10.4 Polarization

Polarization describes the orientation of oscillations in a light wave. Because light is a transverse electromagnetic wave, its electric field can oscillate in different directions perpendicular to the direction it travels. Understanding polarization explains everything from glare on a lake to how LCD screens form images, and it connects wave optics to deeper ideas in electromagnetism and even quantum mechanics.

Nature of polarization

Light from most sources (the sun, a lightbulb) is unpolarized, meaning the electric field oscillates in random directions that are all perpendicular to the wave's travel. Polarized light, by contrast, has its electric field oscillating in a predictable pattern.

Types of polarization

• Linear polarization: The electric field oscillates back and forth in a single plane. This is the simplest and most common type you'll work with.
• Circular polarization: The electric field vector rotates as the wave propagates, tracing out a helix. The magnitude stays constant, but the direction continuously changes.
• Elliptical polarization: A more general case where the electric field traces an ellipse. Linear and circular polarization are actually special cases of elliptical polarization.
• Unpolarized light: A mix of waves oscillating in all possible planes with no preferred direction.

Polarization vs unpolarized light

Natural light sources produce unpolarized light because they consist of huge numbers of atoms emitting independently, each with a random polarization direction. To get polarized light, you need to filter or transform unpolarized light using one of several methods: passing it through a polarizer, reflecting it off a surface at the right angle, or scattering it (which is why the sky is partially polarized).

The key practical difference: polarized light interacts with polarizing filters in predictable, angle-dependent ways, while unpolarized light does not.

Malus's law

When linearly polarized light passes through a second polarizer (often called an analyzer), the transmitted intensity depends on the angle between the light's polarization direction and the polarizer's transmission axis:

$I = I_0 \cos^2 \theta$

• $I_0$ is the intensity of the incoming polarized light
• $\theta$ is the angle between the polarization direction and the transmission axis

When $\theta = 0°$, all the light passes through ($I = I_0$). When $\theta = 90°$, no light passes through ($I = 0$). This is called complete extinction and is the basis for "crossed polarizers."

A useful detail: when unpolarized light hits a single ideal polarizer, the transmitted intensity is always $\frac{I_0}{2}$, regardless of the polarizer's orientation. Malus's law only applies once the light is already polarized.

Polarization techniques

Linear polarizers

A linear polarizer transmits only the component of light oscillating along its transmission axis and blocks the perpendicular component. Two common types:

• Dichroic polarizers (like Polaroid film) absorb one polarization component while transmitting the other. These are the most common in everyday applications.
• Wire-grid polarizers use closely spaced parallel conducting wires. Waves with their electric field parallel to the wires are absorbed or reflected; waves perpendicular to the wires pass through.

Placing two linear polarizers in series with their axes at an adjustable angle lets you continuously vary the transmitted intensity, which is a direct application of Malus's law.

Circular polarizers

A circular polarizer is built from two elements in sequence:

1. A linear polarizer selects one polarization direction.
2. A quarter-wave plate (oriented at 45° to the linear polarizer's axis) introduces a 90° phase shift between the two orthogonal components of the now-linearly-polarized light.

The result is circularly polarized light, where the electric field vector rotates either clockwise (right-handed) or counterclockwise (left-handed) depending on the orientation. Circular polarizers are used in camera filters to reduce reflections without interfering with autofocus systems, and in 3D glasses.

Polarization by reflection

When unpolarized light reflects off a non-metallic surface (water, glass, a road), the reflected light becomes partially polarized. The reflected component is preferentially polarized parallel to the surface. This is why glare from a lake or road is mostly horizontally polarized, and why polarized sunglasses with a vertical transmission axis cut that glare so effectively.

The transmitted (refracted) light becomes partially polarized in the perpendicular direction.

Brewster's angle

At one specific angle of incidence, the reflected light becomes completely polarized. This is Brewster's angle:

$\tan \theta_B = \frac{n_2}{n_1}$

• $n_1$ is the refractive index of the medium the light is coming from
• $n_2$ is the refractive index of the medium it's hitting

At Brewster's angle, the reflected and refracted rays are exactly 90° apart. For an air-glass interface ($n_2 \approx 1.5$), Brewster's angle is about 56.3°.

This principle is used in designing anti-reflection coatings and polarizing prisms (like Brewster windows in lasers).

Birefringence

Some crystalline materials have a different refractive index depending on the polarization direction of the light passing through them. This property is called birefringence (or double refraction).

Double refraction

When unpolarized light enters a birefringent crystal (calcite is the classic example), it splits into two rays:

• The ordinary ray (o-ray) obeys Snell's law normally and experiences one refractive index.
• The extraordinary ray (e-ray) does not follow Snell's law in the usual way and experiences a different refractive index.

These two rays are polarized perpendicular to each other. If you place a piece of calcite on top of printed text, you'll see a double image because the two rays exit the crystal at slightly different positions.

Ordinary vs extraordinary rays

The distinction comes down to the crystal's optic axis, a special direction in the crystal where both rays travel at the same speed:

• The o-ray has a constant refractive index regardless of its propagation direction. Its polarization is perpendicular to the plane containing the optic axis and the wave vector.
• The e-ray has a refractive index that varies depending on the angle between its propagation direction and the optic axis. Its polarization lies in the plane of the optic axis and the wave vector.

Along the optic axis itself, there is no splitting, and the crystal behaves like an ordinary material.

Wave plates and retarders

Wave plates are thin slabs of birefringent material cut so that the two polarization components travel through different refractive indices, accumulating a controlled phase difference:

• Quarter-wave plate: Introduces a 90° ($\frac{\lambda}{4}$) phase shift. Converts linearly polarized light to circularly polarized light (or vice versa) when oriented at 45° to the incoming polarization.
• Half-wave plate: Introduces a 180° ($\frac{\lambda}{2}$) phase shift. Rotates the plane of linearly polarized light by twice the angle between the incoming polarization and the plate's fast axis.

These are essential components in laser systems, optical communication, and polarimetry.

Polarization in everyday life

Polarized sunglasses

Glare from horizontal surfaces (water, roads, snow) is predominantly horizontally polarized. Polarized sunglasses contain a vertically oriented polarizing filter that blocks this horizontal component, significantly reducing glare while still transmitting most of the useful, unpolarized ambient light.

A neat side effect: if you look at tempered glass (like a car's rear window) through polarized sunglasses, you can often see colorful stress patterns that are invisible to the naked eye.

LCD displays

Liquid crystal displays rely on polarization to control each pixel:

1. Backlight passes through a first polarizer.
2. The polarized light enters a liquid crystal layer whose molecular orientation can be changed by applying an electric field.
3. The liquid crystals rotate the polarization of the light by a controllable amount.
4. A second polarizer (the analyzer) either transmits or blocks the light depending on how much the polarization was rotated.

By varying the voltage across each pixel's liquid crystal layer, the display controls how much light gets through, producing the image you see.

Stress analysis

When a transparent material is under mechanical stress, it becomes temporarily birefringent. This phenomenon is called photoelasticity. If you place a stressed sample between crossed polarizers, the internal stress distribution shows up as colorful fringe patterns. Engineers use this technique to identify stress concentrations and weak points in structural components and prototypes.

Optical activity

Certain materials rotate the plane of polarization of linearly polarized light as it passes through them. This is called optical activity, and it arises from molecular chirality (asymmetry) in the material.

Rotation of plane-polarized light

• Dextrorotatory (+) substances rotate the polarization plane clockwise (as seen looking toward the light source).
• Levorotatory (-) substances rotate it counterclockwise.

The rotation angle depends on the substance itself, the path length through the sample, and (for solutions) the concentration. This is measured with an instrument called a polarimeter.

Chiral molecules

A chiral molecule cannot be superimposed on its mirror image, much like your left and right hands. Two mirror-image forms of the same molecule are called enantiomers, and they rotate polarized light by equal amounts in opposite directions.

Chirality is extremely common in biochemistry. Amino acids, sugars, and many drug molecules are chiral. In pharmaceuticals, one enantiomer of a drug may be therapeutic while the other is inactive or even harmful, which is why chiral separation is so important.

Specific rotation

To compare the optical activity of different substances on equal footing, chemists use specific rotation:

$[\alpha]_\lambda^T = \frac{\alpha}{l \cdot c}$

• $\alpha$ = observed rotation angle (in degrees)
• $l$ = path length in decimeters (dm)
• $c$ = concentration in g/mL

The superscript $T$ and subscript $\lambda$ indicate that specific rotation depends on temperature and the wavelength of light used. Specific rotation is a characteristic property of a substance, useful for identification and purity analysis.

Polarization in electromagnetic waves

Polarization isn't unique to visible light. It applies to all electromagnetic waves, from radio waves to X-rays. This has major practical implications for antenna design, radar, and satellite communication.

Transverse waves

Electromagnetic waves are transverse: the electric and magnetic fields oscillate perpendicular to the direction of propagation, and perpendicular to each other. Polarization specifically describes the direction of the electric field oscillations.

This is why sound waves (which are longitudinal) cannot be polarized. Only transverse waves have a meaningful polarization direction.

Polarization states

Any polarization state can be described as a combination of two orthogonal linear polarization components with specific amplitudes and a phase difference between them:

• Phase difference = 0° or 180°: linear polarization
• Phase difference = 90°, equal amplitudes: circular polarization
• Any other combination: elliptical polarization

Unpolarized light is a statistical mixture of all polarization states with no fixed phase relationship between them.

Jones vectors

For fully polarized light, the Jones vector provides a compact mathematical description. It's a two-component complex vector representing the amplitude and phase of the electric field in two orthogonal directions:

• Linear horizontal: $\begin{pmatrix} 1 \\ 0 \end{pmatrix}$
• Linear vertical: $\begin{pmatrix} 0 \\ 1 \end{pmatrix}$
• Linear at 45°: $\frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ 1 \end{pmatrix}$
• Right circular: $\frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ -i \end{pmatrix}$

Optical elements (polarizers, wave plates) are represented by 2×2 Jones matrices, and you find the output polarization by multiplying the Jones matrix by the input Jones vector. This formalism is powerful for analyzing chains of optical elements.

Applications of polarization

Polarimetry

Polarimetry measures how a sample rotates the plane of polarized light. The basic setup:

1. Pass light through a polarizer to produce linearly polarized light.
2. Send it through the sample.
3. Use a second polarizer (analyzer) to measure the rotation angle.

This technique is widely used to determine the concentration of sugar solutions (saccharimetry), to identify unknown optically active compounds, and for quality control in the pharmaceutical and food industries.

Optical communications

Modern fiber-optic networks use polarization in several ways:

• Polarization-maintaining fibers preserve the polarization state over long distances, which is critical for certain sensing and communication applications.
• Polarization-division multiplexing sends two independent data streams on orthogonal polarizations through the same fiber, effectively doubling capacity.
• Polarization-mode dispersion (PMD) compensation corrects for signal degradation caused by slight birefringence in standard optical fibers over long distances.

3D cinema technology

Most modern 3D cinemas use circular polarization. Two projectors (or one projector alternating rapidly) display the left-eye and right-eye images with opposite circular polarizations (one left-handed, one right-handed). Viewers wear glasses with matching circular polarizer filters so each eye sees only its intended image.

Circular polarization is preferred over linear because it still works if you tilt your head, unlike linear polarization systems where tilting causes crosstalk between the two images.

Quantum aspects of polarization

At the single-photon level, polarization takes on quantum mechanical properties that have no classical analog.

Photon polarization

Each individual photon has a polarization state. When a photon encounters a polarizer, the outcome is binary: it either passes through or it doesn't. The probability of passing is given by $\cos^2 \theta$ (Malus's law at the single-photon level), but for any individual photon, the result is all-or-nothing.

A photon can exist in a quantum superposition of polarization states. Only upon measurement does it "collapse" to a definite state. This property makes photon polarization a natural realization of a qubit in quantum information science.

Polarization entanglement

Two photons can be entangled so that their polarization states are correlated in a way that has no classical explanation. Measuring the polarization of one photon instantly determines the polarization of the other, regardless of the distance between them.

Entangled photon pairs are commonly produced through spontaneous parametric down-conversion, where a single high-energy photon splits into two lower-energy photons in a nonlinear crystal. Polarization entanglement is the foundation of quantum key distribution (quantum cryptography) and quantum teleportation protocols.

Bell's inequalities

Bell's inequalities provide a testable boundary: if the correlations between entangled photon measurements exceed a certain limit, then no classical "hidden variable" theory can explain the results.

The most commonly tested form is the CHSH inequality:

$|S| \leq 2$

for any classical system. Quantum mechanics predicts violations up to $|S| = 2\sqrt{2} \approx 2.83$, and experiments consistently confirm this quantum prediction. These results, first demonstrated by Alain Aspect in the 1980s, provide strong evidence that quantum mechanics is fundamentally non-local.