angle $θ$ , the Jones matrix transforms as $T_{θ} = R (θ) T R (- θ)$ , where $R (θ)$ 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:

T_{total} = T_{N} \dots T_{2} 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 $C$ is the outer product of the Jones vector with itself:

C = (\begin{array}{cc} ⟨ ∣ E_{x} ∣^{2} ⟩ & ⟨ E_{x} E_{y}^{*} ⟩ \\ ⟨ E_{y} E_{x}^{*} ⟩ & ⟨ ∣ E_{y} ∣^{2} ⟩ \end{array}) .

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

S_{0} = C_{11} + C_{22}, S_{1} = C_{11} - C_{22},

S_{2} = 2 Re (C_{12}), S_{3} = 2 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).

The Jones calculus calculator multiplies the matrices of up to four polarizers, wave plates, retarders and rotators and draws the output polarization ellipse.

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.