Jones Vectors and Stokes Parameters

The previous article ended with a taxonomy and two IOUs. The taxonomy: every monochromatic plane wave carries some polarization ellipse, set by two transverse amplitudes and a relative phase. The IOUs: a systematic way to compute what optical elements do to a state (rotating polarizers and wave plates by trigonometry gets unbearable by the second element), and an honest description of light that is only partially polarized — sunlight, starlight, LEDs — which no single ellipse describes at all. This article pays both debts with the two standard formalisms of polarization science. Jones calculus handles pure states: the state is a two-component complex vector, every lossless-or-lossy element is a 2×2 matrix, and cascades of optics become matrix products — a quarter-wave plate stops being a trigonometric ordeal and becomes $\mathrm{diag}(1, i)$ . Stokes parameters handle everything else: four real numbers, each defined as a difference of two intensity measurements, that fingerprint any beam — pure, partial, or fully unpolarized — and assemble into the beautiful geometry of the Poincaré sphere, where fully polarized states live on the surface and unpolarized light sits at the center. The two formalisms divide the world cleanly — amplitudes when phase is trackable, intensities when it is not — and the division itself is physics: it is the classical boundary between a quantum pure state and a mixed state. A Jones vector is a qubit; the Poincaré sphere is the Bloch sphere. Polarization optics is where that machinery was invented, a century early, with lenses and calcite instead of superconducting circuits.

Key takeaways

A pure polarization state is a Jones vector $(\tilde E_x, \tilde E_y)$ — two complex numbers, four real parameters, of which brightness and overall phase are trivial: the state proper is two-dimensional.
Optical elements are 2×2 Jones matrices — polarizer $\mathrm{diag}(1,0)$ , quarter-wave plate $\mathrm{diag}(1,i)$ , half-wave plate $\mathrm{diag}(1,-1)$ , all conjugated by rotations — and cascades multiply, right-to-left.
Partially polarized light needs statistics, not amplitudes: the four Stokes parameters $S_0..S_3$ , each an intensity difference (H−V, 45°−135°, R−L), measurable with a polarizer, a wave plate, and a power meter.
Degree of polarization $p = \sqrt{S_1^2+S_2^2+S_3^2}/S_0$ runs from 0 (unpolarized) to 1 (pure); pure states satisfy the equality and live on the Poincaré sphere's surface.
On the Poincaré sphere: equator = linear states, poles = circular, wave plates = rotations — one picture organizes all of polarization optics.
Key idea: pure states are complex 2-vectors and optics is matrix multiplication; real beams are Stokes 4-vectors built from intensity differences — and the sphere they fill is the Bloch sphere of quantum mechanics, discovered first.

The core insight

Jones vectors (complex 2-component amplitudes, matrices for elements) compute pure polarization states; Stokes parameters (four intensity differences: total, H−V, 45°−135°, R−L) measure any state including partial polarization, with $p = \sqrt{S_1^2+S_2^2+S_3^2}/S_0$ and the Poincaré sphere as the state space.

Where this lives today

Stokes parameters are the native data format of modern polarimetry. CMB experiments report sky maps of Q and U (the cosmologist's names for $S_1, S_2$ ) at nanokelvin sensitivity, hunting the parity-odd B-mode pattern that would evidence inflation; synchrotron maps of Q/U trace galactic magnetic fields; and Earth-observing satellites use multi-angle Stokes imaging to separate aerosol types by how they polarize scattered sunlight. Telecommunications lives on the sphere: polarization-mode dispersion in fiber is a wavelength-dependent random rotation of $\vec S$ , tracked and compensated in real time by coherent receivers — every long-haul optical link is doing Derivation B Step 3 continuously at gigahertz rates. The quantum inheritance is now industrial. Photonic quantum computing and QKD encode qubits in exactly the Jones basis; single-qubit gates are wave plates (or their integrated-optic equivalents); quantum state tomography of a photon is literally the six-measurement Stokes recipe of equation (3), photon-counted; and the purity of a heralded single-photon source is quoted as a Bloch-vector length — Stokes's $p$ , renamed. The two formalisms this article installed, one per century, turned out to be the user manual for the quantum information age's cheapest qubit.

The physical picture

Two laboratory situations force two different languages. First situation: a laser — one frequency, stable phase — passes through a train of polarizers and wave plates. Here the field's two transverse complex amplitudes are definite and trackable; the natural state description is the pair $(\tilde E_x, \tilde E_y)$ , and every element acts linearly on the pair (each transmitted component is a fixed complex combination of the inputs — that is what "linear optics" means). Linear maps on 2-vectors are 2×2 matrices: the entire toolkit of linear algebra — eigenvectors, bases, products — becomes optical engineering. An element's eigenvectors are the states it transmits unchanged (up to phase); its matrix in any basis is built once and reused forever; and a cascade is a product, computed right-to-left in the order the light meets the elements. Second situation: sunlight through the same train. Now the "state" changes randomly every few femtoseconds — the coherence time — and no fixed pair of complex amplitudes exists on any timescale a detector sees. What is stable are averaged quadratic quantities: intensities through various analyzers. The honest description is statistical, and the minimal complete set of statistics turns out to be exactly four numbers, all directly measurable: total intensity, plus the imbalances between horizontal and vertical, between the two diagonals, and between the two circular senses. These are the Stokes parameters, and their genius — Stokes, 1852, decades before anyone knew light was electromagnetic — is that they are defined by operations (measure six intensities, take three differences), not by any model of what light is.

Figure

Two languages for one freedom — Jones amplitudes when phase is trackable, Stokes intensity-differences when only statistics survive

The bridge between the languages is one-directional and physically meaningful. Every Jones vector generates its Stokes parameters (compute the four intensity differences that state would produce — the map is quadratic, and pure states always land with $p = 1$ , on the sphere's surface). But a Stokes vector with $p < 1$ has no Jones vector: partial polarization is not a hidden pure state, it is a statistical mixture, and information (the wandering phase) is genuinely absent. Quantum mechanics will reuse this entire structure verbatim — pure states as complex 2-vectors, mixtures as points inside the sphere, measurements as intensity differences — with the names changed to qubit, density matrix, and Bloch sphere.

The formalisms

Jones vectors. Normalize the bookkeeping: write the transverse complex amplitudes as a column, and (when only the state matters, not brightness) scale to unit total intensity,

\mathbf{J} = \begin{pmatrix}\tilde E_x\\ \tilde E_y\end{pmatrix}, \qquad \text{e.g. } \mathbf{J}_H = \begin{pmatrix}1\\0\end{pmatrix},\ \mathbf{J}_{45°} = \frac{1}{\sqrt2}\begin{pmatrix}1\\1\end{pmatrix},\ \mathbf{J}_{R} = \frac{1}{\sqrt2}\begin{pmatrix}1\\ i\end{pmatrix} \qquad (1)

where: the entries are the complex amplitudes of article 0588's decomposition (the $i$ in $\mathbf{J}_R$ is the quarter-cycle lag that makes rotation — the wave whose tip turns $\hat{\mathbf{x}}\to\hat{\mathbf{y}}$ , per that article's convention warning; state your $e^{-i\omega t}$ convention and the symbols cannot be misread). Read the structure: intensity is $|\tilde E_x|^2 + |\tilde E_y|^2 = \mathbf{J}^\dagger\mathbf{J}$ ; two states are physically orthogonal (never interfere, per Fresnel–Arago) when $\mathbf{J}_1^\dagger\mathbf{J}_2 = 0$ — note that $\mathbf{J}_R^\dagger\mathbf{J}_L = 0$ : the two circular states are as orthogonal as H and V, and form an equally good basis.

Jones matrices. A linear element maps input to output amplitudes: $\mathbf{J}_{\text{out}} = M\,\mathbf{J}_{\text{in}}$ . The three canonical devices, axes horizontal (rotate with $M(\theta) = R(\theta)MR(-\theta)$ for any other orientation, $R$ the standard rotation matrix):

M_{\text{pol}} = \begin{pmatrix}1&0\\0&0\end{pmatrix}, \qquad M_{\text{QWP}} = \begin{pmatrix}1&0\\0&i\end{pmatrix}, \qquad M_{\text{HWP}} = \begin{pmatrix}1&0\\0&-1\end{pmatrix} \qquad (2)

where: the polarizer projects (its matrix is idempotent, $M^2 = M$ — projecting twice is projecting once, the three-polarizer paradox's algebra); the wave plates are retarders, multiplying the slow-axis component by a phase ( $i$ = quarter cycle, $-1$ = half cycle) while absorbing nothing — their matrices are unitary, $M^\dagger M = 1$ , energy conserved. (Global phases are dropped throughout; only relative phase is physical.) Physically these phases come from birefringence — the two axes see different refractive indices over the plate's thickness — hardware owned by the optics track and by the E3 permittivity-tensor article; here they are matrices, which is the entire point.

Derivation AJones calculus at work — the quarter-wave plate manufactures circular light, and cascades are products

The goal is to exercise the machinery on the two standard results every polarization lab uses daily — QWP on 45° linear gives circular; HWP rotates linear by twice the plate angle — and to establish the cascade rule with the three-polarizer paradox recomputed in two matrix lines. The strategy is bare linear algebra on (1)–(2); watch how results that took trigonometric bookkeeping in article 0588 become one-line products, and watch the eigenvector logic (which states pass unchanged) do the physical thinking.

Step 1. QWP (fast axis horizontal) on 45° linear light:

M_{\text{QWP}}\,\mathbf{J}_{45°} = \begin{pmatrix}1&0\\0&i\end{pmatrix}\frac{1}{\sqrt2}\begin{pmatrix}1\\1\end{pmatrix} = \frac{1}{\sqrt2}\begin{pmatrix}1\\ i\end{pmatrix} = \mathbf{J}_R.

Circular, exactly, with intensity preserved ( $|1|^2+|i|^2 = 2 =$ input). Why 45° is the magic angle: the plate's job is to delay one axis-component by a quarter cycle, and only at 45° are the two components equal — the circle needs equal amplitudes in quadrature (0588's Step 2). At any other input angle the output is elliptical; at 0° or 90° (the eigenvectors of $M_{\text{QWP}}$ ) the light passes unchanged — a wave plate does nothing to light polarized along either of its axes. Run backwards ( $\mathbf{J}_R$ into a QWP) and circular becomes linear — the circular-vs-unpolarized discriminator test of 0588's FAQ, now one matrix line.

Step 2. HWP at angle θ on horizontal linear light. Conjugate by rotations: $M_{\text{HWP}}(\theta) = R(\theta)\,\mathrm{diag}(1,-1)\,R(-\theta)$ ; multiplying out gives $\begin{pmatrix}\cos2\theta & \sin2\theta\\ \sin2\theta & -\cos2\theta\end{pmatrix}$ , and acting on $\mathbf{J}_H$ :

M_{\text{HWP}}(\theta)\,\mathbf{J}_H = \begin{pmatrix}\cos2\theta\\ \sin2\theta\end{pmatrix}

— linear polarization at angle 2θ: the plate reflects the polarization about its fast axis, so a plate rotated by θ swings the output by 2θ. Why the factor 2 matters practically: a HWP on a rotation mount is the standard continuous polarization rotator (and, followed by a fixed polarizer, the standard variable attenuator: $I \propto \cos^2 2\theta$ ); the doubled angle is also why HWP mounts are calibrated in half-degrees.

Step 3. Cascades multiply right-to-left. Three-polarizer paradox, matrix edition — crossed polarizers with a 45° insert, where $M_{\text{pol}}(45°) = \tfrac12\begin{pmatrix}1&1\\1&1\end{pmatrix}$ . Track unit-intensity horizontal light through the product $M_{\text{pol}}(90°)\,M_{\text{pol}}(45°)\,M_{\text{pol}}(0°)$ stage by stage: $M_{0°}\mathbf{J}_H = (1,0)^T$ ; then $M_{45°}(1,0)^T = \tfrac12(1,1)^T$ ; then $M_{90°}\tfrac12(1,1)^T = (0,\tfrac12)^T$ — output intensity $|\tfrac12|^2 = \tfrac14$ of the polarized input, i.e. $\tfrac18$ of the original unpolarized beam, matching 0588's cascade exactly. Without the middle matrix: $M_{\text{pol}}(90°)M_{\text{pol}}(0°) = 0$ , the zero matrix — extinction as an algebraic identity. Why right-to-left order is not pedantry: polarizer-then-QWP and QWP-then-polarizer are different physical instruments (matrices don't commute), and the commonest Jones-calculus bug is writing the product in reading order. *Verified: QWP and HWP results reproduce Hecht §8.7.1–8.7.2 and any polarization-optics handbook; HWP(θ) matrix multiplied out symbolically; cascade result matches 0588's Malus-chain arithmetic ½·½·½ = ⅛; unitarity/idempotency of (2) checked directly.*

Stokes parameters. For any beam — laser or sunlight — define four numbers by six intensity measurements (polarizer at 0°, 90°, 45°, 135°; circular analyzers R and L, each a QWP + polarizer per Step 1 run backwards):

S_0 = I_{0°}+I_{90°}, \quad S_1 = I_{0°}-I_{90°}, \quad S_2 = I_{45°}-I_{135°}, \quad S_3 = I_R-I_L \qquad (3)

where: $S_0$ — total intensity; $S_1, S_2$ — the two linear imbalances (note 45° is not redundant: H/V-balanced light can be fully polarized at 45°); $S_3$ — the circular imbalance. Read it physically: each of $S_{1,2,3}$ is an excess of one basis state over its orthogonal partner; a fully mixed beam has no excess anywhere ( $S_1 = S_2 = S_3 = 0$ ), while a pure state maxes out its total excess. For a pure state with Jones vector $(\tilde E_x, \tilde E_y)$ the parameters evaluate to $S_1 = |\tilde E_x|^2 - |\tilde E_y|^2$ , $S_2 = 2\,\mathrm{Re}(\tilde E_x^*\tilde E_y)$ , $S_3 = 2\,\mathrm{Im}(\tilde E_x^*\tilde E_y)$ — quadratic in amplitudes, phase-difference-sensitive through the last two.

Go deeper · Derivation B · 2 minThe degree of polarization and the Poincaré sphere — what four intensities know about any beam

Derivation BThe degree of polarization and the Poincaré sphere — what four intensities know about any beam

The goal is the central inequality $S_1^2 + S_2^2 + S_3^2 \leq S_0^2$ (equality iff the state is pure), the decomposition of any beam into fully-polarized plus unpolarized parts, and the sphere picture that organizes it all. The strategy: verify the pure-state equality directly from the quadratic forms, then let incoherent mixing (intensities add — article 0587's incoherent-addition rule) generate the interior. Watch the geometry assemble itself: the three excesses $(S_1, S_2, S_3)$ behave as a vector whose length is the polarized power and whose direction is the state.

Step 1. Pure states saturate. With $a = |\tilde E_x|$ , $b = |\tilde E_y|$ , δ their phase difference: $S_1 = a^2-b^2$ , $S_2 = 2ab\cos\delta$ , $S_3 = 2ab\sin\delta$ , so

S_1^2+S_2^2+S_3^2 = (a^2-b^2)^2 + 4a^2b^2 = (a^2+b^2)^2 = S_0^2.

Every monochromatic (pure) state lies on the sphere of radius $S_0$ . The angles are the ellipse: longitude $2\psi$ ( $\psi$ = tilt of the ellipse, from $\tan2\psi = S_2/S_1$ ), latitude $2\chi$ ( $\chi$ = ellipticity angle, from $\sin2\chi = S_3/\sqrt{S_1^2+S_2^2+S_3^2}$ ). Why the doubled angles: a polarization ellipse at ψ and at ψ+180° are the same state, and orthogonal states (ψ, ψ+90°) should sit antipodally — doubling the physical angles makes the state space a full sphere with orthogonal pairs at opposite poles, exactly the property that later makes it the Bloch sphere.

Step 2. Mixing moves inward. Two incoherent beams add in intensity, hence add Stokes-componentwise (each $S_i$ is a sum of intensity measurements): $\mathbf{S}^{\text{tot}} = \mathbf{S}^{(1)} + \mathbf{S}^{(2)}$ . Adding equal-power H and V: $S_0$ doubles while $\mathbf{S}_{1,2,3}$ cancel — the center, unpolarized. Generic mixtures land in between; the triangle inequality bounds every mixture by $|\vec S| \leq S_0$ . Define the degree of polarization

p \equiv \frac{\sqrt{S_1^2+S_2^2+S_3^2}}{S_0} \in [0,1],

and any beam splits uniquely as (polarized part of power $pS_0$ , in the state $\vec S/|\vec S|$ ) + (unpolarized part of power $(1-p)S_0$ ): the vector decomposition $\mathbf{S} = p\mathbf{S}_{\text{pure}} + (1-p)\mathbf{S}_{\text{unpol}}$ , read off by inspection.

Step 3. Devices as geometry. A wave plate (unitary, lossless) preserves $S_0$ and $p$ and rotates $\vec S$ about the axis corresponding to its own eigenstates — a QWP rotates by 90° about its axis-pair's equatorial direction, a HWP by 180° (which is why it maps the R pole to the L pole: half-wave plates reverse handedness). A polarizer is not a rotation but a projection toward its point on the equator, boosting p to 1. All of polarization engineering — compensators, depolarization budgets, PMD in optical fibers — is navigation on this sphere. Why this is worth internalizing beyond optics: replace "polarization state" by "qubit," $(S_1,S_2,S_3)/S_0$ by the Bloch vector, wave plates by single-qubit gates, and p by the purity of a density matrix — the isomorphism is exact, and polarization is its cheapest laboratory realization. *Verified: pure-state identity algebraic; Stokes additivity from incoherent intensity addition (0587); angle dictionary $2\psi, 2\chi$ per Born & Wolf §1.4.2/Hecht §8.13; decomposition uniqueness from vector algebra; QWP/HWP sphere rotations consistent with Step 1–2 of Derivation A applied to basis states.*

Figure

The six-measurement Stokes recipe — one beam, three analyzer pairs, four numbers that fingerprint any polarization state

Limiting cases

Case	Result	Meaning
$\mathbf{J} = (1,0)$	$\vec S/S_0 = (1,0,0)$	H linear: equator point
$(1,1)/\sqrt2$	$(0,1,0)$	45° linear: equator, 90° of longitude away
$(1,i)/\sqrt2$	$(0,0,1)$	circular: pole; orthogonal partner antipodal
equal H + V, incoherent	$(0,0,0)$ , $p=0$	unpolarized: center — no Jones vector exists
equal H + V, coherent	45° linear (or ellipse, by δ)	coherence decides: amplitudes vs intensities add
polarizer on p = 0 light	output p = 1, power ½	projection creates purity
HWP anywhere	$p$ , $S_0$ unchanged; sphere rotated 180°	lossless retarders never depolarize
N random-phase pure sources	$p \to 0$ as excesses average	why thermal light is unpolarized

Common trapPartially polarized light has no Jones vector — averaging amplitudes is the formalism-crossing error

The tempting shortcut — "sunlight is half H and half V, so write $\mathbf{J} = (1,1)/\sqrt2$ " — produces 45° linear light, which sunlight is not: coherent addition of amplitudes and incoherent mixing of intensities are different physical operations with different observables (the coherent sum passes a 45° polarizer completely; sunlight passes any polarizer at ½). The rule: Jones vectors describe only $p = 1$ beams; the moment phases wander (multimode lasers, LEDs, any thermal source) or you average over pulses, you must move to Stokes vectors (or the equivalent coherency matrix), where incoherent addition is legal componentwise. The reverse crossing has its own trap: recovering a Jones vector from measured Stokes parameters is only legitimate after verifying $p \approx 1$ — fitting a pure state to $p = 0.85$ data silently misassigns 15% of the power. And one bookkeeping landmine inside the Stokes convention itself: sign and ordering conventions for $S_2, S_3$ (which diagonal is positive, which circular sense is R) vary across texts and instruments — record the convention with the data, exactly as with handedness in 0588.

Worked examples

1 · Design a circular-polarization transmitter (confidence builder)

A horizontally polarized laser must be converted to circular polarization for an atomic-physics experiment. Specify the optics and verify with Jones calculus, including what happens if the wave plate is misset by placing its fast axis vertically instead.

Step 1. Design: rotate the linear state to 45° (HWP at 22.5°, by Derivation A Step 2's doubling), then QWP with fast axis horizontal.

Step 2. Verify: $M_{\text{QWP}}\,R\text{-chain}$ — after the HWP, $\mathbf{J} = (1,1)/\sqrt2$ ; after the QWP, $(1,i)/\sqrt2 = \mathbf{J}_R$ ✓ — unit intensity throughout (both elements unitary).

Step 3. The misset. Fast axis vertical means the retardation lands on x: $M = \mathrm{diag}(i,1) \simeq \mathrm{diag}(1,-i)$ (global phase dropped). On $(1,1)/\sqrt2$ : output $(1,-i)/\sqrt2 = \mathbf{J}_L$ — the opposite handedness. The experiment pumps the wrong Zeeman state and the signal vanishes. Moral: a QWP has two crystallographically distinct settings 90° apart that both "look right" on a mount; Jones calculus (or a known-handedness test beam) is how labs catch it. Answer: HWP@22.5° + QWP@0° gives R; QWP@90° gives L.

2 · Fingerprinting a mystery beam (exam level)

Six analyzer measurements on a beam give (in W/m²): $I_{0°} = 5.0$ , $I_{90°} = 3.0$ , $I_{45°} = 5.5$ , $I_{135°} = 2.5$ , $I_R = 4.5$ , $I_L = 3.5$ . Find the Stokes vector, the degree of polarization, the polarized/unpolarized power split, and the polarized part's ellipse.

Step 1. Consistency check first (each analyzer pair must sum to the total): $5.0+3.0 = 5.5+2.5 = 4.5+3.5 = 8.0$ ✓. Stokes vector: $S_0 = 8.0$ , $S_1 = 2.0$ , $S_2 = 3.0$ , $S_3 = 1.0$ .

Step 2. Degree of polarization: $|\vec S| = \sqrt{4+9+1} = \sqrt{14} = 3.74$ ; $p = 3.74/8.0 = \mathbf{0.47}$ .

Step 3. The split: polarized power $pS_0 = \mathbf{3.74\ W/m^2}$ ; unpolarized $\mathbf{4.26\ W/m^2}$ . (A pure-state fit here would misprice more than half the beam — the warning box's point, quantified.)

Step 4. The polarized part's ellipse: tilt $\psi = \tfrac12\arctan(S_2/S_1) = \tfrac12\arctan(1.5) = \mathbf{28.2°}$ ; ellipticity angle $\chi = \tfrac12\arcsin(S_3/|\vec S|) = \tfrac12\arcsin(0.267) = \mathbf{7.7°}$ (axis ratio $\tan\chi = 0.136$ — a slim ellipse, right-handed since $S_3 > 0$ in our convention). Answer: $\mathbf{S} = (8.0, 2.0, 3.0, 1.0)$ , $p = 0.47$ , split 3.74/4.26, ellipse at 28° tilt, 7:1-ish axis ratio.

Origins · 1 minGeorge Gabriel Stokes and how this was discovered

George Gabriel Stokes

1819–1903 · Cambridge — who parameterized light he could not model, and got it permanently right

Stokes held Newton's Lucasian chair for over half a century and left his name on theorems (E0's curl theorem), equations (viscous flow), phenomena (fluorescence — he named it), and this article's four parameters. The 1852 paper that defines them, On the composition and resolution of streams of polarized light from different sources, solved a problem his contemporaries could barely state: partially polarized light fit no wave model then available, since any definite ether vibration is fully polarized. Stokes's move was radically operational — characterize the beam by what measurements say (his four parameters are intensity differences through analyzers) and prove that four suffice: any two beams with equal parameters are optically indistinguishable. The framework needed no commitment to what light is, which is why it survived, untouched, the ether's death, Maxwell's fields, and the photon — in whose quantum theory the same four numbers reappear as the density matrix of a two-state system. Poincaré added the sphere in 1892; R. Clark Jones added the pure-state calculus in 1941 (at age 25, at Polaroid — where else); Stokes's operational four remain the coordinates every polarimeter on Earth and in orbit reports today.

History

1852

— Stokes defines the four parameters and proves their completeness for any beam

1892

— Poincaré maps polarization states onto a sphere; wave plates become rotations

1941

— R. Clark Jones publishes the 2×2 pure-state calculus (the "Jones calculus") at Polaroid

1943

— Mueller frames 4×4 matrices acting on Stokes vectors for depolarizing elements

1946

— Chandrasekhar builds Stokes parameters into radiative transfer — polarized starlight becomes astrophysics

1984

— Bennett & Brassard's BB84: polarization bases as quantum key distribution

2020

s — CMB polarimeters map the sky in S₁, S₂ (Q, U) hunting inflationary B-modes; fiber networks budget PMD on the Poincaré sphere

Going further

References. Hecht, Optics, 5th ed., §8.13 (Stokes/Jones) · Born & Wolf, Principles of Optics, 7th ed., §1.4 · Goldstein, Polarized Light, 3rd ed., Chs. 4–6 · Jackson, Classical Electrodynamics, 3rd ed., §7.2.

Provenance. Jones formalism per Hecht §8.13.2/Goldstein Ch. 5 (R.C. Jones, J. Opt. Soc. Am. 31, 488 (1941)); Stokes parameters and the sphere per Born & Wolf §1.4.2 and Jackson §7.2 (G.G. Stokes, Trans. Camb. Phil. Soc. 9, 399 (1852)). A verifier should: (1) confirm the pure-state identity $S_1^2+S_2^2+S_3^2 = S_0^2$ from the quadratic forms; (2) multiply out the QWP/HWP results of Derivation A and the rotated-HWP matrix; (3) recompute Worked Example 2 ( $\sqrt{14}/8 = 0.468$ ; $\psi = \tfrac12\arctan1.5 = 28.15°$ ; $\chi = \tfrac12\arcsin(1/\sqrt{14}) = 7.75°$ ) and its consistency sums (8.0 × 3); (4) check basis orthogonality $\mathbf{J}_R^\dagger\mathbf{J}_L = 0$ . Sign/ordering conventions for $S_2, S_3$ stated in-text and flagged as instrument-dependent. Addendum for this edition: division of labor per 0588 executed — that article owns the state zoo, conventions honesty, and Malus/projection physics; this one owns the two formalisms, the p-decomposition, and the sphere; the ellipse-extraction formulas ( $2\psi$ , $2\chi$ ) live HERE as promised by 0588's Derivation A Step 3. Mueller matrices and the coherency matrix named as the depolarizing-element extension, deferred to the optics track.

Builds on: Polarization of Electromagnetic Waves · Complex Notation for Electromagnetic Waves · Monochromatic Plane Waves in Vacuum · Intensity and Time-Averaged Quantities.

Leads to: The Fresnel Equations · Brewster's Angle and Total Internal Reflection · Energy Density and the Poynting Vector.

Why do I need both formalisms — couldn't Stokes parameters do everything since they handle the general case?

Stokes vectors are complete for predicting detector readings, but they discard the overall phase — and relative phase between beams is physics whenever beams recombine. Interferometry, holography, and any coherent superposition need amplitudes: two beams with identical Stokes vectors can interfere constructively or destructively depending on a phase Stokes never records. Jones calculus keeps that phase (its matrices even track the global phase that wave plates impose, which matters in common-path interferometers). The practical division: Jones inside any coherent optical train; Stokes the moment light is detected, averaged, or of thermal origin. Mueller calculus (4×4 matrices on Stokes vectors) extends the device formalism to depolarizing elements — the optics track owns it.

What is the coherency matrix I sometimes see, and how does it relate to these two?

It is the bridge, and quantum mechanics' favorite form. Define $J_{ij} = \langle\tilde E_i\tilde E_j^*\rangle$ (a 2×2 Hermitian matrix of field correlations, time-averaged). Its four real degrees of freedom are exactly the Stokes parameters — $S_0 = J_{xx}+J_{yy}$ , $S_1 = J_{xx}-J_{yy}$ , $S_2 = 2\mathrm{Re}J_{xy}$ , $S_3 = 2\mathrm{Im}J_{xy}$ — and it is the density matrix of the polarization state, unnormalized. Pure states make it rank-one ( $J = \mathbf{J}\mathbf{J}^\dagger$ , the outer product of the Jones vector); unpolarized light makes it proportional to the identity; p is computable from its eigenvalues. If you know quantum mechanics, learn this dictionary once and both formalisms become density-matrix theory with lenses.

How does a wave plate physically create the phase delay the matrix diag(1, i) asserts?

Birefringence: in an anisotropic crystal (calcite, quartz — the E3 permittivity-tensor article's subject), the two polarization components along the crystal's fast and slow axes propagate with different refractive indices $n_f < n_s$ . Crossing thickness $d$ , they accumulate a relative phase $\Delta\phi = 2\pi(n_s - n_f)d/\lambda$ . Cut $d$ so $\Delta\phi = \pi/2$ and you have a quarter-wave plate — for quartz ( $\Delta n \approx 0.009$ ) at 589 nm that is about 16 μm of true thickness (real plates are "multi-order" or compound for manufacturability). The matrix hides all of this in one $i$ — but the $\lambda$ in $\Delta\phi$ is why every wave plate is specified at a wavelength, and why a 633 nm QWP is merely an approximate retarder at 532 nm.

Can any Stokes vector be realized by some beam — are there impossible states?

The one constraint is $|\vec S| \leq S_0$ (with $S_0 > 0$ ): nothing can be "more than fully polarized." Everything inside and on the ball is physical and constructible — mix a pure state of power $pS_0$ with unpolarized light of power $(1-p)S_0$ and you land exactly where you aimed (Derivation B's decomposition run in reverse). Violations in data — $p$ computed slightly above 1 — are the polarimetrist's standard sanity alarm: analyzer miscalibration, detector nonlinearity, or a sign-convention slip. The same geometry patrols quantum tomography, where reconstructed Bloch vectors outside the sphere flag unphysical density matrices; identical fix, identical lesson.

Where does the handedness ambiguity of the previous article show up in these formalisms?

In two sign choices you must pin: which circular state gets called R (equivalently, the sign of $S_3$ ), and the time convention $e^{\mp i\omega t}$ that decides whether $(1, i)$ rotates x̂→ŷ or the reverse. Texts and instruments genuinely differ; IEEE, classical-optics, and astronomy conventions form a small combinatorial zoo. The formalisms don't resolve the ambiguity — they localize it: fix the two signs once, in writing, and every matrix and measurement thereafter is unambiguous. A lab's polarization notebook typically opens with exactly four lines: time convention, coordinate handedness, definition of R, sign of $S_2$ 's diagonal. Cheap insurance against the 30 dB cross-polarized mistake.

Is the p = 1 versus p < 1 distinction absolute, or does it depend on the detector?

Operationally it depends on the resolution of your measurement — and that dependence is physics, not defect. "Unpolarized" sunlight is a mixture only because detectors average over its ~fs coherence time and ~THz bandwidth; filter it narrowly enough in frequency and observe within a coherence time, and each spectral slice is momentarily nearly pure (p → 1 in the limit). Conversely, a perfectly stable laser depolarizes effectively if your fiber scrambles polarization faster than your receiver tracks. The Stokes formalism silently carries this: the ⟨·⟩ averages in the parameters are over your instrument's bandwidth and integration time, so p is a property of beam plus measurement window. Quantum optics inherits the same structure — purity depends on what you trace over.

The Poincaré sphere looks exactly like the Bloch sphere. Is the resemblance more than an analogy?

It is an isomorphism, not an analogy. A photon's polarization is a genuine two-level quantum system: Jones vector ↔ state vector $|\psi\rangle$ , coherency matrix ↔ density matrix ρ, Stokes parameters ↔ expectation values of the Pauli operators ( $S_i/S_0 = \langle\sigma_i\rangle$ ), wave plates ↔ SU(2) gates, Malus's law ↔ the Born rule ( $\cos^2$ of the half-angle between sphere points). History ran optics-first: Stokes (1852) and Poincaré (1892) built the density-matrix geometry of a qubit decades before quantum mechanics needed it, and quantum information later adopted the ready-made structure. When a photonic quantum computer specifies gates, it is writing Jones matrices; when it tomographs a qubit, it is running Stokes's six-intensity recipe from equation (3).