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.

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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.

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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

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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, ...

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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.

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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.

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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°

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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.

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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.

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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.

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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.

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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.

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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

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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

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Single Slit, Double Slit, and Grating Patterns

The three patterns have distinctive appearances.

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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.

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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.

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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.

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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.

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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.

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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

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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

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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.

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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

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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.

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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.

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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

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.