Linear Polarization

Linear polarization is the simplest and most useful polarization state, and it is the one every other state is measured against. A transverse wave carries two independent field components; when they rise and fall in step — relative phase exactly 0 or π — their vector sum points along one fixed line at every instant. The tip of the electric-field vector does not orbit; it grows, shrinks, and reverses along a single direction, the polarization azimuth ψ. That is all linear polarization is: the degenerate case of the general polarization ellipse, collapsed to a segment. Because the field has a definite direction, linear light is the natural input and output of a polarizer, which passes only the component along its transmission axis. The surviving amplitude is the projection $E_0\cos\theta$ , so the transmitted intensity falls as $\cos^2\theta$ — Malus's law, owned in full by malus-law (1463) and previewed here because it is how linear light is produced, oriented, and measured in every laboratory. Two representations make linear light easy to compute with. The Jones vector $(\cos\psi,\sin\psi)$ is a real two-component column (up to an overall phase) — the realness is the algebraic signature of "in phase." And a linear state is an equal-weight superposition of two counter-rotating circular states, which is precisely why a quarter-wave plate can convert linear into circular and back. Linear polarization is easy to make (dichroic sheets, Brewster reflection, most laser cavities) and easy to analyze (a rotatable polarizer), which is why it is the workhorse state for spectroscopy, displays, stress analysis, and precision metrology, and the right place to start the study of polarized light after the hub article.

Key takeaways

Linear polarization is a transverse wave whose two field components oscillate in phase (δ = 0) or exactly out of phase (δ = π), so the field stays on a fixed line.
The line's orientation — the azimuth ψ — is a free parameter; there is a continuous family of linear states, one per angle in [0, π).
Its Jones vector $(\cos\psi, \sin\psi)$ is real (up to a global phase); on the Poincaré sphere every linear state sits on the equator (S₃ = 0).
A linear state equals two equal, counter-rotating circular states — the reason wave plates interconvert linear and circular light.
A polarizer transmits linear light as $I = I_0\cos^2\theta$ (Malus), the standard way to produce, orient, and measure it.
Key idea: linear polarization is the in-phase, degenerate reading of the two-component field — the electric vector oscillates along one line at azimuth ψ, and its Jones vector is the real column (cos ψ, sin ψ).

The core insight

Linear polarization is the state in which the two transverse components are in phase (δ = 0 or π), so the electric field oscillates back and forth along one fixed line at azimuth ψ. Its Jones vector (cos ψ, sin ψ) is real; it is the degenerate ellipse and an equal sum of two opposite circular states.

Where this lives today

Linear polarization is the default state of engineered light. Most laser cavities emit it cleanly because a Brewster window inside the cavity gives one linear state a decisive gain advantage — essential for nonlinear frequency conversion and for coupling into polarization-sensitive optics. Ellipsometry measures thin-film thickness to sub-nanometre precision by tracking how a known linear input becomes elliptical on reflection. Polarization spectroscopy uses a linearly polarized probe through a nearly crossed analyzer to pull sub-Doppler atomic resonances out of the noise. Liquid-crystal displays begin and end each pixel with a linear polarizer. And in fibre-optic and free-space communication, "polarization-division multiplexing" doubles a link's capacity by sending independent data on two orthogonal linear states — the modern payoff of a state first seen in a reflected image of a palace window.

The physical picture

Shake a taut rope straight up and down: every point oscillates vertically, and the wave is vertically linearly polarized. Shake diagonally and the polarization tilts to that diagonal. The defining feature is that your hand — and the field — never traces a loop; it moves along one line. In field language, the two transverse components $E_x = E_1\cos(kz-\omega t)$ and $E_y = E_2\cos(kz-\omega t)$ share the same phase, so their ratio $E_y/E_x = E_2/E_1$ is a constant: a straight line of slope $\tan\psi = E_2/E_1$ . The azimuth ψ is the one number that names a linear state. The punchline figure fixes a concrete case — ψ = 30° — and reads its Jones vector straight off the geometry, because that decomposition into $x$ and $y$ amplitudes is exactly what a polarizer, a beamsplitter, or a detector responds to.

Figure

Punchline — a linear state at azimuth ψ = 30° and its Jones vector (0.866, 0.500), read straight off the component projections

A second picture explains why linear light and circular light are so easily swapped. Imagine two vectors of equal length spinning at the same rate about the origin, one clockwise and one counter-clockwise, starting together pointing up. At every instant their horizontal parts cancel and their vertical parts add: the sum points straight up and down — a line. A linear state is the sum of two opposite circular states, and a wave plate that slows one component relative to the other is exactly what unbalances that sum into an ellipse or a circle.

Figure

Linear = two equal counter-rotating circular states; and two orthogonal linears sum to a tilted linear

The core equations

A linearly polarized plane wave has its field along one fixed real direction; the whole state is one unit vector plus an amplitude.

\mathbf E(z,t) = \hat{\mathbf e}\,E_0\cos(kz-\omega t+\phi_0), \qquad \hat{\mathbf e} = \cos\psi\,\hat{\mathbf x} + \sin\psi\,\hat{\mathbf y} \qquad (1)

where: $\hat{\mathbf e}$ — the fixed real polarization direction; ψ — azimuth from the $x$ -axis, $0 \le \psi < \pi$ ; $E_0$ — amplitude; $\phi_0$ — overall phase, irrelevant to the state. Read it physically: because $\hat{\mathbf e}$ does not change with time, the field just oscillates along that line — the tip never leaves it. The compact bookkeeping is the Jones vector, and its realness is the algebraic fingerprint of "in phase."

\mathbf J = \begin{pmatrix}\cos\psi \\ \sin\psi\end{pmatrix} \qquad (2)

where: both entries are real (up to a common phase); a complex ratio $E_y/E_x$ would signal an elliptical state. Normalized so $|\mathbf J|^2 = \cos^2\psi + \sin^2\psi = 1$ : the vector carries direction, not brightness. A polarizer reads this state by projection, which is why linear light is produced and measured with the same device.

I = I_0\cos^2\theta \qquad (3)

where: θ — angle between the incident linear polarization and the transmission axis; the intensity is the square of the projected amplitude. Full derivation and the three-polarizer paradox are owned by malus-law (1463). Two further identities place linear light in the wider family. In the Stokes description linear states have no circular content, so they lie on the Poincaré equator; and any linear state is an equal sum of two opposite circular states.

(S_0,S_1,S_2,S_3) = (I,\ I\cos2\psi,\ I\sin2\psi,\ 0), \qquad \hat{\mathbf e}_x = \tfrac{1}{\sqrt2}\big(\hat{\mathbf e}_+ + \hat{\mathbf e}_-\big) \qquad (4)

where: $S_3 = 0$ is the linear signature (equator); the factor $2\psi$ is the Poincaré double-angle; $\hat{\mathbf e}_\pm = (\hat{\mathbf x} \pm i\hat{\mathbf y})/\sqrt2$ are the two circular basis states (which one is "right-handed" is fixed by circular-polarization (1461)).

Derivations

Derivation AThe straight-line trajectory forced by in-phase components

The claim to establish is precise: if the two transverse components share a phase (δ = 0 or π), the tip of the electric-field vector is confined to a straight line through the origin, so the general polarization ellipse degenerates. This is the defining property of linear polarization, and proving it from the two-component field — rather than asserting it — shows exactly which condition on δ is doing the work, and prepares the ground for circular and elliptical states, which relax that condition.

Step 1. Write the general transverse field with $\tau = kz-\omega t$ : $E_x = E_1\cos\tau$ , $E_y = E_2\cos(\tau+\delta)$ . Why keep δ general: the whole point is to see what happens as δ is dialed away from 0, so it must appear explicitly.

Step 2. Eliminate τ. Expanding $\cos(\tau+\delta)$ and removing τ between the two equations gives the polarization ellipse

\left(\frac{E_x}{E_1}\right)^2 + \left(\frac{E_y}{E_2}\right)^2 - 2\frac{E_xE_y}{E_1E_2}\cos\delta = \sin^2\delta.

Why an ellipse in general: this is a bounded quadratic form in $(E_x,E_y)$ , hence a conic inscribed in the $2E_1\times2E_2$ box — the full derivation is owned by elliptical-polarization (1462).

Step 3. Set δ = 0. The right side vanishes and the left collapses to a perfect square: $\big(E_x/E_1 - E_y/E_2\big)^2 = 0$ , i.e. $E_y = (E_2/E_1)E_x$ — a line of slope $\tan\psi = E_2/E_1$ . For δ = π the cross term flips sign and the slope becomes negative. Why this is linear polarization: a line through the origin is a one-parameter locus, and the parameter is the azimuth ψ; nothing orbits. The ellipse has collapsed to a segment exactly when δ ∈ {0, π}. Verified: the δ = 0 collapse to a perfect square and the slope $\tan\psi = E_2/E_1$ are checked symbolically in verify_linear-polarization.py.

Go deeper · Derivation B · 1 minThe Jones vector is real, and rotates like a vector

Derivation BThe Jones vector is real, and rotates like a vector

Two facts about the Jones representation deserve a proof rather than a table lookup: that a linear state has a real Jones vector, and that rotating the physical polarization by an angle φ rotates the Jones vector by the ordinary 2×2 rotation matrix. Together they make "linear polarization at azimuth ψ" a genuinely geometric object you can manipulate with matrix algebra, and they underpin every later use of Jones calculus for polarizers and wave plates.

Step 1. From $E_x = A_x$ , $E_y = A_y e^{i\varphi}$ (common global phase removed), linear polarization requires the ratio $E_y/E_x$ to be real. That forces $\varphi = 0$ or π, so $\mathbf J \propto (A_x, \pm A_y)$ — a real column. Why realness ⇔ linear: a complex ratio would mean the components peak at different instants, opening the locus into an ellipse.

Step 2. Normalize and set $A_x = \cos\psi$ , $A_y = \sin\psi$ : $\mathbf J = (\cos\psi, \sin\psi)$ . Why this parametrization: it makes $|\mathbf J| = 1$ automatic and ψ the physical azimuth.

Step 3. Rotate the physical field by φ. The azimuth becomes ψ + φ, so

\mathbf J' = \begin{pmatrix}\cos(\psi+\varphi)\\ \sin(\psi+\varphi)\end{pmatrix} = \begin{pmatrix}\cos\varphi & -\sin\varphi\\ \sin\varphi & \cos\varphi\end{pmatrix}\begin{pmatrix}\cos\psi\\ \sin\psi\end{pmatrix}.

Why the ordinary rotation matrix: the angle-addition identities are exactly the entries of $R(\varphi)$ , so a physical rotation is a matrix rotation — the reason Jones calculus works. *Verified: $|\mathbf J|^2 = 1$ , and $R(\varphi)(\cos\psi,\sin\psi)^\top = (\cos(\psi+\varphi),\sin(\psi+\varphi))^\top$ are checked in verify_linear-polarization.py.*

Go deeper · Derivation C · 1 minLinear light is an equal sum of two opposite circular states

Derivation CLinear light is an equal sum of two opposite circular states

The identity $\hat{\mathbf e}_x = (\hat{\mathbf e}_+ + \hat{\mathbf e}_-)/\sqrt2$ looks like a formal rewrite, but it is the physical reason a quarter-wave plate can turn linear light into circular light. This derivation establishes it directly from the circular basis vectors and interprets each step, so that the linear↔circular conversion in circular-polarization (1461) reads as a consequence rather than a new fact.

Step 1. Take the circular basis $\hat{\mathbf e}_\pm = (\hat{\mathbf x} \pm i\hat{\mathbf y})/\sqrt2$ , the eigenvectors of rotation about $\hat{\mathbf z}$ . Why complex: a rotating (circular) field is naturally an $e^{\pm i\omega t}$ object, and these vectors carry that rotation.

Step 2. Add them: $\hat{\mathbf e}_+ + \hat{\mathbf e}_- = \tfrac{1}{\sqrt2}(2\hat{\mathbf x}) = \sqrt2\,\hat{\mathbf x}$ . The imaginary $\hat{\mathbf y}$ parts cancel. Why cancellation matters: the surviving field points along a fixed axis — no rotation, hence linear.

Step 3. Rearrange: $\hat{\mathbf x} = (\hat{\mathbf e}_+ + \hat{\mathbf e}_-)/\sqrt2$ . Equal weight, in phase. Unbalance the weights or add a relative phase and the sum becomes elliptical or circular. Why this is the whole story of wave plates: a retarder changes the relative phase of $\hat{\mathbf e}_+$ and $\hat{\mathbf e}_-$ , sliding the state along the linear→elliptical→circular family. *Verified: $\hat{\mathbf e}_+ + \hat{\mathbf e}_- = \sqrt2\,\hat{\mathbf x}$ and orthonormality of the circular basis are checked in verify_linear-polarization.py.*

Limiting cases

Case	Condition	Meaning
In phase	δ = 0	Linear at ψ = arctan(E₂/E₁), first/third quadrant.
Anti-phase	δ = π	Linear at −ψ; slope sign flips into second/fourth quadrant.
One component zero	E₂ = 0	Trivially linear along x for any δ — only one oscillation exists.
Quarter phase, equal amps	δ = ±π/2, E₁ = E₂	Circular — the opposite extreme, showing linear needs δ ∈ {0, π}.
On the Poincaré sphere	S₃ = 0	All linear states lie on the equator; azimuth ψ ↔ longitude 2ψ.

Worked examples

1 · Components and Malus test of a 30° linear state (confidence builder)

A helium–neon laser emits linearly polarized light at azimuth ψ = 30°. (a) What are its $x$ and $y$ field components as fractions of $E_0$ ? (b) It passes an analyzer with its axis at 75°. What fraction of the intensity emerges?

Step 1. (a) The components are $E_x = E_0\cos\psi = E_0\cos30^\circ = \mathbf{0.866\,E_0}$ and $E_y = E_0\sin\psi = E_0\sin30^\circ = \mathbf{0.500\,E_0}$ . Its Jones vector is $(0.866, 0.500)$ , and $|\mathbf J|^2 = 0.75 + 0.25 = 1$ . Why the sum of squares is one: the Jones vector carries direction, not brightness.

Step 2. (b) The analyzer is at 75°, the polarization at 30°, so the relative angle is $\theta = 75^\circ - 30^\circ = 45^\circ$ . Malus's law gives $I/I_0 = \cos^2 45^\circ = \mathbf{0.5}$ . Why the difference angle: Malus uses the angle between the incident polarization and the transmission axis, not either absolute angle. Half the intensity passes.

2 · Rotating a linear state with a matrix (exam level)

Vertical linear light, Jones vector $(0, 1)$ , passes an element that rotates the polarization by −30°. What is the output state, and at what azimuth?

Step 1. Apply $R(-30^\circ) = \begin{pmatrix}\cos30^\circ & \sin30^\circ\\ -\sin30^\circ & \cos30^\circ\end{pmatrix}$ to $(0,1)$ : $\mathbf J' = (\sin30^\circ, \cos30^\circ) = \mathbf{(0.500, 0.866)}$ . Why this row combination: matrix–vector multiplication picks the second column since the input is $(0,1)$ .

Step 2. The azimuth is $\psi' = \arctan(0.866/0.500) = 60^\circ$ — the vertical state (90°) has rotated by −30° to 60°, still linear. The output is real, confirming a rotation preserves linearity.

Physics of the portrait

Origins · 1 minlinear polarization and how this was discovered

linear polarization

the workhorse state · named through the 1800s · the first polarization anyone ever made — by reflection, by crystals, and eventually by a plastic sheet

Linear polarization has no single discoverer because it was the first polarization state ever isolated, again and again by different means. Malus produced it by reflection in 1808 and gave it a law in 1809. David Brewster (1815) found the special angle at which reflection makes it perfectly. William Nicol (1828) cut a calcite crystal into the Nicol prism, the first practical device to deliver a clean linear beam by throwing away the orthogonal component through total internal reflection. For a century, producing linear light meant crystals or reflection — expensive, small, and fragile. The democratizing step came from a nineteen-year-old: Edwin Land, who in 1929 patented a sheet of stretched, iodine-stained polymer whose aligned long-chain molecules absorb one component and transmit the other — dichroism in a material you could cut with scissors. Marketed as Polaroid, it put a linear polarizer in every camera bag and pair of sunglasses. The state that once required Iceland spar became a strip of plastic; the physics — project the field onto one axis — never changed.

History

1808

— Malus produces linearly polarized light by reflection

1815

— Brewster finds the polarizing angle tan θ_B = n

1828

— Nicol builds the calcite Nicol prism, the first practical linear polarizer

1852

— Stokes formalizes the parameters that place linear states on the equator

1892

— Glazebrook and others refine calcite polarizing prisms (Glan–Thompson type)

1929

— Land patents the first sheet polarizer (dichroic Polaroid)

1938

— Land's H-sheet iodine polarizer reaches the mass market

Going further

Texts. Hecht, Optics, 5th ed., ch. 8 · Saleh & Teich, Fundamentals of Photonics, 3rd ed., ch. 6 · Born & Wolf, Principles of Optics, ch. 1 · Loudon, The Quantum Theory of Light, 3rd ed.

Provenance. The straight-line locus follows from the polarization-ellipse equation at δ = 0, π (Hecht ch. 8); the ellipse itself is owned by elliptical-polarization (1462). The Jones vector $(\cos\psi,\sin\psi)$ , the rotation matrix, and the Stokes form $(I, I\cos2\psi, I\sin2\psi, 0)$ with $S_3 = 0$ follow Saleh & Teich ch. 6. The decomposition $\hat{\mathbf e}_x = (\hat{\mathbf e}_+ + \hat{\mathbf e}_-)/\sqrt2$ with $\hat{\mathbf e}_\pm = (\hat{\mathbf x}\pm i\hat{\mathbf y})/\sqrt2$ follows Loudon; the handedness labels are fixed by circular-polarization (1461). Malus's law is previewed here and owned by malus-law (1463). Numerics are exact: $\cos30^\circ = 0.866$ , $\cos^2 45^\circ = 0.5$ .

Builds on: Polarization of Light (1459) · Plane Electromagnetic Waves. Leads to: Circular Polarization · Elliptical Polarization · Malus's Law · Polarizers and Extinction Ratio · Jones Calculus · Stokes Parameters and the Poincaré Sphere · Wave Plates and Retarders.

Is "plane polarized" the same as "linearly polarized"?

Yes — the two terms are synonyms. "Plane polarized" refers to the fact that the electric field stays in a fixed plane containing the propagation direction (the plane spanned by $\hat{\mathbf e}$ and $\hat{\mathbf z}$ ), while "linearly polarized" refers to the straight-line trajectory of the field tip in the transverse plane. Older texts, especially in the British tradition, favour "plane polarized"; most modern optics texts use "linear." They describe the identical state, δ = 0 or π.

If linear light is a sum of two circular states, why does it look nothing like circular light?

Because the two circular states are present in equal amounts and locked in phase. At every instant their rotating parts add to a vector on a fixed line — the rotations cancel and only the oscillation survives. Circular light, by contrast, is one of the two states alone; then nothing cancels the rotation and the field sweeps a circle. The difference between linear and circular is entirely in the balance and relative phase of the same two building blocks, which is why a wave plate — a device that shifts that relative phase — slides you continuously between them.

How do I know a beam is linearly polarized and not partially polarized or elliptical?

Rotate a single analyzer and watch the transmitted intensity. Linear light gives a clean $\cos^2$ curve that dips to exactly zero at the crossed orientation. Partially polarized light dips to a nonzero minimum (its degree of polarization V < 1). Elliptical light also never reaches zero through a single polarizer. To separate elliptical from partially polarized, insert a quarter-wave plate first: it converts a definite elliptical state into linear (which then extinguishes), but does nothing coherent to the unpolarized part. The full analyzer/wave-plate recipe is in polarizers-and-extinction-ratio (1464).

Why do lasers so often come out linearly polarized?

Most gas and many solid-state lasers contain a surface tilted at Brewster's angle — a Brewster window or Brewster-cut rod end. At that angle the reflection loss is zero for the linear polarization lying in the plane of incidence and nonzero for the orthogonal one. Over many cavity round trips, the low-loss polarization builds up gain while the lossy one dies, so the laser self-selects a clean linear state. Lasers without any polarizing element (some fibre and microchip lasers) can emit unpolarized or randomly drifting polarization instead.

Does rotating a linear polarizer change the light's azimuth or just filter it?

It filters. An ideal polarizer transmits the projection of whatever comes in onto its axis and resets the output polarization to lie along that axis. So after any polarizer the light is linear along the transmission axis, at reduced intensity $\cos^2\theta$ . What changes the azimuth without loss is a rotator (an optically active element or a half-wave plate), which turns the polarization direction while keeping the full intensity. Confusing "polarizer" (projects and attenuates) with "rotator" (turns, lossless) is a common source of error in optical-system design.

Two orthogonal linear beams overlap. When do they add to a linear state, and when to something else?

It depends entirely on their relative phase and coherence. Two coherent orthogonal linears that are in phase add to a single tilted linear state whose azimuth is set by their amplitude ratio (the second panel of the figure above); with a ±π/2 relative phase and equal amplitudes they make circular light; any other phase makes an ellipse. But two incoherent orthogonal linears — from independent sources, with a randomly drifting relative phase — do not combine into any fixed state at all; they average to unpolarized (or partially polarized) light. So "H plus V" can be linear, circular, elliptical, or unpolarized depending on the phase relationship, which is exactly why coherence and polarization must be tracked together.