Diffraction

Single-slit patterns, diffraction gratings, and the equation d sin θ = nλ

SPEC 3.3.2.2 • WAVES

What is Diffraction?

Definition

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

Diffraction occurs with all types of waves, including light, sound, and water waves. The effect is most noticeable when:

Key Condition

Diffraction is most significant when the gap width is comparable to the wavelength of the wave.

Gap width ≈ λ → Maximum diffraction

Plane waves spread out after passing through a narrow gap

Exam Tip

The wavelength is unaffected during diffraction. Only the direction of wave propagation changes - the wave spreads out, but its wavelength remains the same.

Factors Affecting Diffraction

The amount of spreading depends on the relationship between gap width and wavelength:

Key Rule

Diffraction increases when: (1) gap width decreases, or (2) wavelength increases

Maximum diffraction occurs when gap width ≈ wavelength

Single-Slit Diffraction

When monochromatic laser light passes through a narrow single slit, a distinctive diffraction pattern is observed on a screen:

Pattern Features

Central maximum: Bright, wide band in the middle

Secondary maxima: Dimmer bands either side

Minima: Dark regions between bright bands

Central maximum is twice as wide as secondary maxima

Single-Slit Pattern

Intensity vs angle for single-slit diffraction

Bright Fringes

Caused by constructive interference between wavelets from different parts of the slit

Dark Fringes

Caused by destructive interference - wavelets cancel out completely

Remember

The intensity of secondary maxima decreases rapidly as you move away from the centre. The central maximum is much brighter and wider than the others.

Effect of Slit Width

The width of the central maximum depends on the slit width:

Key Relationship

Narrower slit → Wider central maximum

This is because a narrower slit causes more diffraction (greater spreading of the waves).

Common Mistake

Students often think a narrower slit gives a narrower pattern - it's the opposite! More diffraction = more spreading.

Effect of Wavelength

The width of the central maximum also depends on the wavelength of light used:

Key Relationship

Longer wavelength → Wider central maximum

Red light (λ ≈ 700 nm) diffracts more than blue light (λ ≈ 450 nm).

Factor ChangedEffect on Central Maximum Width

Slit width decreasedWidth increases (more spreading)

Wavelength increasedWidth increases (more spreading)

Screen distance increasedWidth increases (pattern magnified)

White Light Diffraction

When white light passes through a single slit, each wavelength diffracts by a different amount:

White light single-slit diffraction pattern

Pattern Features

Central fringe: White (all colours overlap at centre)

Outer fringes: Coloured spectra

Violet: Closest to centre (shortest λ, least diffraction)

Red: Furthest from centre (longest λ, most diffraction)

Worked Example01

A laser beam passes through a single slit. The slit width is halved. Describe and explain the change to the diffraction pattern.

1

The central maximum becomes wider.

2

Explanation: A narrower slit causes more diffraction (greater spreading of the waves).

3

The intensity of the central maximum decreases.

4

Explanation: The same amount of light energy is spread over a larger area.

Key point: Narrower slit → wider pattern, lower intensity

Diffraction Gratings

Definition

A diffraction grating consists of many equally-spaced parallel slits. Light diffracts through each slit and the diffracted beams interfere.

Grating Specification

Gratings are specified by the number of lines per millimetre (e.g., 300 lines/mm).

The slit spacing (distance between adjacent slits) is:

d = \frac{1}{N}

where N = number of lines per metre

Grating vs Double Slit

A diffraction grating produces a pattern similar to Young's double slit, but with important differences:

FeatureDouble SlitDiffraction Grating

MaximaBroad, diffuseSharp, narrow lines

IntensityLowerHigher (more slits contributing)

MeasurementHarder to measure accuratelyEasier to measure precisely

Why Sharper?

More slits means more beams interfering. Constructive interference only occurs at very specific angles, making the maxima narrower and brighter.

Deriving the Grating Equation

You need to know how to derive the diffraction grating equation:

Step 1: Coherent Sources

Light enters the grating as parallel rays. It diffracts through each slit. The slits act as coherent sources (in phase, same wavelength). These diffracted waves interfere with each other.

Step 2: First Order Maximum

The 1st order maximum occurs at an angle where the path difference between adjacent slits equals one wavelength λ (for constructive interference). Call this angle θ.

Step 3: Create Triangle

We know: the distance between slits is d, the path difference is λ, and the angle is θ.

Grating Equation Derivation

Step 4: Use Trigonometry

From the triangle, using trigonometry:

path difference = d sin θ

Step 5: Generalise for All Orders

Maxima occur when path difference = whole number of wavelengths (nλ). Therefore:

nλ = d sin θ

Grating Equation

d sin θ = nλ

d = slit spacing (m)

θ = angle of diffraction from normal

n = order number (0, 1, 2, 3...)

λ = wavelength (m)

Understanding Orders

n = 0: Zero order (straight through, θ = 0°)

n = 1: First order maxima (one each side)

n = 2: Second order maxima, etc.

Maximum Visible Order

Since sin θ cannot exceed 1, there is a maximum order that can be observed:

Maximum Order

From

d sin θ = nλ

, the maximum occurs when

sin θ = 1

(i.e., θ = 90°)

n_{ma x} = \frac{d}{λ}

Take the integer part (round down)

Example

If d = 2.0 μm and λ = 600 nm:

n_{ma x} = \frac{2.0 \times 1 0 ^{- 6}}{600 \times 1 0 ^{- 9}} = \frac{2000}{600} = 3.33

Maximum order = 3 (take integer part)

Orders visible: n = 0, 1, 2, 3 (on each side)

Total Maxima

Total number of maxima = 2n_max + 1

(n_max on each side, plus the zero order in the middle)

Common Mistake

Don't forget that diffraction occurs on both sides of the central axis. Each order (n = 1, 2, 3...) appears twice - once on each side of the zero order.

Worked Example02

A diffraction grating has 500 lines per mm. Monochromatic light of wavelength 589 nm is incident normally on the grating. Calculate the angle of the first-order maximum.

1

Calculate slit spacing: d = 1/(500 × 10³) = 2.0 × 10⁻⁶ m

2

Use grating equation: d sin θ = nλ

3

For n = 1: sin θ = λ/d = (589 × 10⁻⁹)/(2.0 × 10⁻⁶)

4

sin θ = 0.2945

5

θ = sin⁻¹(0.2945) = 17.1°

Answer: θ = 17.1°

Worked Example03

A grating with 300 lines per mm is used with light of wavelength 650 nm. Calculate (a) the maximum order visible, and (b) the total number of maxima observed.

1

Calculate slit spacing: d = 1/(300 × 10³) = 3.33 × 10⁻⁶ m

2

Part (a): n_max = d/λ = (3.33 × 10⁻⁶)/(650 × 10⁻⁹) = 5.13

3

Maximum order = 5 (integer part)

4

Part (b): Total maxima = 2n_max + 1 = 2(5) + 1 = 11

Answer: (a) 5, (b) 11 maxima

Worked Example04

Light passes through a diffraction grating with 600 lines per mm. The second-order maximum is observed at an angle of 43.2°. Calculate the wavelength of the light.

1

Calculate slit spacing: d = 1/(600 × 10³) = 1.67 × 10⁻⁶ m

2

Rearrange grating equation: λ = d sin θ / n

3

λ = (1.67 × 10⁻⁶ × sin 43.2°) / 2

4

λ = (1.67 × 10⁻⁶ × 0.684) / 2 = 5.71 × 10⁻⁷ m

Answer: λ = 571 nm

Worked Example05

A grating produces a first-order maximum at 15.0° for red light (λ = 650 nm) and at 11.5° for blue light (λ = 480 nm). Verify these results are consistent with the same grating.

1

For red light: d = nλ/sin θ = (1 × 650 × 10⁻⁹)/sin 15.0°

2

d = (650 × 10⁻⁹)/0.259 = 2.51 × 10⁻⁶ m

3

For blue light: d = (1 × 480 × 10⁻⁹)/sin 11.5°

4

d = (480 × 10⁻⁹)/0.199 = 2.41 × 10⁻⁶ m

5

The values are close (within ~4%), confirming same grating (small difference due to measurement uncertainty)

Answer: Yes, consistent (d ≈ 2.5 μm)

Applications of Diffraction Gratings

Analysing Light (Spectroscopy)

Diffraction gratings separate wavelengths in light from different substances. The wavelengths present reveal information about the substance - used to analyse starlight and identify elements.

Wavelength Measurement

Precisely measuring wavelengths of light sources by measuring diffraction angles. More accurate than prisms.

X-ray Crystallography

Atoms in crystals act like diffraction gratings. X-rays diffract through the gaps between atoms. By analysing the diffraction pattern, we can determine atomic spacing in the crystal structure.

CD/DVD Surfaces

The tracks on CDs and DVDs act as reflection gratings, creating rainbow patterns when white light is reflected.

Why Gratings Over Prisms?

More accurate wavelength measurement

Linear dispersion (angles proportional to wavelength)

Can separate very close wavelengths (high resolution)

Practice

Question 1: A diffraction grating has 400 lines per mm. Light of wavelength 550 nm is incident normally. Calculate the angle of the second-order maximum.

1

d = 1/(400 × 10³) = 2.5 × 10⁻⁶ m

2

sin θ = nλ/d = (2 × 550 × 10⁻⁹)/(2.5 × 10⁻⁶) = 0.44

3

θ = sin⁻¹(0.44) = 26.1°

Practice

Question 2: A grating produces a third-order maximum at 62.5° for monochromatic light. The slit spacing is 1.80 μm. Calculate the wavelength of the light.

1

λ = d sin θ / n

2

λ = (1.80 × 10⁻⁶ × sin 62.5°) / 3

3

λ = (1.80 × 10⁻⁶ × 0.887) / 3 = 5.32 × 10⁻⁷ m

4

λ = 532 nm (green light)

Practice

Question 3: A grating with 750 lines per mm is illuminated with light of wavelength 620 nm. How many complete orders can be observed on one side of the central maximum?

1

d = 1/(750 × 10³) = 1.33 × 10⁻⁶ m

2

n_max = d/λ = (1.33 × 10⁻⁶)/(620 × 10⁻⁹) = 2.15

3

Maximum order = 2 (take integer part)

4

Orders visible on one side: n = 1, 2

Summary

Key Equations

EquationDescription

d sin θ = nλDiffraction grating equation

d = 1/NSlit spacing (N = lines per metre)

n_max = d/λMaximum order (integer part)

Key Concepts

Single slit: Narrower slit or longer λ → wider central maximum

White light: Central white fringe, coloured spectra outside

Grating: Produces sharp, bright maxima at specific angles

Orders: n = 0, 1, 2, 3... with maximum limited by sin θ ≤ 1

Exam Tips

Always convert lines/mm to slit spacing in metres first

Remember n_max is the INTEGER part of d/λ

For white light, red diffracts most (longest λ)