---
title: "Mueller Matrix Polarimeter Interactive Lab | Sandbox Physics"
description: "How can intensity measurements reveal whether a sample scrambles polarization?"
canonical_url: "https://sandboxphysics.com/labs/mueller-polarimeter/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/mueller-polarimeter/"
last_modified: "2026-11-03"
---

IN21 · What does a sample do to every polarization?

# Mueller Matrix Polarimeter

**How can intensity measurements reveal whether a sample scrambles polarization?** A polarimeter sends several known polarization states through a sample and measures each output through several analyzers. Together, these intensities reveal how the sample changes polarization. Reduced polarization can arise when the detector averages light from differently oriented regions.

Rotate the sample, cycle six input and analyzer states, and recover a Mueller matrix from calibrated photon counts.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

Model assumptions

A monochromatic transmission model at 633 nanometres uses incoherent sums of passive Jones elements. Independently known six-state references calibrate the analyzer, followed by sample-free generator characterization; this is not eigenvalue calibration. Independent air and retarder checks are withheld. Poisson counts, a shared dark measurement and all calibration counts enter a parametric count bootstrap. Reference-state definitions and monitor fraction are assumed exact. Reported uncertainties omit alignment drift and standard uncertainty. Finite detector saturation and rank loss with linear-only states invalidate inversion. The optical bench, domains and photon speed are schematic. Physicality checks test coherency positivity and both passive-throughput inequalities with count uncertainty; fitted matrices are never projected onto a physical model.

### A retained polarization measurement

$M_{ij}\quad(i,j=0,1,2,3)\qquad B_{ab}\quad(a,b=1,\ldots,6)$

Raw counts, references and physicality

Rows cycle analyzer states; columns cycle prepared input states. H, V, D, A denote horizontal, vertical and the two diagonal commands. C+ and C− denote the declared signed circular commands; real calibrated states include the instrument errors.

Sample signal photoelectron counts

|  | H | V | D | A | C+ | C− |
| --- | --- | --- | --- | --- | --- | --- |

Retained measurements

| Record | Calibration | Validity |
| --- | --- | --- |

CSV and JSON include the sample signal and monitor counts, shared dark samples, six known reference definitions, generator characterization, independent air and retarder checks, command angles, units, metadata and fitted results. The ideal sample reference is separate and is excluded from reconstruction. No clipping or physicality projection changes the fitted matrix.

Physics tutorial

## Measure what happens to every input

**Background** A quarter-wave plate can turn diagonal linear polarization into circular polarization. A detector that sees only total power misses that change.

**Why it matters** Many small regions can each preserve polarization while their unresolved sum loses it. A full set of input and output measurements reveals this distinction.

Start with the essentials

- **Focus question**: How can intensity measurements reveal whether a sample scrambles polarization?
- **One-sentence intuition**: Different prepared inputs and analyzer states turn hidden polarization changes into different intensities.

### Core mathematical model

#### From intensity table to sample matrix

$$
B=AMW,\qquad M=A^{+}BW^{+}
$$

Rows of the calibrated analyzer matrix A describe six output measurements. Columns of W describe six prepared inputs. Their inverses exist on all four Stokes coordinates only when both sets span them. B contains monitor-normalized dark-corrected intensity observations.

#### Polarization fraction

$$
P=\frac{\sqrt{S_1^2+S_2^2+S_3^2}}{S_0}
$$

The first Stokes coordinate is total intensity; the other three measure paired linear and circular contrasts. The right-handed transverse basis uses the negative complex time convention and S3 equals minus twice the imaginary cross correlation.

### Common difficulties

#### A single ellipse cannot represent a mixed beam

**Typical misconception** Every output must have one polarization ellipse.

**Better mental model** An unresolved incoherent mixture needs intensity and polarization contrasts. Its polarized fraction can fall below one.

#### Small negative eigenvalues need count uncertainty

**Typical misconception** Any negative fitted physicality eigenvalue proves an impossible sample.

**Better mental model** Finite counts perturb the fitted matrix. The count bootstrap includes shared calibration fluctuations and reports a resolved violation only beyond its declared three-standard-deviation threshold.

### Run the experiment

1. 01

   #### Compare air and a wave plate

   Choose the air and phase-change presets.

   **What to observe:** The intensity table changes with polarization even when the ideal sample preserves power. A wave plate keeps a fully polarized input fully polarized.

2. 02

   #### Mix unresolved regions

   Select four unresolved domains in Sample and lower the polarization-retention control.

   **What to observe:** The diagonal polarization responses shrink. The detector averages mutually incoherent contributions from four regions.

3. 03

   #### Find the hidden difference

   Choose the two-domain preset and inspect the matrix.

   **What to observe:** Horizontal input loses polarization, but diagonal input remains polarized. One input alone would miss this directional response.

4. 04

   #### Remove the circular states

   Choose the linear-only preset, then restore complete coverage and measure standards.

   **What to observe:** Linear inputs cannot determine the circular sector. The instrument reports rank loss instead of completing an unsupported matrix.

Physical conclusion

A Mueller matrix describes an averaged intensity transformation; unresolved mixtures can depolarize even when each region preserves polarization.

## Related experiments and articles

How physics instruments measure · Reading path

### How physics instruments measure

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **How a polarimeter reveals mixed polarization**](https://sandboxphysics.com/stories/instruments/mueller-polarimeter/)

Waves & optics · Starter path

## From one ray to interference measurement

Start with refraction, find where the ray picture stops working, then use interference to measure tiny distances.

[Start here → **Optical Boundary Explorer** Turn the incident angle and watch reflected and transmitted power trade off. Then switch to the critical-angle scene. Observe · 11 min](https://sandboxphysics.com/labs/optical-boundary-explorer/)

1. [1 Optical Boundary Explorer](https://sandboxphysics.com/labs/optical-boundary-explorer/)
2. [2 Rainbow Caustic Observatory](https://sandboxphysics.com/labs/rainbow-laboratory/)
3. [3 Camera Obscura](https://sandboxphysics.com/labs/camera-obscura/)
4. [4 Coherent Wave Composer](https://sandboxphysics.com/labs/coherent-wave-composer/)
5. [5 Young Double-Slit Bench](https://sandboxphysics.com/labs/young-double-slit-bench/)
6. [6 Michelson Precision Lab](https://sandboxphysics.com/labs/michelson-precision-lab/)

[View the full path →](https://sandboxphysics.com/topics/waves-and-optics/#starter-path)