Elliptical Polarization
Watch the tip of the electric-field vector of a fully polarized light wave, at one fixed plane, over a single optical period. In the special cases it runs back and forth along a line, or sweeps a perfect circle. But those are the exceptions. The generic thing it does — for two transverse components of unequal amplitude and any relative phase — is trace a tilted ellipse, once per cycle, and return. Elliptical polarization is therefore not an exotic sub-case to memorize after linear and circular; it is the whole of fully polarized light, of which linear and circular are the two symmetric corners. An ellipse is fixed by more than a single angle, and that richness is exactly why the general state needs its own bookkeeping. Three numbers plus a sign describe it: the overall size (set by the intensity), the tilt of the major axis in the transverse plane (the azimuth ψ), the shape or fatness (the ellipticity angle χ, with tan χ = ±b/a the axis ratio), and the handedness (which way the field circulates, carried by the sign of sin δ). These four map cleanly onto the two amplitudes and the phase difference of the underlying components, and — up to overall size and phase — onto the two-component Jones vector. Elliptical light appears the instant two coherent orthogonal components acquire a relative phase, which is precisely what a birefringent wave plate or a reflecting surface does. Send linear light through a quarter-wave plate at a generic angle and you get an ellipse of tunable shape; bounce light off a thin film and the reflected state is elliptical, its exact ellipse encoding the film's thickness to a fraction of a nanometre. The two complementary calculi — Jones vectors for pure states, Stokes parameters for partial and mixed light — exist to track these ellipses through optical systems, and the Poincaré sphere turns the whole subject into rigid geometry: every pure ellipse is one point on a sphere, and every wave plate is a rotation of it.
- A fully polarized monochromatic wave has its electric-field vector trace an ellipse in the transverse plane; linear (χ = 0) and circular (χ = ±π/4) polarization are the degenerate corners of this general state.
- The ellipse is fixed by two amplitudes E₀ₓ, E₀ᵧ and the phase difference δ = φ_y − φ_x, or equivalently by an azimuth ψ (major-axis tilt) and an ellipticity angle χ (with tan χ = ±b/a).
- Eliminating the travelling phase from the two components gives the ellipse directly: (Eₓ/E₀ₓ)² + (E_y/E₀ᵧ)² − 2(EₓE_y/E₀ₓE₀ᵧ)cos δ = sin²δ.
- The tilt obeys tan 2ψ = 2E₀ₓE₀ᵧ cos δ /(E₀ₓ² − E₀ᵧ²) and the shape obeys sin 2χ = 2E₀ₓE₀ᵧ sin δ /(E₀ₓ² + E₀ᵧ²); the sign of sin δ is the handedness.
- The four Stokes parameters (S₀, S₁, S₂, S₃) encode intensity, orientation, and handedness, with S₀² = S₁² + S₂² + S₃² for any fully polarized ellipse — a single point on the Poincaré sphere.
- A quarter-wave plate at angle θ to its axes turns linear light into an ellipse of ellipticity χ = θ, tuning continuously from linear (θ = 0) to circular (θ = 45°).
- Key idea: fully polarized light is an ellipse in the transverse plane; its azimuth ψ and ellipticity χ (equivalently the amplitudes and phase δ) capture every pure state, with linear and circular as the χ = 0 and χ = ±π/4 limits.
A fully polarized monochromatic wave is completely described by an ellipse in the transverse plane. Its azimuth ψ and ellipticity angle χ — equivalently the two amplitudes and the phase difference δ — capture every pure polarization state, with linear (χ = 0) and circular (χ = ±π/4) as the two symmetric limits, and every such state is a single point on the Poincaré sphere.
Where this lives today
Elliptical polarization is the working state of precision optical metrology. In ellipsometry, linear light reflects off a layered surface and comes back elliptical, because the s and p components pick up different amplitude and phase changes; measuring the reflected ellipse's two angles (the ellipsometric Ψ and Δ, with tan Ψ e^{iΔ} = r_p/r_s) yields film thickness and refractive index with sub-nanometre sensitivity — the daily workhorse of semiconductor fabs. In polarimetry and remote sensing, the full Stokes vector of scattered light distinguishes surfaces, aerosols, and biological tissue, and astronomers measure the faint elliptical polarization of starlight to map magnetic fields. In quantum information, an arbitrary elliptical state on the Poincaré sphere is a qubit encoded in a single photon, prepared and read with exactly the wave-plate-and-polarizer rotations this article describes. And the geometric (Pancharatnam–Berry) phase that a beam picks up when its polarization is cycled around a loop on the Poincaré sphere now underlies flat "metasurface" lenses and vortex-beam generators.
The physical picture
Picture two children on swings hung side by side, one swinging along x, one along y, both at the same frequency. If they move in step, a bead tied between them slides along a fixed diagonal — linear polarization. If they are a quarter-period out of step and swing equally hard, the bead goes round a circle. For any other combination — a different phase offset, or unequal swing amplitudes — the bead traces a tilted ellipse. The relative phase δ controls the shape and the sense of circulation; the amplitude ratio controls the elongation and, together with δ, the tilt. The subtlety that makes the ellipse interesting is that its major axis need not point along x or y. Two components along the coordinate axes, with a phase between them, generally produce an ellipse tilted at some azimuth ψ that depends on both the amplitude ratio and cos δ. So you cannot read the ellipse's orientation directly off the x and y amplitudes — the tilt is a genuine second piece of information, which is why the general state needs two shape angles, ψ and χ, not one. There is a cleaner way to see the two angles as independent, and it is the key to the whole subject: decompose the ellipse into right- and left-circular parts. Any ellipse is a superposition of the two circular states with definite amplitudes and a relative phase. The ratio of the two circular amplitudes fixes the ellipticity χ (equal parts give a line, one part alone gives a circle), while the relative phase of the two circular parts fixes the azimuth ψ — rotating one against the other simply turns the whole ellipse without changing its shape. This is the algebra behind the Poincaré sphere, where latitude is 2χ and longitude is 2ψ.
On the Poincaré sphere the whole story becomes geometry you can turn in your hand: the equator holds every linear state, the two poles the two circular states, and every latitude in between an ellipse of fixed fatness. A wave plate rotates the state-point about an axis set by the plate's fast axis; designing a target ellipse becomes a matter of choosing rotations, which is why polarimeters are built from a rotatable wave plate and an analyzer.
The core equations
Take a wave along +z with two transverse components of the same frequency but a fixed relative phase. Written as real fields:
where: E₀ₓ, E₀ᵧ ≥ 0 — the two component amplitudes; δ = φ_y − φ_x — the phase of the y component relative to x (the only phase that affects the shape); k = ω/c. Physically, only three quantities matter for the ellipse: the two amplitudes and δ. The overall phase and overall size do not change the shape. Eliminate the common travelling phase between the two equations and the field tip is confined to a fixed conic — the polarization ellipse:
where: the cross term carries cos δ and the right-hand side sin²δ sets the ellipse's "opening." When sin δ = 0 the right side vanishes and the conic degenerates to a pair of lines (linear light); when the cross term vanishes and the amplitudes are equal, it is a circle. Everything else is a genuine, tilted ellipse. The tilt of the major axis is the azimuth ψ, obtained by rotating axes to kill the cross term:
where: ψ is measured from the x-axis, −π/2 < ψ ≤ π/2. Note the double angle: the formula returns 2ψ, so extracting ψ needs care about which quadrant you are in — a common slip. When E₀ₓ = E₀ᵧ the denominator vanishes and ψ is 45° (or undefined for a circle), consistent with the symmetry. The fatness of the ellipse is the ellipticity angle χ, with tan χ = ±b/a the ratio of semi-minor to semi-major axis:
where: −π/4 ≤ χ ≤ π/4, and the sign of χ (carried by sin δ) is the handedness — positive for right-elliptical in our convention (δ > 0), matching the circular limit χ = +π/4 at δ = +π/2. Linear light is χ = 0; circular is χ = ±π/4. The compact modern description is the four Stokes parameters, which work for partial polarization too and place the state on the Poincaré sphere:
where: S₀ is the total intensity, S₁ the excess of x over y linear, S₂ the excess of +45° over −45° linear, S₃ the excess of right over left circular. For a fully polarized wave S₀² = S₁² + S₂² + S₃², so the state lies on the sphere; a real, partially polarized beam lies inside it, with degree of polarization P = √(S₁² + S₂² + S₃²)/S₀ < 1. The sphere coordinates are S₁ = S₀ cos 2χ cos 2ψ, S₂ = S₀ cos 2χ sin 2ψ, S₃ = S₀ sin 2χ: longitude 2ψ, latitude 2χ.
The goal is to prove that two orthogonal linear oscillations of common frequency but arbitrary amplitudes and relative phase confine the field tip to an ellipse, and to see exactly when that ellipse degenerates to a line or a circle. The strategy is to eliminate the shared travelling phase τ = kz − ωt between the two component equations, leaving a phase-free relation between Eₓ and E_y — a conic — and then to read off the degenerate limits from the value of sin δ.
Step 1. Normalize the components: set u = Eₓ/E₀ₓ = cos τ and v = E_y/E₀ᵧ = cos(τ + δ). Expanding the second with the angle-addition identity, v = cos τ cos δ − sin τ sin δ = u cos δ − sin τ sin δ. Why normalize first: dividing out the amplitudes puts both components on the same footing, so the surviving relation is purely about the phase δ, which is what shapes the ellipse.
Step 2. Isolate the sin τ term: v − u cos δ = −sin τ sin δ. Square both sides and use sin²τ = 1 − cos²τ = 1 − u²: (v − u cos δ)² = (1 − u²) sin²δ. Why squaring is legitimate here: we are after the locus of the tip, not the time order, so eliminating τ by squaring loses nothing about the shape — it only forgets which way the point moves around the curve.
Step 3. Expand and collect: v² − 2uv cos δ + u² cos²δ = sin²δ − u² sin²δ. The u² cos²δ and u² sin²δ terms combine to u², giving u² + v² − 2uv cos δ = sin²δ — the ellipse of Eq. (2). Why this is an ellipse and not a hyperbola: the discriminant of the quadratic form is 4cos²δ − 4 < 0 whenever sin δ ≠ 0, which is the condition for a bounded, closed conic.
Step 4. Read the limits. If δ = 0 or π the right side vanishes and the form factors into (u ∓ v)² = 0, the straight lines E_y = ±(E₀ᵧ/E₀ₓ)Eₓ — linear polarization. If in addition E₀ₓ = E₀ᵧ and δ = ±π/2, the cross term dies and u² + v² = 1 — a circle. Why both special cases fall out of one formula: linear and circular are not separate physics but the two ways this single conic can degenerate, which is the unifying content of the ellipse picture. *Verified: the identity u² + v² − 2uv cos δ = sin²δ holds for all τ; δ = 0 gives a line and equal amplitudes with δ = π/2 give a circle — verify_elliptical-polarization.py.*
Go deeper · Derivation B · 1 minEllipticity from the circular basis
The aim is to show that the ellipse's shape is nothing but the ratio of its right- and left-circular content, which is why circular decomposition separates the two shape angles so cleanly. The method is to expand a general pure state in the orthonormal circular basis and connect the amplitudes of the two circular parts to the ellipticity angle χ and their relative phase to the azimuth ψ.
Step 1. Take the orthonormal circular basis (same convention as the circular-polarization article): ê_R = (1, i)ᵀ/√2 and ê_L = (1, −i)ᵀ/√2, with ⟨ê_R|ê_L⟩ = 0. Write a general pure state as J = c_R ê_R + c_L ê_L with complex weights. Why this basis: right- and left-circular are the eigenstates of rotation about the beam, so a rotation of the ellipse acts on them by a pure phase — the cleanest possible handle on the azimuth.
Step 2. The relative phase of the two weights sets the tilt: ψ = ½(arg c_L − arg c_R). Rotating the ellipse by an angle is exactly a relative phase shift of the two circular components, changing ψ but not the shape. Why phase becomes rotation: multiplying ê_R and ê_L by opposite phases rigidly rotates their vector sum, which is the ellipse — so orientation lives entirely in the circular relative phase.
Step 3. The amplitude ratio sets the shape: tan χ = (|c_L| − |c_R|)/(|c_L| + |c_R|), equivalently sin 2χ = (|c_L|² − |c_R|²)/(|c_L|² + |c_R|²). Equal magnitudes give χ = 0 (linear); one weight zero gives χ = ±π/4 (circular). Why the ratio controls fatness: a pure circle needs all of one handedness and none of the other, while a line needs an equal, phase-locked mix — the imbalance between the two circular parts is precisely how "round" the ellipse is. *Verified: x-linear light has |c_R| = |c_L| (χ = 0); a single circular component gives χ = ±π/4 — verify_elliptical-polarization.py.*
Go deeper · Derivation C · 1 minA quarter-wave plate turns linear light into an ellipse of ellipticity χ = θ
The purpose is to prove the everyday laboratory fact that a quarter-wave plate converts linearly polarized input, oriented at angle θ to the plate's fast axis, into an elliptical state whose ellipticity angle equals θ — so one rotatable plate sweeps continuously from linear to circular. The approach is a direct Jones-matrix application followed by reading χ from the Stokes formula.
Step 1. The quarter-wave plate with fast axis along x is J = diag(1, i) up to a global phase, imposing a π/2 lag on the y (slow) component. Why diagonal in the plate axes: the fast and slow axes are the crystal's two independent propagation directions, so the plate only delays one relative to the other and mixes nothing.
Step 2. Feed in linear light at angle θ to the fast axis, J_in = (cos θ, sin θ)ᵀ. Applying the matrix: J_out = (cos θ, i sin θ)ᵀ. The y component now leads by δ = +π/2, so the output has amplitudes (cos θ, sin θ) and phase δ = π/2. Why the output is elliptical, not circular in general: only when cos θ = sin θ (that is, θ = 45°) are the two amplitudes equal, giving a circle; every other θ leaves unequal amplitudes and hence an ellipse.
Step 3. Read the ellipticity from Eq. (4): sin 2χ = 2 cos θ sin θ /(cos²θ + sin²θ) = sin 2θ, so χ = θ. The output ellipticity equals the input angle: θ = 0 gives linear (χ = 0), θ = 45° gives circular (χ = 45°), and intermediate angles give ellipses. Why this makes the plate a tunable ellipse generator: rotating the input (or the plate) dials χ continuously across the whole range, which is how any target point on a meridian of the Poincaré sphere is reached. *Verified: diag(1, i)·(cos θ, sin θ)ᵀ gives χ = θ for θ = 0°, 15°, 30°, 45° — verify_elliptical-polarization.py.*
Limiting cases
The general ellipse contains every pure state as a special amplitude ratio and phase:
| Case | Condition | Meaning | ||||
|---|---|---|---|---|---|---|
| Linear | δ = 0 or π (any amplitudes) | Ellipse collapses to a line at azimuth ψ = arctan(E₀ᵧ/E₀ₓ); χ = 0; on the Poincaré equator | ||||
| Circular | E₀ₓ = E₀ᵧ, δ = ±π/2 | a = b; χ = ±π/4; azimuth undefined (any diameter is an axis); poles of the sphere | ||||
| Axis-aligned ellipse | δ = ±π/2, E₀ₓ ≠ E₀ᵧ | Axes along x, y; ψ = 0; a = E₀ₓ, b = E₀ᵧ (or swapped) | ||||
| General ellipse | 0 < \ | δ\ | < π, δ ≠ ±π/2 | Tilted: ψ ≠ 0 and 0 < \ | χ\ | < π/4; needs the full ψ, χ formulas |
| Partial polarization | incoherent mixture | S₀² > S₁²+S₂²+S₃²; point inside the sphere; DOP < 1 |
Worked examples
Example 1 — Reading off a real ellipse (compute the shape from the components). A wave has E₀ₓ = 1, E₀ᵧ = 0.5 (in the same units) and a phase difference δ = 60°. Find its azimuth, ellipticity, and Stokes vector, and confirm it is fully polarized.
Step 1. Azimuth from Eq. (3): tan 2ψ = 2(1)(0.5)cos 60° /(1² − 0.5²) = (1)(0.5)/0.75 = 0.667. So 2ψ = arctan(0.667) = 33.7°, giving ψ = 16.85° — the major axis is tilted about 17° from x.
Step 2. Ellipticity from Eq. (4): sin 2χ = 2(1)(0.5)sin 60° /(1² + 0.5²) = (1)(0.866)/1.25 = 0.693. So 2χ = 43.8°, giving χ = 21.9° and an axis ratio tan χ = 0.40 — a moderately fat ellipse, right-handed since δ > 0.
Step 3. Stokes vector from Eq. (5): S₀ = 1 + 0.25 = 1.25, S₁ = 1 − 0.25 = 0.75, S₂ = 2(0.5)cos 60° = 0.50, S₃ = 2(0.5)sin 60° = 0.866. Check S₁² + S₂² + S₃² = 0.5625 + 0.25 + 0.75 = 1.5625 = (1.25)² = S₀², so the degree of polarization is exactly 1 — a pure, fully polarized ellipse. The answer: ψ = 16.85°, χ = 21.9°, S = (1.25, 0.75, 0.50, 0.866), DOP = 1.
Example 2 — A rotatable quarter-wave plate as an ellipse dial (design a target state). You have linearly polarized light and a quarter-wave plate, and you need an ellipse with axis ratio b/a = 0.40. At what angle θ should the plate's fast axis sit relative to the input polarization, and what happens as you rotate it?
Step 1. From the QWP derivation, the output ellipticity equals the input angle: χ = θ. The axis ratio is tan χ, so you need tan θ = 0.40, i.e. θ = 21.8° between the input polarization and the plate's fast axis.
Step 2. Check the endpoints as you rotate: at θ = 0° the light passes as linear (χ = 0, b/a = 0); at θ = 45° it emerges circular (χ = 45°, b/a = 1). Your target b/a = 0.40 sits between, at θ ≈ 22°, and rotating the plate dials the ellipse continuously from a line to a circle.
Step 3. Sanity check against the Poincaré sphere: rotating the plate traces a meridian from the equator (linear) to a pole (circular), and θ = 22° puts you at latitude 2χ ≈ 44° — a fat ellipse two-fifths of the way up. The answer: set the fast axis at θ ≈ 22° to the input; the plate is a continuous linear-to-circular ellipse dial.
Physics of the portrait
The general ellipse was the state that finally organized polarization into a coordinate system. For decades after Fresnel, the phenomenology — reflection, retarders, crystals — outran the bookkeeping, and every problem seemed to need its own trigonometry. George Gabriel Stokes cut through it in 1852 by observing that any beam, fully or partially polarized, is completely characterized by just four measurable intensities — the parameters that now bear his name — and that these combine and transform by simple rules through optical elements. Forty years later Henri Poincaré noticed that Stokes' three normalized parameters are the coordinates of a point on a sphere, turning polarization optics into spherical geometry: linear states on the equator, circular at the poles, ellipses everywhere between, and every wave plate a rotation. Together they gave elliptical polarization the two things a physical quantity needs — a set of numbers you can measure and a space in which those numbers move simply.
Origins · 1 minelliptical polarization and how this was discovered
The description of the general elliptical state is the joint legacy of George Gabriel Stokes (1819–1903) and Henri Poincaré (1854–1912). In an 1852 paper on the optics of partially polarized light, Stokes introduced four intensity-based parameters (S₀…S₃) that fully specify any beam — polarized, unpolarized, or in between — and transform through optical components by linear rules; crucially, they are all measurable with a polarizer and a retarder, needing no access to the field's phase. In 1892 Poincaré recognized that the three normalized Stokes parameters are Cartesian coordinates on a unit sphere, so every pure polarization state is a point on its surface, partially polarized states lie inside, and each birefringent element rotates the state-point about a fixed axis. The Poincaré sphere remains the working picture of polarization optics, from designing wave-plate combinations to visualizing the Pancharatnam–Berry geometric phase.
Going further
Texts. Hecht, Optics, 5th ed., §8.1 (the nature of polarized light) · Born & Wolf, Principles of Optics, §1.4 (the polarization ellipse) · Saleh & Teich, Fundamentals of Photonics, 3rd ed., ch. 6 (Jones and Stokes) · Loudon, The Quantum Theory of Light, 3rd ed., ch. 4 (circular decomposition) · Collett, Field Guide to Polarization.
Provenance. The ellipse equation, azimuth and ellipticity formulas, Jones/Stokes representations, and the Poincaré-sphere coordinates are cross-checked against Born & Wolf §1.4, Hecht ch. 8, and Saleh & Teich ch. 6. The circular-basis derivation uses the same convention as the sibling circular-polarization article (ê_R = (1, i)/√2, e^{−iωt}), so the two articles are mutually consistent; the worked ellipse (E₀ₓ = 1, E₀ᵧ = 0.5, δ = 60° → ψ = 16.85°, χ = 21.9°, DOP = 1) and the QWP result χ = θ are recomputed in verify_elliptical-polarization.py. No non-standard constants are used.
Builds on: Polarization of Light · Linear Polarization · Circular Polarization. Leads to: Jones Calculus · Stokes Parameters and the Poincaré Sphere · Wave Plates and Retarders · Birefringence.
Why does an ellipse need two angles (ψ and χ) when a circle or line needs only one?
Because an ellipse carries two independent pieces of shape information that a line or circle lacks. A line needs only its orientation (one angle); a circle needs nothing at all, being symmetric. A general ellipse needs both its tilt (the azimuth ψ, where the major axis points) and its fatness (the ellipticity χ, the axis ratio) — and these are genuinely independent: you can rotate an ellipse without changing its shape, or fatten it without turning it. That is exactly why the Poincaré sphere is two-dimensional (latitude 2χ, longitude 2ψ): it takes two angles to name every pure state.
Is the handedness of an ellipse a real physical property or just a convention?
The sense in which the field circulates is physically real and measurable — a right-handed ellipse and its mirror image are genuinely different states, and a chiral molecule or a Faraday medium responds to them differently. What is conventional is only the label "right" versus "left," which depends on your viewing direction and phasor sign convention, exactly as for circular light. So: the fact that the field goes one way rather than the other is physics; whether you call that "right-handed" is bookkeeping. Always state the convention, and if in doubt, specify the sign of δ = φ_y − φ_x directly.
How is elliptical light actually produced in the lab?
The standard route is a birefringent retarder — a wave plate. Linearly polarized light entering a plate at a generic angle to its fast axis splits into two components that travel at different speeds, emerging with a relative phase; that phase turns the input line into an ellipse whose shape you tune by rotating the plate (a quarter-wave plate at angle θ gives ellipticity χ = θ). Reflection produces it too: light bouncing off a metal or a thin film has its s and p components changed unequally, so an incident linear beam reflects as elliptical — the basis of ellipsometry. Any coherent process that gives two orthogonal components a relative phase makes an ellipse.
What is the difference between the Jones and Stokes descriptions of an ellipse?
Jones vectors describe pure, fully polarized states with two complex numbers (a two-component vector), tracking the amplitude and phase of each component; they are ideal for coherent, deterministic optics. Stokes parameters describe any beam — including partially polarized or unpolarized light — with four real, directly measurable intensities, at the cost of discarding absolute phase. For a fully polarized wave the two carry the same information, and S₀² = S₁² + S₂² + S₃². When incoherence or depolarization enters (a real beam, scattering, a rough surface), only the Stokes/Mueller description survives, and the state moves inside the Poincaré sphere with degree of polarization below one.
Why does the tilt formula have a 2ψ, and what goes wrong if I ignore it?
The ellipse is unchanged when you rotate it by 180°, so its orientation is only defined modulo 180°, and the natural variable that captures that is 2ψ — which ranges over a full 360°. If you compute tan 2ψ and take a plain arctangent (range −90° to 90°), you get 2ψ only up to a 180° ambiguity, which means ψ up to 90° — enough to confuse a nearly-horizontal ellipse with a nearly-vertical one. The fix is a two-argument arctangent that uses the signs of the numerator 2E₀ₓE₀ᵧcos δ and denominator E₀ₓ² − E₀ᵧ² separately, pinning 2ψ to the correct quadrant before halving.
Where do the semi-axis lengths a and b come from?
They follow from the amplitudes and phase through two conserved combinations: a² + b² = E₀ₓ² + E₀ᵧ² (the total intensity is the sum of the squared semi-axes) and a·b = |E₀ₓE₀ᵧ sin δ| (a kind of conserved "area" of the ellipse). Solving these two together gives the individual semi-axes, and their ratio is tan χ. The second relation shows why sin δ controls the fatness: when sin δ → 0 the product a·b → 0, so one axis collapses and the ellipse becomes a line; when sin δ = 1 with equal amplitudes, a = b and it is a circle.