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.

Ask the sample six ways

Current preview

Intensity response $0.816$

Polarization retention index $0.577$

Independent check RMS $0.0119$

State-set condition number $1.77$

SampleSample axis

\theta/{}^\circ

25

Sample retardance

\delta/{}^\circ

90

Polarization retention / first-domain fraction

q

0.55

Sample transmission factor

T

0.82

Exposure per state

t/\mathrm{s}

1

Drag the sample rim. The moving optics cycle through the same input and analyzer commands as the count table.

State coverage, alignment and assumptions

Input and analyzer statesGenerator alignment trim

\epsilon_G/{}^\circ

0

Analyzer alignment trim

\epsilon_A/{}^\circ

0

Instrument wave-plate error

\epsilon_\delta/{}^\circ

0

Acquisition seed

s

194

Poisson count noise

A monochromatic transmission bench at 633 nanometres. The signal travels along world positive x; transverse x and y correspond to world positive y and z. Optical dimensions, source rays and domain areas are schematic. Each unresolved region contributes intensity incoherently.

S_3=-2\operatorname{Im}(E_xE_y^*),\qquad e^{\mathrm{i}(kz-\omega t)}

For the two-region sample, the second region is a half-wave plate whose fast axis is forty-five degrees beyond the sample reference axis; the mixture control gives the first region’s intensity weight. For the four-region sample, it gives isotropic polarization retention.

Six independently known Stokes references calibrate the analyzer; an empty-path measurement then determines generated states. Air and a known retarder remain independent checks. The monitor split and known reference states are assumed exact. This is known-reference calibration, not eigenvalue calibration.

Shot counts use exact Poisson sampling below fifty counts and a rounded Gaussian approximation above it. Sixty-four count-bootstrap realizations share the same resampled dark within each realization and resample all calibration frames. Three-standard-deviation physicality flags omit standard uncertainty, drift and alignment instability; they do not certify a material.

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.

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 misconceptionEvery output must have one polarization ellipse.

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

Small negative eigenvalues need count uncertainty

Typical misconceptionAny negative fitted physicality eigenvalue proves an impossible sample.

Better mental modelFinite 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

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.
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.
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.
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.

Mueller Matrix Polarimeter

A retained polarization measurement

Measure what happens to every input

Core mathematical model

From intensity table to sample matrix

Polarization fraction

Common difficulties

A single ellipse cannot represent a mixed beam

Small negative eigenvalues need count uncertainty

Run the experiment

Compare air and a wave plate

Mix unresolved regions

Find the hidden difference

Remove the circular states

Related experiments and articles

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