---
title: "Polarization State Explorer Interactive Lab | Sandbox Physics"
description: "How can a Jones vector, an ellipse and Stokes measurements all describe the same polarization state? They describe one state in different ways."
canonical_url: "https://sandboxphysics.com/labs/polarization-state-explorer/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/polarization-state-explorer/"
last_modified: "2026-10-27"
---

Optics 073 · Polarization, anisotropy, and modulation

# Polarization State Explorer

**How can a Jones vector, an ellipse and Stokes measurements all describe the same polarization state?** They describe one state in different ways. The Jones vector lists the two field components and their relative phase; the ellipse is the path the field tip traces in one cycle; the Stokes parameters are built from intensities measured through polarizers. Linear polarizers alone cannot show which way the field turns. That handedness needs a circular analysis.

Rotate polarizers on a bench to see Malus’s law and the third-polarizer effect, then measure Stokes parameters from counts and see when the handedness becomes ambiguous.

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

The polarizers are ideal and absorptive, the light is described by a monochromatic coherency matrix, and the circular analyzer is fixed and calibrated. All exposures are equal, and Poisson noise is optional. Leakage through crossed polarizers, alignment errors and self-calibration are not included.

Physics tutorial

## Audit the physics of Polarization State Explorer

**Background** A linear polarizer passes the projection of the field onto its axis; it does not act like a slot shaped for one direction. For a general mixture of polarizations, the transmitted power is $I_\alpha=\tfrac12(S_0+S_1\cos2\alpha+S_2\sin2\alpha)$. For purely linear input this becomes Malus’s law. For unpolarized or circular input every ideal linear analyzer transmits one half.

**Why it matters** How can a Jones vector, an ellipse and Stokes measurements all describe the same polarization state?

Start with the essentials

- **Focus question**: How can a Jones vector, an ellipse and Stokes measurements all describe the same polarization state?
- **One-sentence intuition**: Linear projections cannot see handedness: $I_\alpha(S_3)=I_\alpha(-S_3)$. Finding the missing component takes the two calibrated exposures through circular analyzers. A flat analyzer curve never tells you what it is.

### Core mathematical model

#### Jones field and the partial mixture

$$
\begin{aligned}\mathbf j&=\cos\chi\,\mathbf a_\psi+i\sin\chi\,\mathbf b_\psi\\\mathbf a_\psi&=(\cos\psi,\sin\psi)^{\mathsf T}\\\mathbf b_\psi&=(-\sin\psi,\cos\psi)^{\mathsf T}\\\mathbf C&=p\mathbf j\mathbf j^\dagger+\tfrac{1-p}{2}\mathbf1\\\mathbf E(t)&=\Re\{\mathbf j e^{-i\omega t}\}\end{aligned}
$$

The source power is one. The horizontal basis vector points along negative Z in the scene and the vertical one along Y, so their cross product points along the beam in the X direction. Angles are counted counterclockwise as you look back toward the source. In the NIST convention used here, positive ellipticity is left-handed.

#### Ideal projection and Malus’s limit

$$
\begin{aligned}\mathbf C'&=\mathbf P_\alpha\mathbf C\mathbf P_\alpha^\dagger\\\mathbf P_\alpha&=\mathbf a_\alpha\mathbf a_\alpha^{\mathsf T}\\\mathbf a_\alpha&=(\cos\alpha,\sin\alpha)^{\mathsf T}\\I_{\rm linear}/I_0&=\cos^2(\alpha-\psi)\end{aligned}
$$

The orthogonal component is absorbed. The light that leaves is purely linear along the transmission axis, and the opaque mount around the plate stays opaque.

#### Three projections add no energy

$$
\begin{aligned}I_D/I_0&=\tfrac12\cos^2\theta_2\cos^2(\alpha-\theta_2)\\\alpha&=\tfrac\pi2,\quad\theta_2=\tfrac\pi4\\I_D/I_0&=\tfrac18\end{aligned}
$$

This expression assumes unpolarized input and a first plate fixed horizontal. Without the middle plate, the crossed end plates transmit nothing.

#### Estimating the Stokes parameters from counts

$$
\begin{aligned}C_b&\sim{\rm Poisson}(N_{\rm in}I_b)\\\widehat S_0&=\tfrac{\sum_b C_b}{3N_{\rm in}}\\\widehat S_1&=\tfrac{C_H-C_V}{N_{\rm in}}\\\widehat S_2&=\tfrac{C_D-C_A}{N_{\rm in}}\\\widehat S_3&=\tfrac{C_L-C_R}{N_{\rm in}}\end{aligned}
$$

The estimate uses six independent, equal exposures, a calibrated expected incident count and unit efficiency. With only the four linear records, the total intensity comes from two pair sums and the circular Stokes parameter stays unknown. With noise turned off, the records are expected values and can be fractional.

#### Physical states and noisy estimates

$$
p=\frac{\sqrt{S_1^2+S_2^2+S_3^2}}{S_0}\le1
$$

Every true coherency matrix satisfies this inequality, but an unconstrained estimate from finite counts can break it. A flat scan with a linear analyzer could come from circular light, unpolarized light or a mixture, so it does not identify one state.

### Common difficulties

#### More polarizers must mean less light?

**Typical misconception** Inserting a middle plate into crossed end plates cannot brighten the detector.

**Better mental model** The middle plate projects the light onto a new direction, which changes the state that reaches the last plate. With an unpolarized source, a diagonal middle plate lets through one eighth of the power; without it, nothing gets through. Every plate still absorbs power, so no energy is created.

### Run the experiment

1. 01

   #### Scene 1: Malus-law analyzer

   Start with a single analyzer. Sweep its axis through a half-turn, then compare linear, circular and unpolarized input.

   **What to observe:** A Jones vector describes only the polarized part. The unpolarized remainder is carried by a coherency matrix, and no definite ellipse is drawn for it.

2. 02

   #### Scene 2: Three-polarizer paradox

   Choose three polarizers and keep the final plate crossed. Remove the diagonal middle plate and insert it again, and watch its physical mount move out of the beam.

   **What to observe:** All the ellipse panels share one fixed amplitude scale. The zero of local phase is arbitrary, and the slow field animation shows neither photon travel nor a measured optical phase.

3. 03

   #### Scene 3: Stokes polarimetry

   Measure the Stokes parameters. Select each exposure to see the analyzer setting it really used. Remove the circular records, reverse the ellipticity, then add them back. Also try low Poisson counts.

   **What to observe:** The six records are separate, equal exposures of the same input, not six output branches. The count estimator works only from the counts and never sees the reference state.

Physical conclusion

Predict the transmitted power, tell a field ellipse apart from a statistical mixture, and know exactly which recorded projections make each Stokes component measurable.

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)