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.
E-field oscillates in a fixed plane at angle α. Created by a linear polarizer or PBS.
E-field rotates in a circle. RCP: clockwise looking into beam. Created by linear + QWP at 45°.
Most general case. Described by orientation ψ and ellipticity χ. Linear/circular are special cases.
Splits light into orthogonal linear polarizations. Extinction ratio: ~1:1,000 (cube) or ~1:100,000 (GT PBS).
Transmit Horizontal Reflect Vertical
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!
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.
Scenario 1, Linear polarization input:
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:
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.
Total beam power. For normalized states, S₀ = 1.
+1 = fully horizontal, −1 = fully vertical, 0 = equal.
+1 = fully +45°, −1 = fully −45°, 0 = symmetric.
+1 = fully RCP, −1 = fully LCP. Cannot be measured with HWP+PBS alone!
Every fully polarized state maps to a point on the unit sphere. Drag to rotate.
| 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) |
DOP = 1.0 for fully polarized; DOP = 0.0 for unpolarized light.
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.
Directly measure I with a linear polarizer at 0°, 45°, 90°, 135°, and with a QWP for circular.
S₃ requires a QWP at 45° before the detector.
No S₃ term — circular component is invisible. For circular input [S₁=S₂=0]: I(θ) = S₀/2 = flat line.
Using trig identities (Schaefer 2007 expansion):
where:
Recovery:
| 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.
Or use sliders (ψ=orientation, χ=ellipticity):
Raw measurement signal. The Plotly toolbar (top-right) lets you download as PNG or SVG.
Recovered via Fourier analysis. Compare with input.
Current state (cyan dot). Drag to rotate.
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].
Note: S₃ → −S₃ (flips circular handedness, but PBS still loses it).
This matrix generates the S₃ → sin(2θ) term in I(θ).
For $[E_x,\,E_y e^{i\delta}]$ normalized to $S_0=1$:
Inverse (S₀=1):
| 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 |
Place rotating QWP (motorized or manual) followed by a fixed horizontal linear polarizer. Connect output to a power meter or photodiode.
Rotate QWP 0°→360°, recording intensity at each angle. Need ≥50 points for good SNR. More points = better Fourier resolution.
Fit I(θ) = a₀ + b₂sin2θ + a₄cos4θ + b₄sin4θ. Apply: S₁=4a₄, S₂=4b₄, S₃=−2b₂, S₀=2a₀−2a₄.
Write a single photon's polarization state as a qubit, $|\psi\rangle = \alpha|H\rangle+\beta|V\rangle$. Its density matrix $\hat\rho = |\psi\rangle\langle\psi|$ (or a mixed state for partially polarized light) is a $2\times2$ Hermitian, unit-trace matrix, exactly the object the Poincaré sphere in the Stokes Parameters tab already visualizes. Expand it in the Pauli basis $\hat\sigma_0=\hat I,\hat\sigma_1, \hat\sigma_2,\hat\sigma_3$ and the Stokes parameters fall out as the expansion coefficients:
The normalized Stokes vector $(S_1/S_0,S_2/S_0,S_3/S_0)$ is the Bloch vector. Degree of polarization is Bloch-vector length; a fully mixed (unpolarized) beam sits at the Poincaré sphere's center, exactly like the maximally mixed qubit state $\hat\rho = \hat I/2$.
The continuous rotating-QWP method above is built for a classical beam with a power meter, plenty of photons to average over every angle. A single photon gives one click per shot, so photonic qubit tomography instead uses four fixed projective measurements:
$\mu_0$ (both PBS outputs summed) fixes the total count for normalization; $\mu_1,\mu_2,\mu_3$ are projections onto H, +45° diagonal, and right-circular, the same three axes this page already calls the H/V, D/A, and R/L bases. Count $n_i$ photons in each setting, then:
Feed these into $\hat\rho = \tfrac{1}{2}\sum_i (S_i/S_0)\hat\sigma_i$ above and you have a full single-qubit tomographic reconstruction, four measurement settings, no rotation stage required.
With finite photon counts, statistical noise on $n_0..n_3$ can push the linear-inversion $\hat\rho$ above to a matrix with a negative eigenvalue, not a valid quantum state. James et al.'s fix, now the field standard, is maximum-likelihood estimation: parametrize $\hat\rho = \hat T^\dagger\hat T/\mathrm{Tr}(\hat T^\dagger\hat T)$ for a lower-triangular $\hat T$ (this form is positive semidefinite by construction for any $\hat T$) and numerically maximize the likelihood of the observed counts over $\hat T$'s free parameters. The result is always a physical density matrix.
If you've built the rotating-QWP polarimeter above, you already understand atomic-qubit state tomography. Swap $\mu_1$ (H projection) for a bare population measurement in $\{|0\rangle,|1\rangle\}$; swap $\mu_2,\mu_3$ (D and R projections) for a population measurement preceded by a resonant $\pi/2$ pulse at phase $0$ and phase $\pi/2$, rotating the X- or Y-axis of the atomic Bloch sphere onto the measurement axis. Same four measurement settings, same $\hat\rho = \tfrac12\sum_i (S_i/S_0)\hat\sigma_i$ reconstruction, same maximum-likelihood fix for shot-noise-induced unphysical estimates.
The +45° preset in the Interactive Simulator tab has Stokes vector $(S_1,S_2,S_3)/S_0=(0,1,0)$. Its density matrix is $\hat\rho=\tfrac12\!\begin{pmatrix}1&1\\1&1\end{pmatrix}$, a pure state, $\mathrm{Tr}(\hat\rho^2)=1$, eigenvalues $\{1,0\}$. Try the Elliptical preset, $(S_1,S_2,S_3)/S_0=(0.5,0.5,0.707)$: since $S_1^2+S_2^2+S_3^2 \approx S_0^2$, it is still (to rounding) a pure state sitting on the Poincaré sphere's surface, not a partially mixed one inside it.