Jones Matrices and Polarization States

How to describe, predict, and manipulate the polarization of light with nothing more than 2-component vectors and 2×2 matrices.

Key Takeaways

Polarization describes the direction the electric field oscillates. It can be linear, circular, or elliptical.
A Jones vector (two complex numbers $[E_{x}, E_{y}]$ ) completely describes a polarization state. A Jones matrix (a 2×2 matrix) describes what an optical element does to that state.
To find what comes out of a chain of optical elements, multiply the matrices together and apply them to the input vector.
Stokes parameters and the Poincaré sphere let you visualize polarization, including partially polarized light that Jones vectors can't represent.

Polarization Is Everywhere (You Just Need the Right Filter)

You're trying to photograph a fish through the water, but all you see is glare. You rotate a polarizing filter on your camera lens and the fish appears. Or you're aligning a laser and need to know whether its output is linearly or circularly polarized. Or you're designing an optical isolator to protect a laser from back-reflections.

All of these require understanding polarization, and more specifically, being able to calculate what happens to polarized light as it passes through optical elements. That's what Jones calculus does: it gives you a simple matrix algebra for predicting the polarization state at every point in an optical system.

What Is Polarization?

Light is a wave where the electric field oscillates perpendicular to the direction it travels. If the wave moves along the z-axis, the electric field lives in the x-y plane.

Polarization is the pattern the electric field traces in that plane as the wave passes a fixed point. There are three kinds:

Linear polarization: The field oscillates along a fixed line: horizontal (along x), vertical (along y), or any angle in between. This is what you get from a polarizer.

Circular polarization: The field vector rotates in a circle at constant amplitude. If it rotates clockwise as you look into the oncoming beam, it's right-hand circular (RCP). Counterclockwise is left-hand circular (LCP). This is what you get when you put a quarter-wave plate after a polarizer at the right angle.

Elliptical polarization: The general case. The field traces out an ellipse. Linear and circular are just special cases: a degenerate ellipse is a line, and a symmetric ellipse is a circle.

Polarization State Explorer

Angle: 0°

Jones Vector

E_x = 1.000

E_y = 0.000

I = 1.000 · Linear

Jones Vectors: Describing Polarization with Two Numbers

To do calculations with polarization, you need a mathematical representation. The Jones vector is the simplest one. A Jones vector is a column of two complex numbers representing the x and y components of the electric field:

E = [\begin{array}{c} E_{x} \\ E_{y} \end{array}]

Each component has an amplitude and a phase. Horizontal polarization has all its field along x:

H = [\begin{array}{c} 1 \\ 0 \end{array}]

Vertical has all its field along y:

V = [\begin{array}{c} 0 \\ 1 \end{array}]

Diagonal (+45°) has equal parts x and y, in phase:

D = \frac{1}{\sqrt{2}} [\begin{array}{c} 1 \\ 1 \end{array}]

Now here's where complex numbers earn their keep. The imaginary unit $i$ represents a phase delay of a quarter wavelength: it is how we encode the timing difference between the x and y components.

Right circular polarization has equal amplitudes in x and y, but the y-component lags the x-component by 90°:

R C P = \frac{1}{\sqrt{2}} [\begin{array}{c} 1 \\ - i \end{array}]

The $- i$ means "same amplitude as x, but delayed by a quarter wavelength." That quarter-cycle delay is what makes the field rotate in a circle instead of oscillating in a line.

The Six Standard Polarization States

State	Jones vector	What it means
H (horizontal)	$[1, 0]$	Field along x
V (vertical)	$[0, 1]$	Field along y
D (diagonal, +45°)	$[1, 1] / \sqrt{2}$	Equal x and y, in phase
A (anti-diagonal, −45°)	$[1, - 1] / \sqrt{2}$	Equal x and y, opposite phase
RCP (right circular)	$[1, - i] / \sqrt{2}$	y lags x by 90°
LCP (left circular)	$[1, + i] / \sqrt{2}$	y leads x by 90°

The intensity of a Jones vector is the sum of the squared amplitudes: $I = ∣ E_{x} ∣^{2} + ∣ E_{y} ∣^{2}$ . For all six states above, $I = 1$ (normalized to unit intensity).

Jones Matrices: What Optical Elements Do

Each optical element has a 2×2 Jones matrix that transforms the input polarization into the output:

E_{out} = J \cdot E_{in}

This is ordinary matrix-vector multiplication. A chain of elements is a chain of multiplications: if light passes through element 1, then 2, then 3:

E_{out} = J_{3} \cdot J_{2} \cdot J_{1} \cdot E_{in}

Note the order: the first element the light hits is on the right, closest to the input vector. Here are the most important optical elements, all shown with the element's axis aligned horizontally (0°).

Jones Matrix for a Linear Polarizer

Transmits light along its transmission axis and blocks the perpendicular component:

J_{pol} = [\begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array}]

Jones Matrix for a Quarter-Wave Plate (QWP)

Introduces a 90° phase delay between fast and slow axes. This converts linear to circular polarization:

J_{QWP} = [\begin{array}{cc} 1 & 0 \\ 0 & i \end{array}]

Jones Matrix for a Half-Wave Plate (HWP)

Introduces a 180° phase delay. Rotates the direction of linear polarization: if the fast axis is at angle θ, the output is rotated by 2θ:

J_{HWP} = [\begin{array}{cc} 1 & 0 \\ 0 & - 1 \end{array}]

Rotator

A pure rotation of the polarization direction by angle α:

J_{rot} (α) = [\begin{array}{cc} \cos α & - \sin α \\ \sin α & \cos α \end{array}]

Rotating Any Element to an Arbitrary Angle

All the matrices above are shown at 0°. To rotate any element to angle θ, use:

J (θ) = R (θ) \cdot J_{0} \cdot R (- θ)

The idea: rotate the coordinate system so the element's axis is aligned with x, apply the element's matrix, then rotate back. The Polarization Calculator handles this rotation automatically for every element.

Quick Reference

Element	Matrix (at 0°)	What it does
Linear polarizer	$[[1, 0], [0, 0]]$	Passes x, blocks y
QWP	$[[1, 0], [0, i]]$	90° phase delay on slow axis
HWP	$[[1, 0], [0, - 1]]$	180° phase delay (sign flip)
Rotator(α)	$[[\cos α, - \sin α], [\sin α, \cos α]]$	Rotates by α

Try It: Jones Matrix Chain Builder

Build your own optical element chain and see the Jones vector, Stokes parameters, and Poincaré sphere update in real time. Start with the default preset (QWP: Linear → Circular) or load any of 7 built-in configurations.

Open Polarization Calculator

Malus's Law: The First Calculation

The most basic polarization experiment: shine horizontally polarized light through a polarizer rotated to angle θ. What's the transmitted intensity?

E_{out} = J_{pol} (θ) \cdot H

The result: $I = \cos^{2} θ$ . This is Malus's law.

Angle	$\cos^{2} θ$	Transmitted
0°	1.000	100%
30°	0.750	75%
45°	0.500	50%
60°	0.250	25%
90°	0.000	0% (extinction)

At 90°, the polarizer is perpendicular to the input: complete extinction. This is the "crossed polarizers" configuration.

Malus's Law: I = I₀ cos²θ

Polarizer angle: 0°

Transmitted intensity100.0%

Output

Open Malus's Law preset in Polarization Calculator →

Worked Example: Making Circular Light

Problem: You need right-hand circular polarization. You have a horizontal laser and a box of optical elements. What do you use?

Solution: Place a QWP with its fast axis at 45° after the laser.

Step 1: Write the input. Horizontal laser: $E_{in} = [1, 0]$ .

Step 2: The QWP matrix at 45° (using the rotation formula) works out to:

J = \frac{1}{2} [\begin{array}{cc} 1 + i & 1 - i \\ 1 - i & 1 + i \end{array}]

Step 3: Multiply: $E_{out} = \frac{1}{2} [1 + i, 1 - i]$

Step 4: Check. Both components have magnitude $1 / \sqrt{2}$ , so $I = 1.0$ (100% transmission). The phase of $E_{y}$ relative to $E_{x}$ is −90°. This is RCP.

Verify it yourself in the Polarization Calculator: select H input, add a QWP at 45°, and watch the output ellipse become a circle with the Poincaré sphere point jumping to the north pole (R).

Worked Example: Rotating Polarization with a Half-Wave Plate

Problem: You have a vertically polarized beam and you need it horizontal. You can't rotate the laser.

Solution: Use an HWP at 45°. An HWP at angle θ rotates the polarization direction by 2θ. Setting θ = 45° gives a rotation of 90°: V becomes H.

HWP (45 °) = [\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}]

E_{out} = [\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}] [\begin{array}{c} 0 \\ 1 \end{array}] = [\begin{array}{c} 1 \\ 0 \end{array}] = H

Full intensity, pure horizontal. The HWP rotated the polarization by 90° without losing any light. This is how lasers with fixed polarization are adapted to different experimental geometries: you rotate the wave plate, not the laser.

Stokes Parameters: When Jones Vectors Aren't Enough

Jones vectors describe fully polarized light perfectly. But real light is often partially polarized: sunlight, LED light, light scattered off a rough surface. Jones vectors can't represent this.

Stokes parameters can. A Stokes vector has four real components, each answering a measurable question:

$S_{0}$ = total intensity
$S_{1}$ = preference for horizontal over vertical
$S_{2}$ = preference for +45° over −45°
$S_{3}$ = preference for RCP over LCP

For fully polarized light, $S_{0}^{2} = S_{1}^{2} + S_{2}^{2} + S_{3}^{2}$ . For partially polarized light, the right side is smaller. The degree of polarization (DOP) is:

DOP = \frac{\sqrt{S_{1}^{2} + S_{2}^{2} + S_{3}^{2}}}{S_{0}}

DOP = 1 means fully polarized. DOP = 0 means completely unpolarized. Our Polarization Calculator converts Jones to Stokes automatically in the Stokes Parameters tab.

State	Stokes vector
H	$[1, + 1, 0, 0]$
V	$[1, - 1, 0, 0]$
RCP	$[1, 0, 0, + 1]$
Unpolarized	$[1, 0, 0, 0]$

The Poincaré Sphere: Seeing Polarization in 3D

The normalized Stokes parameters $[S_{1} / S_{0}, S_{2} / S_{0}, S_{3} / S_{0}]$ define a point in three-dimensional space. For fully polarized light, that point lies on the surface of a unit sphere: the Poincaré sphere.

It gives you an intuitive map of every possible polarization state:

Equator: All linear polarizations. H at one end, V at the opposite. D and A are 90° away.
North pole ( $S_{3} = + 1$ ): Right circular polarization.
South pole ( $S_{3} = - 1$ ): Left circular polarization.
Between equator and poles: Elliptical polarization.
Inside the sphere: Partially polarized light. The center is completely unpolarized.

Optical elements move the point around the sphere. A QWP rotates the point by 90° around an axis set by its fast-axis orientation. An HWP rotates by 180°. The worked example above (H through QWP at 45° to RCP) traces a 90° arc from the equator straight up to the north pole.

Explore: Interactive Poincaré Sphere

Drag to rotate the sphere. Add optical elements and watch the polarization state trace a path across the surface. The default preset shows horizontal light converted to RCP by a quarter-wave plate at 45°.

Open Polarization Calculator

The Poincaré sphere is more than a visualization tool. It tells you the "distance" between polarization states and predicts how elements will transform them. Two states on opposite sides of the sphere are orthogonal: they interfere destructively if combined. Two states close together on the sphere are nearly identical.

Putting It All Together

Tool	What it describes	When to use it
Jones vector	Polarization state (fully polarized)	Laser beams, clean optical setups
Jones matrix	Effect of an optical element	Polarizers, wave plates, rotators
Stokes parameters	Polarization state (any)	Measurements, partially polarized light
Poincaré sphere	Geometric picture	Visualizing transformations, system design

For fully polarized light in a clean optical chain, Jones calculus is all you need: write down the input vector, multiply by each element's matrix, read off the output. For real-world measurements involving partial polarization, you need Stokes and Mueller matrices. Our Polarization Calculator implements Jones calculus with full Stokes conversion, so you can work in whichever picture is most natural.

Build your own optical chain, or start with one of the built-in presets: crossed polarizers, circular polarizer, optical isolator, Malus's law, and more.

Try it now in the Photizon Polarization Calculator.

Chain optical elements, see polarization states transform in real time, and explore the Poincaré sphere. Free, no signup required.

Open Polarization Calculator

References

E. Hecht, Optics, 5th ed. (Pearson, 2017), Ch. 8: "Polarization." Standard treatment of Jones and Stokes formalisms.
B.E.A. Saleh and M.C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley, 2019), Ch. 6. Excellent coverage of Jones matrices and the Poincaré sphere.
D. Goldstein, Polarized Light, 3rd ed. (CRC Press, 2011). Comprehensive reference on polarization measurement and manipulation.
R.C. Jones, "A New Calculus for the Treatment of Optical Systems," J. Opt. Soc. Am. 31, 488 (1941). The original paper.