---
title: "How a polarimeter reveals mixed polarization"
description: "Equal light intensity can hide different polarization. Several input states and analyzer settings reveal how a sample transforms the light."
canonical_url: "https://sandboxphysics.com/stories/instruments/mueller-polarimeter/"
language: "en"
page_type: "story"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/stories/instruments/mueller-polarimeter/"
last_modified: "2026-11-03"
---

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Inside the instrument

# How a polarimeter reveals mixed polarization

A Mueller polarimeter sends several polarization states into a sample and measures each output through several analyzers. These intensities distinguish a common rotation of every contribution from a mixture whose unresolved contributions reduce the overall polarization.

Sandbox Physics · 2026-11-03

## Turn a polarizing filter and brightness becomes a question

Rotate polarizing sunglasses in front of an LCD screen. The screen changes brightness because its electric field favors one direction and the sunglasses transmit one direction more strongly. An ordinary photodetector measures total intensity. Add a rotating filter and it can also ask which field direction is favored.

The difference between horizontal and vertical intensities gives one linear-polarization contrast. Two diagonal settings give another. A wave plate changes the relative timing of perpendicular field components so an analyzer can also distinguish the two circular states. Total intensity and these three contrasts are the four Stokes parameters. Each coordinate comes from an intensity comparison.

Sources: [1]

## A wave plate can change polarization at constant power

Send diagonally polarized light through an ideal quarter-wave plate. Its perpendicular field components begin in step. After the plate, one is a quarter-cycle behind the other, so their combined field rotates in a circle. The plate absorbs no light in this ideal example. Total intensity remains unchanged even though polarization changes.

Before the sample, a polarimeter uses polarizing elements to prepare different inputs. After the sample, analyzer elements turn different output polarizations into measurable intensities. Cycling both sets produces a table. A four-by-four Mueller matrix recovered from that table describes how the sample transforms the four input Stokes coordinates into the four output coordinates.

One input asks about one direction in this four-coordinate description. Linear states alone leave part of the circular response undetermined. Both preparation and analysis must span all four coordinates. Additional measurements provide repeated information that can be compared.

Sources: [2]

## Small regions can preserve polarization while their sum loses it

Imagine two equal ideal wave-plate regions. One leaves polarization unchanged; the other turns horizontal polarization into vertical polarization. Horizontal illumination then produces equal horizontal and vertical contributions. If the regions have no stable relative phase, their intensities add. A detector that cannot resolve the regions receives a beam with no favored polarization. This averaging is called depolarization.

Now use a diagonal input that both regions send to the same diagonal output. Their sum remains fully polarized. The difference between the two experiments reveals a directional property of the sample. Horizontal illumination alone would suggest a more general scrambling than actually occurs.

A uniform wave plate gives a definite relation between two field components. An unresolved incoherent mixture needs the averaged intensity description supplied by a Mueller matrix. Spatial averaging of differently transformed light is enough to produce depolarization, even for a sample that stays still.

Sources: [2] [3]

## The instrument transforms polarization too

Polarizers leak some light along the rejected direction. Real wave plates miss the exact intended delay. Treating a commanded angle as an exact state would assign these instrument errors to the sample. Independently known reference light can characterize each analyzer response. An empty-path measurement then determines the states the generator actually sends.

Fresh air and known-wave-plate measurements test predictions beyond the fitted reference observations. Photon counts fluctuate, so small departures of a fitted matrix from physicality must be compared with count uncertainty. A resolved departure from passive-optics requirements calls for checking alignment, calibration or the model. The reference light and the unresolved regions together determine what the final measurement describes.

Sources: [1] [3]

## [Ask the same sample with several input states](https://sandboxphysics.com/labs/mueller-polarimeter/)

Compare air, a uniform plate and unresolved regions, rotate the sample, and inspect the matrix recovered from counts.

The model uses monochromatic transmission and incoherent Jones mixtures with known-reference calibration, independent checks and count uncertainty.

## Sources and further reading

1. [NIST — Optical Polarization Metrology](https://www.nist.gov/programs-projects/optical-polarization-metrology)

   Four Stokes coordinates describe intensity and polarization; modulation of input and analyzer reveals sample structure.

2. [NIST SCATMECH — MuellerMatrix](https://pages.nist.gov/SCATMECH/docs/muellermatrix.htm)

   Mueller transformations, depolarization measures and the distinction between cone mapping and physically realizable Jones ensembles.

3. [San José and Gil — Characterization of passivity in Mueller matrices (2019 preprint)](https://arxiv.org/abs/1909.10077)

   A physically realizable Mueller matrix must also satisfy both passivity inequalities to represent a mixture of passive components.

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