1. Double Slit Diffraction

Double Slit Diffraction

The double slit experiment is one of the most important demonstrations of wave behaviour in physics. When waves pass through two narrow openings, each opening acts as a source of new waves. These waves spread out, overlap, and interfere.

For light, this produces a pattern of alternating bright and dark fringes on a screen.

The pattern can be explained using two fundamental wave ideas:

• diffraction — waves spread after passing through narrow openings.
• interference — overlapping waves combine according to the principle of superposition.

The experiment provides strong evidence for the wave nature of light. Remarkably, similar interference patterns can also be produced using particles such as electrons, demonstrating the wave-like behaviour of matter.

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Diffraction at the Slits

Diffraction is the spreading of a wave as it passes through an opening or around an obstacle.

When light reaches two sufficiently narrow slits, it does not simply continue as two narrow straight beams.

Instead, light emerging from each slit spreads into the region beyond the barrier.

The two spreading waves overlap.

Where they overlap, they interfere.

Therefore, both processes are important:

narrow slits → diffraction → overlapping waves → interference pattern

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Why Two Slits Are Needed

A single narrow slit produces diffraction.

Two narrow slits allow the diffracted waves from the two openings to overlap.

The slits effectively behave as two closely spaced sources of coherent waves.

At different points on the screen, the waves have travelled slightly different distances.

This difference in distance determines whether they arrive:

• in phase
• partly out of phase
• completely out of phase

The result is an interference pattern.

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Coherent Waves

For a clear and stable interference pattern, the waves from the two slits must be coherent.

Coherent waves have:

• the same frequency
• the same wavelength
• a constant phase relationship

In the double slit experiment, both slits are illuminated by the same source.

This helps ensure that the waves emerging from the slits maintain a stable phase relationship.

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Monochromatic Light

A clear double slit pattern is easiest to observe using monochromatic light.

Monochromatic light contains essentially one wavelength.

A laser is therefore commonly used.

If many wavelengths are present simultaneously, each wavelength produces a slightly different interference pattern, causing the fringes to overlap.

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Principle of Superposition

When the two diffracted waves overlap, their displacements combine according to the principle of superposition.

At any point:

resultant displacement = displacement from slit 1 + displacement from slit 2

Depending on the relative phase of the two waves, this produces constructive or destructive interference.

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Constructive Interference

Constructive interference occurs when the waves arrive in phase.

For example:

crest + crest → larger amplitude

or:

trough + trough → larger amplitude

For light, constructive interference produces a bright fringe.

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Condition for Bright Fringes

Suppose the difference between the distances travelled from the two slits is called the path difference, Δd.

Constructive interference occurs when:

Δd = nλ

where:

• Δd = path difference
• λ = wavelength
• n = 0, 1, 2, 3, ...

Therefore, bright fringes occur when the path difference is:

0, λ, 2λ, 3λ, ...

The waves then arrive in phase.

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The Central Bright Fringe

At the exact center of the screen, the distance from each slit is the same.

Therefore:

Δd = 0

Since:

0 = 0λ

the waves arrive in phase.

Constructive interference occurs.

The center of the pattern is therefore a bright fringe.

This is called the central maximum.

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Destructive Interference

Destructive interference occurs when the waves arrive out of phase.

A crest from one wave meets a trough from the other.

If their amplitudes are equal:

crest + trough → zero resultant amplitude

For light, this produces a dark fringe.

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Condition for Dark Fringes

Complete destructive interference occurs when the path difference is an odd number of half-wavelengths:

Δd = (n + ½)λ

where:

n = 0, 1, 2, 3, ...

Therefore, dark fringes occur for:

λ/2, 3λ/2, 5λ/2, 7λ/2, ...

At these positions, the waves arrive 180° out of phase.

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The Interference Pattern

The screen displays alternating regions:

bright → dark → bright → dark → bright

The central fringe is bright.

On either side are:

• first dark fringes
• first-order bright fringes
• second dark fringes
• second-order bright fringes
• and so on

For an ideal double slit with narrow identical slits, the fringes are approximately equally spaced near the center of the screen.

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Why Bright and Dark Fringes Form

Consider a point on the screen.

Light from both slits reaches that point.

If the two waves travel distances differing by exactly one wavelength:

Δd = λ

they arrive in phase.

Result:

constructive interference → bright

If the difference is half a wavelength:

Δd = λ/2

they arrive out of phase.

Result:

destructive interference → dark

The entire pattern is produced by this changing path difference across the screen.

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Path Difference and Geometry

Let:

• d = separation between the two slits
• θ = angle from the central axis
• λ = wavelength

The path difference is approximately:

Δd = d sin θ

For constructive interference:

d sin θ = nλ

For destructive interference:

d sin θ = (n + ½)λ

These equations determine the directions in which bright and dark fringes occur.

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Bright Fringe Equation

For bright fringes:

d sin θ = nλ

where:

• d = slit separation
• θ = angle of the fringe
• n = fringe order
• λ = wavelength

For the central maximum:

n = 0

so:

θ = 0°

For the first bright fringe:

n = 1

For the second:

n = 2

and so on.

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Worked Example: Bright Fringe Angle

Light has wavelength:

λ = 600 nm

Convert:

λ = 600 × 10⁻⁹ m

Slit separation:

d = 0.20 mm

d = 2.0 × 10⁻⁴ m

For the first-order bright fringe:

n = 1

Using:

d sin θ = nλ

sin θ = λ/d

sin θ = (600 × 10⁻⁹)/(2.0 × 10⁻⁴)

sin θ = 0.0030

Therefore:

θ ≈ 0.17°

The first bright fringe appears at a small angle from the center.

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Fringe Spacing

In many double slit experiments, the screen is relatively far from the slits.

For small angles:

sin θ ≈ tan θ ≈ y/L

where:

• y = distance from the central maximum
• L = distance from slits to screen

Combining this with:

d sin θ = nλ

gives:

y_n = nλL/d

This gives the position of the nth bright fringe.

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Fringe Separation Equation

The distance between neighbouring bright fringes, or neighbouring dark fringes, is called the fringe spacing.

It is commonly represented by w:

w = λL/d

where:

• w = fringe spacing
• λ = wavelength
• L = slit-to-screen distance
• d = slit separation

This relationship is extremely important in double slit calculations.

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What Determines Fringe Spacing?

From:

w = λL/d

we can see:

w ∝ λ

Longer wavelength produces wider fringe spacing.

w ∝ L

Greater screen distance produces wider fringe spacing.

w ∝ 1/d

Greater slit separation produces narrower fringe spacing.

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Effect of Increasing Wavelength

Suppose all other variables remain constant.

If wavelength increases:

λ ↑ → w ↑

The fringes move farther apart.

Therefore, red light generally produces wider fringe spacing than blue light in the same double slit arrangement because red light has a longer wavelength.

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Effect of Screen Distance

If the screen is moved farther from the slits:

L ↑ → w ↑

The interference pattern spreads over a larger distance.

For example, doubling L approximately doubles the fringe spacing.

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Effect of Slit Separation

If the distance between the slits increases:

d ↑ → w ↓

The fringes become closer together.

This inverse relationship is sometimes surprising.

Moving the slits farther apart makes the interference fringes more closely spaced.

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Summary of Fringe Spacing

These relationships follow directly from:

w = λL/d

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Worked Example: Calculating Fringe Spacing

A laser has wavelength:

λ = 500 nm

Slit separation:

d = 0.25 mm

Screen distance:

L = 2.0 m

Convert units:

λ = 5.00 × 10⁻⁷ m

d = 2.5 × 10⁻⁴ m

Using:

w = λL/d

w = (5.00 × 10⁻⁷)(2.0)/(2.5 × 10⁻⁴)

w = 0.0040 m

Therefore:

w = 4.0 mm

Neighbouring bright fringes are approximately 4.0 mm apart.

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Worked Example: Changing the Screen Distance

Using the previous experiment:

w = 4.0 mm

at:

L = 2.0 m

Suppose the screen is moved to:

L = 4.0 m

The distance has doubled.

Since:

w ∝ L

the fringe spacing also doubles:

w = 8.0 mm

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Worked Example: Changing Slit Separation

Suppose:

λ = 600 nm

L = 2.0 m

d = 0.30 mm

Then:

w = λL/d

w = (6.00 × 10⁻⁷)(2.0)/(3.0 × 10⁻⁴)

w = 0.0040 m = 4.0 mm

Now reduce the slit separation to:

d = 0.15 mm

The slit separation has been halved.

Therefore, the fringe spacing doubles:

w = 8.0 mm

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Finding Wavelength

The fringe equation can be rearranged:

λ = wd/L

Suppose:

w = 5.0 mm

d = 0.20 mm

L = 2.0 m

Convert:

w = 5.0 × 10⁻³ m

d = 2.0 × 10⁻⁴ m

Then:

λ = (5.0 × 10⁻³)(2.0 × 10⁻⁴)/2.0

λ = 5.0 × 10⁻⁷ m

λ = 500 nm

The double slit experiment can therefore be used to measure the wavelength of light.

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Finding Slit Separation

The same equation can be rearranged:

d = λL/w

Suppose:

λ = 650 nm

L = 1.5 m

w = 3.25 mm

Convert:

λ = 6.50 × 10⁻⁷ m

w = 3.25 × 10⁻³ m

Then:

d = (6.50 × 10⁻⁷)(1.5)/(3.25 × 10⁻³)

d = 3.0 × 10⁻⁴ m

or:

d = 0.30 mm

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Measuring Several Fringes

Measuring only one fringe spacing can produce a relatively large percentage uncertainty.

A more reliable method is to measure across several fringes.

For example, suppose the distance across 10 fringe spacings is:

42 mm

Then:

w = 42/10

w = 4.2 mm

Measuring a larger distance and dividing by the number of spacings can reduce the effect of measurement uncertainty.

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Fringe Order

The integer n describes the order of a bright fringe.

For:

n = 0

central maximum.

For:

n = 1

first-order bright fringe.

For:

n = 2

second-order bright fringe.

The pattern is symmetrical, so fringes can occur on both sides:

n = ±1, ±2, ±3 ...

The sign indicates the side of the central maximum.

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Position of a Bright Fringe

Using the small-angle approximation:

y_n = nλL/d

Suppose the fringe spacing is:

w = 3 mm

The first bright fringe is:

3 mm

from the center.

The second is:

6 mm

from the center.

The third is:

9 mm

from the center.

This demonstrates why the fringes are approximately equally spaced under the usual experimental conditions.

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Position of Dark Fringes

For dark fringes:

d sin θ = (n + ½)λ

Using the small-angle approximation:

y_dark = (n + ½)λL/d

The first dark fringe corresponds to:

n = 0

Therefore:

y_dark = λL/(2d)

This places the first dark fringe halfway between the central bright fringe and the first bright fringe.

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Bright and Dark Fringe Positions

If the bright-fringe spacing is w:

Bright fringes occur at approximately:

0, w, 2w, 3w ...

on either side.

Dark fringes occur at:

w/2, 3w/2, 5w/2 ...

This creates the regular alternating pattern.

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Why Narrow Slits Are Important

The slits need to be narrow enough for significant diffraction to occur.

If each opening is too wide relative to the wavelength:

• diffraction is reduced
• the beams spread less
• overlap may be reduced
• the interference pattern may be less useful

Narrow openings allow the waves from the two slits to spread and overlap across the screen.

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Slit Width vs Slit Separation

These quantities should not be confused.

Slit width describes how wide each individual opening is.

Slit separation describes the distance between the two slits, usually measured between their centers.

The double slit fringe equation:

w = λL/d

uses the slit separation d, not the individual slit width.

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Double Slit Diffraction and Single Slit Diffraction

A real double slit experiment actually involves both diffraction and interference.

Each individual slit produces a diffraction pattern.

The two slits together produce interference fringes.

As a result, the brightness of the double slit fringes can be modified by a broader single-slit diffraction envelope.

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The Diffraction Envelope

The narrow bright and dark interference fringes are contained within a broader diffraction pattern.

This means the bright fringes are not necessarily all equally intense.

Fringes near the center are often stronger.

Farther from the center, their intensity may decrease according to the diffraction envelope.

This explains why a real double slit pattern can look more complicated than a simple series of identical bright lines.

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Light Intensity

Constructive interference produces greater resultant amplitude.

Destructive interference produces smaller resultant amplitude.

For electromagnetic waves:

intensity ∝ amplitude²

Therefore, changes in amplitude caused by interference produce changes in observed brightness.

Bright fringes correspond to high intensity.

Dark fringes correspond to very low or ideally zero intensity.

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Why the Experiment Shows That Light Behaves as a Wave

Interference and diffraction are characteristic wave behaviours.

If light simply behaved like classical particles travelling along straight paths, we might expect two bright regions corresponding directly to the two slits.

Instead, we observe many alternating bright and dark fringes.

The pattern is naturally explained by:

diffraction + superposition + interference

Therefore, the double slit experiment provides powerful evidence for the wave behaviour of light.

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Young's Double Slit Experiment

The double slit experiment is strongly associated with the work of Thomas Young in the early nineteenth century.

Young's observations demonstrated that light could produce interference.

This provided important evidence supporting a wave description of light.

The experiment became one of the most famous demonstrations in the history of physics.

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Light Also Behaves Like Particles

Modern physics shows that light cannot be described completely as only a classical wave.

Light also interacts with matter in discrete packets called photons.

This creates an important idea:

light displays both wave-like and particle-like behaviour

The double slit experiment is especially important because its interference pattern reveals the wave-like aspect of light.

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Double Slit Experiment with Individual Photons

An extraordinary version of the experiment can be performed using very weak light so that photons pass through the apparatus one at a time.

Individual detections appear as separate points on the screen.

At first, the points may appear scattered.

As many photons are detected, however, an interference pattern gradually develops.

This demonstrates that the probability distribution of photon detection contains wave-like interference.

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Matter Waves

The double slit experiment is not limited to light.

Particles of matter can also display interference.

Experiments have produced interference patterns using particles such as:

• electrons
• neutrons
• atoms
• molecules

This demonstrates that matter can exhibit wave-like properties.

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de Broglie Wavelength

The wave behaviour of matter can be described using the de Broglie wavelength:

λ = h/p

where:

• λ = de Broglie wavelength
• h = Planck's constant
• p = momentum

For non-relativistic motion:

p = mv

so:

λ = h/(mv)

Matter with momentum therefore has an associated wavelength.

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Why Matter Waves Are Usually Not Obvious

For everyday objects, mass and momentum are enormous compared with microscopic particles.

Since:

λ = h/p

large momentum produces an extremely small wavelength.

Therefore, the wave properties of ordinary objects are usually impossible to observe directly.

For particles such as electrons, however, the wavelength can be large enough to produce measurable diffraction and interference.

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Electron Double Slit Experiment

Electrons can be directed toward two narrow openings.

After many electrons reach a detector, an interference pattern can appear.

This is remarkable because electrons are detected as individual particles, yet their overall distribution shows wave-like interference.

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Wave-Particle Duality

The behaviour observed in double slit experiments contributes to the concept of wave-particle duality.

Quantum objects can display:

• particle-like detection events
• wave-like interference patterns

This applies to both light and matter.

The double slit experiment therefore connects classical wave physics with quantum physics.

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Which-Path Information

An especially important result occurs when an experiment is arranged to determine which slit a quantum particle passes through.

Obtaining reliable which-path information changes the experimental conditions, and the interference pattern can disappear.

This demonstrates that quantum interference cannot be interpreted simply as tiny classical particles following ordinary definite paths while an unrelated wave passes through both slits.

The quantum description is more subtle.

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Double Slit Experiment with Water Waves

The same basic interference pattern can be demonstrated with water waves.

Two openings in a barrier act as wave sources.

Beyond the openings:

• waves spread
• waves overlap
• constructive interference occurs in some directions
• destructive interference occurs in others

This provides a visible model for understanding the same basic wave principle.

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Double Slit Experiment with Sound

Sound waves can also demonstrate two-source interference.

Two coherent speakers can act like two wave sources.

At different locations:

• constructive interference produces louder sound
• destructive interference produces quieter sound

Although sound experiments are usually arranged differently from optical double slit experiments, the underlying superposition principle is the same.

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Comparing Different Waves

The physical observations differ, but the mathematical principle of interference is shared.

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Effect of Using Red and Blue Light

Suppose the same apparatus is used first with blue light and then with red light.

Red light has a longer wavelength than blue light.

Since:

w = λL/d

red light produces a larger fringe spacing.

Therefore:

red light → wider fringes

blue light → narrower fringes

assuming L and d remain unchanged.

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Effect of White Light

White light contains many wavelengths.

At the central fringe:

Δd = 0

All wavelengths can interfere constructively there.

The central fringe can therefore appear white.

Away from the center, different wavelengths satisfy constructive interference at different positions.

This can produce coloured fringes.

Farther from the center, patterns from different wavelengths increasingly overlap.

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Worked Comparison: Two Wavelengths

Suppose:

L = 2.0 m

d = 0.25 mm

For blue light:

λ_blue = 450 nm

w_blue = λL/d

w_blue = (450 × 10⁻⁹)(2.0)/(2.5 × 10⁻⁴)

w_blue = 3.6 mm

For red light:

λ_red = 650 nm

w_red = (650 × 10⁻⁹)(2.0)/(2.5 × 10⁻⁴)

w_red = 5.2 mm

The red fringes are farther apart.

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Experimental Setup

A typical classroom double slit experiment uses:

• laser
• double slit
• screen
• ruler or measuring scale
• known slit separation

The laser illuminates both slits.

The screen is positioned some distance away.

The fringe spacing is measured.

The wavelength can then be calculated using:

λ = wd/L

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Improving Experimental Accuracy

Accuracy can be improved by:

• measuring across several fringes
• using a large slit-to-screen distance
• measuring L carefully
• keeping the screen perpendicular to the central beam
• identifying fringe centers consistently
• using accurately known slit separation
• repeating measurements
• reducing unwanted background light

These methods reduce uncertainty.

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Example Experimental Calculation

Suppose the distance across 8 fringe spacings is:

32.0 mm

Therefore:

w = 32.0/8

w = 4.00 mm

Given:

d = 0.300 mm

L = 2.00 m

Convert:

w = 4.00 × 10⁻³ m

d = 3.00 × 10⁻⁴ m

Using:

λ = wd/L

λ = (4.00 × 10⁻³)(3.00 × 10⁻⁴)/2.00

λ = 6.00 × 10⁻⁷ m

Therefore:

λ = 600 nm

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Assumptions in the Fringe Equation

The simple equation:

w = λL/d

works well when:

• the screen is far from the slits compared with slit separation
• fringe angles are relatively small
• the slits are narrow
• the light is coherent
• wavelength is well defined

For larger angles or more precise analysis, the exact angular relationship:

d sin θ = nλ

should be used.

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Why the Pattern Is Symmetrical

The double slit arrangement is symmetrical around the central axis.

For a fringe at angle +θ, there is a corresponding fringe at −θ.

Therefore, bright and dark fringes appear on both sides of the central maximum.

The central bright fringe acts as the center of symmetry.

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Applications of Interference

The physics demonstrated by the double slit experiment has many applications.

Interference principles contribute to:

• interferometers
• precision measurement
• spectroscopy
• optical testing
• wavelength measurement
• astronomy
• microscopy
• quantum experiments
• measurement of very small distances

The double slit experiment therefore introduces ideas used in advanced science and engineering.

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A Reliable Problem-Solving Strategy

For double slit calculations:

Identify the quantities.

Usually:

• λ = wavelength
• L = screen distance
• d = slit separation
• w = fringe spacing

Convert everything to SI units.

Common conversions:

1 nm = 10⁻⁹ m

1 μm = 10⁻⁶ m

1 mm = 10⁻³ m

Choose the equation.

For fringe spacing:

w = λL/d

For bright fringe angles:

d sin θ = nλ

For dark fringes:

d sin θ = (n + ½)λ

Substitute carefully.

Check the result.

Visible-light fringe spacings in classroom experiments are often measured in millimetres, while wavelengths are typically hundreds of nanometres.

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Common Mistakes

Confusing diffraction with interference

Diffraction causes waves from each slit to spread. Interference occurs when those waves overlap.

Using slit width instead of slit separation

The equation w = λL/d uses the distance between the slits.

Thinking dark fringes contain "black light"

A dark fringe results from destructive interference and very low resultant intensity.

Thinking the central fringe is dark

At the center, the path difference is zero, so constructive interference produces a bright fringe.

Forgetting to convert units

Nanometres and millimetres must normally be converted to metres before calculation.

Thinking larger slit separation produces wider fringes

Since w ∝ 1/d, larger slit separation produces narrower fringe spacing.

Thinking shorter wavelengths produce wider fringes

Longer wavelengths produce wider fringe spacing.

Assuming diffraction proves only that light is a wave and nothing more is needed

Modern physics shows that light also has particle-like properties.

Thinking electrons cannot produce interference because they are particles

Quantum particles such as electrons can produce interference patterns and therefore display wave-like behaviour.

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Key Terms

Double slit experiment — experiment in which waves pass through two narrow openings and interfere.

Diffraction — spreading of waves after passing through an opening or around an obstacle.

Interference — combination of overlapping waves according to superposition.

Superposition — principle stating that overlapping wave displacements add.

Constructive interference — interference that increases resultant amplitude.

Destructive interference — interference that reduces resultant amplitude.

Bright fringe — high-intensity region produced by constructive interference.

Dark fringe — low-intensity region produced by destructive interference.

Central maximum — bright fringe at the center of the interference pattern.

Path difference — difference between the distances travelled by two waves to a point.

Fringe spacing — distance between neighbouring bright fringes or neighbouring dark fringes.

Coherent waves — waves with the same frequency and a stable phase relationship.

Monochromatic light — light consisting essentially of one wavelength.

Diffraction envelope — broader intensity pattern that modifies the brightness of double slit interference fringes.

Fringe order — integer describing a particular constructive interference maximum.

Matter wave — wave-like behaviour associated with a material particle.

de Broglie wavelength — wavelength associated with a particle's momentum, λ = h/p.

Wave-particle duality — principle that quantum objects can display both wave-like and particle-like behaviour.

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Key Takeaways

• Double slit diffraction involves both diffraction and interference.
• Each narrow slit causes waves to spread.
• Waves from the two slits overlap and interfere.
• Coherent waves are required for a stable interference pattern.
• Constructive interference produces bright fringes.
• Destructive interference produces dark fringes.
• Bright fringes occur when Δd = nλ.
• Dark fringes occur when Δd = (n + ½)λ.
• The central fringe is bright because its path difference is zero.
• The exact bright-fringe condition is d sin θ = nλ.
• For small angles, fringe spacing is w = λL/d.
• Increasing wavelength increases fringe spacing.
• Increasing screen distance increases fringe spacing.
• Increasing slit separation decreases fringe spacing.
• Red light produces wider fringes than blue light in the same apparatus.
• Measuring several fringes can improve experimental precision.
• Real double slit patterns are modified by the diffraction pattern from each individual slit.
• The double slit experiment provides strong evidence for the wave behaviour of light.
• Individual photons can collectively build an interference pattern.
• Electrons and other particles can also produce interference, demonstrating the wave nature of matter.
• The experiment is one of the most important links between classical wave physics and quantum physics.

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Check Your Understanding

1. Explain the roles of both diffraction and interference in the double slit experiment.
2. Why must the waves from the two slits overlap for an interference pattern to form?
3. State the path-difference condition for a bright fringe.
4. State the path-difference condition for a dark fringe.
5. Explain why the central fringe in a double slit experiment is bright.
6. Light of wavelength 600 nm passes through slits separated by 0.30 mm. The screen is 2.0 m away. Calculate the fringe spacing.
7. A double slit pattern has a fringe spacing of 5.0 mm. The slit separation is 0.25 mm, and the screen is 2.0 m away. Calculate the wavelength of the light.
8. Explain what happens to the fringe spacing if the screen distance is doubled while all other quantities remain constant.
9. Explain what happens to the fringe spacing if the slit separation is doubled.
10. The same double slit is illuminated first with 450 nm blue light and then with 650 nm red light. Which produces the wider fringe spacing? Explain your answer.
11. A bright fringe is produced where the path difference is 1.20 μm using light with wavelength 600 nm. Determine the order of the bright fringe.
12. Explain the difference between slit width and slit separation and identify which quantity appears in the basic double slit fringe-spacing equation.
13. Explain why measuring the distance across ten fringe spacings can give a more reliable result than measuring a single fringe spacing.
14. Describe how the double slit experiment provides evidence that light behaves as a wave.
15. Electrons are sent through a double slit apparatus one at a time, yet an interference pattern gradually appears after many detections. Explain why this result is important for our

2. Single Slit Diffraction

Single Slit Diffraction

Single slit diffraction occurs when a wave passes through a narrow opening and spreads into the region beyond it. Rather than producing a sharp beam with perfectly defined edges, the wave forms a pattern containing a broad central maximum surrounded by alternating dark and weaker bright fringes.

For light, the effect becomes especially noticeable when the width of the slit is comparable to the wavelength.

The pattern forms because waves emerging from different parts of the same slit interfere with one another.

The most important relationship for the dark fringes, or diffraction minima, is:

a sin θ = nλ

where:

• a = slit width
• θ = angle from the center
• n = 1, 2, 3, ...
• λ = wavelength

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What Is Diffraction?

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

Diffraction can occur with many types of waves, including:

• water waves
• sound waves
• light
• radio waves
• microwaves
• matter waves

The amount of diffraction depends strongly on the relationship between the wavelength and the size of the opening.

A useful general rule is:

wavelength comparable to opening size → significant diffraction

wavelength much smaller than opening → less noticeable diffraction

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Diffraction Through a Wide Opening

Suppose waves approach an opening that is much wider than their wavelength.

Most of the wave continues approximately forward.

Some spreading occurs near the edges, but the overall diffraction is relatively small.

Therefore:

a >> λ → relatively little spreading

where a is the width of the opening.

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Diffraction Through a Narrow Opening

Now reduce the width of the opening.

The spreading becomes more noticeable.

As the slit becomes narrower relative to the wavelength, the wave spreads through a greater range of angles.

Therefore:

smaller slit width → greater diffraction

This relationship is extremely important in single slit diffraction.

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Diffraction and Wavelength

The wavelength also affects the amount of spreading.

For the same slit:

longer wavelength → greater diffraction

shorter wavelength → less diffraction

This explains why long-wavelength radio waves can diffract significantly around large structures while visible light generally produces much less obvious diffraction around everyday objects.

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Why a Single Slit Produces an Interference Pattern

At first, it may seem surprising that interference can occur with only one slit.

The key idea is that the slit is not treated as a single point source.

According to the wave model, different parts of the opening can be considered sources of secondary wavelets.

These wavelets travel slightly different distances to points on the screen.

They therefore arrive with different phases.

At some positions they reinforce each other.

At other positions they cancel.

The result is:

single opening → diffraction → internal interference → bright and dark pattern

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Huygens' Principle

Huygens' principle provides a useful way of understanding diffraction.

It states that every point on a wavefront can be considered a source of secondary wavelets.

As the incoming wave reaches the slit, many points across the opening act as sources.

The secondary wavelets spread beyond the slit.

Their superposition produces the observed diffraction pattern.

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Constructive and Destructive Interference

The waves coming from different parts of the slit can interfere.

When their displacements reinforce one another:

constructive interference → greater intensity

When their displacements cancel:

destructive interference → lower intensity

At certain angles, the contributions from across the slit cancel almost completely.

These locations form the diffraction minima, which appear as dark fringes.

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The Single Slit Pattern

A typical single slit diffraction pattern contains:

• one wide, bright central maximum
• a dark minimum on each side
• weaker secondary maxima
• additional minima
• progressively weaker fringes farther from the center

The pattern is symmetrical about its center.

---

The Central Maximum

The large bright region at the center is called the central maximum.

It occurs around:

θ = 0°

At the exact center, waves from across the slit travel approximately equal distances to the screen.

Their contributions therefore reinforce strongly.

This produces the brightest part of the diffraction pattern.

---

Why the Central Maximum Is Bright

At the center of the screen:

path differences ≈ 0

The waves emerging from different parts of the slit therefore arrive nearly in phase.

Their amplitudes combine constructively.

Since light intensity is related to the square of amplitude:

I ∝ A²

strong constructive interference produces a high-intensity central region.

---

Why the Central Maximum Is Wider

The first dark minimum occurs on either side of the central maximum.

The central maximum therefore extends:

from the first minimum on one side to the first minimum on the other side

The neighbouring secondary maxima lie between successive minima.

For small angles, the central maximum is approximately twice as wide as each secondary bright fringe.

This is one of the most recognizable features of single slit diffraction.

---

Central Maximum vs Secondary Maxima

The secondary maxima also become progressively weaker farther from the center.

---

Diffraction Minima

The dark regions in the diffraction pattern are called minima.

They occur where waves from different parts of the slit cancel through destructive interference.

The condition for single slit minima is:

a sin θ = nλ

where:

• a = slit width
• θ = angle of the minimum
• λ = wavelength
• n = 1, 2, 3, ...

Notice that:

n = 0 is not a minimum

At θ = 0°, the central maximum occurs instead.

---

First Diffraction Minimum

For the first minimum:

n = 1

Therefore:

a sin θ₁ = λ

or:

sin θ₁ = λ/a

This angle determines the edge of the central maximum.

The first minima occur symmetrically at:

+θ₁ and −θ₁

---

Second Diffraction Minimum

For the second minimum:

n = 2

Therefore:

a sin θ₂ = 2λ

or:

sin θ₂ = 2λ/a

Additional minima follow for n = 3, 4, and so on, provided the required value of sin θ is physically possible.

---

Why the First Minimum Occurs

The first minimum can be understood by dividing the slit into two equal halves.

For the first minimum:

a sin θ = λ

The path difference between light from the very top and very bottom of the slit is one wavelength.

Now pair each point in the upper half with a corresponding point in the lower half.

The path difference within each pair is:

λ/2

A half-wavelength difference corresponds to a phase difference of:

180°

Each pair cancels through destructive interference.

The total intensity therefore reaches a minimum.

---

Higher-Order Minima

For higher minima, the slit can be divided into additional sections whose contributions cancel in pairs.

The general result remains:

a sin θ = nλ

This relationship predicts the dark fringes rather than the exact positions of the secondary bright fringes.

That distinction is important.

---

Worked Example: First Minimum

Light with wavelength:

λ = 600 nm

passes through a slit of width:

a = 0.20 mm

Convert to metres:

λ = 6.00 × 10⁻⁷ m

a = 2.00 × 10⁻⁴ m

For the first minimum:

a sin θ = λ

sin θ = λ/a

sin θ = (6.00 × 10⁻⁷)/(2.00 × 10⁻⁴)

sin θ = 0.00300

Therefore:

θ ≈ 0.172°

The first minima occur approximately 0.172° on either side of the center.

---

Worked Example: Second Minimum

Using the same slit and wavelength:

a = 2.00 × 10⁻⁴ m

λ = 6.00 × 10⁻⁷ m

For:

n = 2

a sin θ = 2λ

sin θ = 2λ/a

sin θ = 0.00600

Therefore:

θ ≈ 0.344°

The second minimum occurs farther from the center.

---

Finding Slit Width

The minimum condition can be rearranged:

a = nλ/sin θ

Suppose the first minimum occurs at:

θ = 0.20°

and:

λ = 500 nm

Then:

a = (1)(500 × 10⁻⁹)/sin(0.20°)

Using:

sin(0.20°) ≈ 0.00349

a ≈ 1.43 × 10⁻⁴ m

Therefore:

a ≈ 0.143 mm

---

Finding Wavelength

The same equation can be rearranged:

λ = a sin θ/n

Suppose:

a = 0.10 mm

The second minimum occurs at:

θ = 0.60°

Then:

a = 1.0 × 10⁻⁴ m

n = 2

λ = (1.0 × 10⁻⁴)sin(0.60°)/2

Using:

sin(0.60°) ≈ 0.01047

λ ≈ 5.24 × 10⁻⁷ m

Therefore:

λ ≈ 524 nm

---

From Angle to Position on a Screen

Experiments often measure the distance of a minimum from the center rather than its angle.

If:

• y = distance from the center
• L = slit-to-screen distance

then:

tan θ = y/L

For small angles:

sin θ ≈ tan θ ≈ y/L

Therefore, the minimum condition becomes approximately:

ay/L = nλ

and:

y_n = nλL/a

This gives the approximate position of the nth dark fringe.

---

Worked Example: Position of the First Minimum

Suppose:

λ = 500 nm

a = 0.25 mm

L = 2.0 m

Convert:

λ = 5.00 × 10⁻⁷ m

a = 2.50 × 10⁻⁴ m

For the first minimum:

y₁ = λL/a

y₁ = (5.00 × 10⁻⁷)(2.0)/(2.50 × 10⁻⁴)

y₁ = 0.0040 m

Therefore:

y₁ = 4.0 mm

The first dark fringes appear approximately 4.0 mm on either side of the center.

---

Width of the Central Maximum

The first minima occur at:

+y₁ and −y₁

Therefore, the full width of the central maximum is:

W = 2y₁

Using:

y₁ = λL/a

we obtain:

W = 2λL/a

for the small-angle approximation.

This is an important relationship.

---

Worked Example: Central Maximum Width

Suppose:

λ = 600 nm

L = 3.0 m

a = 0.30 mm

Convert:

λ = 6.00 × 10⁻⁷ m

a = 3.00 × 10⁻⁴ m

Using:

W = 2λL/a

W = 2(6.00 × 10⁻⁷)(3.0)/(3.00 × 10⁻⁴)

W = 0.012 m

Therefore:

W = 12 mm

The central maximum extends approximately 6 mm to either side of the center.

---

Effect of Slit Width

From:

sin θ₁ = λ/a

a smaller slit width produces a larger diffraction angle.

Therefore:

a decreases → θ increases → pattern spreads

Conversely:

a increases → θ decreases → pattern narrows

This inverse relationship is fundamental to diffraction.

---

Effect of Wavelength

Again:

sin θ₁ = λ/a

A larger wavelength produces a larger diffraction angle.

Therefore:

λ increases → diffraction increases

For the same slit:

• red light spreads more than blue light
• longer-wavelength waves diffract more than shorter-wavelength waves

---

Red Light vs Blue Light

Suppose:

λ_red = 650 nm

λ_blue = 450 nm

Both pass through the same slit.

Because:

650 nm > 450 nm

the red light has the larger diffraction angle.

Therefore:

red light → wider central maximum

blue light → narrower central maximum

---

Worked Comparison of Two Wavelengths

Suppose:

a = 0.20 mm

L = 2.0 m

For blue light:

λ = 450 nm

Central width:

W = 2λL/a

W = 2(450 × 10⁻⁹)(2.0)/(2.0 × 10⁻⁴)

W = 9.0 mm

For red light:

λ = 650 nm

W = 2(650 × 10⁻⁹)(2.0)/(2.0 × 10⁻⁴)

W = 13 mm

The longer-wavelength red light produces the broader central maximum.

---

Effect of Screen Distance

From:

W = 2λL/a

increasing L increases the physical width of the pattern on the screen.

Therefore:

L increases → central maximum becomes wider on the screen

However, changing L does not change the diffraction angle produced by the slit.

It simply allows the spreading wave to travel farther, producing a larger physical separation on the screen.

---

Angular Width of the Central Maximum

The first minima occur at:

−θ₁ and +θ₁

Therefore, the angular width of the central maximum is:

2θ₁

For small angles:

θ₁ ≈ λ/a

when θ is measured in radians.

Therefore:

angular width ≈ 2λ/a

A narrower slit produces a larger angular width.

---

Intensity Pattern

The single slit pattern is not a set of equally bright fringes.

The central maximum is the strongest.

The secondary maxima become progressively weaker farther from the center.

The intensity distribution has a characteristic shape:

• large central peak
• zero intensity at first minima
• much smaller secondary peak
• another minimum
• still smaller secondary peak
• continuing outward

---

Mathematical Intensity Distribution

For a uniformly illuminated ideal slit, the intensity can be described by:

I = I₀(sin β/β)²

where:

β = πa sin θ/λ

and:

• I = intensity at angle θ
• I₀ = intensity at the center
• a = slit width
• λ = wavelength

The full equation is more advanced than the minimum condition, but it explains why the central maximum is much brighter than the secondary maxima.

At the minima:

sin β = 0

which leads to:

a sin θ = nλ

for non-zero integers n.

---

Single Slit vs Double Slit

Single slit and double slit experiments both involve wave interference, but their patterns are different.

In a real double slit experiment, each individual slit also diffracts the light, so the double slit interference pattern is contained inside a broader diffraction envelope.

---

Slit Width vs Slit Separation

This distinction is essential.

For single slit diffraction:

a = width of one slit

and:

a sin θ = nλ

For double slit interference:

d = separation between the two slits

and bright fringes satisfy:

d sin θ = nλ

Using d instead of a is a common calculation error.

---

Diffraction of Sound

Diffraction is not unique to light.

Sound wavelengths are often comparable to the dimensions of:

• doors
• windows
• walls
• corridors

As a result, sound can diffract noticeably around obstacles and through openings.

This is one reason you can sometimes hear someone speaking from another room even when you cannot see them directly.

---

Diffraction of Radio Waves

Radio wavelengths can be much longer than visible-light wavelengths.

Some radio waves can therefore diffract significantly around:

• buildings
• hills
• other obstacles

The exact behaviour depends on wavelength, obstacle size, atmospheric conditions, and other propagation effects.

Diffraction is therefore important in communication engineering.

---

Why Light Usually Appears to Travel in Straight Lines

Visible-light wavelengths are extremely small compared with most everyday openings.

For example, visible wavelengths are roughly hundreds of nanometres, while a doorway is about a metre wide.

Therefore:

a >> λ

and diffraction through the doorway is extremely small.

Light consequently appears to travel approximately in straight lines in ordinary situations.

---

Diffraction Limits

Diffraction places a fundamental limit on how narrowly waves can be confined and how precisely optical systems can resolve small details.

This affects:

• microscopes
• telescopes
• cameras
• optical instruments

Even a perfect lens cannot eliminate diffraction.

The wave nature of light therefore creates fundamental limits on image resolution.

---

Diffraction and Telescopes

Light entering a telescope passes through a finite aperture.

Because of diffraction, light from a distant point does not form a perfectly infinitesimal point in the image.

Instead, it forms a small diffraction pattern.

Larger apertures generally reduce diffraction spreading and allow finer details to be resolved.

This is one reason large telescope apertures are scientifically valuable.

---

Diffraction and Microscopes

Microscopes also face diffraction limits.

Very small objects can only be distinguished if the imaging system can resolve their separate diffraction patterns.

Using shorter wavelengths can improve resolution.

This is one reason electron microscopy can achieve much higher resolution than ordinary visible-light microscopy: electrons can have extremely short de Broglie wavelengths.

---

Single Slit Diffraction and Matter

Matter particles can also diffract.

According to de Broglie's relationship:

λ = h/p

where:

• λ = matter wavelength
• h = Planck's constant
• p = momentum

Electrons, neutrons, atoms, and other particles can exhibit diffraction when interacting with structures of an appropriate size.

This provides further evidence for the wave nature of matter.

---

Experimental Investigation

A basic optical investigation can use:

• monochromatic laser
• narrow single slit
• screen
• ruler
• known slit width

The pattern on the screen can be measured.

Useful quantities include:

• central maximum width
• distance to first minimum
• slit-to-screen distance
• wavelength
• slit width

The relationship:

y₁ = λL/a

can then be tested experimentally.

---

Improving Measurements

A good diffraction experiment should:

• measure L carefully
• identify the center of the pattern
• measure between dark minima when possible
• use a sufficiently large screen distance
• repeat measurements
• use consistent units
• keep the slit and screen aligned

The central maximum width can often be measured more reliably than the position of one edge alone.

---

Worked Experimental Problem

A single slit produces a central maximum:

16.0 mm wide

The screen is:

2.50 m

from the slit.

The light wavelength is:

640 nm

Find the slit width.

Using:

W = 2λL/a

Rearrange:

a = 2λL/W

Convert:

λ = 6.40 × 10⁻⁷ m

W = 1.60 × 10⁻² m

Then:

a = 2(6.40 × 10⁻⁷)(2.50)/(1.60 × 10⁻²)

a = 2.00 × 10⁻⁴ m

Therefore:

a = 0.200 mm

---

A Complete Diffraction Problem

Light of wavelength:

λ = 550 nm

passes through a slit:

a = 0.10 mm

The screen is:

L = 1.5 m

First find the first minimum.

a sin θ₁ = λ

sin θ₁ = (550 × 10⁻⁹)/(1.0 × 10⁻⁴)

sin θ₁ = 0.0055

Therefore:

θ₁ ≈ 0.315°

Using the small-angle approximation:

y₁ = λL/a

y₁ = (550 × 10⁻⁹)(1.5)/(1.0 × 10⁻⁴)

y₁ = 8.25 mm

Therefore, the central maximum width is:

W = 2y₁

W = 16.5 mm

---

Predicting Changes

The equation:

W = 2λL/a

makes several useful predictions.

If λ doubles:

W doubles

If L doubles:

W doubles

If a doubles:

W halves

If a is halved:

W doubles

These proportional relationships can often solve problems without a full calculation.

---

Example: Multiple Changes

Suppose an experiment is changed so that:

• wavelength doubles
• slit width is halved
• screen distance remains unchanged

Since:

W ∝ λ/a

doubling λ gives:

×2

Halving a gives another:

×2

Therefore:

W_new = 4W_original

The central maximum becomes four times as wide.

---

Limits on Diffraction Orders

The minimum condition is:

a sin θ = nλ

Since:

sin θ ≤ 1

we must have:

nλ ≤ a

Therefore, only a limited number of diffraction minima can exist for a particular slit width and wavelength.

If a proposed value of n gives:

sin θ > 1

that diffraction minimum cannot physically occur.

---

Example: Possible Minima

Suppose:

a = 2.5λ

The minimum condition gives:

sin θ = nλ/a

For n = 1:

sin θ = 1/2.5 = 0.4

Possible.

For n = 2:

sin θ = 2/2.5 = 0.8

Possible.

For n = 3:

sin θ = 3/2.5 = 1.2

Impossible.

Therefore, only the first and second minima occur on each side before reaching 90°.

---

A Reliable Problem-Solving Strategy

First identify:

• wavelength λ
• slit width a
• screen distance L
• diffraction order n
• angle θ
• screen position y

For angular minima, use:

a sin θ = nλ

For small-angle screen positions:

y_n = nλL/a

For central maximum width:

W = 2λL/a

Always convert units before calculating.

Useful conversions include:

1 mm = 10⁻³ m

1 μm = 10⁻⁶ m

1 nm = 10⁻⁹ m

Finally, check whether the result makes physical sense.

A narrower slit or longer wavelength should produce more, not less, spreading.

---

Common Mistakes

Thinking diffraction only occurs when a slit is extremely narrow

Diffraction always occurs, but it becomes more noticeable when the opening size is comparable to the wavelength.

Thinking the single slit pattern requires two separate slits

Different parts of the same opening contribute waves that interfere.

Using n = 0 for a dark fringe

The minimum equation uses n = 1, 2, 3, ...

At θ = 0°, there is a central maximum.

Thinking the central maximum has the same width as the other bright fringes

It is approximately twice as wide as the secondary maxima under the usual small-angle conditions.

Thinking all bright fringes have equal intensity

The secondary maxima become progressively weaker.

Thinking a wider slit causes more diffraction

A narrower slit produces greater angular spreading.

Thinking shorter wavelengths diffract more

For the same slit, longer wavelengths produce greater diffraction.

Confusing slit width with slit separation

Single slit diffraction uses slit width a. Double slit interference commonly uses slit separation d.

Using W = λL/a for the full central width

λL/a gives the approximate distance from the center to the first minimum. The full central maximum width is 2λL/a.

---

Key Terms

Diffraction — spreading of a wave as it passes through an opening or around an obstacle.

Single slit diffraction — diffraction pattern produced when waves pass through one narrow opening.

Huygens' principle — model in which points on a wavefront act as sources of secondary wavelets.

Superposition — addition of overlapping wave displacements.

Interference — combination of overlapping waves.

Constructive interference — interference that increases resultant amplitude.

Destructive interference — interference that decreases resultant amplitude.

Central maximum — broad, bright central region of a single slit diffraction pattern.

Secondary maximum — weaker bright region on either side of the central maximum.

Diffraction minimum — dark region produced by destructive interference.

Slit width — width a of the opening through which the wave passes.

Diffraction angle — angle θ between the central direction and a feature in the diffraction pattern.

Wavelength — distance between corresponding points on successive wave cycles.

Intensity — wave power transferred per unit area; for light it determines the brightness of the pattern.

Angular width — angle between the two sides of a diffraction feature such as the central maximum.

---

Key Takeaways

• Diffraction is the spreading of waves through openings and around obstacles.
• Single slit diffraction occurs because contributions from different parts of one slit interfere.
• A single slit pattern contains a broad bright central maximum surrounded by dark minima and weaker secondary maxima.
• The central maximum is the brightest part of the pattern.
• The central maximum is approximately twice as wide as neighbouring secondary maxima.
• Secondary maxima become progressively weaker farther from the center.
• Single slit minima satisfy a sin θ = nλ, where n = 1, 2, 3, ...
• The first minimum defines one edge of the central maximum.
• Narrower slits produce greater diffraction.
• Longer wavelengths produce greater diffraction.
• For small angles, the position of a minimum is approximately y_n = nλL/a.
• The central maximum width is approximately W = 2λL/a.
• Increasing screen distance increases the physical width of the pattern but does not change its angular diffraction pattern.
• Red light produces greater diffraction than blue light through the same slit.
• Single slit diffraction is different from double slit interference, although both involve superposition.
• Slit width a must not be confused with double slit separation d.
• Diffraction places fundamental limits on the resolving ability of optical instruments.
• Sound, radio waves, light, and matter waves can all exhibit diffraction.
• The diffraction pattern provides strong evidence for the wave nature of light.

---

Check Your Understanding

1. Define diffraction and explain when it becomes particularly noticeable.
2. Explain how interference can occur even though a single slit experiment contains only one opening.
3. Describe the appearance of a typical single slit diffraction pattern.
4. Why is the central maximum brighter than the surrounding secondary maxima?
5. Explain why the central maximum is approximately twice as wide as the neighbouring bright fringes.
6. State the condition for a single slit diffraction minimum and explain each symbol.
7. Light of wavelength 500 nm passes through a slit 0.20 mm wide. Calculate the angle of the first minimum.
8. Using the same values from Question 7, calculate the angle of the second minimum.
9. A 600 nm laser passes through a 0.30 mm slit onto a screen 2.0 m away. Calculate the approximate distance from the center to the first minimum.
10. Using your result from Question 9, calculate the full width of the central maximum.
11. Predict what happens to the diffraction pattern if the slit width is reduced by half. Explain using the diffraction equation.
12. Red light and blue light pass separately through the same slit. Which produces the wider central maximum? Explain why.
13. A central maximum is 20 mm wide on a screen 2.5 m from a slit. The wavelength is 500 nm. Calculate the slit width.
14. Explain the difference between the roles of slit width in single slit diffraction and slit separation in double slit interference.
15. A student says, "Making a slit narrower should make the light beam narrower." Evaluate this statement using diffraction and the relationship between slit width, wavelength, and the angular width of the central maximum.

3. Diffraction Gratings

4. Interference and Diffraction Patterns

Interference and Diffraction Patterns

Interference and diffraction are closely related wave phenomena. Both arise from the superposition of waves, but they are usually associated with different experimental arrangements.

Interference occurs when two or more coherent waves overlap. Depending on their relative phase, they reinforce or cancel one another.

Diffraction is the spreading of waves as they pass through an opening or around an obstacle. In a single slit, waves from different parts of the same opening interfere and produce a diffraction pattern.

The patterns produced by single slits, double slits, and diffraction gratings look different, but all can be understood using the same fundamental principle:

when waves overlap, their displacements add according to the principle of superposition.

---

The Principle of Superposition

The principle of superposition states that when two or more waves overlap, the resultant displacement is the sum of their individual displacements.

In simple form:

resultant displacement = displacement₁ + displacement₂ + ...

The waves do not permanently destroy or alter one another. They combine while overlapping and then continue propagating.

Superposition produces two particularly important effects:

• constructive interference
• destructive interference

These effects create the bright and dark regions observed in optical interference and diffraction patterns.

---

Constructive Interference

Constructive interference occurs when waves arrive in phase.

For example:

crest + crest → larger crest

trough + trough → larger trough

For light, the increased wave amplitude produces greater intensity.

Therefore:

constructive interference → high intensity → bright region

For two coherent sources, constructive interference occurs when the path difference is:

Δd = nλ

where:

• Δd = path difference
• n = 0, 1, 2, 3, ...
• λ = wavelength

---

Destructive Interference

Destructive interference occurs when waves arrive out of phase.

A crest from one wave can overlap with a trough from another.

If the waves have equal amplitudes and are exactly 180° out of phase, complete cancellation occurs.

For light:

destructive interference → low or zero intensity → dark region

For two coherent sources, complete destructive interference occurs when:

Δd = (n + ½)λ

where:

n = 0, 1, 2, 3, ...

---

Amplitude and Intensity

The brightness of an interference or diffraction pattern depends on the intensity of the light.

For a wave:

intensity ∝ amplitude²

This means that changes in resultant amplitude can produce significant changes in brightness.

If two equal waves interfere constructively, their amplitudes add.

If each has amplitude A:

resultant amplitude = 2A

Since intensity depends on amplitude squared:

I ∝ (2A)² = 4A²

This explains why constructive interference can produce strong intensity maxima.

---

Path Difference

Path difference is the difference between the distances travelled by two waves before reaching the same point.

Suppose one wave travels distance r₁ and another travels r₂.

Then:

Δd = |r₂ − r₁|

Path difference determines the relative phase of the waves when they meet.

For waves of the same wavelength:

Δd = nλ → constructive interference

Δd = (n + ½)λ → destructive interference

Path difference is therefore one of the most important ideas for interpreting interference patterns.

---

Phase Difference

Interference can also be described using phase difference.

If two waves arrive with:

phase difference = 0°, 360°, 720° ...

they interfere constructively.

If they arrive with:

phase difference = 180°, 540°, 900° ...

they interfere destructively.

A path difference of one wavelength corresponds to a phase difference of:

360°

A path difference of half a wavelength corresponds to:

180°

---

What Is a Diffraction Pattern?

A diffraction pattern forms when waves spread through an opening and waves from different parts of that opening interfere.

A single slit produces:

• broad central maximum
• dark minima
• weaker secondary maxima

The central maximum is usually the brightest and widest part of the pattern.

---

Single Slit Diffraction

Consider monochromatic light passing through one narrow slit.

Different parts of the slit can be treated as sources of secondary wavelets.

These wavelets:

• spread beyond the slit
• overlap
• interfere

At some angles, the contributions reinforce.

At other angles, they cancel.

This produces the characteristic single slit diffraction pattern.

---

Single Slit Minima

For a slit of width a, the dark minima satisfy:

a sin θ = nλ

where:

• a = slit width
• θ = angle from the central axis
• λ = wavelength
• n = 1, 2, 3, ...

The first minimum occurs when:

a sin θ = λ

Notice that n begins at 1.

At:

θ = 0°

there is a bright central maximum, not a minimum.

---

The Single Slit Central Maximum

The central maximum extends between the first minimum on one side and the first minimum on the other.

For small angles:

y₁ = λL/a

where y₁ is the distance from the center to the first minimum.

Therefore, the full width of the central maximum is approximately:

W = 2λL/a

where:

• W = central maximum width
• L = slit-to-screen distance

The central maximum is approximately twice as wide as the neighbouring secondary maxima.

---

Why Secondary Maxima Are Weaker

Away from the center, contributions from different parts of the slit increasingly cancel.

Complete cancellation occurs at the minima.

Between minima, some constructive interference occurs, creating secondary maxima.

However, the contributions are not as strongly in phase as at the center.

Therefore:

central maximum → brightest

secondary maxima → progressively weaker

This produces a characteristic intensity distribution rather than a series of equally bright fringes.

---

Double Slit Interference

A double slit contains two narrow openings.

Each slit diffracts the incoming wave.

The waves from the two slits overlap and interfere.

For ideal narrow slits, the bright interference fringes satisfy:

d sin θ = nλ

where:

• d = slit separation
• θ = angle
• n = 0, 1, 2, ...
• λ = wavelength

For small angles, the spacing between neighbouring fringes is:

w = λL/d

---

The Ideal Double Slit Pattern

An idealized double slit pattern consists of regularly spaced:

bright → dark → bright → dark → bright

regions.

The central bright fringe occurs at:

θ = 0°

because the path difference from the two slits is zero.

For very narrow identical slits, the interference fringes are approximately equally spaced near the center.

---

Real Double Slits

Real slits have a finite width.

Therefore, each individual slit also produces its own diffraction pattern.

The result is that the double slit interference fringes are contained inside a broader diffraction envelope.

This is one of the most important connections between interference and diffraction.

---

The Diffraction Envelope

The diffraction envelope describes the broad variation in intensity caused by diffraction from each individual slit.

Inside this envelope are the much narrower double slit interference fringes.

The pattern therefore has two different scales:

broad pattern → single slit diffraction

narrow fringes → double slit interference

The diffraction envelope controls how bright the interference fringes can be at different positions.

---

Why Fringe Brightness Changes

If double slit interference alone were considered, the bright fringes might be expected to have similar maximum intensities.

But because each slit has finite width, diffraction also occurs.

The interference maxima are therefore multiplied by the diffraction intensity distribution.

Near the center:

diffraction intensity high → bright interference fringes

Farther from the center:

diffraction intensity lower → weaker interference fringes

At a diffraction minimum:

diffraction intensity ≈ 0

so an interference maximum cannot appear strongly there.

---

Interference Inside an Envelope

A useful way to visualize a real double slit pattern is:

interference determines where the narrow fringes occur

while:

diffraction determines the overall brightness envelope

Therefore, the two phenomena do not compete with each other.

They occur simultaneously.

---

Missing Orders

Sometimes a predicted double slit interference maximum falls exactly at a single slit diffraction minimum.

The diffraction envelope has zero intensity at that position.

The expected bright interference fringe therefore disappears.

This is called a missing order.

For double slit maxima:

d sin θ = nλ

For single slit minima:

a sin θ = mλ

If both conditions occur at the same angle, the interference maximum is suppressed.

---

Example of a Missing Order

Suppose:

d = 3a

A diffraction minimum occurs when:

a sin θ = mλ

Multiply by 3:

d sin θ = 3mλ

But double slit maxima satisfy:

d sin θ = nλ

Therefore:

n = 3m

So every third interference order coincides with a diffraction minimum.

Those maxima are missing from the observed pattern.

This is a good example of how interference and diffraction must sometimes be analyzed together.

---

Diffraction Gratings

A diffraction grating contains many closely spaced slits or grooves.

Like a double slit, waves from different openings interfere.

However, because many openings contribute, constructive interference occurs strongly only at very specific angles.

The bright maxima satisfy:

d sin θ = nλ

This looks like the double slit equation, but the resulting pattern is noticeably different.

---

Why Grating Maxima Are Narrow

Consider two slits.

A small change in angle produces only a relatively small phase difference between the two waves.

Now consider hundreds or thousands of slits.

A small change in angle produces increasingly large phase differences between waves coming from widely separated slits.

Away from the exact constructive-interference angle, cancellation therefore becomes much stronger.

The result is:

many slits → narrower, sharper maxima

---

Single Slit, Double Slit, and Grating Patterns

The three patterns have distinctive appearances.

---

Recognizing a Single Slit Pattern

Look for:

• one very broad central maximum
• central maximum approximately twice the width of secondary maxima
• secondary maxima that become progressively weaker
• dark minima separating the maxima

If these features dominate, the pattern is probably produced by a single slit.

---

Recognizing a Double Slit Pattern

Look for:

• many regularly spaced interference fringes
• alternating bright and dark regions
• a broader variation in brightness across the pattern
• interference fringes contained inside a diffraction envelope

A real double slit pattern therefore combines two scales of structure.

---

Recognizing a Diffraction Grating Pattern

Look for:

• very narrow bright maxima
• large dark regions between maxima
• sharply defined diffraction orders
• strong wavelength separation if several wavelengths are present

Compared with a double slit, grating maxima are generally much sharper.

---

Comparing Intensity Graphs

Intensity graphs provide a useful way to distinguish the patterns.

A single slit graph has:

one large broad central peak + smaller side peaks

A double slit graph has:

many closely spaced peaks inside a broad envelope

A diffraction grating graph has:

very narrow, sharp peaks separated by regions of very low intensity

The number and width of openings therefore affect both the positions and shapes of the intensity maxima.

---

Effect of Wavelength

For single slit diffraction:

a sin θ = nλ

For double slit interference:

d sin θ = nλ

For diffraction gratings:

d sin θ = nλ

In every case, increasing wavelength generally moves corresponding features to larger angles.

Therefore:

larger λ → pattern spreads farther

This is why red light generally produces greater angular separation than blue light in the same apparatus.

---

Effect of Single Slit Width

For single slit minima:

a sin θ = nλ

If a decreases:

sin θ increases

Therefore:

narrower slit → wider diffraction pattern

The central maximum becomes wider.

If a increases:

wider slit → narrower diffraction pattern

---

Effect of Double Slit Separation

For double slit maxima:

d sin θ = nλ

Increasing d decreases the angle of a particular interference order.

For small angles:

w = λL/d

Therefore:

larger slit separation → closer interference fringes

smaller slit separation → wider fringe spacing

---

Effect of Individual Slit Width in a Double Slit

The separation d controls the spacing of the interference fringes.

The individual slit width a controls the diffraction envelope.

Therefore:

d controls fringe spacing

while:

a controls envelope width

This distinction is extremely important when interpreting a real double slit pattern.

---

Worked Example: Single Slit Pattern

A 600 nm laser passes through a slit:

a = 0.20 mm

onto a screen:

L = 2.0 m

For the first minimum:

y₁ ≈ λL/a

y₁ = (600 × 10⁻⁹)(2.0)/(2.0 × 10⁻⁴)

y₁ = 0.0060 m

y₁ = 6.0 mm

Therefore, the central maximum width is:

W = 12 mm

---

Worked Example: Double Slit Pattern

The same 600 nm light passes through two slits separated by:

d = 0.40 mm

with:

L = 2.0 m

Fringe spacing:

w = λL/d

w = (600 × 10⁻⁹)(2.0)/(4.0 × 10⁻⁴)

w = 0.0030 m

w = 3.0 mm

Therefore, interference fringes occur every 3.0 mm.

---

Combining the Two Results

Suppose each slit in the double slit experiment is:

a = 0.20 mm

wide, while their separation is:

d = 0.40 mm

The diffraction envelope central width is:

12 mm

The interference fringe spacing is:

3 mm

Therefore, several narrow interference fringes fit inside the broad central diffraction envelope.

This demonstrates how the two patterns combine.

---

Worked Example: Changing the Slit Width

Suppose the individual slit width is reduced from:

0.20 mm

to:

0.10 mm

while d remains unchanged.

Because:

W ∝ 1/a

the diffraction envelope becomes twice as wide.

But because d has not changed:

w = λL/d

the spacing of the interference fringes remains unchanged.

Therefore:

narrower individual slits → wider envelope, same fringe spacing

---

Worked Example: Changing Slit Separation

Now keep the slit width constant but double d.

Since:

w ∝ 1/d

the interference fringe spacing halves.

But because a has not changed, the diffraction envelope remains approximately the same width.

Therefore:

larger slit separation → closer fringes, same envelope

This distinction is often tested in pattern-analysis questions.

---

Diffraction Grating Example

Suppose a diffraction grating contains:

500 lines/mm

First calculate d:

500 lines/mm = 500,000 lines/m

d = 1/500,000

d = 2.00 × 10⁻⁶ m

For:

λ = 600 nm

and first order:

d sin θ = λ

sin θ = (600 × 10⁻⁹)/(2.00 × 10⁻⁶)

sin θ = 0.300

Therefore:

θ ≈ 17.5°

Instead of producing a broad bright fringe, the many grating lines produce a sharply defined maximum near this angle.

---

Interpreting Changes in a Pattern

When analyzing an unfamiliar pattern, ask:

How wide is the central maximum?

A very broad central maximum suggests single slit diffraction.

Are there many regularly spaced narrow fringes?

This suggests double slit interference.

Are the maxima extremely sharp and widely separated?

This suggests a diffraction grating.

Do the interference fringes become weaker toward the edges?

This indicates a diffraction envelope.

Are some expected fringes missing?

An interference maximum may coincide with a diffraction minimum.

---

Water Wave Patterns

The same principles can be observed with water waves.

Two coherent water-wave sources can produce alternating regions of constructive and destructive interference.

A narrow opening produces diffraction.

Therefore, optical patterns are not unique to light.

They are examples of general wave behaviour.

---

Sound Interference and Diffraction

Sound also demonstrates both effects.

Two coherent speakers can produce regions where sound is:

• louder due to constructive interference
• quieter due to destructive interference

Sound can also diffract through doorways and around obstacles.

Because sound wavelengths are much longer than visible-light wavelengths, diffraction is often obvious on everyday scales.

---

Light as a Wave

Interference and diffraction provide powerful evidence for the wave nature of light.

A simple ray model cannot fully explain:

• bright and dark interference fringes
• diffraction spreading
• diffraction minima
• diffraction envelopes

These observations require a wave description involving phase and superposition.

---

Matter Waves

Particles such as electrons can also produce interference and diffraction patterns.

According to de Broglie:

λ = h/p

where:

• λ = matter wavelength
• h = Planck's constant
• p = momentum

Electron diffraction and interference provide evidence that matter can display wave-like behaviour.

The same ideas used to analyze light patterns can therefore extend into quantum physics.

---

Applications of Interference and Diffraction

These phenomena are used in:

• spectroscopy
• wavelength measurement
• astronomical observations
• interferometers
• optical coatings
• communication technology
• microscopy
• crystal structure analysis
• precision measurement
• laser technology

Understanding patterns allows scientists to obtain information that may not be visible directly.

---

Diffraction and Resolution

Diffraction also limits the ability of optical instruments to distinguish small details.

Every finite aperture produces diffraction.

This affects:

• telescopes
• microscopes
• cameras

Larger apertures generally produce less angular diffraction and therefore can provide better resolution.

Diffraction is therefore both a useful scientific tool and a fundamental limitation.

---

Experimental Analysis

Suppose students are given an unknown laser pattern.

A useful analysis procedure is:

• identify the central maximum
• compare its width with surrounding maxima
• examine whether narrow fringes appear inside a broad envelope
• measure fringe spacing
• measure distances to minima
• determine whether the maxima are broad or extremely sharp
• compare observations with predicted equations

This allows the apparatus to be identified from the pattern itself.

---

Choosing the Correct Equation

For single slit minima:

a sin θ = nλ

For the small-angle central width:

W = 2λL/a

For double slit bright fringes:

d sin θ = nλ

For small-angle double slit fringe spacing:

w = λL/d

For diffraction grating maxima:

d sin θ = nλ

The equations may look similar, so always identify what a and d represent before calculating.

---

A Complete Pattern Analysis

Suppose a pattern contains narrow interference fringes separated by:

2.0 mm

The broad central diffraction envelope has width:

20 mm

The narrow fringes indicate double slit interference.

The broad intensity variation indicates diffraction from the individual slits.

Approximately:

20/2 = 10

fringe spacings fit across the width of the central envelope.

The pattern therefore provides information about both:

• slit separation d
• individual slit width a

A single experimental image can contain evidence of both interference and diffraction.

---

Common Mistakes

Thinking interference and diffraction are completely unrelated

Both result from superposition. Diffraction patterns themselves involve interference between wave contributions.

Thinking diffraction requires two slits

A single opening can produce a diffraction pattern.

Thinking a double slit produces only interference

Real double slits also produce diffraction because each slit has finite width.

Confusing a with d

The individual slit width a controls the single slit diffraction envelope. The slit separation d controls double slit fringe spacing.

Thinking all double slit fringes must have equal brightness

A real double slit pattern is modified by the diffraction envelope.

Thinking the central single slit maximum has the same width as the secondary maxima

It is approximately twice as wide.

Thinking more grating slits simply produce more equally wide fringes

Increasing the number of slits makes the principal maxima much sharper.

Assuming a dark region means no waves arrive there

Waves can arrive but cancel through destructive interference.

Thinking destructive interference destroys energy

Interference redistributes energy spatially. Regions of reduced intensity are accompanied by regions of increased intensity.

---

Key Terms

Superposition — principle stating that overlapping wave displacements add.

Interference — combination of overlapping waves.

Constructive interference — interference that increases resultant amplitude.

Destructive interference — interference that decreases resultant amplitude.

Path difference — difference in distances travelled by interfering waves.

Phase difference — difference in the stage of oscillation of two waves.

Diffraction — spreading of waves through an opening or around an obstacle.

Diffraction pattern — distribution of maxima and minima produced by diffraction and interference.

Central maximum — broad, bright central region of a single slit diffraction pattern.

Secondary maximum — weaker bright region outside the central maximum.

Diffraction minimum — low-intensity region produced by destructive interference in a diffraction pattern.

Interference fringe — bright or dark region produced by interference.

Diffraction envelope — broad intensity distribution that controls the brightness of interference fringes from finite-width slits.

Missing order — interference maximum suppressed because it coincides with a diffraction minimum.

Slit width (a) — width of an individual opening.

Slit separation (d) — distance between corresponding points of neighbouring slits, usually their centers.

Diffraction grating — device containing many regularly spaced openings or grooves.

Intensity — wave power per unit area; related to the square of wave amplitude.

---

Key Takeaways

• Interference and diffraction are both consequences of wave superposition.
• Constructive interference produces increased amplitude and bright regions.
• Destructive interference produces reduced amplitude and dark regions.
• Path difference determines whether waves interfere constructively or destructively.
• Single slit diffraction produces a broad bright central maximum surrounded by weaker secondary maxima.
• Single slit minima satisfy a sin θ = nλ.
• The central single slit maximum is approximately twice as wide as neighbouring secondary maxima.
• A double slit produces regularly spaced interference fringes.
• Double slit bright fringes satisfy d sin θ = nλ.
• Double slit fringe spacing is approximately w = λL/d.
• Real double slit patterns are contained within a broader diffraction envelope.
• Slit width controls the diffraction envelope.
• Slit separation controls interference fringe spacing.
• An interference maximum can disappear when it coincides with a diffraction minimum.
• Diffraction gratings use many openings to produce extremely sharp principal maxima.
• Increasing the number of contributing slits generally makes the principal maxima narrower and more distinct.
• Longer wavelengths generally produce greater angular spreading.
• Interference and diffraction occur with water, sound, light, and matter waves.
• Pattern analysis can reveal information about wavelength and the dimensions of the experimental apparatus.
• Interference and diffraction provide strong evidence for the wave nature of light and form the basis of many important optical technologies.

---

Check Your Understanding

1. Explain how the principle of superposition leads to constructive and destructive interference.
2. State the path-difference conditions for constructive and destructive interference between two coherent waves.
3. Describe the main features of a single slit diffraction pattern.
4. Explain why the central maximum in a single slit pattern is wider and brighter than the secondary maxima.
5. Describe the main features of an ideal double slit interference pattern.
6. Explain what is meant by a diffraction envelope and how it affects a real double slit pattern.
7. A 600 nm laser passes through a single slit 0.15 mm wide onto a screen 2.0 m away. Calculate the approximate width of the central maximum.
8. The same laser is used with two slits separated by 0.50 mm. The screen remains 2.0 m away. Calculate the interference fringe spacing.
9. Explain what happens to a single slit diffraction pattern when the slit becomes narrower.
10. Explain what happens to a double slit pattern when the slit separation increases but the individual slit widths remain unchanged.
11. A student reduces the individual width of both slits but keeps their separation unchanged. Predict what happens to the interference fringe spacing and the diffraction envelope.
12. Explain why some interference orders can be missing from a real double slit pattern.
13. Compare the intensity patterns produced by a single slit, a double slit, and a diffraction grating.
14. Explain why diffraction gratings produce sharper principal maxima than double slits.
15. An unknown optical pattern contains many narrow, equally spaced fringes whose brightness gradually decreases inside a broad central envelope. Identify the most likely arrangement that produced the pattern and explain how both interference and diffraction contribute to its appearance.

5. Factors Affecting Diffraction and Interference

 

Factors Affecting Diffraction and Interference

Diffraction and interference patterns are controlled by several important experimental variables. Changing the wavelength, slit width, slit separation, number of slits, screen distance, or coherence of the waves can significantly change what is observed.

The main relationships are:

Single slit diffraction minima:

a sin θ = nλ

Approximate single slit central maximum width:

W = 2λL/a

Double slit interference maxima:

d sin θ = nλ

Approximate double slit fringe spacing:

w = λL/d

These equations reveal an important overall pattern:

• longer wavelength → more spreading
• narrower slit → more diffraction
• smaller double-slit separation → wider fringe spacing
• greater screen distance → greater physical spacing on the screen
• more grating slits → sharper principal maxima
• greater coherence → clearer, more stable interference

---

The Role of Wavelength

Wavelength is one of the most important factors affecting both diffraction and interference.

For single slit diffraction:

a sin θ = nλ

For a fixed slit width a, increasing λ increases θ.

Therefore:

longer wavelength → greater diffraction angle

For double slit interference:

w = λL/d

Therefore:

longer wavelength → greater fringe spacing

Wavelength affects both the spreading of diffraction patterns and the separation of interference fringes.

---

Longer Wavelengths

Suppose all other experimental conditions remain constant.

If wavelength increases:

• diffraction angles increase
• the single slit central maximum becomes wider
• double slit fringes become farther apart
• diffraction-grating maxima move to larger angles

Therefore:

λ ↑ → pattern spreads out

---

Shorter Wavelengths

If wavelength decreases:

• diffraction angles decrease
• the central diffraction maximum becomes narrower
• double slit fringes become closer together
• grating maxima occur at smaller angles

Therefore:

λ ↓ → pattern contracts

---

Red Light vs Blue Light

Red visible light has a longer wavelength than blue light.

For the same apparatus:

λ_red > λ_blue

Therefore, red light generally produces:

• greater single slit diffraction
• wider double slit fringe spacing
• larger diffraction-grating angles

Blue light produces a more closely spaced pattern.

---

Worked Example: Changing Wavelength

Suppose a double slit experiment uses:

λ₁ = 450 nm

d = 0.30 mm

L = 2.0 m

Using:

w = λL/d

Convert:

λ₁ = 4.50 × 10⁻⁷ m

d = 3.00 × 10⁻⁴ m

Then:

w₁ = (4.50 × 10⁻⁷)(2.0)/(3.00 × 10⁻⁴)

w₁ = 3.0 mm

Now replace the light with:

λ₂ = 600 nm

Then:

w₂ = (6.00 × 10⁻⁷)(2.0)/(3.00 × 10⁻⁴)

w₂ = 4.0 mm

The longer wavelength produces wider fringe spacing.

---

Wavelength and Diffraction

The importance of wavelength can also be understood by comparing it with the size of an opening.

If:

a >> λ

diffraction is relatively small.

If a becomes comparable to λ:

a ≈ λ

diffraction becomes very significant.

This principle applies to all types of waves, not just light.

---

Why Sound Diffracts Easily

Sound wavelengths can be comparable to the dimensions of:

• doorways
• windows
• corridors
• buildings

Sound therefore diffracts noticeably around everyday objects.

Visible-light wavelengths are extremely small compared with these objects.

As a result, visible-light diffraction around everyday openings is much less noticeable.

---

Slit Width and Single Slit Diffraction

The width of a single slit is represented by a.

The minima satisfy:

a sin θ = nλ

For the first minimum:

sin θ₁ = λ/a

Therefore:

smaller a → larger θ₁

A narrower slit produces a broader diffraction pattern.

---

Narrowing the Slit

When the slit becomes narrower:

• diffraction increases
• first minima move farther from the center
• central maximum becomes wider
• light is spread over a greater angular range

For small angles:

W = 2λL/a

Therefore:

W ∝ 1/a

Halving the slit width approximately doubles the central maximum width.

---

Widening the Slit

If the slit becomes wider:

a ↑ → diffraction angle ↓

Therefore:

• central maximum becomes narrower
• minima move toward the center
• wave spreads less

When the opening is enormously larger than the wavelength, diffraction may become difficult to observe.

---

Slit Width and Intensity

Slit width also affects the amount of wave energy passing through the opening.

A wider slit allows more light through than a narrower slit, assuming the same incident illumination.

Therefore, changing slit width can affect both:

• the width of the diffraction pattern
• the intensity of the pattern

A narrower slit generally allows less total light through while spreading that transmitted light over a greater angular range.

As a result, the central maximum becomes wider but may also become less intense.

---

Width vs Brightness

It is important to distinguish these two effects.

Narrowing the slit causes:

greater diffraction → wider pattern

but also:

less transmitted light → reduced total transmitted power

Therefore, a wider diffraction pattern does not necessarily mean a brighter pattern.

---

Worked Example: Changing Slit Width

A single slit experiment initially has:

a = 0.20 mm

λ = 600 nm

L = 2.0 m

Central maximum width:

W = 2λL/a

W = 2(600 × 10⁻⁹)(2.0)/(2.00 × 10⁻⁴)

W = 12 mm

Now reduce the slit width to:

a = 0.10 mm

The slit width has been halved.

Since:

W ∝ 1/a

the new central width is:

W = 24 mm

The pattern becomes twice as wide.

---

Double Slit Separation

For double slit interference, the distance between the centers of the two slits is represented by d.

The bright maxima satisfy:

d sin θ = nλ

For small angles, the fringe spacing is:

w = λL/d

Therefore:

w ∝ 1/d

This means slit separation has the opposite relationship to fringe spacing.

---

Increasing Slit Separation

If the slits are moved farther apart:

d ↑ → w ↓

The fringes move closer together.

For example, doubling d approximately halves the fringe spacing.

---

Decreasing Slit Separation

If the slits are moved closer together:

d ↓ → w ↑

The fringes become farther apart.

This is a common source of confusion because students sometimes expect wider-separated slits to produce wider-separated fringes.

The equation shows the opposite.

---

Worked Example: Changing Slit Separation

Suppose:

λ = 500 nm

L = 2.0 m

d = 0.20 mm

Then:

w = λL/d

w = (5.00 × 10⁻⁷)(2.0)/(2.00 × 10⁻⁴)

w = 5.0 mm

Now double the slit separation:

d = 0.40 mm

Then:

w = (5.00 × 10⁻⁷)(2.0)/(4.00 × 10⁻⁴)

w = 2.5 mm

Doubling the slit separation halves the fringe spacing.

---

Slit Width vs Slit Separation

In a real double slit experiment, both quantities can matter.

Individual slit width a controls the broad diffraction envelope.

Slit separation d controls the spacing between the narrow interference fringes.

Therefore:

change a → change envelope width

change d → change interference fringe spacing

These effects should not be confused.

---

Changing Slit Width in a Double Slit Experiment

Suppose the slit separation remains unchanged but each individual slit becomes narrower.

Because d is unchanged:

w = λL/d

so the interference fringe spacing remains approximately unchanged.

However, narrowing each slit increases diffraction.

Therefore:

narrower individual slits → wider diffraction envelope

More interference fringes may then fit inside the central diffraction envelope.

---

Changing Slit Separation Without Changing Width

Suppose the individual slit width remains unchanged while the separation d increases.

The diffraction envelope remains approximately unchanged because a has not changed.

But:

w ∝ 1/d

so the interference fringes move closer together.

Therefore:

same envelope + more closely spaced fringes

---

Worked Combined Example

Suppose a double slit arrangement has:

a = 0.10 mm

d = 0.50 mm

Now:

• a is halved
• d is doubled

Halving a makes the diffraction envelope:

2 times wider

Doubling d makes the interference fringe spacing:

½ as large

The new pattern therefore has:

a wider envelope containing more closely spaced fringes

This type of reasoning is useful when analyzing experimental changes without doing detailed calculations.

---

Number of Slits

The number of slits has an important effect on interference.

A double slit contains:

N = 2

A diffraction grating can contain hundreds or thousands of illuminated slits or grooves.

Increasing the number of slits changes the shape and sharpness of the interference maxima.

---

Increasing the Number of Slits

When more equally spaced coherent slits contribute:

• principal maxima become narrower
• principal maxima become more sharply defined
• intensity becomes more concentrated around the principal maxima
• regions between principal maxima generally become much darker

Therefore:

more slits → sharper interference pattern

This is why diffraction gratings are excellent for precise wavelength measurements.

---

Why More Slits Produce Sharper Maxima

At the exact constructive-interference angle, waves from all slits arrive in phase.

Their amplitudes reinforce strongly.

Move slightly away from this angle and small phase differences occur between neighbouring slits.

Across a large number of slits, these phase differences accumulate.

Waves from different parts of the grating then cancel strongly.

Therefore, only a narrow range of angles around the correct value produces strong constructive interference.

---

Number of Slits and Maximum Intensity

If N identical coherent slits contribute equal amplitudes at a principal maximum, their amplitudes can add approximately as:

A_total ∝ N

Since:

I ∝ A²

the ideal principal maximum intensity can scale approximately as:

I_max ∝ N²

This assumes comparable illumination per slit and ideal coherence.

The key qualitative result is:

more coherent slits → much stronger and sharper principal maxima

---

Two Slits vs Many Slits

---

Grating Spacing

For a diffraction grating:

d sin θ = nλ

where d is the distance between neighbouring grating lines.

For the same wavelength and order:

smaller d → larger θ

Therefore, increasing the number of lines per millimetre generally increases the angular separation of wavelengths because the grating spacing becomes smaller.

---

Lines per Millimetre

A grating may be specified as:

600 lines/mm

Convert to lines per metre:

600 × 1000 = 600,000 lines/m

Then:

d = 1/600,000

d = 1.67 × 10⁻⁶ m

A greater line density corresponds to a smaller value of d.

---

Number of Illuminated Slits vs Line Density

These ideas are related but not identical.

Line density describes how closely spaced the grating lines are.

It affects the angles of the maxima through:

d sin θ = nλ

The number of illuminated slits affects how narrow and sharply defined the maxima are.

Therefore:

smaller spacing d → greater angular separation

while:

larger number of illuminated slits N → sharper maxima

Both can improve a grating's usefulness in spectroscopy, but in different ways.

---

Screen Distance

Screen distance L affects the physical size of patterns observed on a screen.

For double slit interference:

w = λL/d

Therefore:

w ∝ L

If the screen is moved farther away:

fringes become farther apart on the screen

---

Screen Distance and Single Slit Diffraction

For the central maximum:

W = 2λL/a

Therefore:

W ∝ L

Moving the screen farther away makes the central maximum physically wider on the screen.

However, the diffraction angles do not change.

The pattern simply has more distance over which to spread.

---

Angular Pattern vs Screen Pattern

This distinction is important.

Changing L does not change the angles determined by:

a sin θ = nλ

or:

d sin θ = nλ

Instead, it changes where those angles intersect the screen.

Therefore:

larger L → same angles, greater physical separation

---

Worked Example: Changing Screen Distance

Suppose double slit fringes have spacing:

w = 3.0 mm

when:

L = 1.5 m

The screen is moved to:

L = 3.0 m

Since L has doubled:

w_new = 6.0 mm

The fringe spacing doubles.

The interference angles themselves remain unchanged.

---

Coherence

A stable interference pattern requires the waves to have a predictable phase relationship.

Such waves are described as coherent.

Coherent waves have:

• the same frequency
• a stable phase relationship

Without sufficient coherence, the positions of constructive and destructive interference change rapidly.

The pattern can then become blurred or disappear.

---

Why a Single Source Is Useful

In a double slit experiment, both slits are usually illuminated by the same source.

This helps create coherent waves because both slit waves originate from the same incoming wavefront.

A laser is especially useful because it provides light with:

• a narrow range of wavelengths
• strong spatial coherence
• strong directionality

This makes interference fringes easier to observe.

---

What Happens When Coherence Decreases?

If the relative phase between the two waves changes randomly:

• constructive regions move
• destructive regions move
• the detector averages many different patterns
• fringe contrast decreases

Eventually, distinct interference fringes may disappear.

Therefore:

greater coherence → clearer fringes

poor coherence → washed-out fringes

---

Fringe Visibility

The contrast between bright and dark fringes is sometimes called fringe visibility.

High visibility means:

• bright maxima are clearly bright
• dark minima are clearly dark

Poor visibility means the difference between bright and dark regions is small.

Coherence strongly affects fringe visibility.

---

Equal Intensities and Fringe Contrast

The clearest interference pattern is generally obtained when the waves from the two slits have similar intensities.

If one slit produces a much stronger wave than the other, destructive interference cannot cancel the stronger wave completely.

The dark fringes are therefore less dark.

This reduces the contrast of the pattern.

---

Monochromatic vs White Light

Monochromatic light has essentially one wavelength.

It produces a regular pattern associated with that wavelength.

White light contains many wavelengths.

Because fringe spacing depends on λ:

w = λL/d

different wavelengths produce different fringe spacings.

Their patterns therefore overlap.

---

White Light Double Slit Pattern

At the center:

Δd = 0

so all visible wavelengths can interfere constructively.

The central fringe is therefore approximately white.

Away from the center, different wavelengths reach maxima at different positions.

This produces coloured fringes.

Farther out, overlapping patterns can make the fringes increasingly difficult to distinguish.

---

Wavelength Range and Pattern Clarity

A source with a very narrow wavelength range produces clearer fringes.

A source containing many wavelengths produces overlapping patterns.

Therefore:

narrow wavelength range → clearer interference pattern

broad wavelength range → greater overlap and reduced visibility

This is one reason lasers are convenient for interference experiments.

---

Alignment

Experimental alignment also affects the pattern.

The source, slits, and screen should be positioned carefully.

Poor alignment can:

• shift the pattern
• reduce symmetry
• reduce brightness
• make measurements more difficult

Accurate alignment is especially important in precise optical experiments.

---

Distance Between Source and Slits

The illumination of the slits should be sufficiently uniform and coherent.

If one slit receives much more light than the other, fringe visibility decreases.

Optical arrangements often use additional components to control the incoming wavefront before it reaches the slits.

---

Medium and Wavelength

Wave behaviour also depends on the medium.

For a wave:

v = fλ

If wave speed changes while frequency remains constant, wavelength changes.

Because diffraction depends on wavelength, changing the medium can change diffraction behaviour.

This is especially important when comparing waves travelling through different materials.

---

Predicting Pattern Changes

A useful strategy is to identify which equation contains the changed variable.

For a single slit:

W = 2λL/a

Therefore:

• λ ↑ → W ↑
• L ↑ → W ↑
• a ↑ → W ↓

For a double slit:

w = λL/d

Therefore:

• λ ↑ → w ↑
• L ↑ → w ↑
• d ↑ → w ↓

These relationships allow many questions to be solved without numerical calculation.

---

Proportional Reasoning

Suppose wavelength increases by 50%.

Then:

λ_new = 1.5λ

For the same double slit apparatus:

w_new = 1.5w

The fringe spacing increases by 50%.

For a single slit:

W_new = 1.5W

The central maximum width also increases by 50%.

---

Multiple Changes

Sometimes several variables change simultaneously.

Suppose:

• λ doubles
• L triples
• d doubles

For double slit interference:

w = λL/d

Therefore:

w_new/w_old = (2)(3)/2

w_new/w_old = 3

The fringe spacing becomes three times larger.

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Another Multiple-Change Example

For single slit diffraction:

W = 2λL/a

Suppose:

• wavelength doubles
• screen distance halves
• slit width becomes one quarter of its original value

Then:

W_new/W_old = (2)(1/2)/(1/4)

W_new/W_old = 4

Therefore:

the central maximum becomes four times wider

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Pattern Analysis Table

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Interpreting an Experimental Pattern

Suppose a student observes a double slit pattern and then makes one change.

After the change:

• fringes remain equally bright
• fringe spacing increases
• diffraction envelope remains the same angular width

A likely explanation is that:

slit separation d decreased

because decreasing d increases fringe spacing without directly changing the individual slit diffraction envelope.

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Another Pattern Analysis

Suppose:

• fringe spacing stays the same
• diffraction envelope becomes wider

This suggests:

individual slit width a decreased

while d remained unchanged.

The narrow interference fringes retain their spacing, but more of them may fit inside the broader envelope.

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A Third Pattern Analysis

Suppose:

• the physical fringe spacing doubles
• angular spacing remains unchanged

This suggests that:

screen distance L doubled

The interference geometry has not changed, but the screen intercepts the same angular pattern farther away.

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A Fourth Pattern Analysis

Suppose the positions of the fringes remain approximately unchanged, but the pattern becomes blurred and the dark fringes are no longer clearly dark.

A likely cause is:

reduced coherence or unequal wave intensities

The geometry may remain the same while fringe visibility decreases.

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Designing a Wider Double Slit Pattern

To increase double slit fringe spacing:

w = λL/d

A student could:

• use a longer wavelength
• move the screen farther away
• reduce the slit separation

Any of these changes increases w.

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Designing a Wider Single Slit Pattern

To increase the width of a single slit central maximum:

W = 2λL/a

A student could:

• use a longer wavelength
• move the screen farther away
• use a narrower slit

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Designing a Sharper Grating Spectrum

To obtain sharply defined diffraction maxima:

• illuminate more grating lines
• use coherent light
• maintain good optical alignment
• use a high-quality regularly spaced grating

To increase angular separation between wavelengths, a smaller grating spacing can also be useful.

This explains why diffraction gratings are valuable in spectroscopy.

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Experimental Trade-Offs

Changes that improve one property can create another challenge.

For example:

narrower slit

produces more diffraction but transmits less light.

larger screen distance

produces wider physical spacing but can make the pattern dimmer and require more laboratory space.

more grating lines

can produce sharper maxima but requires precise manufacturing and alignment.

Good experimental design therefore involves balancing several factors.

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A Reliable Problem-Solving Strategy

First identify the pattern:

• single slit
• double slit
• diffraction grating

Then identify the variable being changed:

• λ
• a
• d
• L
• N
• coherence

Next choose the appropriate relationship.

For single slit minima:

a sin θ = nλ

For central maximum width:

W = 2λL/a

For double slit maxima:

d sin θ = nλ

For double slit fringe spacing:

w = λL/d

For grating maxima:

d sin θ = nλ

Then predict the direction of change before calculating.

This provides a useful check on the final answer.

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Common Mistakes

Thinking shorter wavelengths diffract more

Longer wavelengths generally produce greater diffraction for the same opening.

Thinking a wider slit produces more spreading

A narrower slit produces a wider diffraction pattern.

Thinking wider-separated double slits produce wider-separated fringes

The relationship is inverse: increasing d decreases fringe spacing.

Confusing slit width a with slit separation d

Slit width controls the diffraction envelope; slit separation controls double slit fringe spacing.

Thinking moving the screen changes the diffraction angle

It changes the physical spacing measured on the screen, not the angles established by the slit geometry.

Thinking more grating slits mainly move the maxima

The number of illuminated slits mainly affects their sharpness and intensity. Grating spacing determines their angular positions.

Confusing number of slits with lines per millimetre

The total number of illuminated slits and their spacing are different properties.

Ignoring coherence

A geometrically correct setup can still produce poor interference fringes if the waves do not maintain a stable phase relationship.

Thinking wider patterns must be brighter

A narrower slit can produce a wider diffraction pattern while allowing less total light through.

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Key Terms

Wavelength (λ) — distance between corresponding points on successive wave cycles.

Slit width (a) — width of an individual opening, especially important in single slit diffraction.

Slit separation (d) — distance between neighbouring slits, usually measured center to center.

Screen distance (L) — distance from the slit system to the observation screen.

Fringe spacing (w) — distance between neighbouring interference maxima or minima.

Central maximum — broad central bright region of a single slit diffraction pattern.

Diffraction angle — angle through which a wave spreads relative to the central direction.

Diffraction envelope — broad intensity distribution produced by the finite width of individual slits.

Coherence — stable phase relationship between waves.

Monochromatic light — light consisting of essentially one wavelength.

Fringe visibility — degree of contrast between bright and dark interference fringes.

Line density — number of grating lines per unit length.

Grating spacing — distance between neighbouring lines or slits of a diffraction grating.

Principal maximum — strong, narrow maximum produced by constructive interference from many grating slits.

Resolving ability — ability to distinguish closely spaced wavelengths or features.

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Key Takeaways

• Wavelength affects both diffraction spreading and interference fringe spacing.
• Longer wavelengths produce greater diffraction.
• Longer wavelengths produce wider double slit fringe spacing.
• Narrower single slits produce broader diffraction patterns.
• The approximate central maximum width is W = 2λL/a.
• Narrowing a slit can also reduce the total transmitted light.
• Double slit fringe spacing is w = λL/d.
• Increasing slit separation decreases interference fringe spacing.
• Decreasing slit separation increases fringe spacing.
• In a real double slit, individual slit width controls the diffraction envelope while slit separation controls the interference fringes.
• Increasing screen distance increases the physical size of patterns on the screen without changing their angular positions.
• Increasing the number of coherent grating slits produces sharper principal maxima.
• Grating line spacing determines the angular positions of the maxima.
• The number of illuminated grating lines affects their sharpness and resolving ability.
• Good coherence is required for clear, stable interference fringes.
• Unequal wave intensities reduce fringe contrast.
• A broad range of wavelengths can cause overlapping patterns and reduced visibility.
• Multiple experimental changes can be analyzed using proportional reasoning.
• Pattern changes can often reveal which experimental variable was altered.
• Understanding these factors allows diffraction and interference experiments to be designed, optimized, and interpreted quantitatively.

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Check Your Understanding

1. Explain how increasing wavelength affects a single slit diffraction pattern.
2. Explain how increasing wavelength affects the fringe spacing in a double slit experiment.
3. Why does red light generally diffract more than blue light when both pass through the same slit?
4. A single slit is made half as wide while wavelength and screen distance remain unchanged. Predict the change in the central maximum width.
5. A single slit is widened from 0.10 mm to 0.20 mm. If the original central maximum was 16 mm wide, calculate its new approximate width.
6. Explain how narrowing a slit can make the diffraction pattern wider while also reducing the amount of transmitted light.
7. A double slit experiment has a fringe spacing of 4.0 mm. The slit separation is doubled. Calculate the new fringe spacing.
8. Explain the different roles of individual slit width a and slit separation d in a real double slit experiment.
9. The wavelength and slit separation remain unchanged, but the screen distance is tripled. What happens to the physical fringe spacing? What happens to the angular fringe spacing?
10. Explain why increasing the number of illuminated slits in a diffraction grating produces narrower, sharper principal maxima.
11. Distinguish between increasing the number of illuminated grating lines and increasing the number of lines per millimetre. How does each affect the pattern?
12. A double slit experiment uses λ = 600 nm, d = 0.30 mm, and L = 1.5 m. Calculate the fringe spacing.
13. In Question 12, the wavelength is changed to 450 nm and the screen distance is doubled. Calculate the new fringe spacing and compare it with the original.
14. An interference pattern becomes blurred even though the slit geometry has not changed. Give two possible explanations involving the properties of the waves.
15. A student wants a double slit experiment with widely separated, clearly visible fringes. Recommend changes involving wavelength, slit separation, screen distance, coherence, and illumination, and explain why each change would improve the observed pattern.