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Polarization Visualization

Build an ordered bench of polarizers, wave plates, birefringent crystal plates, rotators and depolarizers. Follow the Jones vector, Stokes vector and Poincaré-sphere point after every element, including partially polarized and unpolarized light (Mueller calculus).

Every slider has a number box • Drag or use arrow keys on the Poincaré sphere • Settings are kept in the page URL
I 1.000 Output I/I₀
P 1.000 Output DoP
📐 Linear Output state
θ 45.0° Orientation
χ 0.0° Ellipticity angle
↻ None (linear) Handedness
# 0 / 5 Elements on bench

Field ellipse input → output

Transverse field components Ex/E₀, Ey/E₀ (E₀ = input amplitude) seen facing the source; +z (⊙) points at you. Dashed: state entering the inspected element; solid: state leaving it. Only the polarized part has an ellipse.

Poincaré sphere S/S₀

Normalized Stokes vector (S₁, S₂, S₃)/S₀ after each element; its length is the degree of polarization, so partial light sits inside the sphere. Arcs show how each element moves the state. Drag, or focus and use the arrow keys, to rotate; double-click to reset.

Bench: field along z schematic spacing

Oblique view of the instantaneous field E(z) of the polarized part, with amplitude to scale and propagation along +z (left to right). Element spacing and the number of drawn wavelengths are schematic; elements are thin planes.

Birefringent retarder: o and e components A-plate

Uniaxial A-plate: optic axis c in the plate plane, light along z ⊥ c, so no walk-off. The field entering the plate splits into components along the fast and slow axes; inside the plate the slow one falls behind by δ(z) = 2π Δn z/λ. The common carrier is drawn with 5 schematic cycles (the real plate holds ≈ n d/λ of them); the relative lag between the two lanes and the δ(z) plot are to scale.

Measurement: rotate the inspected element

Output intensity I/I₀ after the whole bench as the inspected element's angle is swept; the cursor marks the current setting.

Retardance vs wavelength

Γ/2π = Δn d/λ for the inspected crystal plate (Δn held constant). Dashed: Γ reduced mod one wave, which is all the Jones matrix sees.

Output I/I₀
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Output DoP
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Output state
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Orientation θ
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Ellipticity χ
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Inspected element
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Inspected element: Jones and Mueller matrices

Jones matrix M (acts on (Ex, Ey))
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Mueller matrix (acts on (S₀, S₁, S₂, S₃))

State after each element

Stage Jones vector (Ex, Ey)/E₀ S₀ S₁ S₂ S₃ DoP State θ χ

Handedness convention: the observer faces the source (wave travels toward the viewer along +z, x right, y up). Counter-clockwise rotation, S₃ > 0, is called right-handed here (IEEE / positive helicity). Born & Wolf and Hecht name the same state left-handed. A Jones vector is listed only while the light is fully polarized; after a depolarizer or for p < 1 the state is carried by the Stokes vector (Mueller calculus).

💡 How to use

Learn with this tool

Learning objectives

  • Predict the output of an ordered chain of polarizers and retarders with Jones matrices, and explain why the order of elements matters.
  • Describe partially polarized light with Stokes vectors and Mueller matrices, and show which elements change intensity, which change the degree of polarization, and which only move the state on the Poincaré sphere.
  • Design a wave plate from a crystal's birefringence, Γ = 2π Δn d/λ, and identify its fast axis from the crystal geometry.

Prerequisites

  • Complex phasors and plane waves E = Re{E₀ ei(kz − ωt)}
  • 2×2 matrix multiplication and rotation matrices
  • Fresnel coefficients (polarization-dependent reflection, Brewster's angle)
  • Two-beam interference (why a phase difference matters)

Model

A monochromatic plane wave travels along +z through thin, ideal, lossless (except for the polarizer's absorbed component) elements at normal incidence. Fully polarized light is a Jones vector J = (Ex, Ey); each element is a 2×2 matrix and a chain acts as Jout = MN ⋯ M2 M1 Jin (element 1 met first, so its matrix is on the right). Partially polarized light is the Stokes vector S = (S₀, S₁, S₂, S₃) with S₀ ≥ √(S₁² + S₂² + S₃²); each element is a real 4×4 Mueller matrix, Sout = 𝕄N ⋯ 𝕄1 Sin. An element with axis at θ is R(−θ) M₀ R(θ).

ψ, δ
amplitude angle (tan ψ = |Ey|/|Ex|) and phase difference φy − φx of the input's polarized part
p
degree of polarization √(S₁² + S₂² + S₃²)/S₀, 0 ≤ p ≤ 1
θ
transmission axis (polarizer) or fast axis (retarder), from +x toward +y
Γ
retardance: extra phase of the slow-axis component, Jones matrix diag(1, eiΓ) in the element frame
Δn, d, λ
birefringence ne − no, plate thickness, vacuum wavelength; Γ = 2π|Δn| d/λ
ρ
rotation angle of a rotator (optical activity or Faraday rotation), positive from x toward y
D
depolarization of an ideal depolarizer, 𝕄 = diag(1, 1 − D, 1 − D, 1 − D)
θe, χ
ellipse orientation ½ atan2(S₂, S₁) and ellipticity angle ½ asin(S₃/√(S₁²+S₂²+S₃²)); axis ratio |tan χ|

Assumptions and validity: monochromatic light (so the retardance is a single number), ideal elements with no reflection loss or finite extinction ratio, normal incidence, uniaxial A-plates with constant indices (no dispersion), and incoherent summation only through the Stokes description. Crystal indices are near-589 nm values: quartz 1.5443/1.5534, MgF₂ 1.3777/1.3895, sapphire 1.7681/1.7599, calcite 1.6584/1.4864 (no/ne).

Derivation: from Jones matrices to Mueller matrices and the Poincaré sphere

Define the coherency matrix C = ⟨J J†⟩. For fully polarized light C has rank 1; incoherent mixtures add their C matrices, which is how unpolarized light (C = ½ I₀ 𝟙) is represented. With σ₀ = 𝟙, σ₁ = diag(1, −1), σ₂ = [[0, 1], [1, 0]], σ₃ = [[0, −i], [i, 0]] the Stokes parameters are Si = tr(C σi), and C = ½ Σ Si σi.

A Jones element maps C → M C M†. Substituting C gives S′i = Σj 𝕄ij Sj with 𝕄ij = ½ tr(σi M σj M†). The tool computes element Mueller matrices from closed forms and checks this identity in its tests.

For a retarder with fast axis along x, M = diag(1, eiΓ) multiplies Ex*Ey by eiΓ, so (S₂ + iS₃) → eiΓ(S₂ + iS₃) while S₀ and S₁ are unchanged: a rotation of the sphere by Γ about the S₁ axis. Rotating the element by θ rotates (S₁, S₂) by 2θ, so the rotation axis becomes (cos 2θ, sin 2θ, 0). Rotations preserve the length √(S₁² + S₂² + S₃²), hence the degree of polarization. A polarizer at θ gives S₀′ = ½(S₀ + S₁ cos 2θ + S₂ sin 2θ); for unpolarized light that is S₀/2 for every θ.

In a uniaxial A-plate the two eigen-polarizations are E ∥ c (extraordinary, index ne) and E ⊥ c (ordinary, no). After thickness d they have phases 2π n d/λ, so their difference is δ(d) = 2π(nslow − nfast) d/λ. In a positive crystal (ne > no, quartz) the o-wave is fast and the fast axis is ⊥ c; in a negative crystal (calcite) the fast axis is ∥ c.

Exercise 1 — Three polarizers and unpolarized light

  1. Unpolarized light meets polarizers at 0° and 90°. Predict I/I₀. Now insert a third polarizer at 45° between them. Predict I/I₀ again, and the middle angle that maximizes it.
  2. Load “Three polarizers”, then remove the middle element (✕) and add it back, moving it to position 2 with ↑.
  3. Read I/I₀ in the table after each element. Inspect the middle polarizer and read the scan plot at 22.5°, 45° and 67.5°.
  4. Explain why adding an absorbing element increases the transmitted power, and derive the scan curve.
Show answer

Crossed pair: 0. With the middle polarizer at θ: ½ · cos²θ · cos²(90° − θ) = ⅛ sin² 2θ, so 1/8 = 0.125 at 45° and 1/16 at 22.5° and 67.5°. The first polarizer passes ½ for any orientation (Malus averaged over an unpolarized ensemble). The middle one projects the state onto a new direction that has a non-zero component along the last axis: it does not “add” light, it changes which component the final polarizer receives.

Exercise 2 — Quarter-wave plate, handedness and the Poincaré sphere

  1. Horizontal light (the H point on the sphere) meets a QWP with fast axis at +45°. On the sphere, which axis does the state rotate about, by how much, and does it end at the north (R) or south (L) pole?
  2. Load “QWP → circular”, then set the plate to −45°, then to 20°.
  3. Record S₃ and χ for each angle; watch the arc on the sphere.
  4. Explain why a QWP at 20° gives an elliptical state with χ = ½ asin(−sin 2·20°) and the ellipse axis along the fast or slow axis.
Show answer

The rotation axis is the +45° point (0, 1, 0); rotating H = (1, 0, 0) by +90° about +S₂ gives (0, 0, −1): S₃ = −1, left-handed in this tool's convention (clockwise facing the source). At −45° the axis is (0, −1, 0) and the state goes to R (S₃ = +1). At 20° the component of H along the axis is cos 40°; a quarter turn moves the rest onto S₃, giving S₃ = −sin 40° ≈ −0.643 and χ ≈ −20°: the ellipse is aligned with the plate axis and its axis ratio is tan 20° ≈ 0.364.

Exercise 3 — Limiting cases: p → 0 and Γ → 2π

  1. (a) With p = 0, what does any sequence of retarders and rotators do to the output? (b) A quartz plate is twice the quarter-wave thickness, then four times it. What does each do to horizontal light when its optic axis is at 45°?
  2. Load “Quartz QWP (crystal)”. Set p = 0; then return p = 1 and type thickness 34.78 µm, then 69.56 µm, in the plate's number box.
  3. Read DoP and S; for the plate read Γ/2π in the crystal panel and the output state.
  4. Why is the Stokes vector the right description for (a), and why is a full-wave plate indistinguishable from no plate at the design wavelength but not at 532 nm?
Show answer

(a) S = (1, 0, 0, 0) is a fixed point of every retarder and rotator (they are rotations of the sphere about its centre): the output stays unpolarized with I = 1; only a polarizer changes it. (b) 2 × quarter-wave = half-wave: horizontal becomes vertical (linear at 90°). 4 × quarter-wave = one full wave: the output equals the input. At 532 nm the same plate has Γ/2π = 0.0091 × 69.56 µm / 0.532 µm ≈ 1.19 waves, i.e. an effective 0.19 wave retarder, and the output is elliptical. The full-wave condition holds only at the design wavelength, which is why full-wave plates are used as wavelength filters between polarizers.

Worked example: designing a quarter-wave plate from quartz at 633 nm

Problem. A He-Ne laser (λ = 633 nm) is horizontally polarized. Make it circular with a crystalline-quartz A-plate (no = 1.5443, ne = 1.5534). Find the zero-order thickness, the orientation of the optic axis, the handedness, and the retardance error if the same plate is used at 532 nm.

Solution. Δn = ne − no = +0.0091 (positive uniaxial), so the o-wave (E ⊥ c) is fast. Quarter-wave needs Δn d/λ = ¼: d = λ/(4Δn) = 633 nm/(4 × 0.0091) ≈ 17.39 µm. The fast axis must be at ±45° to the input; with c at +45° the fast axis is at 135° ≡ −45°, which rotates H about the −45° point to S₃ = +1 (right-handed here). At 532 nm, Γ/2π = 0.0091 × 17.39/0.532 ≈ 0.297 waves (107° instead of 90°); S₃ = sin 107° ≈ 0.956 and the axis ratio is tan(½ asin 0.956) ≈ 0.74 — noticeably elliptical. A third-order plate (d ≈ 17.39 + 3 × 69.56 = 226.1 µm) is circular at 633 nm but its retardance error at 532 nm is (3.25)(633/532 − 1) ≈ 0.62 waves: multi-order plates are far more wavelength-sensitive. Load “Quartz QWP (crystal)” and move λ to check each number.

When the model fails

  • Real polarizers have finite extinction ratios (10²–10⁶), wavelength-dependent transmission and reflections; here crossed polarizers give exactly zero.
  • Dispersion: Δn depends on λ (quartz 0.0092 at 546 nm, 0.0090 at 1 µm); the tool holds the 589 nm indices fixed, so broad-band retardance is only approximate.
  • Oblique incidence and thick crystals: off-normal rays see a different ne(θ), the e-ray walks off, and the plate becomes a displacer (Wollaston/Savart). Only A-plates at normal incidence are modelled.
  • Coherence: if the retardance exceeds the source coherence length (Δn d > λ²/Δλ), the two components no longer interfere and a multi-order plate depolarizes — Jones calculus then fails; the Mueller depolarizer is only a phenomenological stand-in.
  • Optical activity vs Faraday rotation: the rotator matrix is the same for both in one pass, but a double pass cancels optical activity and doubles Faraday rotation (non-reciprocity); this is not modelled.

References

  • E. Hecht, Optics, 5th ed., ch. 8 (polarization, retarders, Jones and Mueller calculus).
  • B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 3rd ed., ch. 6 (Jones matrices, anisotropic media, Poincaré sphere).
  • M. Born and E. Wolf, Principles of Optics, 7th ed., §10.8–10.9 (coherency matrix, partial polarization).
  • D. H. Goldstein, Polarized Light, 3rd ed. (Stokes/Mueller calculus, depolarization, handedness conventions).
  • G. Ghosh, “Dispersion-equation coefficients for the refractive index and birefringence of calcite and quartz crystals”, Opt. Commun. 163, 95 (1999).

📚 Physics background

🌊 Polarization ellipse

For a wave travelling along +z the transverse field is E = Re{J ei(kz − ωt)} with J = E₀(cos ψ, sin ψ eiδ). At fixed z the tip of E traces an ellipse:

Ex = E₀ cos ψ cos(kz − ωt), Ey = E₀ sin ψ cos(kz − ωt + δ)

tan 2θ = tan 2ψ cos δ, sin 2χ = sin 2ψ sin δ

  • Linear: S₃ = 0 (δ = 0 or ±180°, or a single component ψ = 0°/90° for any δ).
  • Circular: S₁ = S₂ = 0 (ψ = 45°, δ = ±90°); δ = +90° rotates counter-clockwise facing the source: right-handed in this tool (IEEE/helicity).
  • Elliptical: everything else, with axis ratio |tan χ|.

🌟 Stokes parameters and degree of polarization

S₀ = |Ex|² + |Ey|², S₁ = |Ex|² − |Ey|²

S₂ = 2 Re(Ex* Ey), S₃ = 2 Im(Ex* Ey) (time/ensemble averages)

p = √(S₁² + S₂² + S₃²)/S₀

S₁ compares horizontal and vertical, S₂ compares +45° and −45°, S₃ compares right and left circular power, each measurable with a polarizer and a quarter-wave plate. Any partially polarized beam is the incoherent sum of an unpolarized part (1 − p)S₀ and a fully polarized part pS₀.

🧮 Element matrices

Polarizer (axis θ): M = [[cos²θ, cos θ sin θ], [cos θ sin θ, sin²θ]]

Retarder (fast axis θ): M = R(−θ) diag(1, eiΓ) R(θ); HWP Γ = π, QWP Γ = π/2

Rotator: M = [[cos ρ, −sin ρ], [sin ρ, cos ρ]]

Depolarizer: no Jones matrix; 𝕄 = diag(1, 1 − D, 1 − D, 1 − D)

  • Malus's law: linear light at α through an analyzer at θ transmits cos²(θ − α); unpolarized light transmits ½.
  • Half-wave plate: mirrors a linear state about the fast axis (rotation by 2(θ − α)) and reverses handedness. Two HWPs at 0 and ρ/2 act as a rotator by ρ.
  • Retarders and rotators are unitary: they preserve intensity and the degree of polarization.

💎 Birefringent crystals

In a uniaxial crystal the ordinary wave (E ⊥ optic axis c) sees no and the extraordinary wave sees ne(θ), equal to ne when the light travels ⊥ c. A plate cut with c in its face (an A-plate) therefore has two linear eigen-polarizations with a phase difference Γ = 2π|ne − no| d/λ and no beam walk-off at normal incidence. Zero-order plates (Γ < 2π) are thin — 17 µm of quartz or under 1 µm of calcite for a quarter wave at 633 nm — so commercial plates are often multi-order or compound.

Related effects not simulated here: Brewster-angle polarization by reflection (see Fresnel coefficients), stress birefringence (photoelasticity), and Faraday isolators.