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

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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.

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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

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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

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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

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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.

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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.

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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.

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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.

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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

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Two Slits vs Many Slits

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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.

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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.

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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.

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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

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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.

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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

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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.

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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.

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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.

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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

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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%.

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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.