Circular Polarization
Send linearly polarized light through a thin sliver of quartz cut so that one transverse direction lags the other by exactly a quarter of a cycle, and something strange comes out the far side: light whose electric field never grows or shrinks, but instead spins. The tip of the field vector traces a perfect circle at the optical frequency — roughly 5×10¹⁴ turns every second — while the whole corkscrew glides forward at the speed of light. This is circular polarization, the most symmetric of all the fully polarized states, and the one that forces us to be honest about a question linear light lets us dodge: which way is the field turning, and from whose point of view? Circular light is not a new kind of wave. It is the same two-component transverse field that gives us every polarization state, read at one special setting: two orthogonal linear oscillations of equal amplitude with a relative phase of exactly ±90°. Because the two components are permanently a quarter-cycle out of step, when one is at full stretch the other is passing through zero, and their vector sum sweeps a circle of constant length. That constant length is why a circularly polarized beam carries a perfectly steady instantaneous intensity — unlike a linear wave, whose field magnitude pulses at twice the optical frequency. The steadiness hides a second, deeper fact. A rotating field carries angular momentum: each photon of circular light delivers exactly one unit, ±ħ, aligned with its direction of travel. That is not a metaphor. Spin ±ħ per photon is what lets circular light pump atoms up a ladder of magnetic sublevels, twist tiny particles in optical tweezers, and read the handedness of the molecules of life. From a quarter-wave plate on a lab bench to the σ± selection rules of atomic spectroscopy, circular polarization is where the geometry of a spinning field meets the physics of angular momentum.
- Circular polarization is the state in which the electric-field vector keeps a constant magnitude and rotates uniformly at the optical frequency ω; freeze time and it is a helix of one-wavelength pitch.
- It is built from two orthogonal, equal-amplitude linear components with a relative phase of exactly δ = ±π/2 — equivalently, an equal superposition of the two circular basis states seen from the linear side.
- Handedness (right vs left) is the sign of that rotation, and it is genuinely convention-dependent: the optics convention (observer facing the source) and the physics/IEEE conventions label the same field oppositely, so the convention must always be named.
- A quarter-wave plate at 45° to its axes converts linear light to circular and back; the plate imposes a π/2 retardance, which for zero-order quartz at 550 nm means a slice just d = λ/(4Δn) ≈ 15 µm thick.
- Each circularly polarized photon carries spin angular momentum ±ħ along k — the origin of the σ± = Δm ±1 selection rules and of optical pumping.
- A giveaway test: circular light transmits exactly one half through any linear polarizer, at every orientation — it looks like unpolarized light to a single analyzer, but a wave plate unmasks it.
- Key idea: circular polarization is two equal-amplitude orthogonal linear fields a quarter-cycle out of phase; the field spins at constant length, sits at a pole of the Poincaré sphere (S₃ = ±S₀), and carries spin ±ħ per photon.
Circular polarization = two equal-amplitude orthogonal linear fields with a ±π/2 relative phase. The field vector rotates at constant magnitude, sits at a pole of the Poincaré sphere with S₃ = ±S₀, and carries spin angular momentum ±ħ per photon along the direction of travel.
Where this lives today
Circular polarization is a routine tool wherever a definite handedness or a unit of spin matters. In atomic physics, a beam of σ⁺ light — circularly polarized along the quantization axis — drives only Δm = +1 transitions, so repeated absorption pumps an atom into a single stretched state; this optical pumping is the heart of atomic magnetometers, alkali atomic clocks, and the spin-polarized targets of nuclear physics. In chemistry and biology, chiral molecules absorb left- and right-circular light unequally, and the difference — circular dichroism — reports on protein secondary structure and the handedness of pharmaceuticals. In display and imaging technology, circular-polarizer glasses (a quarter-wave plate laminated to a linear polarizer) give stereo 3D cinema its tolerance to head tilt, because circular states, unlike linear ones, are not extinguished by rotating the glasses. And in astronomy, the small circular polarization of light from magnetized regions maps stellar and interstellar magnetic fields through the same Faraday and Zeeman physics that Faraday first glimpsed in 1846.
The physical picture
Freeze a circularly polarized wave at one plane and let the clock run. The electric-field vector is a hand of fixed length sweeping a circle at the optical angular frequency, ω ≈ 3.5×10¹⁵ rad/s for green light. Now freeze time instead and look down the beam: the field vectors at successive planes are rotated a little more each step, so along the propagation axis they thread out a smooth corkscrew — a helix whose pitch is exactly one wavelength. The wave is the helix sliding rigidly forward at c; a detector at a fixed plane simply watches the local arm of that helix rotate. The defining feature is what the field magnitude does: nothing. For a linear wave, E = E₀ x̂ cos(kz − ωt), the length pulses between 0 and E₀ twice per cycle. For circular light the two equal, quadrature components fill in each other's gaps — when the x-component passes through zero the y-component is at full stretch — so |E|² = E₀²(cos² + sin²) = E₀² is flat. Circular light is the polarization state with a perfectly steady instantaneous intensity, and the only fully polarized state with no preferred transverse axis. That axis-free symmetry is exactly why circular states are the natural basis for anything with rotational symmetry about the beam: an optically active sugar solution, a medium magnetized along k, an atom absorbing a photon of definite spin. In each case the two circular states propagate or couple independently, while a linear state — which is an equal mix of both — gets scrambled between them.
The mechanical intuition for making circular light is a race between the two components. Start with a diagonal linear field, equal parts x and y, marching in step. Slip a plate into the beam that slows the y-component by exactly a quarter cycle relative to the x-component. Now, everywhere along the beam, y lags x by 90°: the in-step diagonal has become the quadrature pair that spins. That plate is the quarter-wave plate, and it is the workhorse of every polarization bench.
The core equations
A circularly polarized wave travelling along +z is the sum of two orthogonal linear components with equal amplitude and a quarter-cycle phase offset. Written as a real field, with the shorthand τ = kz − ωt for the travelling phase:
where: E₀ — the common amplitude of the two components; τ = kz − ωt — the travelling phase; the minus sign in front of ŷ (equivalently a relative phase δ = +π/2, since cos(τ + π/2) = −sin τ) selects one sense of rotation. Reading it physically: at any fixed plane the two components are always a quarter-cycle apart, so as one crosses zero the other peaks, and the resultant vector — of fixed length E₀ — sweeps a circle once per optical period. The compact bookkeeping of a pure polarization state is the two-component Jones vector. Adopting the phasor convention that the real field is Re{ J e^{i(kz−ωt)} } (an e^{−iωt} time factor), and the phase δ = φ_y − φ_x, the two circular states are:
where: ê_R — right-circular (δ = +π/2 in this convention); ê_L — left-circular; the factor ±i encodes the 90° phase between the y and x components. Both are unit vectors and they are orthogonal, ⟨ê_R|ê_L⟩ = 0, so they form a complete basis: any polarization state is a superposition of the two. A caution to bank now and unpack below: Hecht's Jones table uses the opposite time convention (e^{+iωt}) and therefore writes right-circular as (1, −i)/√2. The field and its handedness are identical — only the sign of i tracks the chosen phasor convention. On the Poincaré sphere, the compact map of every state, circular light sits at the poles. In terms of the four Stokes parameters,
where: S₀ — total intensity; S₁, S₂ — the linear content (both zero for circular); S₃ — the circular content. With equal amplitudes and δ = +π/2, S₁ = S₂ = 0 and S₃ = +S₀: the north pole. The opposite handedness (δ = −π/2) gives S₃ = −S₀, the south pole. Every latitude between is elliptical; the equator is linear. A quarter-wave plate is the operator that moves a state from equator to pole. With its fast axis along x̂ and a π/2 retardance, its Jones matrix (up to a global phase) is diagonal:
where: the factor i on the slow (y) axis is the quarter-cycle lag; Γ is the retardance, set by the plate's birefringence Δn (the material physics belongs to birefringence) and thickness d. A true zero-order plate is the thinnest slab that delivers exactly Γ = π/2. The retardance — not the plate — is what matters: any Γ = π/2 element interconverts 45°-linear and circular light. Finally, the rotating field is not just geometry; it carries angular momentum. Each photon of circular light carries a spin projection along the propagation axis of
where: ħ = 1.0546×10⁻³⁴ J·s and the sign follows the handedness. Classically the same content is an angular-momentum density proportional to S₃; quantum-mechanically it is one unit of spin per photon, the fact behind the σ± selection rules and optical pumping.
The goal is to show, with no hand-waving, that the two conditions "equal amplitudes" and "δ = ±π/2" are exactly what turn the generic two-component field into a circle of constant radius, and to read the sense of rotation straight off the field. We start from the most general superposition of two orthogonal linear components and impose the two conditions one at a time, watching the locus of the field tip close into a circle and the intensity go flat.
Step 1. Write the general transverse field as Eₓ = E₀ₓ cos τ and E_y = E₀ᵧ cos(τ + δ) with τ = kz − ωt. This already contains every polarization state; the amplitude ratio and the phase δ = φ_y − φ_x choose which one. Why start here: circular light is not a separate phenomenon but one setting of this single two-parameter family, so any honest derivation must show which setting.
Step 2. Impose equal amplitudes, E₀ₓ = E₀ᵧ ≡ E₀, and δ = +π/2. Then E_y = E₀ cos(τ + π/2) = −E₀ sin τ, so the pair becomes Eₓ = E₀ cos τ, E_y = −E₀ sin τ. Why these two conditions and not others: unequal amplitudes give an ellipse squashed along an axis, and a phase that is not a quarter cycle tilts and opens the ellipse — only equal amplitude and exact quadrature remove both defects at once.
Step 3. Eliminate the phase τ by squaring and adding: Eₓ² + E_y² = E₀²(cos²τ + sin²τ) = E₀². The locus of the field tip is a circle of radius E₀, independent of time and position. Why this is the whole content: a fixed radius means the field magnitude never changes, which is the operational signature of circular light and the reason its instantaneous intensity is constant.
Step 4. Read the sense of rotation. The polar angle of the field is φ = arctan(E_y/Eₓ) = arctan(−tan τ) = −τ = ωt − kz. At a fixed plane φ increases with t: the vector turns steadily. Choosing δ = −π/2 instead flips the sign and gives the opposite handedness. Why the sign of δ is the whole handedness story: it is the only thing that distinguishes the two circular states, which is why naming it — and the viewpoint — is non-negotiable. *Verified: |E|² = E₀² for δ = ±π/2 and equal amplitudes; rotation sense flips with the sign of δ — verify_circular-polarization.py.*
Go deeper · Derivation B · 1 minA quarter-wave plate at 45° makes circular light
The aim is to prove that the quarter-wave plate is the exact operator that carries a 45°-linear state to a circular one, using the Jones calculus, and to pin down its thickness for a real crystal. The strategy is to write the plate as a diagonal retarder, feed it the equal-amplitude diagonal input, and identify the output with a pole of the Poincaré sphere; then convert the π/2 retardance condition into a thickness.
Step 1. Take the plate's fast axis along x̂; a retardance Γ makes it J = diag(1, e^{iΓ}). For a quarter-wave plate Γ = π/2, so J = diag(1, i). Why diagonal in the crystal axes: the fast and slow axes are precisely the directions along which the crystal's two indices act independently, so the plate does nothing but delay one relative to the other.
Step 2. Feed in light linearly polarized at 45° to the axes, ê₄₅ = (1, 1)ᵀ/√2, so both axes carry equal amplitude. Applying the matrix: J ê₄₅ = (1, i)ᵀ/√2 = ê_R. The output is right-circular. Why 45° specifically: only when the two axes receive equal amplitude does the π/2 lag produce the equal-amplitude quadrature pair that is a circle; at any other input angle the output is an ellipse (the subject of the sibling article).
Step 3. Convert the retardance to a thickness. Γ = (2π/λ)Δn·d = π/2 gives d = λ/(4Δn). For zero-order quartz at λ = 550 nm with Δn = n_e − n_o = 0.00915, d = 550 nm /(4 × 0.00915) = 15.0 µm. Why "zero order" matters: a plate that instead delivers Γ = π/2 + 2πm (m ≥ 1) is far thicker and far more sensitive to wavelength and temperature — the thin zero-order slice is the robust one. *Verified: diag(1, i)·(1,1)ᵀ/√2 = (1, i)ᵀ/√2 = ê_R; d = λ/(4Δn) = 15.0 µm and Γ = π/2 — verify_circular-polarization.py.*
Go deeper · Derivation C · 1 minCircular light is invisible to a single polarizer — but a wave plate unmasks it
The purpose here is to prove the practical test that distinguishes circular light from unpolarized light: both pass exactly half their intensity through any single linear polarizer, so one analyzer cannot tell them apart, yet a quarter-wave plate followed by a polarizer separates them cleanly. The method is a direct Jones projection at an arbitrary analyzer angle, then a second pass through a plate.
Step 1. Project the right-circular state onto a linear analyzer at angle θ, with Jones row (cos θ, sin θ). The transmitted amplitude is (cos θ·1 + sin θ·i)/√2, whose squared modulus is (cos²θ + sin²θ)/2 = 1/2. Why the answer is independent of θ: circular light has no preferred transverse axis, so rotating the analyzer changes nothing — exactly the behaviour of unpolarized light, which is why one polarizer cannot distinguish them.
Step 2. Now precede the analyzer with a quarter-wave plate whose fast axis is at 0°. It sends ê_R = (1, i)ᵀ/√2 to diag(1,−i)ᵀ... apply J = diag(1, −i) for the reverse-oriented plate: J ê_R = (1, i·(−i))ᵀ/√2 = (1, 1)ᵀ/√2, which is linear at 45°. Why the plate breaks the symmetry: it converts the quarter-cycle phase back into an amplitude difference, turning the axis-free circle into a definite linear direction the analyzer can then extinguish.
Step 3. The recovered 45° linear state is fully extinguished by an analyzer at −45° and fully passed at +45° — a modulation from 0 to 1 as the analyzer rotates. Unpolarized light, by contrast, still gives a flat 1/2 after the plate, because it has no fixed phase to convert. Why this is the definitive test: only a genuinely coherent quarter-cycle phase survives the plate; the plate is a phase-to-amplitude converter that circular light passes and unpolarized light fails. *Verified: |⟨analyzer(θ)|ê_R⟩|² = 1/2 for all θ; plate + analyzer recovers full 0→1 modulation — verify_circular-polarization.py.*
Limiting cases
Circular polarization is the sharp centre of the general two-component family. Vary the amplitude ratio and the relative phase δ = φ_y − φ_x and it dissolves into its neighbours:
| Case | Condition | Meaning | ||
|---|---|---|---|---|
| Right-circular | E₀ₓ = E₀ᵧ, δ = +π/2 | Field spins clockwise seen facing the source; S₃ = +S₀ (north pole); Jones (1, i)/√2 | ||
| Left-circular | E₀ₓ = E₀ᵧ, δ = −π/2 | Opposite sense; S₃ = −S₀ (south pole); Jones (1, −i)/√2 | ||
| Unequal amplitude or δ ≠ ±π/2 | E₀ₓ ≠ E₀ᵧ, or generic δ | General ellipse; 0 < \ | S₃\ | < S₀; the circle deforms |
| δ = 0 or π (any amplitudes) | in step or anti-phase | Linear polarization; S₃ = 0; field collapses onto a line | ||
| Detector view | any single linear analyzer | Transmits exactly ½ at every orientation — looks unpolarized until a wave plate is added |
Worked examples
Example 1 — A zero-order quartz quarter-wave plate for a green laser (build something real). You need to turn the 550 nm linear output of a diode-pumped laser into circular light. Quartz has ordinary and extraordinary indices differing by Δn = n_e − n_o = 0.00915 near this wavelength. How thick must the plate be?
Step 1. The plate must impose a retardance Γ = π/2 between its axes: Γ = (2π/λ)Δn·d = π/2. Solve for the thickness: d = λ/(4Δn). This is the zero-order thickness — the thinnest slab that works.
Step 2. Substitute: d = 550×10⁻⁹ m /(4 × 0.00915) = 550×10⁻⁹ / 0.0366 = 15.0 µm. A true zero-order quartz quarter-wave plate at 550 nm is a slice about fifteen microns thick — so thin it is usually cemented to a thick glass window for support, or replaced by a multi-order plate that adds a whole number of extra waves.
Step 3. Sanity check the physics: 15 µm is roughly 27 wavelengths of green light, and the quarter-wave difference between the two axes accumulates as the ~0.9% index difference acts over those 27 waves — 0.00915 × 15 µm / 550 nm ≈ 0.25 wave, exactly a quarter. The answer is d ≈ 15.0 µm.
Example 2 — How much twist does a circular beam carry? (angular momentum with numbers). A P = 1.0 W beam of pure circular light at λ = 633 nm shines onto a small absorbing target. What spin angular momentum does it deposit per second?
Step 1. Each photon carries energy ħω = hc/λ and spin ħ. The photon arrival rate is the power divided by the photon energy: rate = Pλ/(hc) = (1.0)(633×10⁻⁹)/[(6.626×10⁻³⁴)(2.998×10⁸)] = 3.19×10¹⁸ photons/s.
Step 2. Every photon delivers one unit of spin ħ = 1.0546×10⁻³⁴ J·s, so the angular-momentum flux is ħ × rate = (1.0546×10⁻³⁴)(3.19×10¹⁸) = 3.36×10⁻¹⁶ J·s per second — that is 3.19×10¹⁸ units of ħ every second.
Step 3. Put it in perspective: this is the torque (in J·s/s = N·m) that a perfect circular absorber feels. It is tiny, but it is exactly what Beth measured in 1936 by suspending a wave plate on a fine fibre, and it is what spins micron-scale particles in optical tweezers today. The answer is 3.36×10⁻¹⁶ N·m of steady torque per watt of circular light.
Physics of the portrait
Circular polarization earned its place not as a curiosity but as the evidence that settled what light is. When Malus and Arago found in the 1810s that reflected light and calcite images carried a hidden direction, the reigning picture — light as longitudinal pressure waves in an ether — had no place for it. It was Augustin-Jean Fresnel who, having accepted (with Young) that the vibration must be transverse, worked out that two transverse components a quarter-cycle apart would spin, and identified circular and elliptical light as the natural family of a transverse wave. His prism of glass — the Fresnel rhomb — used two total-internal reflections to impose the π/2 phase shift and produce circular light without any crystal at all, proving the idea in glass. Circular polarization is the state a wave theory must allow and a particle-of-light-in-ether theory cannot — its very existence is a fingerprint of transversality.
Origins · 1 mincircular polarization and how this was discovered
The classification of polarized light into linear, circular, and elliptical states is the achievement of Augustin-Jean Fresnel (1788–1827), working from the transverse-wave hypothesis he and Thomas Young had reached independently around 1817. Fresnel realized that a transverse wave has two independent components, and that a quarter-cycle phase between equal components makes the field spin — circular polarization. He demonstrated it around 1823 with the Fresnel rhomb, a glass prism whose two total internal reflections each add a 45° phase shift, converting linear light to circular with no birefringent crystal. The insight closed a decade of confusion: circular light cannot exist for a longitudinal wave, so its reality was direct evidence that light's vibrations are perpendicular to its travel. The same geometry underlies today's σ± atomic selection rules, where the photon's spin ±ħ is the quantum echo of Fresnel's spinning field.
Going further
Texts. Hecht, Optics, 5th ed., §8.1 (the nature of polarized light) and §8.4 (retarders) · Saleh & Teich, Fundamentals of Photonics, 3rd ed., ch. 6 (polarization optics) · Loudon, The Quantum Theory of Light, 3rd ed. (photon polarization and angular momentum) · Foot, Atomic Physics, §7 (σ± transitions and optical pumping) · Born & Wolf, Principles of Optics, §1.4.
Provenance. The field decomposition, Jones vectors, and Stokes values for circular light follow Hecht §8.1/§8.4 and Saleh & Teich ch. 6; the spin-per-photon result and its Beth-experiment confirmation follow Loudon and Foot §7. This upgrade fixes an internal-convention inconsistency in the migrated base, which labelled the e^{−iωt} Jones vector (1, −i)/√2 as "right-circular"; under a consistent e^{−iωt} convention that vector is left-circular, so the article uses (1, i)/√2 for right-circular and flags Hecht's opposite (e^{+iωt}) table sign explicitly (see the warning). The quartz thickness d = λ/(4Δn) = 15.0 µm uses Δn = 0.00915 at 550 nm; the spin-flux example uses h = 6.626×10⁻³⁴ J·s, c = 2.998×10⁸ m/s, λ = 633 nm. Every equation and number is recomputed in verify_circular-polarization.py.
Builds on: Polarization of Light · Linear Polarization · Elliptical Polarization. Leads to: Jones Calculus · Stokes Parameters and the Poincaré Sphere · Birefringence · Optical Activity.
Is circular polarization "really" two linear waves, or is that just a mathematical trick?
Both descriptions are exactly equivalent, and which is "real" depends on what you do next. The decomposition into two orthogonal linear components a quarter-cycle apart is complete and correct — it reproduces every measurable property. But you could equally build the same field from two circular components, or measure it directly with a circular analyzer, in which case the single circular amplitude is the natural object. Polarization states form a two-dimensional space; linear and circular are just two convenient bases for it, like Cartesian versus polar coordinates. The field itself — a vector of fixed length spinning at ω — is what is real; the basis is your bookkeeping choice.
How can I tell circularly polarized light from unpolarized light? Both give half through a polarizer.
A single linear polarizer genuinely cannot distinguish them: both transmit I₀/2 at every orientation. The trick is to add a quarter-wave plate before the polarizer. Circular light has a fixed, coherent quarter-cycle phase between its components; the plate converts that phase into a definite linear direction, which then shows full modulation (0 to maximum) as you rotate the polarizer. Unpolarized light has no fixed phase for the plate to work on, so it still reads a flat I₀/2. So: flat through a polarizer alone but fully modulating once a wave plate is inserted means circular; flat in both cases means unpolarized.
Why does the handedness "flip" when I look it up in different books?
Because three independent conventions all affect the label and different fields fix them differently. The optics tradition names handedness from the receiver's point of view (looking back toward the source); the physics/quantum tradition names it by the photon's spin along the propagation direction; and the engineering/IEEE tradition uses yet another viewpoint. On top of that, the sign of i in the Jones vector depends on whether you write phasors as e^{−iωt} or e^{+iωt}. The spinning field in space is unambiguous — what differs is only the name. Always state which convention you use, and check that a source's real-field expression, not just its label, matches yours.
Does a photon of circular light literally spin?
It carries a well-defined spin angular momentum of ±ħ along its direction of travel, which is a genuine, measurable mechanical quantity — Beth demonstrated it in 1936 by letting circular light exert a torque on a suspended wave plate, and optical tweezers spin micron particles with it today. Whether the photon "spins" like a top is a picture, not a claim; what is rigorous is that a circularly polarized mode is an eigenstate of the angular-momentum operator along k with eigenvalue ±ħ. Absorbing one photon changes an atom's magnetic quantum number by exactly ±1 (the σ± selection rule), which is angular-momentum conservation made visible.
Why is a quarter-wave plate so thin — only about 15 microns?
Because the two crystal axes differ in refractive index by only a small amount (Δn ≈ 0.009 for quartz), and you only need to accumulate a quarter-wave of relative delay. That delay builds up as Δn times the number of wavelengths of path, so d = λ/(4Δn) works out to about 15 µm for green light — roughly 27 wavelengths of glass. A slab that thin is fragile, so a "true zero-order" plate is usually bonded to a thick support window, or replaced by a multi-order plate that adds a whole number of extra full waves (much thicker, but far more sensitive to wavelength and temperature). The thin zero-order plate is the one that stays a quarter-wave over a broad band.
If I add a right- and a left-circular beam, what do I get?
Two equal-amplitude, coherent circular beams of opposite handedness add up to linear polarization. Adding (1, i)/√2 and (1, −i)/√2 gives (2, 0)/√2 ∝ (1, 0) — light linearly polarized along x. Shifting the relative phase of the two circular components rotates the resulting linear direction, which is exactly how optical rotation and Faraday rotation work: a medium that gives the two circular states slightly different speeds re-phases them, and the linear polarization plane they add up to turns. So linear light is "made of" two circular states, just as circular light is "made of" two linear ones.
Can circular polarization exist for sound or other longitudinal waves?
No, and that impossibility is historically decisive. Circular polarization requires two independent transverse directions for the field to trade between; a longitudinal wave (like ordinary sound in a fluid) oscillates only along its direction of travel and has no transverse plane to spin in. When circular and elliptical light were confirmed in the 1820s, they proved that light's vibration must be transverse — the death blow to the longitudinal-ether picture. (Transverse elastic waves in solids, such as shear seismic waves, can be circularly polarized, precisely because they do have two transverse directions.)