Polarization of Light

Polarization describes the orientation of the oscillations of a transverse wave. For light, polarization refers to the direction in which the electric field oscillates as the electromagnetic wave travels.

Ordinary light from sources such as the Sun or a lamp is usually unpolarized. Its electric field oscillates in many different directions perpendicular to the direction of travel.

Polarized light has a more restricted pattern of electric-field oscillation. In linearly polarized light, the electric field oscillates along one particular direction.

Polarization is important because it provides strong evidence that light behaves as a transverse wave and has many practical applications in sunglasses, photography, displays, microscopy, communication, and scientific measurement.

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Light as a Transverse Wave

Light is an electromagnetic wave consisting of oscillating electric and magnetic fields.

The electric field E and magnetic field B are:

• perpendicular to each other
• perpendicular to the direction of wave travel

Therefore, light is a transverse wave.

When discussing the polarization of light, we normally describe the orientation of the electric field.

For example, if light travels horizontally while its electric field oscillates vertically, we describe the light as vertically polarized.

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Transverse Waves

In a transverse wave, the oscillations are perpendicular to the direction in which the wave travels.

Imagine a wave travelling in the x-direction.

Its oscillations might occur:

• vertically
• horizontally
• diagonally
• in another direction perpendicular to x

Because there are different possible directions of oscillation, a transverse wave can be polarized.

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Why Longitudinal Waves Cannot Be Polarized

In a longitudinal wave, the oscillations occur parallel to the direction of wave travel.

A sound wave travelling through air is a common example.

The air molecules oscillate back and forth along the same direction in which the wave propagates.

There is no independent sideways direction of oscillation to select.

Therefore:

longitudinal waves cannot be polarized in the same way as transverse waves.

This makes polarization an important test of transverse wave behaviour.

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Polarization as Evidence for the Nature of Light

Historically, scientists debated whether light waves were longitudinal or transverse.

The observation that light could be polarized provided important evidence that light was transverse.

If light behaved as an ordinary longitudinal wave, linear polarization could not occur.

Therefore:

polarization provides evidence for the transverse nature of light.

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Unpolarized Light

Most everyday light sources produce approximately unpolarized light.

Examples include:

• the Sun
• incandescent lamps
• flames
• many LEDs

In unpolarized light, the electric field does not remain aligned along one fixed transverse direction.

Instead, its orientation changes rapidly and randomly.

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Linearly Polarized Light

In linearly polarized light, the electric field oscillates along one fixed line.

For example:

vertical polarization

means the electric field oscillates vertically.

horizontal polarization

means the electric field oscillates horizontally.

The light itself still travels forward.

Polarization describes the orientation of the electric field, not the direction of propagation.

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Comparing Polarized and Unpolarized Light

Both are electromagnetic waves and travel at the same speed in the same medium.

The difference is the organization of their electric-field oscillations.

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Polarizing Filters

One of the simplest ways to produce polarized light is with a polarizing filter, or polarizer.

A polarizer has a transmission axis.

It preferentially transmits the component of the electric field parallel to this axis.

The perpendicular component is strongly reduced.

Therefore:

unpolarized light → polarizer → linearly polarized light

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What Happens Inside a Polarizer?

Suppose unpolarized light reaches a vertical polarizer.

The incoming electric field contains components in many transverse directions.

The polarizer preferentially transmits the vertical component.

The emerging light is therefore predominantly:

vertically polarized

Rotating the polarizer changes the direction of polarization of the transmitted light.

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Polarization Does Not Mean the Light Stops Oscillating

A common misconception is that a polarizing filter somehow removes the wave's oscillation.

It does not.

The transmitted electric field continues oscillating.

The difference is that its oscillations are restricted to a particular orientation.

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Two Polarizing Filters

Now place a second polarizer after the first.

The first polarizer creates linearly polarized light.

The second polarizer is often called an analyzer.

The amount of light transmitted by the analyzer depends on the angle between:

• the polarization direction of the incoming light
• the transmission axis of the analyzer

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Parallel Polarizers

Suppose the first polarizer produces vertical polarization.

If the second polarizer is also vertical, their axes are parallel.

The second filter transmits much of the already polarized light.

Therefore:

parallel polarizers → high transmission

Some light may still be lost because real polarizers are not perfectly transparent.

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Crossed Polarizers

Now rotate the second polarizer by:

90°

Its transmission axis is perpendicular to the polarization direction of the incoming light.

Ideally:

transmitted intensity → 0

The polarizers are described as crossed.

This dramatic change in brightness demonstrates that polarization has a direction.

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Malus's Law

For linearly polarized light passing through an ideal analyzer:

I = I₀ cos² θ

where:

• I = transmitted intensity
• I₀ = intensity before the analyzer
• θ = angle between the incoming polarization direction and the analyzer's transmission axis

This relationship is called Malus's law.

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Worked Example: Malus's Law

Linearly polarized light has intensity:

I₀ = 100 W/m²

It reaches a polarizer whose transmission axis is at:

θ = 60°

Using:

I = I₀ cos² θ

I = 100 cos² 60°

Since:

cos 60° = 0.5

I = 100(0.5)²

I = 25 W/m²

Therefore:

I = 25 W/m²

Only 25% of the original polarized intensity is transmitted.

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Another Malus's Law Example

Suppose:

I₀ = 80 W/m²

θ = 45°

Then:

I = 80 cos² 45°

Since:

cos² 45° = 0.5

I = 40 W/m²

Therefore:

I = 40 W/m²

Half of the polarized intensity passes through the analyzer.

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Important Malus's Law Angles

This assumes ideal polarizers and already linearly polarized incident light.

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Unpolarized Light Through One Polarizer

For ideal unpolarized light passing through one ideal polarizer, the average transmitted intensity is:

I = I₀/2

Therefore, an ideal polarizer transmits approximately half of the incident unpolarized intensity.

The emerging light is linearly polarized.

This result comes from averaging the different incoming polarization orientations.

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Worked Example: Unpolarized Light

Unpolarized light has intensity:

I₀ = 200 W/m²

After one ideal polarizer:

I₁ = I₀/2

I₁ = 100 W/m²

Now it passes through a second polarizer at 60°.

I₂ = I₁ cos² 60°

I₂ = 100(0.25)

I₂ = 25 W/m²

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Three Polarizers

An interesting result occurs when a third polarizer is inserted between two crossed polarizers.

Suppose the first and third polarizers are:

90° apart

Normally almost no light passes through both.

Now place another polarizer between them at:

45°

After the first polarizer, the light is polarized in one direction.

The middle polarizer transmits a component at 45° and establishes a new polarization direction.

The final polarizer can then transmit a component of this new polarization.

Therefore, inserting another polarizer can actually increase the amount of light emerging from two crossed polarizers.

This is a striking demonstration of polarization.

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Worked Three-Polarizer Example

Suppose unpolarized light has intensity:

I₀

After the first polarizer:

I₁ = I₀/2

The middle polarizer is at 45°:

I₂ = I₁ cos²45°

I₂ = I₀/4

The third polarizer is another 45° from the middle one:

I₃ = I₂ cos²45°

I₃ = I₀/8

Therefore, one eighth of the original idealized intensity emerges.

Without the middle polarizer, the first and last are crossed and ideally transmit no light.

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Polarization by Reflection

Light can also become partially polarized when it reflects from a surface.

Reflected light from:

• water
• glass
• roads
• snow
• other non-metallic surfaces

can contain a strong polarization preference.

This effect is particularly important in glare.

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Glare from Horizontal Surfaces

Light reflected from a horizontal surface such as water or a road tends to become strongly horizontally polarized under suitable conditions.

This produces intense glare.

Polarized sunglasses are designed to reduce this glare.

Their transmission axis is usually approximately vertical.

Therefore, they strongly reduce the horizontally polarized component.

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Polarized Sunglasses

Polarized sunglasses can be especially useful around:

• roads
• lakes
• oceans
• snow
• wet surfaces

They reduce reflected glare while allowing much of the differently polarized light from the surrounding scene to pass.

This can improve:

• visibility
• contrast
• visual comfort

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Brewster's Angle

At a particular incidence angle called Brewster's angle, reflected light from a dielectric boundary can become strongly linearly polarized.

Brewster's law is:

tan θ_B = n₂/n₁

where:

• θ_B = Brewster's angle
• n₁ = refractive index of incident medium
• n₂ = refractive index of second medium

At this angle, the reflected and refracted rays are perpendicular to one another.

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Worked Example: Brewster's Angle

Light travels from air into glass.

Suppose:

n₁ = 1.00

n₂ = 1.50

Then:

tan θ_B = 1.50/1.00

tan θ_B = 1.50

Therefore:

θ_B ≈ 56.3°

At approximately 56°, the reflected light is strongly polarized.

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Polarization by Scattering

Light can also become partially polarized through scattering.

Sunlight interacting with molecules in Earth's atmosphere is scattered in different directions.

The scattered light can have a preferred polarization direction.

Therefore, different regions of the sky contain partially polarized light.

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Polarization of Skylight

The polarization pattern of the sky depends on the position of the Sun and the viewing direction.

This polarization is normally invisible to human vision.

However, some animals can detect it and use it as a navigational cue.

Polarizing filters can also reveal changes in the appearance of the sky.

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Polarization in Nature

Polarization occurs naturally in several ways.

Examples include:

• reflected sunlight from water
• reflected light from leaves and other surfaces
• atmospheric scattering
• light interacting with biological structures
• reflections from wet surfaces

Some animals can detect polarization patterns that humans cannot normally perceive directly.

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Animal Navigation

Certain animals can use polarized skylight as part of their navigation systems.

Examples occur among some:

• insects
• aquatic animals
• birds

Polarization can provide directional information even when the Sun itself is not directly visible.

This shows that polarization is not only a laboratory phenomenon—it can carry useful information in natural environments.

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Polarization in Photography

Photographers use polarizing filters to control reflections and improve image appearance.

A camera polarizer can sometimes:

• reduce reflections from water
• reduce reflections from glass
• reduce glare from foliage
• increase apparent sky contrast
• improve colour saturation

The effect depends strongly on viewing and illumination geometry.

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Looking Through Water

Reflected glare from the surface of water can hide objects below the surface.

A polarizing filter can reduce part of this reflected light.

As a result, a photographer or observer may be able to see more clearly into the water.

This is the same basic principle used by polarized sunglasses.

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Polarization and LCD Screens

Liquid crystal displays (LCDs) rely heavily on polarization.

LCD devices contain polarizing layers and liquid-crystal material.

The liquid crystals can alter how the polarization of light passes through the display system.

Electrical signals control the liquid crystals.

This allows individual pixels to control the amount of transmitted light.

Applications include:

• monitors
• calculators
• instrument displays
• televisions
• some device screens

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Viewing an LCD Through a Polarizer

If you look at some LCD screens through polarized sunglasses and rotate your head, the screen may become:

• brighter
• darker
• nearly black

This occurs because the screen already contains polarized light.

Rotating the sunglasses changes the angle between the light's polarization and the sunglasses' transmission axis.

Malus's law then changes the transmitted intensity.

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Polarization in 3D Cinema

Some 3D cinema systems use polarization to send different images to the two eyes.

Two images are projected with different polarization states.

The glasses contain different polarizing filters for each eye.

Each lens preferentially transmits the image intended for that eye.

The brain combines the two different views to produce the perception of depth.

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Linear and Circular Polarization in 3D Systems

Some systems use different linear polarization directions.

Others use different circular polarization states.

Circular polarization can allow viewers to tilt their heads more freely without losing the separation between the two images.

The exact system depends on the projection technology.

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Circular Polarization

Not all polarized light is linearly polarized.

In circularly polarized light, the direction of the electric field rotates as the wave travels.

The magnitude of the electric field can remain constant while its direction rotates.

Circular polarization can be understood as the combination of two perpendicular oscillations with:

• equal amplitudes
• a quarter-cycle phase difference

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Elliptical Polarization

A more general form is elliptical polarization.

Here, the tip of the electric-field vector traces an ellipse.

Linear and circular polarization can be regarded as special cases of elliptical polarization.

At an introductory level, however, most polarization problems focus on linear polarization.

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Polarization and Wave Plates

A wave plate is an optical device that changes the relative phase of different polarization components.

Common examples include:

• quarter-wave plates
• half-wave plates

A quarter-wave plate can convert suitable linearly polarized light into circularly polarized light.

A half-wave plate can rotate the direction of linear polarization.

These devices are widely used in advanced optical systems.

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Stress Analysis

Some transparent materials become optically anisotropic when stressed.

When viewed between polarizers, different parts of the material can produce colourful patterns.

This technique is called photoelasticity.

It can reveal how mechanical stress is distributed through an object.

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Photoelasticity

Engineers can make transparent models of structures and place them between polarizers.

When forces are applied, stress changes the material's optical properties.

The resulting polarization effects create visible patterns.

These patterns can reveal regions of:

• high stress
• low stress
• stress concentration

This provides a striking connection between optics and mechanical engineering.

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Polarizing Microscopes

Polarization is also used in microscopy.

A polarizing microscope can examine materials whose optical behaviour depends on direction.

It is particularly useful for studying:

• crystals
• minerals
• geological samples
• polymers
• biological structures

Different materials can alter the polarization of transmitted light in characteristic ways.

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Mineral Identification

Many minerals interact with polarized light differently depending on:

• crystal orientation
• composition
• thickness

Geologists can examine thin rock sections between polarizers.

The resulting colours and brightness changes provide information that can help identify minerals and investigate rock structure.

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Polarization in Communication

Polarization can also carry information.

Different polarization states can sometimes be used as separate channels or as part of sophisticated communication and sensing systems.

Polarization control is important in:

• fiber optics
• satellite communication
• radar
• laser systems
• optical measurement

In such systems, uncontrolled polarization changes can also create problems.

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Polarization and Reflection

Polarization affects how strongly light reflects from surfaces.

Different polarization orientations can have different reflection coefficients.

This becomes particularly important at non-normal incidence.

At Brewster's angle, one polarization component has zero ideal reflection at a dielectric boundary.

This relationship is important in optical design.

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Polarization and Interference

Polarization can also affect interference.

For strong visible interference fringes, the electric-field components of the two waves must be able to combine.

If two coherent waves are polarized in the same direction, they can produce strong interference.

If they are polarized at 90° to one another, ordinary intensity interference fringes disappear.

Therefore:

coherence alone is not sufficient—the polarization relationship also matters.

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Why Perpendicular Polarizations Do Not Produce Ordinary Fringes

Suppose one wave is vertically polarized and another is horizontally polarized.

Their electric fields oscillate in perpendicular directions.

They do not reinforce and cancel one another in the same way as parallel field components.

As a result, the usual alternating bright and dark interference pattern is absent when intensity is measured without an additional polarization projection.

This is an important connection between polarization and interference.

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Polarization and Intensity

Polarization does not necessarily mean that light has low intensity.

A highly intense laser beam can be strongly polarized.

A weak beam can also be polarized.

Polarization describes the orientation behaviour of the electric field, while intensity describes the amount of energy transported per unit area per unit time.

These are different properties.

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Polarization and Colour

Polarization is also different from colour.

Colour is primarily associated with wavelength or frequency.

Polarization is associated with electric-field orientation.

Two beams can have:

• the same wavelength
• the same frequency
• the same colour

but different polarization states.

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Analyzing a Polarization Experiment

Suppose unpolarized light passes through a polarizer and then an analyzer.

To analyze the system:

Determine the initial intensity.

For unpolarized light through an ideal first polarizer:

I₁ = I₀/2

Determine the angle θ.

Measure the angle between the first and second transmission axes.

Apply Malus's law.

I₂ = I₁ cos² θ

This procedure can be repeated if additional polarizers are present.

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Complete Worked Example

Unpolarized light has intensity:

I₀ = 240 W/m²

It passes through a vertical polarizer.

After the first polarizer:

I₁ = 240/2

I₁ = 120 W/m²

It then passes through an analyzer at 30°.

Using:

I₂ = I₁ cos²30°

Since:

cos²30° = 0.75

I₂ = 120(0.75)

I₂ = 90 W/m²

Therefore, the final transmitted intensity is:

90 W/m²

for ideal polarizers.

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Common Mistakes

Thinking polarization describes the direction light travels

Polarization describes the orientation of the electric field, not the direction of propagation.

Thinking longitudinal waves can be polarized

Ordinary longitudinal waves oscillate only along their direction of propagation and cannot be linearly polarized in the same sense.

Thinking unpolarized light has no electric field

It has electric-field oscillations, but their transverse orientation varies rapidly.

Thinking a polarizer simply blocks half of every type of light

The I₀/2 result applies to ideal unpolarized light. Already polarized light follows Malus's law.

Using I = I₀ cos θ

Malus's law contains:

cos² θ

Using the angle to the direction of travel

θ is the angle between the polarization direction and the analyzer axis.

Thinking crossed polarizers always transmit light because each transmits half

After the first polarizer, the light is no longer unpolarized. An ideal perpendicular second polarizer transmits zero.

Thinking polarization changes the colour of light

Polarization and wavelength are different properties.

Thinking polarized sunglasses simply make everything darker

Their main advantage is selectively reducing strongly polarized glare.

Assuming all reflected light is completely polarized

Reflection usually produces partial polarization. Complete polarization occurs only under particular ideal conditions such as Brewster's angle.

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Key Terms

Polarization — description of the orientation behaviour of transverse wave oscillations.

Polarized light — light whose electric field has an organized polarization state.

Unpolarized light — light whose electric-field orientation varies randomly with time.

Linear polarization — polarization in which the electric field oscillates along a fixed line.

Circular polarization — polarization in which the electric-field direction rotates while the wave propagates.

Elliptical polarization — general polarization state in which the electric-field tip traces an ellipse.

Polarizer — device that produces or selects a polarization state.

Analyzer — polarizer used to investigate the polarization of incoming light.

Transmission axis — direction of electric-field oscillation preferentially transmitted by a linear polarizer.

Malus's law — relationship I = I₀ cos² θ describing transmission of linearly polarized light through an ideal analyzer.

Crossed polarizers — two linear polarizers whose transmission axes are perpendicular.

Brewster's angle — incidence angle at which reflected light from an ideal dielectric boundary is completely polarized in one linear orientation.

Brewster's law — relationship tan θ_B = n₂/n₁.

Polarization by reflection — production of partially or completely polarized light through reflection.

Polarization by scattering — polarization produced when light is scattered by particles or molecules.

Photoelasticity — use of polarized light to investigate stress in transparent materials.

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Key Takeaways

• Polarization describes the orientation of transverse wave oscillations.
• For light, polarization normally refers to the orientation of the electric field.
• Light is an electromagnetic transverse wave.
• The electric and magnetic fields are perpendicular to the direction of propagation.
• Only transverse waves can be polarized in the ordinary linear-polarization sense.
• The ability of light to be polarized provides evidence for its transverse nature.
• Ordinary light is often approximately unpolarized.
• Linearly polarized light has its electric field oscillating along a fixed direction.
• A polarizing filter can convert unpolarized light into linearly polarized light.
• Ideal unpolarized light loses half its intensity when passing through one ideal polarizer.
• Linearly polarized light passing through an analyzer follows I = I₀ cos² θ.
• Parallel polarizers produce maximum transmission.
• Crossed ideal polarizers produce zero transmission.
• Polarization can be produced by reflection and scattering as well as by polarizing filters.
• Reflected glare from horizontal surfaces is often strongly polarized.
• Polarized sunglasses use this effect to reduce glare.
• At Brewster's angle, reflected light from an ideal dielectric boundary is completely linearly polarized.
• Polarization occurs naturally in scattered skylight.
• Polarization is used in photography, LCDs, 3D cinema, microscopy, stress analysis, communication, and scientific instruments.
• Polarization can also affect whether coherent light waves produce visible interference fringes.

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Check Your Understanding

1. Define polarization in terms of the electric field of a light wave.
2. Explain why light can be polarized but an ordinary sound wave in air cannot.
3. Distinguish between unpolarized light and linearly polarized light.
4. Describe how a polarizing filter produces linearly polarized light from unpolarized light.
5. Unpolarized light has an intensity of 160 W/m². Calculate the intensity after it passes through one ideal polarizer.
6. Linearly polarized light with intensity 120 W/m² reaches an analyzer at 60° to its polarization direction. Calculate the transmitted intensity.
7. Two ideal polarizers have parallel transmission axes. Describe the transmission of already polarized light aligned with their axes.
8. Explain what happens when the second of two ideal polarizers is rotated until the polarizers are crossed.
9. Unpolarized light passes through a vertical polarizer and then through a second polarizer at 45°. What fraction of the original intensity emerges?
10. Explain how inserting a 45° polarizer between two crossed polarizers can cause some light to emerge.
11. Explain how polarization by reflection produces glare and how polarized sunglasses reduce it.
12. Calculate Brewster's angle for light travelling from air, n = 1.00, into a material with n = 1.40.
13. Explain how atmospheric scattering can produce partially polarized skylight and describe one way this polarization is useful in nature.
14. Describe two technological applications of polarization and explain how polarization is used in each.
15. Two coherent light beams have the same wavelength and frequency but are polarized at 90° to one another. Explain why they do not produce the usual bright and dark intensity interference fringes, and describe what this tells us about the role of polarization in interference.