Polarimetry Explorer

Light is an electromagnetic wave. Polarization describes how the electric field vector E oscillates as the wave propagates. Click below to see the E-field for each type.

Linear 0° (Horizontal): E-field oscillates along x-axis. Stokes: S=[1,1,0,0].

Types of Polarization

Linear Polarization

E-field oscillates in a fixed plane at angle α. Created by a linear polarizer or PBS.

S = [1, cos 2α, sin 2α, 0]

Circular Polarization

E-field rotates in a circle. RCP: clockwise looking into beam. Created by linear + QWP at 45°.

S = [1, 0, 0, ±1]

Elliptical Polarization

Most general case. Described by orientation ψ and ellipticity χ. Linear/circular are special cases.

S = [1, cos2χ·cos2ψ, cos2χ·sin2ψ, sin2χ]

Key Optical Components

PBS — Polarizing Beam Splitter

Splits light into orthogonal linear polarizations. Extinction ratio: ~1:1,000 (cube) or ~1:100,000 (GT PBS).

Transmit Horizontal Reflect Vertical

HWP — Half-Wave Plate

Phase retardance π. Rotates linear polarization by 2θ (fast axis at θ). Flips circular handedness.

Used in Scenario 1 & 2 (rotating HWP + PBS). Cannot detect S₃!

QWP — Quarter-Wave Plate

Phase retardance π/2. Converts linear ↔ circular. QWP at 45° converts RCP/LCP to linear.

Central element in the rotating QWP method — detects all four Stokes parameters.

Lab Scenarios (from your notes)

Scenario 1 — Linear polarization input:

SOURCE

Unknown linear

HWP

Half-wave plate

θ: 0→180°

PBS

Fixed horizontal

DETECTOR

Sinusoidal I(θ) curve. Formula: I(θ) = ½[1 + cos(4θ − 2α)]. Maximum at θ = α/2 reveals polarization angle.

Scenario 2 — Circular polarization input:

SOURCE

Circular pol.

QWP

at 45° (fixed)

HWP

rotating

θ: 0→180°

PBS

DETECTOR

Without QWP: circular gives a flat line. QWP at 45° converts RCP/LCP → linear, then HWP+PBS sees a sinusoidal curve.

The Stokes formulation represents polarization using four intensity measurements — no complex math needed. All values are directly measurable with a power meter.

The Four Parameters

S₀ — Total Intensity

Total beam power. For normalized states, S₀ = 1.

S₀ = I_H + I_V

S₁ (Q) — Horizontal vs Vertical

+1 = fully horizontal, −1 = fully vertical, 0 = equal.

S₁ = I(0°) − I(90°)

S₂ (U) — Diagonal Preference

+1 = fully +45°, −1 = fully −45°, 0 = symmetric.

S₂ = I(45°) − I(135°)

S₃ (V) — Circular Handedness

+1 = fully RCP, −1 = fully LCP. Cannot be measured with HWP+PBS alone!

S₃ = I_RCP − I_LCP

Poincaré Sphere

Every fully polarized state maps to a point on the unit sphere. Drag to rotate.

+S₁ equator

Linear 0°

+S₂ equator

Linear +45°

+S₃ north

Right Circular

−S₁ equator

Linear 90°

−S₂ equator

Linear −45°

−S₃ south

Left Circular

Common States Table

State	S₀	S₁	S₂	S₃	Location
Horizontal (0°)	1	+1	0	0	+S₁ equator
Vertical (90°)	1	−1	0	0	−S₁ equator
Linear +45°	1	0	+1	0	+S₂ equator
Linear −45°	1	0	−1	0	−S₂ equator
Right Circular (RCP)	1	0	0	+1	North pole
Left Circular (LCP)	1	0	0	−1	South pole
Linear at angle α	1	cos 2α	sin 2α	0	Equator at 2α
Unpolarized	1	0	0	0	Center (inside)

Fully Polarized Light

S₀² = S₁² + S₂² + S₃² DOP = √(S₁²+S₂²+S₃²) / S₀ 1.0 = fully polarized 0.0 = unpolarized

Ellipse Parameters

ψ = ½ atan2(S₂, S₁) [orientation 0–180°] χ = ½ arcsin(S₃/S₀) [ellipticity ±45°] χ = 0 → Linear χ = ±45° → Circular

Three approaches with increasing sophistication. The rotating QWP method (Schaefer 2007) recovers all four Stokes parameters from a single continuous rotation.

Method A: Classical (6 Separate Measurements)

Directly measure I with a linear polarizer at 0°, 45°, 90°, 135°, and with a QWP for circular.

S₀ = I(0°) + I(90°) S₁ = I(0°) − I(90°) S₂ = I(45°) − I(135°) S₃ = I_RCP − I_LCP ← requires QWP at 45°

Limitation: Six separate alignments. Drift between shots introduces error. Tedious for S₃.

Method B: Rotating HWP + PBS

SOURCE

HWP

θ: 0→360°

PBS

Fixed horizontal

DETECTOR

I(θ) = ½ [S₀ + S₁·cos(4θ) + S₂·sin(4θ)] Note: No S₃ term — circular component is INVISIBLE! For circular input [S₁=S₂=0]: I(θ) = S₀/2 = FLAT LINE

Key limitation: HWP flips S₃ but PBS transmits only linear — so S₃ cancels out. To detect circular, you need a fixed QWP at 45° first (Scenario 2), or use Method C below.

Method C: Rotating QWP + LP (Schaefer 2007) ★

SOURCE

QWP

Fast axis at θ

θ: 0→360°

LP

Linear polarizer

Fixed (horiz.)

DETECTOR

I(θ) = ½ [S₀ + S₁·cos²(2θ) + S₂·sin(2θ)cos(2θ) − S₃·sin(2θ)] Using trig identities: I(θ) = a₀ + b₂·sin(2θ) + a₄·cos(4θ) + b₄·sin(4θ) where: a₀ = S₀/2 + S₁/4 ← DC offset b₂ = −S₃/2 ← 2nd harmonic encodes S₃! a₄ = S₁/4 ← 4th harmonic cosine b₄ = S₂/4 ← 4th harmonic sine Recovery: S₁ = 4·a₄ S₂ = 4·b₄ S₃ = −2·b₂ S₀ = 2·a₀ − 2·a₄

Why this wins: S₃ appears in the 2nd harmonic — unique to the QWP method. A single continuous rotation gives all four parameters via Fourier analysis. Noise averages out across the full sweep.

Comparison

Method	S₁	S₂	S₃	Notes
Classical (6 shots)	✓	✓	✓	6 separate alignments, drift-prone
Rotating HWP + PBS	✓	✓	✗	Cannot detect S₃ (circular)
Rotating QWP + LP ★	✓	✓	✓	Single rotation, Fourier fit, all 4 params

Set any input polarization state, choose the measurement method, and see the exact I(θ) curve live. Fourier analysis extracts all Stokes parameters automatically.

Input Polarization State

Or use sliders (ψ=orientation, χ=ellipticity):

ψ (orient) 0°

χ (ellip.) 0°

DOP 100%

S₀

1.000

S₁

1.000

S₂

0.000

S₃

0.000

Orientation ψ

0.0°

Ellipticity χ

0.0°

DOP

100%

Type

Linear

Measurement Setup

Add measurement noise

Show Fourier fit overlay

E-Field (xy-plane view)

I(θ) — Intensity vs. Waveplate Angle

Raw measurement signal. The Plotly toolbar (top-right) lets you download as PNG or SVG.

Extracted Stokes Parameters

Recovered via Fourier analysis. Compare with input.

Poincaré Sphere

Current state (cyan dot). Drag to rotate.

Fourier Decomposition of I(θ)

The I(θ) curve broken into its harmonic components. Each encodes a Stokes parameter.

Mueller matrices operate on Stokes vectors [S₀,S₁,S₂,S₃]. Jones matrices operate on the complex E-field vector [Ex, Ey·e^iδ].

Mueller Matrices

Horizontal Linear Polarizer

M_LP = ½ | 1 1 0 0 | | 1 1 0 0 | | 0 0 0 0 | | 0 0 0 0 |

HWP (fast axis at θ)

QWP (fast axis at θ)

Jones Vectors & Matrices

H linear: [1, 0]ᵀ V linear: [0, 1]ᵀ +45°: 1/√2 · [1, 1]ᵀ −45°: 1/√2 · [1, −1]ᵀ RCP: 1/√2 · [1, −i]ᵀ LCP: 1/√2 · [1, +i]ᵀ

Jones → Stokes

For [Ex, Ey·e^(iδ)] normalized to S₀=1: S₀ = |Ex|² + |Ey|² S₁ = |Ex|² − |Ey|² S₂ = 2|Ex||Ey|·cos(δ) S₃ = 2|Ex||Ey|·sin(δ) Inverse (S₀=1): |Ex| = √[(1+S₁)/2] |Ey| = √[(1−S₁)/2] δ = atan2(S₃, S₂)

Key Formulas Cheatsheet

Formula	What it gives
I_QWP(θ) = ½[S₀+S₁cos²2θ+S₂sin2θcos2θ−S₃sin2θ]	Rotating QWP + LP intensity
I_HWP(θ) = ½[S₀+S₁cos4θ+S₂sin4θ]	Rotating HWP + PBS (no S₃ term!)
I(θ) = a₀ + b₂sin2θ + a₄cos4θ + b₄sin4θ	QWP Fourier expansion
S₁=4a₄, S₂=4b₄, S₃=−2b₂, S₀=2a₀−2a₄	Stokes from Fourier coefficients
DOP = √(S₁²+S₂²+S₃²)/S₀	Degree of polarization
ψ = ½·atan2(S₂,S₁)	Polarization orientation (0–180°)
χ = ½·arcsin(S₃/S₀·DOP)	Ellipticity angle (±45°)
I_linear(θ)\|_HWP = ½[1+cos(4θ−2α)]	HWP scan for linear pol. at angle α

Step-by-Step: Rotating QWP Method

Step 1 — Setup

Place rotating QWP (motorized or manual) followed by a fixed horizontal linear polarizer. Connect output to a power meter or photodiode.

Step 2 — Measure I(θ)

Rotate QWP 0°→360°, recording intensity at each angle. Need ≥50 points for good SNR. More points = better Fourier resolution.

Step 3 — Fourier Extract

Fit I(θ) = a₀ + b₂sin2θ + a₄cos4θ + b₄sin4θ. Apply: S₁=4a₄, S₂=4b₄, S₃=−2b₂, S₀=2a₀−2a₄.

🔭 Stokes Polarimetry Explorer