Interactive learning tool for optical polarimetry. Visualize any polarization state,
simulate the rotating-QWP measurement (Schaefer et al. 2007), extract Stokes parameters
via Fourier analysis, and understand why S₃ (circular) requires a QWP, not just an HWP.
Every beam in an AMO experiment has a polarization that must be carefully controlled. σ⁺/σ⁻ selection rules determine optical pumping efficiency into stretched states; π beams drive Δm=0 transitions for spectroscopy; the Rydberg excitation polarization selects which magnetic sublevel of the Rydberg manifold is excited. The Stokes parameters (S₀, S₁, S₂, S₃) and the Poincaré sphere provide a complete coordinate system for any polarization state, and understanding how waveplates (λ/4, λ/2) and polarizing beam splitters transform a Stokes vector is an everyday skill for building and aligning AMO optical setups.
Convention and calibration note: RCP/LCP and σ⁺/σ⁻ labels depend on whether you define handedness looking into the beam or along the propagation direction.
This page uses an optics/Stokes convention for visualization. Before mapping to atomic Δm = ±1 selection rules, fix the beam propagation direction, quantization axis, and lab convention.
Real polarimetry also needs calibration of QWP retardance, PBS extinction, detector offsets, and angle zero.
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.
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.
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.
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°
DOP100%
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
SNR (dB)20 dB
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].