Polarization

What is Polarization?

Light is an electromagnetic wave consisting of oscillating electric and magnetic fields perpendicular to each other and to the direction of propagation. Polarization describes the orientation and behavior of the electric field as the wave propagates. Unpolarized light, like sunlight, has electric field oscillations in all directions perpendicular to the propagation direction. Polarized light has electric field oscillations confined to specific directions or following specific patterns.

Light from most sources is unpolarized because atoms emit light randomly with no coordination between photons. Each photon has random polarization. However, when light reflects or transmits through certain materials, or passes through optical elements, it can become polarized. Polarization is a fundamental property of all electromagnetic waves—radio waves, microwaves, and X-rays all can be polarized.

The three main types of polarization describe the electric field's behavior. Linear polarization occurs when the electric field oscillates in one fixed direction. The field vector traces a line perpendicular to the propagation direction. Circular polarization occurs when the electric field magnitude remains constant but rotates, with the field vector tracing a circle in the plane perpendicular to propagation. Elliptical polarization is intermediate—the field vector traces an ellipse. These categories encompass all possible polarization states.

Polarization is not immediately visible to human eyes because our visual system doesn't distinguish polarization. We see only light intensity. Yet polarization is detectable with polarization-sensitive instruments. Cameras with polarizing filters show dramatic differences—sky appears darker when viewed through a polarizer because scattered sunlight is partially polarized. Water surfaces can be viewed without glare by using polarizers that block reflected light, which is strongly polarized.

Types of Polarization

Linear polarization is the simplest and most common type. The electric field oscillates in one fixed direction as the wave propagates. Mathematically, if the wave propagates in the z-direction, the electric field might oscillate only in the x-direction: E = E₀cos(kz - ωt)x̂. All the field's energy is in this x-direction. A polarizer transmitting only x-polarized light acts as a "filter" that blocks other polarizations.

Circular polarization occurs when electric field components in two perpendicular directions have equal magnitude but are out of phase by 90 degrees (quarter wavelength). The result is that the electric field vector rotates as the wave propagates, tracing a circular path. Right-handed circular polarization has the field rotating clockwise (when looking toward the incoming light); left-handed circular polarization rotates counterclockwise. Circular polarization is important in quantum mechanics—photons have helicity (handedness) related to circular polarization.

Elliptical polarization is the general case where electric field components in perpendicular directions have unequal magnitudes and are out of phase by something other than 0 or 90 degrees. The electric field vector traces an ellipse as the wave propagates. Any polarization state can be decomposed into two orthogonal linear polarizations with different amplitudes and phases. This decomposition is mathematically convenient for analysis.

The Stokes parameters provide a mathematical description of any polarization state using four numbers. For linear polarization, one Stokes parameter is non-zero. For circular polarization, a different Stokes parameter is non-zero. For arbitrary elliptical polarization, multiple Stokes parameters are non-zero. Stokes parameters enable precise measurement and manipulation of polarization states experimentally.

Brewster's Angle and Polarization by Reflection

Light reflected from surfaces exhibits polarization-dependent behavior. At most angles of incidence, both s-polarized light (perpendicular to the plane of incidence) and p-polarized light (parallel to the plane of incidence) are partially reflected. But at Brewster's angle, p-polarized light is not reflected at all—only s-polarized light reflects. The reflected light is therefore purely s-polarized (perpendicular to the plane of incidence).

Brewster's angle depends on the refractive indices of the two media: θB = arctan(n₂/n₁). For light traveling from air (n = 1.0) toward glass (n = 1.5), Brewster's angle is arctan(1.5) ≈ 56°. At this angle, the reflected and refracted rays are perpendicular to each other. This geometric condition causes p-polarized light (in the plane containing both rays) to have a zero reflection coefficient. Physically, oscillating charges in the glass are excited by the incident light's electric field. These charges radiate light that interferes destructively with the incident p-polarized light's reflection, canceling it.

Brewster's angle has practical applications. Photographers use Brewster's angle to photograph water surfaces without glare. Water-reflected light is partially p-polarized at Brewster's angle (about 53° for water). Using a camera with a polarizing filter aligned to block reflected p-polarized light eliminates glare. Laser systems use Brewster's angle windows to minimize reflection losses. Windows at Brewster's angle for the laser's wavelength and the material have no reflection for the desired polarization, improving laser efficiency.

The Mathematics of Polarization

Polarizers and Optical Elements

A polarizer is an optical element that transmits light of one polarization and blocks other polarizations. The most common polarizer is a polaroid filter, containing long-chain polymer molecules (like polyvinyl alcohol) doped with iodine, which are stretched and aligned. Light whose electric field oscillates parallel to the chains is strongly absorbed; light perpendicular to the chains transmits.

When unpolarized light passes through a single polarizer, the transmitted light is linearly polarized. The transmitted intensity is half the incident intensity (on average, unpolarized light has equal intensity in all perpendicular directions; the polarizer transmits light in only one direction). When this polarized light passes through a second polarizer (an analyzer), the transmitted intensity depends on the angle θ between the two polarizers' transmission axes: I = I₀cos²(θ). This is Malus's law. With crossed polarizers (θ = 90°), no light transmits.

Wave plates are optical elements that change polarization by introducing phase differences between perpendicular polarization components. A quarter-wave plate (QWP) introduces a 90-degree phase difference. Linear polarized light entering a QWP at 45 degrees to the plate's fast axis emerges as circularly polarized light. A half-wave plate (HWP) introduces a 180-degree phase difference and rotates linear polarization without changing its type.

Circular polarizers, used in 3D cinema and LCD screens, combine linear polarizers and wave plates. They convert unpolarized light into circularly polarized light or separate circularly polarized lights of different handedness. This enables 3D technologies where left and right eyes see different circularly polarized images. The human eye is insensitive to polarization, so the two images reach each eye's photoreceptors without the brain automatically combining them as a single 2D image.

LCD Screens and 3D Cinema

Liquid crystal displays (LCDs) use polarization and liquid crystals to create images. An LCD pixel consists of liquid crystal material sandwiched between two polarizers (polarizer plates) with perpendicular transmission axes. Backlight provides unpolarized light. This light passes through the first polarizer, becoming linearly polarized. It then passes through the liquid crystal layer, which can rotate the polarization, and finally through the second polarizer.

Liquid crystals are organic molecules that respond to electric fields by changing orientation. When oriented, they act as "wave plates," rotating the light's polarization. By applying voltage across the liquid crystal layer, you control its orientation and how much the polarization rotates. If the liquid crystal rotates the polarization by 90 degrees (quarter rotation), light passing through reaches the second polarizer aligned to transmit it—the pixel appears bright. If no voltage is applied, the liquid crystal doesn't rotate the polarization; light is blocked by the crossed second polarizer—the pixel appears dark.

Color LCDs use subpixels—tiny red, green, and blue pixels placed side by side. Each subpixel has a color filter. The eye blends these colors, perceiving the complete spectrum of colors. Modern LCDs use this principle to display millions of colors. The brightness and color at each pixel is independently controlled. This allows displaying full-color images, videos, and text.

3D cinema uses polarization to send different images to each eye. In polarization-based 3D systems, the left and right projectors emit left- and right-circularly polarized light respectively. Viewers wear 3D glasses with polarizing lenses: the left lens transmits left-circular polarization, the right lens transmits right-circular polarization. Each eye sees only its intended image. The brain combines these two images, creating the perception of depth. This technology enables immersive 3D movies and experiences without active shuttering.