Diffraction and Single Slit Patterns

Diffraction is the spreading of waves when they pass through a gap or around an obstacle. It is a defining characteristic of wave behaviour and cannot be explained by a simple particle model.

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What is Diffraction?

When a wave passes through a gap (aperture) or encounters an obstacle, it spreads out into the region beyond. This spreading is called diffraction.

Key observations:

• Diffraction is most significant when the gap width is comparable to the wavelength (gap ≈ λ).
• If the gap is much larger than the wavelength, very little diffraction occurs — the wave passes through almost as a straight beam.
• If the gap is much smaller than the wavelength, the wave spreads out almost equally in all directions (semicircular wavefronts in 2D).

Huygens' Principle

Diffraction can be explained using Huygens' principle:

Every point on a wavefront can be considered as a source of secondary spherical wavelets. The new wavefront is the envelope (tangent surface) of all these secondary wavelets.

When a wave passes through a narrow slit, only the wavelets originating from within the slit opening contribute to the wave on the other side. These wavelets spread out, causing the wave to fan out beyond the slit.

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Single Slit Diffraction Pattern

When monochromatic light passes through a single narrow slit of width a and falls on a distant screen, the following pattern is observed:

Features of the Pattern

1. A broad central maximum — the brightest and widest part of the pattern.
2. Subsidiary (secondary) maxima on either side — much dimmer, decreasing in intensity with distance from the centre.
3. Dark minima between the maxima — positions of zero intensity (destructive interference).
4. The central maximum is twice as wide as each subsidiary maximum.

Positions of the Minima

The first minimum on either side of the central maximum occurs at an angle θ given by:

sin θ = λ/a

More generally, the minima occur at angles where:

sin θ = nλ/a (n = ±1, ±2, ±3, ...)

where:

• θ = angle from the central axis to the minimum
• λ = wavelength of light
• a = slit width
• n = order number (n ≠ 0; n = 0 corresponds to the central maximum)

Why Do the Minima Occur?

At the first minimum (n = 1), the path difference between wavelets from the top and bottom of the slit is exactly one wavelength (λ). The slit can be divided into two halves: every wavelet from the top half has a corresponding wavelet from the bottom half that is exactly λ/2 ahead, so they cancel in pairs. The result is complete destructive interference.

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Worked Examples

Worked Example 1 — Light of wavelength 589 nm passes through a single slit of width 0.10 mm. Calculate the angle of the first minimum.

λ = 589 nm = 5.89 × 10⁻⁷ m a = 0.10 mm = 1.0 × 10⁻⁴ m

sin θ = λ/a = (5.89 × 10⁻⁷)/(1.0 × 10⁻⁴) = 5.89 × 10⁻³

θ = arcsin(5.89 × 10⁻³) = 0.34°

The angle is very small because the slit width is much larger than the wavelength.

Worked Example 2 — In the same experiment, the screen is 3.0 m from the slit. Calculate the width of the central maximum on the screen.

The central maximum extends from the first minimum on one side to the first minimum on the other. Its half-width is y₁ = D tan θ ≈ Dθ (for small angles).

y₁ = D sin θ = 3.0 × 5.89 × 10⁻³ = 1.767 × 10⁻² m

Full width of central maximum = 2y₁ = 2 × 1.767 × 10⁻² = 3.53 × 10⁻² m

Width = 35 mm (to 2 s.f.)