What it does
Light is a wave whose electric field points in a direction across the beam. That direction can stay fixed (linear polarisation), spin round (circular), trace an ellipse (elliptical), or wander randomly (unpolarised). Polarisers, waveplates and other elements change it in ways that are easy to get wrong by hand.
This calculator follows the light through a chain of elements with two standard methods:
- Jones calculus uses a two-number complex vector (Ex, Ey) for fully polarised light and a 2 × 2 matrix for each element.
- Mueller calculus uses the four Stokes parameters (S₀ intensity, S₁ horizontal minus vertical, S₂ +45° minus −45°, S₃ right minus left circular) and 4 × 4 matrices, so it also handles partly polarised and unpolarised light and depolarisers.
The state after every element is drawn on the Poincaré sphere, where linear states lie on the equator and circular states at the poles. A waveplate turns the point around an axis through the sphere; a polariser jumps it to a point on the equator.
How to use it
- Choose the input polarisation, or Elliptical to set the azimuth ψ (tilt of the ellipse) and the ellipticity angle χ (0 linear, ±45° circular). Set the degree of polarisation below 1 for partly polarised light.
- List the elements in order. Angles are measured from horizontal, anticlockwise as you look at the oncoming beam:
polariser 45: an ideal linear polariser with its transmission axis at 45°;qwp 0,hwp 22.5: quarter- and half-wave plates with the fast axis at that angle;retarder 60 30: any waveplate, retardance 60°, fast axis at 30°;rotator 30: optical rotation (sugar solution, Faraday rotator) by 30°;partial 0.9 0.1 0: a partial polariser passing 90 % along 0° and 10 % across;depolariser 0.5: scrambles half of the polarisation.
- Read the output state, intensity, Stokes and Jones vectors, the table after each element, and the whole chain’s Mueller matrix.
- The chart turns the last element through 180° and shows the output intensity: for a polariser after linear light this is Malus’s law, I = I₀ cos²θ.
Worked example
The made-up example starts with horizontal light. A half-wave plate at 22.5° turns it to linear +45° (a half-wave plate rotates linear light by twice the angle between them). A quarter-wave plate at 0° then makes it right-handed circular (S₃ = 1, the north pole of the sphere). Finally a polariser at 30° passes 50 % of the light as linear at 30°. The sweep chart is flat at 50 %: circular light has no preferred direction, so any polariser angle passes half.
Try pol 0, pol 90 (crossed polarisers: no light) and then put pol 45 between them: an eighth of unpolarised light (12.5 %) gets through.
FAQ
Why is the handedness the opposite of my textbook? There are two conventions. This page uses the IEEE one (looking along the beam from behind; right-handed means clockwise, S₃ > 0). Many optics texts look into the beam, which swaps the names. The physics is the same.
Why no Jones vector for partly polarised light? A Jones vector can only describe fully polarised light; the Stokes vector and Mueller matrices cover everything.
Are real waveplates like this? Real ones have a retardance that depends on wavelength and angle of incidence, and polarisers have a finite extinction ratio (use partial for that). For study and planning.
Where the numbers come from
- Jones calculus, Stokes parameters, Mueller matrices (M_ij = ½ Tr(J† σᵢ J σⱼ)) and the Poincaré sphere: standard results, for example Goldstein, D., Polarized Light, and Hecht, E., Optics, chapter 8. Handedness follows the IEEE convention, stated on the page.
- Checked against an independent NumPy implementation (Mueller matrices built as A (J ⊗ J*) A⁻¹) and against Malus's law; no data tables are used. The example chain is made up.
Standard scientific values, such as constants, atomic weights and melting points, are facts found in every textbook; we name where ours were checked. If you are an author or publisher and want a value, a credit or a table changed or removed, tell us and we will act quickly.
Last updated 2026-11-09.