Polarization

Light is a transverse wave: its electric field oscillates perpendicular to the direction of travel, and polarization describes how that oscillation is oriented in the transverse plane. If the field swings back and forth along one fixed line, the light is linearly polarized; if its tip traces a circle or an ellipse once per optical cycle, the light is circularly or elliptically polarized; if the orientation wanders randomly on a time scale shorter than the measurement, as in sunlight, the light is unpolarized. Many lasers emit linearly polarized light, often with a power ratio between the two orthogonal components of 100:1 or more, while thermal sources are close to unpolarized. An ideal linear polarizer transmits 50% of unpolarized light, and a glass surface of index 1.5 reflects no p-polarized light at all at its 56.3° Brewster angle.

Description

For a plane wave travelling along $z$, any fully polarized state can be written as two orthogonal components with a relative phase $\delta$:

$$E_x = E_{0x}\cos(kz - \omega t),$$ $$E_y = E_{0y}\cos(kz - \omega t + \delta).$$

With $\delta = 0$ or $\pi$ the light is linear; with $E_{0x} = E_{0y}$ and $\delta = \pm\pi/2$ it is circular; everything else is elliptical. Classification and handedness conventions are covered under polarization states. Two formalisms turn this into calculation. Jones vectors describe fully polarized, coherent light as a two-component complex vector, and each optical element as a 2 × 2 matrix. Stokes parameters describe partially polarized light with four measurable intensities, $S_0$ to $S_3$, acted on by 4 × 4 Mueller matrices; the degree of polarization is

$$\mathrm{DOP} = \frac{\sqrt{S_1^2 + S_2^2 + S_3^2}}{S_0},$$

1 for fully polarized light, such as a linearly polarized laser beam, and 0 for unpolarized light.

Controlling polarization

Most polarization components rely on one of four physical effects. Selective absorption or reflection of one field direction gives film and wire-grid polarizers, and birefringent crystal prisms separate the two directions by total internal reflection; extinction ratios range from about 10²:1 to 10⁵:1 or better depending on type and wavelength. Transmission through a linear polarizer follows Malus's law, $I = I_0\cos^2\theta$: two ideal polarizers at 30° pass 75% of the light leaving the first. A difference in refractive index between two field directions, birefringence, delays one component relative to the other; a waveplate uses this to convert between linear, circular and elliptical states. Crystal quartz has a birefringence of about 0.0091 at 633 nm, so a zero-order quarter-wave plate for that wavelength is about 17.4 µm thick. Reflection at a dielectric surface treats the s and p components differently, as described by the Fresnel equations; at the Brewster angle the p reflectance vanishes, while s-polarized light at the same 56.3° on glass of index 1.5 is reflected at 14.8%, compared with 4.0% for either polarization at normal incidence. The fourth, magneto-optic rotation in the Faraday effect, is non-reciprocal and is the basis of optical isolators.

Measurement

A rotating polarizer before a power meter gives the orientation and the maximum-to-minimum power ratio. A full measurement of the Stokes parameters, a polarimeter, adds a quarter-wave plate or a set of fixed analyzers to separate circular from unpolarized light, which a polarizer alone cannot distinguish. In fiber and integrated optics the quantities of interest are the polarization extinction ratio of a source or a PM fiber, and the polarization-dependent loss of a component; the article on measuring PER and PDL gives the procedures.

Where it matters

Laser cavities with Brewster windows, and semiconductor lasers whose gain favors the TE mode, emit linearly polarized light. Waveguides on a chip support distinct TE and TM modes with different effective indices, so photonic circuits are usually designed for one polarization and handle the other with a splitter-rotator. In standard single-mode fiber the state drifts randomly along the length, and the differential group delay between the two principal states produces polarization mode dispersion; polarization-maintaining fiber holds a linear state by building in strong birefringence. Coherent transceivers carry independent data on both polarizations. Other uses include LCD panels, stress analysis, ellipsometry of thin films and polarization-resolved microscopy.

Pitfalls

In fiber setups, moving a patch cord changes the polarization state, and any polarization-dependent element downstream converts that into power drift. Beamsplitters and metal mirrors at 45° change both the amplitude ratio and the phase between s and p, so a linear beam that is not purely s- or p-polarized at each fold can arrive elliptical after a few folds.

Common questions

Is unpolarized light the same as circularly polarized light?

No. Both give constant transmission as a linear polarizer is rotated, but circular light becomes linear after a quarter-wave plate and can then be extinguished, while unpolarized light cannot. The Stokes parameter $S_3$ separates them.

How much light passes through three polarizers?

Starting from unpolarized light, ideal polarizers at 0°, 45° and 90° pass 50% × 50% × 50%, or 12.5%, even though the first and last alone pass nothing.

Why do many lasers emit polarized light?

Any element in the cavity with polarization-dependent loss or gain, such as a Brewster window, a crystal gain medium, or the TE-favoring gain of a quantum-well diode, lets one polarization reach threshold first, and it then dominates.

The Jones calculus calculator follows a chosen polarization state through polarizers, wave plates, retarders and rotators.

References: E. Hecht, Optics, 5th ed. (Pearson, 2017), Ch. 8; M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999), Ch. 1 and 15; B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley, 2019).