1. Polarization of Light

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.

---

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.

---

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.

---

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.

---

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.

---

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.

---

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.

---

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.

---

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

---

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.

---

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.

---

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

---

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.

---

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.

---

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.

---

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.

---

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.

---

Important Malus's Law Angles

This assumes ideal polarizers and already linearly polarized incident light.

---

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.

---

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

---

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.

---

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.

---

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.

---

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.

---

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

---

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.

---

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.

---

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.

---

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.

---

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.

---

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.

---

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.

---

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.

---

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

---

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.

---

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.

---

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.

---

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

---

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.

---

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.

---

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.

---

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.

---

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.

---

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.

---

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.

---

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.

---

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.

---

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.

---

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.

---

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.

---

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.

---

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.

---

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.

---

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.

---

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.

---

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.

2. Polarizing Filters

Polarizing Filters

A polarizing filter, or polarizer, controls light according to the orientation of its electric field. It has a preferred direction called its transmission axis.

When unpolarized light passes through an ideal polarizing filter, the emerging light becomes linearly polarized along the filter's transmission axis.

Polarizing filters allow us to investigate one of the most important properties of light: light is a transverse electromagnetic wave.

They also have many practical uses, including:

• polarized sunglasses
• photography
• LCD screens
• scientific instruments
• stress analysis
• optical experiments

---

How a Polarizing Filter Works

Light is an electromagnetic wave.

For unpolarized light, the electric field oscillates in many different directions perpendicular to the direction of travel.

A linear polarizer preferentially transmits the component of the electric field that lies along its transmission axis.

The perpendicular component is strongly reduced.

Therefore:

unpolarized light → polarizer → linearly polarized light

If the transmission axis is vertical, the emerging light is vertically polarized.

If the filter is rotated, the polarization direction of the transmitted light rotates with it.

---

The Transmission Axis

The transmission axis specifies the electric-field direction that an ideal linear polarizer transmits.

For example, consider a vertical polarizer.

The incoming electric field can be resolved into:

• vertical component
• horizontal component

The vertical component is transmitted.

The horizontal component is blocked or strongly absorbed.

The outgoing light is therefore vertically polarized.

---

Unpolarized Light Through One Polarizer

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

I = I₀/2

where:

• I₀ = incident intensity
• I = transmitted intensity

Therefore, approximately half of the incident intensity is transmitted.

The transmitted light is now linearly polarized.

---

Why Half the Intensity Is Transmitted

The electric-field directions in unpolarized light are randomly distributed.

For a particular electric-field orientation θ relative to the transmission axis, the transmitted electric-field component is proportional to:

E cos θ

Intensity is proportional to the square of electric-field amplitude:

I ∝ E²

Therefore, the transmitted fraction involves:

cos² θ

Averaging cos² θ over all possible polarization orientations gives:

1/2

Hence:

I = I₀/2

for ideal unpolarized light.

---

Worked Example: One Polarizer

Unpolarized light has intensity:

I₀ = 300 W/m²

It passes through an ideal polarizer.

Therefore:

I = I₀/2

I = 300/2

I = 150 W/m²

The emerging light has an intensity of 150 W/m² and is linearly polarized along the filter's transmission axis.

---

Two Polarizers

Now suppose polarized light passes through a second polarizing filter.

The second filter is often called an analyzer.

The amount of light transmitted depends on the angle θ between:

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

This relationship is described by Malus's law.

---

Malus's Law

For linearly polarized light passing through an ideal analyzer:

I = I₀ cos² θ

where:

• I = transmitted intensity
• I₀ = intensity entering the analyzer
• θ = angle between the incoming polarization direction and analyzer axis

This equation allows the transmitted intensity to be predicted for any analyzer angle.

---

Parallel Polarizers

Suppose the first and second polarizers have parallel transmission axes.

Then:

θ = 0°

Using Malus's law:

I = I₀ cos² 0°

Since:

cos 0° = 1

then:

I = I₀

Ideally, the second polarizer does not reduce the intensity further.

Real polarizers do absorb or reflect some light, so actual transmission is lower.

---

Polarizers at 30°

If:

θ = 30°

then:

I = I₀ cos² 30°

Since:

cos² 30° = 0.75

then:

I = 0.75I₀

The analyzer transmits 75% of the polarized light reaching it.

---

Polarizers at 45°

If:

θ = 45°

then:

I = I₀ cos² 45°

Since:

cos² 45° = 0.50

then:

I = 0.50I₀

Half of the polarized light entering the second filter is transmitted.

---

Polarizers at 60°

If:

θ = 60°

then:

I = I₀ cos² 60°

Since:

cos² 60° = 0.25

then:

I = 0.25I₀

Only one quarter of the polarized light entering the analyzer is transmitted.

---

Crossed Polarizers

If the two transmission axes are perpendicular:

θ = 90°

Using Malus's law:

I = I₀ cos² 90°

Since:

cos 90° = 0

then:

I = 0

Ideally, no light is transmitted.

Two polarizers arranged at 90° are called crossed polarizers.

---

Rotation and Intensity

As one polarizer is rotated relative to another, the transmitted intensity changes continuously.

The change is not linear.

For example, rotating from 0° to 45° does not reduce the intensity by 45/90 = 50% because of the angle alone.

The result follows:

cos² θ

---

Intensity During a Full Rotation

If the analyzer is rotated through 360°, the intensity follows:

I = I₀ cos² θ

Maximum intensity occurs at:

0°, 180°, 360°

Minimum intensity occurs at:

90°, 270°

The pattern therefore repeats every:

180°

This periodic variation is strong experimental evidence of polarization.

---

Worked Example: Two Polarizers

Unpolarized light has intensity:

I₀ = 400 W/m²

It passes through a vertical ideal polarizer.

After the first polarizer:

I₁ = I₀/2

I₁ = 200 W/m²

The light then passes through another polarizer at 60°.

Using Malus's law:

I₂ = I₁ cos²60°

I₂ = 200(0.25)

I₂ = 50 W/m²

Therefore, the final intensity is:

50 W/m²

---

A Common Calculation Error

Suppose unpolarized light passes through two polarizers.

It is incorrect to immediately write:

I = I₀ cos² θ

using the original unpolarized intensity.

The first polarizer must be considered first:

I₁ = I₀/2

Then apply Malus's law to the second polarizer:

I₂ = I₁ cos² θ

Therefore:

I₂ = (I₀/2)cos² θ

---

Finding the Angle

Malus's law can also be rearranged to determine the angle.

Suppose:

I = 0.25I₀

Then:

0.25 = cos² θ

Take the square root:

cos θ = 0.5

Therefore:

θ = 60°

So an analyzer at 60° transmits 25% of the incident linearly polarized intensity.

---

Worked Example: Finding an Unknown Angle

Polarized light enters an analyzer with intensity:

I₀ = 120 W/m²

The transmitted intensity is:

I = 90 W/m²

Then:

I/I₀ = cos² θ

90/120 = cos² θ

0.75 = cos² θ

cos θ = √0.75

cos θ ≈ 0.866

Therefore:

θ ≈ 30°

---

Three Polarizing Filters

A surprising result occurs when a third polarizer is placed between two crossed polarizers.

Suppose:

• Polarizer A = 0°
• Polarizer C = 90°

Without another filter between them:

I = 0

Now insert Polarizer B at:

45°

Some light can pass through all three filters.

---

Why the Middle Polarizer Allows Light Through

After the first polarizer, the light is polarized at 0°.

The 45° polarizer transmits the component along 45°.

The emerging light is now polarized at 45°.

The final 90° polarizer can transmit the component of this 45°-polarized light along its own axis.

Therefore, the middle polarizer establishes a new polarization direction.

The light is not simply travelling unchanged through each filter.

---

Worked Example: Three Polarizers

Unpolarized light has intensity:

I₀ = 160 W/m²

After the first polarizer:

I₁ = 160/2

I₁ = 80 W/m²

The second polarizer is at 45°:

I₂ = 80 cos²45°

I₂ = 80(0.5)

I₂ = 40 W/m²

The third polarizer is another 45° away:

I₃ = 40 cos²45°

I₃ = 40(0.5)

I₃ = 20 W/m²

Therefore:

I_final = 20 W/m²

Without the middle polarizer, the crossed first and third filters would ideally transmit zero.

---

Polarizing Filters and Glare

Polarizing filters have an important practical application: reducing glare.

Light reflected from surfaces such as:

• water
• wet roads
• glass
• snow

can become strongly polarized.

For a horizontal reflecting surface, much of the glare is polarized approximately horizontally under suitable viewing conditions.

A polarizing filter with a vertical transmission axis can strongly reduce this horizontally polarized light.

---

Polarized Sunglasses

Polarized sunglasses typically contain filters oriented to reduce horizontally polarized glare.

They are particularly useful for:

• driving
• boating
• fishing
• skiing
• outdoor activities near water

Reducing glare can improve:

• visibility
• contrast
• visual comfort

The filters do not simply darken all light equally. They selectively reduce certain polarization components.

---

Why Reflected Light Becomes Polarized

When unpolarized light strikes a dielectric surface at an angle, the two polarization components do not reflect equally.

One component can be reflected more strongly than the other.

As a result, the reflected light becomes partially polarized.

At a particular angle called Brewster's angle, one component has zero ideal reflection and the reflected light is completely linearly polarized.

---

Brewster's Angle

For light travelling from a medium with refractive index n₁ into a medium with refractive index n₂:

tan θ_B = n₂/n₁

For air and glass:

n₁ = 1.00

n₂ = 1.50

tan θ_B = 1.50

Therefore:

θ_B ≈ 56°

Around this angle, reflected glare can be strongly polarized.

---

Polarizing Filters in Photography

Photographers often use polarizing filters in front of camera lenses.

Rotating the filter changes which polarization direction is transmitted.

This can:

• reduce reflections from water
• reduce some reflections from glass
• reduce glare from vegetation
• increase apparent colour saturation
• alter the appearance of the sky

The photographer can rotate the filter while observing the scene to select the desired effect.

---

Why Rotating a Camera Polarizer Matters

Suppose reflected glare is predominantly horizontally polarized.

If the filter's transmission axis is horizontal, much of the glare passes.

Rotate the filter toward vertical and the horizontal component is increasingly suppressed.

Therefore, the amount of glare changes as the filter rotates.

This is a direct real-world application of polarization.

---

Seeing Through Water

Surface reflections can make it difficult to see beneath water.

A correctly oriented polarizing filter can reduce much of the reflected glare.

More of the light coming from objects beneath the surface then contributes to the observed image.

This is useful in:

• fishing
• aquatic photography
• environmental observations
• outdoor recreation

---

Looking Through Glass

A polarizing filter can sometimes reduce reflections from glass.

For example, it may help a photographer see through a shop window by reducing reflected light from the surroundings.

However, the effectiveness depends on:

• viewing angle
• glass properties
• polarization of the reflected light

Polarization does not remove every possible reflection.

---

LCD Screens and Polarizers

Liquid crystal displays use polarizing filters as essential components.

A simplified LCD contains:

• a polarizer
• liquid-crystal material
• another polarizing layer

The liquid crystals affect the polarization of light.

Electrical signals change the behaviour of the liquid crystals.

This controls how much light can pass through the final polarizer.

---

Testing an LCD with a Polarizer

A simple polarization experiment can be performed by viewing an LCD through a polarizing filter.

As the filter rotates:

• the screen may become brighter
• then darker
• sometimes nearly black
• then brighter again

This demonstrates that the light leaving the screen has a strong polarization state.

---

Investigating Polarizers Experimentally

A basic classroom experiment requires:

• light source
• two polarizing filters
• light sensor or lux meter
• rotating mount or angle scale

The first filter produces linearly polarized light.

The second filter acts as the analyzer.

Students can rotate the analyzer and measure transmitted intensity.

---

Experimental Procedure

Place the polarizers in the light path.

Set their transmission axes parallel:

θ = 0°

Measure the transmitted intensity.

Rotate the analyzer in regular steps, such as:

• 0°
• 15°
• 30°
• 45°
• 60°
• 75°
• 90°

Record the intensity at each angle.

Then compare the results with:

I = I₀ cos² θ

---

Expected Experimental Pattern

The intensity should be largest when:

θ = 0°

It should decrease as θ increases.

At:

θ = 45°

the ideal intensity is:

I = 0.5I₀

At:

θ = 90°

the ideal intensity is:

I = 0

A graph of intensity against angle therefore produces a curved, periodic pattern rather than a straight line.

---

Testing Malus's Law

A stronger analysis is to calculate:

cos² θ

for every measured angle.

Then compare I with cos² θ.

According to Malus's law:

I ∝ cos² θ

Therefore, a graph of:

I against cos² θ

should be approximately linear for an ideal experimental setup.

This provides a useful way to test the mathematical model.

---

Example Experimental Results

Suppose the intensity entering an analyzer is:

100 units

Ideal predictions would be:

Real measurements may differ because real filters and detectors are not ideal.

---

Why Real Crossed Polarizers May Not Be Completely Dark

In an ideal model:

I = 0

at 90°.

Real experiments may still show a small amount of light because of:

• imperfect polarizers
• stray light
• reflections
• detector background
• imperfect alignment
• scattering
• incomplete polarization

Therefore, a non-zero experimental value does not automatically disprove Malus's law.

---

Extinction Ratio

The quality of a polarizer can be described partly by its extinction ratio.

This compares how strongly the filter transmits the desired polarization with how much unwanted perpendicular polarization leaks through.

A high-quality polarizer has:

• high transmission along its preferred direction
• very low transmission in the perpendicular direction

Therefore, high-quality crossed polarizers can produce a very dark field.

---

Experimental Variables

In a polarizer experiment, possible variables include:

Independent variable

Angle θ between the polarizers.

Dependent variable

Transmitted light intensity.

Control variables

These might include:

• light-source intensity
• distance from source
• detector position
• polarizer type
• ambient lighting
• wavelength or colour of light

Controlling these variables improves the reliability of the experiment.

---

Sources of Experimental Error

Possible errors include:

• inaccurate angle measurements
• changing source intensity
• ambient light reaching the detector
• polarizers not centered correctly
• detector movement
• imperfect polarizers
• reflections from surrounding surfaces

Repeating measurements and averaging results can improve reliability.

---

Analyzing Unexpected Results

Suppose the intensity at 90° is much larger than expected.

Possible explanations include:

• the filters are not actually at 90°
• one filter is a poor polarizer
• ambient light is reaching the detector
• light is scattering around the filters

A good experimental analysis should compare observations with theory and then explain reasonable causes of differences.

---

A Complete Analysis Example

Unpolarized light has intensity:

I₀ = 240 W/m²

It passes through an ideal horizontal polarizer.

After the first polarizer:

I₁ = I₀/2

I₁ = 120 W/m²

A second polarizer is positioned at 30°.

Using Malus's law:

I₂ = I₁ cos²30°

I₂ = 120(0.75)

I₂ = 90 W/m²

Now rotate the second polarizer to 60°:

I₂ = 120 cos²60°

I₂ = 120(0.25)

I₂ = 30 W/m²

Increasing the relative angle from 30° to 60° reduces the transmitted intensity from 90 W/m² to 30 W/m².

---

Predicting Changes Without Calculating

Malus's law can also be used qualitatively.

If θ increases from:

0° → 90°

intensity generally decreases.

If θ decreases toward:

0°

intensity increases.

At:

0° → maximum

At:

90° → minimum

The pattern is symmetric because:

cos² θ

repeats every 180°.

---

Polarizers and Energy

A polarizer does not simply make the unwanted polarization component disappear without an energy transfer.

Depending on the type of polarizer, unwanted light may be:

• absorbed
• reflected
• redirected

In an absorbing polarizing sheet, much of the rejected component is converted into internal energy.

Energy conservation still applies.

---

Common Mistakes

Thinking a polarizer changes the direction the light travels

A linear polarizer mainly selects electric-field orientation; it does not normally redirect the transmitted beam.

Confusing transmission axis with propagation direction

The transmission axis describes the electric-field direction that passes through the filter.

Using I = I₀ cos θ

Malus's law is:

I = I₀ cos² θ

Applying Malus's law directly to unpolarized light without considering the first polarizer

For ideal unpolarized light:

I₁ = I₀/2

first.

Thinking two polarizers always halve the intensity twice

The second polarizer depends on its angle relative to the first.

Thinking 45° means 25% transmission

cos²45° = 0.5, so the transmitted fraction is 50% of the polarized light entering that analyzer.

Thinking crossed polarizers must transmit half the light

Ideal crossed polarizers transmit zero.

Assuming real experiments must give exactly zero at 90°

Real polarizers, detectors, and experimental environments are imperfect.

Thinking polarized sunglasses simply reduce all light equally

Their main advantage is preferentially reducing polarized glare.

---

Key Terms

Polarizing filter — optical device that preferentially transmits a particular polarization direction.

Polarizer — device used to produce or select polarized light.

Analyzer — polarizer used to determine or control the polarization of incoming light.

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

Unpolarized light — light whose electric-field orientation varies rapidly and randomly.

Linearly polarized light — light whose electric field oscillates along a fixed direction.

Malus's law — relationship I = I₀ cos² θ for linearly polarized light passing through an ideal analyzer.

Crossed polarizers — polarizers whose transmission axes are perpendicular.

Parallel polarizers — polarizers whose transmission axes point in the same direction.

Glare — intense reflected light that can reduce visibility and visual comfort.

Brewster's angle — incidence angle at which reflected light from an ideal dielectric boundary is completely linearly polarized.

Extinction ratio — measure of how effectively a polarizer suppresses the unwanted polarization direction.

Light intensity — optical power passing through a unit area.

---

Key Takeaways

• A polarizing filter selectively transmits a component of the electric field.
• Light emerging from an ideal linear polarizer is linearly polarized along its transmission axis.
• Ideal unpolarized light loses half its intensity through the first ideal polarizer.
• A second polarizer is commonly called an analyzer.
• The intensity through an analyzer follows I = I₀ cos² θ.
• θ is the angle between the incoming polarization direction and the analyzer's transmission axis.
• Parallel ideal polarizers provide maximum transmission.
• Polarizers at 45° transmit half of the polarized intensity entering the analyzer.
• Polarizers at 60° transmit one quarter of the polarized intensity.
• Crossed ideal polarizers transmit zero light.
• Rotating one polarizer causes transmitted intensity to vary periodically.
• Inserting another polarizer between crossed polarizers can allow light to pass.
• Reflected glare from horizontal surfaces is often strongly horizontally polarized.
• Polarized sunglasses use a suitable transmission axis to reduce this glare.
• Camera polarizers can reduce reflections and alter the appearance of outdoor scenes.
• LCD screens use polarization to control transmitted light.
• Malus's law can be tested experimentally by measuring intensity as a polarizer rotates.
• A graph of I against cos² θ should be approximately linear under ideal conditions.
• Real polarizers may transmit a small amount of light even when crossed.
• Polarizer experiments provide a direct way to investigate the transverse nature and intensity behaviour of light.

---

Check Your Understanding

1. Explain how a linear polarizing filter changes unpolarized light.
2. What is meant by the transmission axis of a polarizer?
3. Unpolarized light has an intensity of 500 W/m². Calculate its intensity after passing through one ideal polarizer.
4. State Malus's law and define each variable.
5. Polarized light with intensity 80 W/m² reaches an analyzer at 30°. Calculate the transmitted intensity.
6. Repeat Question 5 for an analyzer angle of 60°.
7. Explain why two parallel ideal polarizers transmit much more light than two crossed polarizers.
8. Unpolarized light with intensity 200 W/m² passes through two ideal polarizers whose axes differ by 45°. Calculate the final intensity.
9. Two ideal polarizers are crossed. Predict the transmitted intensity and explain your reasoning using Malus's law.
10. Explain why inserting a third polarizer at 45° between two crossed polarizers allows some light to pass.
11. Unpolarized light of intensity 320 W/m² passes through polarizers oriented at 0°, 30°, and 90°. Calculate the final intensity.
12. Explain how polarized sunglasses reduce glare from a wet road or the surface of a lake.
13. A student rotates an analyzer from 0° to 180° while recording light intensity. Describe the expected pattern and identify the angles of maximum and minimum intensity.
14. In an experiment, the detector records a small intensity when the polarizers are at 90°. Give three experimental reasons why the measured intensity may not be exactly zero.
15. Design an experiment to test Malus's law. Identify the independent variable, dependent variable, important control variables, measurements you would take, and the graph you could use to determine whether the results support I ∝ cos² θ.

3. Resolution and Optical Instruments

Resolution and Optical Instruments

What Is Resolution?

Resolution is the ability of an optical instrument to distinguish between two objects or details that are very close together.

An instrument with high resolution can show small details clearly and can distinguish two nearby objects as separate. An instrument with low resolution may cause nearby objects to blur together.

For example, imagine looking at two stars that appear very close together:

• With poor resolution, they may appear as one point of light.
• With good resolution, they can be identified as two separate stars.

Resolution is therefore an important measure of the detail that an optical instrument can reveal.

---

Resolution vs Magnification

Resolution and magnification are related, but they are not the same thing.

Magnification tells us how much larger an image appears.

Resolution tells us how much detail can actually be distinguished.

An image can be highly magnified without revealing additional information.

For example, enlarging a blurry photograph makes the photograph bigger, but it does not reveal details that were not captured in the original image.

This is sometimes called empty magnification.

Example

Suppose two microscope images are both enlarged 1000 times.

• Microscope A produces a blurry image in which two nearby cells merge together.
• Microscope B clearly shows the boundaries between the two cells.

Both have the same magnification, but Microscope B has better resolution.

---

Resolving Power

Resolving power describes how well an optical instrument can distinguish between objects that are very close together.

Greater resolving power means that smaller separations can be detected.

A telescope with high resolving power might distinguish two nearby stars that appear as a single star through a poorer telescope.

A microscope with high resolving power can distinguish structures that are extremely close together.

The terms resolution and resolving power are sometimes used slightly differently depending on the field, but both describe an instrument's ability to separate fine details.

---

The Resolution Limit

Every optical instrument has a limit to the amount of detail it can resolve.

This occurs partly because light behaves as a wave.

When light passes through an opening such as a telescope aperture or microscope objective, it undergoes diffraction. Instead of producing a perfectly sharp point, a point source produces a small diffraction pattern.

If two diffraction patterns overlap too strongly, the two objects become difficult or impossible to distinguish.

---

The Rayleigh Criterion

A common method for deciding whether two point sources can be resolved is the Rayleigh criterion.

According to this criterion, two objects are just resolved when the central maximum of one diffraction pattern falls approximately on the first minimum of the other.

For a circular aperture such as a telescope objective:

θ = 1.22λ/D

where:

• θ = minimum angular separation that can be resolved, in radians
• λ = wavelength of the light
• D = diameter of the aperture

A smaller value of θ means better resolution because the instrument can distinguish objects separated by a smaller angle.

Example

Consider two telescopes observing the same wavelength of light.

Telescope A:

D = 0.10 m

Telescope B:

D = 1.0 m

Because Telescope B has an aperture ten times larger, its minimum resolvable angle is approximately ten times smaller.

Therefore, Telescope B has much better resolving power.

---

How Wavelength Affects Resolution

The equation

θ = 1.22λ/D

shows that resolution depends strongly on wavelength.

If λ decreases, θ decreases.

Therefore:

shorter wavelengths generally produce better resolution.

For visible light:

• violet/blue light has a shorter wavelength
• red light has a longer wavelength

An optical system using shorter-wavelength light can therefore potentially distinguish finer details.

This principle extends beyond visible light. Instruments that use very short wavelengths can reveal structures that ordinary visible-light instruments cannot.

---

How Aperture Size Affects Resolution

The diameter of the aperture also affects resolution.

From:

θ = 1.22λ/D

increasing D decreases the minimum resolvable angle.

Therefore:

larger aperture → better resolution

This is one reason astronomical telescopes often have very large mirrors.

A large mirror does more than collect additional light. It also improves the telescope's theoretical ability to distinguish closely spaced objects.

For example, two stars that blur together through a small telescope may be distinguishable using a larger telescope.

---

Resolution in Microscopes

For microscopes, resolution depends on factors including:

• wavelength of the light
• properties of the objective lens
• numerical aperture
• refractive index of the material between the specimen and lens

A simplified relationship is:

d ≈ λ/(2NA)

where:

• d = minimum distance between two resolvable points
• λ = wavelength
• NA = numerical aperture

For this equation, smaller d means better resolution.

Therefore:

• shorter wavelength → better resolution
• larger numerical aperture → better resolution

A typical light microscope has a resolution of roughly 200 nm (0.2 μm) under good conditions.

Objects much smaller than this cannot normally be distinguished clearly using an ordinary visible-light microscope.

---

Light Microscopes and Electron Microscopes

Electron microscopes can achieve much greater resolution than ordinary light microscopes.

This is because electrons can behave as waves, and their effective wavelength can be far shorter than the wavelength of visible light.

Therefore, electron microscopes can reveal extremely small structures that light microscopes cannot resolve.

For example:

The exact resolution of an electron microscope depends on its design and operating conditions.

---

Resolution in Telescopes

Telescopes must distinguish objects separated by very small angles.

Important factors include:

Aperture size — larger apertures provide better theoretical resolution.

Wavelength — shorter wavelengths generally provide better diffraction-limited resolution.

Atmospheric conditions — turbulence in Earth's atmosphere can distort incoming light and reduce image quality.

Optical quality — imperfections in mirrors and lenses can reduce practical resolution.

Detector quality — the sensor must be capable of recording the detail produced by the optical system.

---

Atmospheric Effects

Ground-based telescopes face an additional problem: Earth's atmosphere.

Different layers of air have slightly different temperatures and densities. As light passes through these moving layers, it is refracted in slightly different directions.

This produces atmospheric turbulence, often called astronomical seeing.

Stars may appear to:

• twinkle
• move slightly
• blur
• change shape

Modern observatories can use adaptive optics, in which mirrors are adjusted rapidly to compensate for atmospheric distortion.

Space telescopes avoid most atmospheric distortion because they operate above Earth's atmosphere.

---

Other Factors That Limit Image Quality

The theoretical resolution of an instrument does not guarantee that every image will have that quality.

Practical image quality can also be reduced by:

• incorrect focusing
• lens imperfections
• mirror imperfections
• chromatic aberration
• spherical aberration
• vibrations
• atmospheric turbulence
• poor lighting
• detector pixel size
• electronic noise
• motion of the object or instrument

This means an instrument's theoretical resolution and its actual image quality may be different.

---

Comparing Optical Instruments

Different optical instruments are designed for different purposes.

There is therefore no single "best" optical instrument. The appropriate instrument depends on the size, distance and nature of the object being observed.

---

Evaluating Image Quality

When evaluating an image, do not simply ask:

"How large is the image?"

Instead ask:

"What details can actually be distinguished?"

A good-resolution image should allow closely spaced structures to be identified separately.

For example, if two tiny structures appear as one blurred feature, increasing the display size does not necessarily solve the problem. The original optical system must have captured enough information to resolve them.

---

Worked Example

A telescope has an aperture diameter of 0.25 m and observes light with a wavelength of 550 nm.

First convert the wavelength:

550 nm = 550 × 10⁻⁹ m

Use:

θ = 1.22λ/D

θ = (1.22 × 550 × 10⁻⁹) / 0.25

θ ≈ 2.68 × 10⁻⁶ rad

The telescope can theoretically resolve objects separated by an angle of approximately:

2.7 × 10⁻⁶ rad

If the aperture diameter were increased while the wavelength remained constant, θ would decrease and the resolution would improve.

---

Common Mistakes

Mistake: "Higher magnification always means better resolution."
Magnification only makes an image appear larger. It does not necessarily reveal additional detail.

Mistake: "A larger resolution value means better resolution."
When resolution is expressed as the minimum resolvable distance or angle, a smaller value represents better resolution.

Mistake: "Only the lens determines resolution."
Wavelength, aperture, diffraction, optical quality, atmospheric conditions and detector performance can all affect the final image.

Mistake: "Diffraction only happens with poor-quality equipment."
Diffraction is a fundamental property of waves and occurs even in an ideal optical system.

---

Key Terms

Resolution — the ability to distinguish two closely spaced objects or details.

Resolving power — a measure of an instrument's ability to separate fine details.

Diffraction — the spreading of waves when they pass through an opening or around an obstacle.

Resolution limit — the smallest separation that an optical system can distinguish.

Rayleigh criterion — a commonly used criterion for deciding when two diffraction patterns are just resolvable.

Aperture — the opening through which light enters an optical instrument.

Angular resolution — the smallest angular separation between two objects that can be distinguished.

Numerical aperture (NA) — a measure related to the light-gathering and resolving ability of a microscope objective.

Airy pattern — the diffraction pattern produced when light from a point source passes through a circular aperture.

Atmospheric seeing — distortion of astronomical images caused by turbulence in Earth's atmosphere.

Adaptive optics — technology that compensates for atmospheric distortion in astronomical observations.

Empty magnification — enlargement of an image without an increase in useful detail.

---

Key Takeaways

• Resolution describes an instrument's ability to distinguish fine details or closely spaced objects.
• Resolution and magnification are not the same thing.
• Diffraction places a fundamental limit on the resolution of optical instruments.
• For telescopes, angular resolution can be estimated using θ = 1.22λ/D.
• Larger apertures improve resolution.
• Shorter wavelengths improve resolution.
• Microscopes can improve resolution by using shorter wavelengths and larger numerical apertures.
• Electron microscopes achieve much greater resolution than ordinary light microscopes because electrons can have extremely short effective wavelengths.
• Practical image quality can be worse than theoretical resolution because of aberrations, atmospheric turbulence, focusing, detector limitations and other factors.
• The best way to evaluate resolution is to ask how much real detail can be distinguished, rather than simply how large the image appears.

Check Your Understanding

1. What is meant by the resolution of an optical instrument?
2. Why are magnification and resolution not the same thing?
3. Two telescopes observe the same wavelength. Which should have better resolution: one with a 0.20 m aperture or one with a 2.0 m aperture? Explain.
4. How does decreasing the wavelength of light affect the diffraction-limited resolution?
5. Why can electron microscopes reveal smaller structures than light microscopes?
6. Why might a real telescope produce poorer images than predicted by θ = 1.22λ/D?
7. A photograph is enlarged five times but becomes increasingly blurry. Has its resolution improved? Explain.
8. Explain why large telescope mirrors improve both light collection and resolving power.

4. Diffraction Limits

5. Applications of Polarization and Resolution