Circular polarization

Light is circularly polarized when its electric field vector, observed at a fixed point, has constant length and rotates through a full circle once per optical period: $2.1$ fs for 633 nm light. It is the special case of elliptical polarization in which the two orthogonal field components have equal amplitude and a phase difference of exactly 90°. If the amplitudes differ, or the phase differs from 90°, the tip of the field traces an ellipse instead; linear polarization is the other limit, at 0° or 180°. The general classification is covered under polarization states.

Field, Jones vector and Stokes parameter

For a wave travelling along $z$, circular polarization has components

$$E_x = E_0\cos(kz-\omega t)$$ $$E_y = \pm E_0\sin(kz-\omega t)$$

with the upper sign for right-handed light in the convention this site follows (below). The corresponding normalized Jones vectors are $(1, \mp i)/\sqrt{2}$ in the $e^{i(kz - \omega t)}$ convention the Jones vectors entry uses, so right circular is $(1, -i)/\sqrt{2}$. In Stokes parameters, fully circular light has $S_1 = S_2 = 0$ and $S_3 = \pm S_0$, positive for right-handed light; $S_3$ is often written $V$. A polarimeter reports the degree of circular polarization $S_3/S_0$, which is 1 or $-1$ for pure circular light and 0 for linear or unpolarized light.

Handedness conventions

Two opposite conventions are in use, and papers do not always state which they follow. In the traditional optics convention of Born & Wolf and Hecht, which this site follows, light is right-circularly polarized when the field rotates clockwise as seen by an observer looking toward the source. In the IEEE convention used in radio and antenna engineering, right-hand circular polarization (RHCP) is defined by the right-hand rule with the thumb along the direction of propagation, which is counter-clockwise for the same observer. A further sign ambiguity comes from writing the wave as $e^{i(kz-\omega t)}$ or $e^{i(\omega t-kz)}$, which swaps which Jones vector is which. Specifying the rotation sense relative to the propagation direction avoids the problem.

Making and analysing circular light

A quarter-wave waveplate delays one linear component by 90° relative to the other. Linearly polarized light entering at 45° to the fast axis splits into two equal components, and the quarter-wave retardance turns them into circular light; rotating the input to −45° reverses the handedness.

Errors show up as ellipticity. If the input sits at 40° to the fast axis instead of 45°, the output ellipse has an axial ratio of $\tan 40^\circ = 0.84$ and $S_3/S_0 = \sin 80^\circ = 0.985$. A zero-order quarter-wave plate designed for 633 nm and used at 532 nm has about 0.297 waves of retardance, ignoring dispersion of the birefringence, which gives $S_3/S_0 = 0.956$ and an axial ratio of 0.74.

To measure handedness, pass the light through a quarter-wave plate, which converts it back to linear polarization at ±45° to the plate axes, then through a polarizer: which of the two orientations transmits reveals the handedness.

Angular momentum

A circularly polarized photon carries spin angular momentum $\pm\hbar$ along its direction of travel. A beam of power $P$ and angular frequency $\omega$ carries angular momentum flux $P/\omega$. For 1 W at 1064 nm, $\omega = 1.77\times10^{15}$ rad/s, so a plate that absorbs the beam, or converts it to linear polarization, experiences a torque of $5.6\times10^{-16}$ N·m. Beth measured this torque with a suspended waveplate in 1936.

Where it is used

• 3D cinema. Left and right images are projected in opposite circular states and separated by circular-polarizer glasses. Unlike linear systems, separation does not degrade when the viewer tilts the head.
• Photography. A camera "circular polarizer" filter is a linear polarizer followed by a quarter-wave plate. The linear element removes glare; the waveplate turns the output circular, so polarization-sensitive autofocus and metering beamsplitters behind the lens still work.
• Glare and back-reflection suppression. Normal-incidence reflection from a mirror reverses the handedness relative to the propagation direction, so light passing a linear polarizer and quarter-wave plate twice returns crossed and is blocked. Displays use this against ambient reflections, and optical setups use it as a simple, wavelength-sensitive substitute for a Faraday optical isolator.
• Chirality. Chiral molecules absorb left and right circular light differently, which is circular dichroism, and rotate linear polarization through circular birefringence (optical activity). The Faraday effect is magnetically induced circular birefringence.
• Satellite and radio links. GPS transmits RHCP, and many satellite links use circular polarization so that reception does not depend on antenna orientation or on Faraday rotation in the ionosphere.

Common questions

What is the difference between circular and elliptical polarization?

In both, the field tip rotates. In circular polarization the two orthogonal components have equal amplitude and a 90° phase difference, so the field length stays constant. Any other combination of amplitude and phase, apart from 0° and 180°, gives an ellipse.

Is circularly polarized light unpolarized?

No. Both pass a linear polarizer at any angle with 50 % transmission, but circular light is fully polarized: a quarter-wave plate followed by a polarizer can transmit all of it or block all of it, which is impossible for unpolarized light.

The Jones calculus calculator shows how a quarter-wave plate turns linear light into circular light of either handedness, using the same convention as this entry.

References: M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999), Sec. 1.4; E. Hecht, Optics, 5th ed. (Pearson, 2017), Ch. 8; B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley, 2019), Ch. 6; R. A. Beth, Physical Review 50, 115 (1936).