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