Single-Slit Diffraction

Spec mapping (OCR H556 Module 4.5): Describe and explain the single-slit diffraction pattern: bright wide central maximum, dimmer secondary maxima, dark minima at $\sin\theta \approx n\lambda/a$ . Identify that the central maximum is twice as wide as the side-maxima, and contrast the single-slit envelope with the equally-spaced fringes of the double-slit experiment.

Diffraction is the spreading of a wave as it passes through a gap or around an obstacle. It happens to all waves — water, sound, radio, light — whenever the size of the gap or obstacle is comparable to the wavelength. The single-slit diffraction pattern is a striking example of wave superposition: a narrow, bright central maximum flanked by progressively dimmer secondary maxima separated by dark minima.

In the OCR H556 specification you are not asked to compute the full intensity distribution of the single-slit pattern (which is a $\mathrm{sinc}^2$ function — beyond A-Level), but you must be able to describe the pattern qualitatively, explain why it arises from superposition of waves from different parts of the slit, apply the first-minimum condition $a\sin\theta \approx \lambda$ , explain why the central maximum is twice the width of subsequent maxima, and contrast the pattern with double-slit and grating patterns.

Diffraction at a Single Slit

Consider a plane wave — for example, a monochromatic laser beam — hitting a narrow slit of width $a$ , with a screen placed at distance $D$ beyond.

If the slit were infinitely narrow, every point in it would act as a single point source and the wave would spread out spherically (Huygens's principle), giving a uniformly-lit screen with no fringes at all. Real slits have a finite width, so waves from different points across the slit arrive at any point on the screen having travelled slightly different distances. This internal path difference causes interference between the wavelets emerging from different parts of the slit — and produces the characteristic single-slit diffraction pattern.

flowchart LR
    L[Monochromatic plane wave] --> SL[Single slit of width a]
    SL --> H[Each point in slit emits a Huygens wavelet]
    H --> SC[Screen at distance D]
    SC --> CM[Bright central maximum: wide and intense]
    SC --> SM[Secondary maxima: 4.5 percent, 1.6 percent, ...]
    SC --> Min[Dark minima at sin theta equals n lambda over a]

The Shape of the Pattern

The single-slit diffraction pattern has these distinctive features:

A very bright, wide central maximum directly in line with the slit. It is twice as wide as the other maxima and contains the overwhelming majority of the diffracted light energy.
A series of secondary maxima symmetrically on each side of the centre. The first secondary maximum has about $4.5\%$ of the central intensity; the second only $\sim 1.6\%$ . Intensity drops rapidly with order.
Dark minima between maxima, where the intensity is effectively zero.
The pattern is symmetric about the central maximum.

This is strikingly different from the double-slit pattern of Young's experiment, in which all bright fringes are of approximately equal intensity and equal spacing. The single-slit envelope places most of its energy in the centre. This is one of the key contrasts OCR examiners test.

The horizontal axis runs in $\sin\theta$ ; the first minima sit at $\sin\theta = \pm\lambda/a$ , successive minima at $\pm 2\lambda/a$ , $\pm 3\lambda/a$ , ... The central maximum is therefore the region $|\sin\theta| < \lambda/a$ — width $2\lambda/a$ — while each subsequent bright region spans only $\lambda/a$ in $\sin\theta$ , i.e. half the central width.

Why the Central Maximum Is So Wide

In the direction straight through the slit ( $\theta = 0$ ), all the Huygens wavelets from across the slit travel equal distances to the screen and arrive in phase. This gives a strong central maximum.

As you move slightly off-axis, path differences across the slit remain small — the waves still mostly reinforce each other. The path difference between the top edge of the slit and the bottom edge is $a\sin\theta$ . Provided this is less than $\lambda$ , the wavelets do not yet cancel completely. Only when the total path difference across the slit reaches one full wavelength does the contribution from every point in the slit pair-cancel against another, giving the first dark minimum.

The First Minimum

The condition for the first dark minimum is:

$a \sin\theta = \lambda$

For small angles ( $\sin\theta \approx \theta$ ):

$\theta_{\min,1} \approx \frac{\lambda}{a}$

For a narrow slit ( $a$ comparable to $\lambda$ ), $\sin\theta$ is large — central maximum spreads widely (in extreme cases beyond $90°$ , meaning no minimum exists and the wave diffracts into the whole half-space). For a wide slit ( $a \gg \lambda$ ), $\sin\theta$ is tiny and the pattern collapses to a sharp narrow spot — effectively no visible diffraction.

This is why diffraction is everyday-obvious for sound (wavelength $\sim 1\text{ m}$ , similar to typical doorways) but almost imperceptible for light (wavelength $\sim 500\text{ nm}$ , far smaller than any ordinary aperture).

Higher Minima

Subsequent dark minima occur at:

$a \sin\theta = n\lambda, \quad n = \pm 1, \pm 2, \pm 3, \ldots$

with $n = 0$ excluded (the central maximum, not a minimum). Between consecutive minima are the secondary maxima, each spanning $\lambda/a$ in $\sin\theta$ — half the width of the central peak.

Note. Do not confuse this formula with the grating equation $d\sin\theta = n\lambda$ . The grating equation gives the angles of bright maxima; the single-slit formula above gives the angles of dark minima. The two formulas look identical because both are saying "path difference across the relevant scale equals an integer number of wavelengths", but the consequence — bright vs dark — is opposite.

Effect of Slit Width on the Pattern

Three things change as you vary the slit width $a$ :

Slit width	First-minimum angle	Central-max width	Pattern character
Very narrow ( $a \sim \lambda$ )	Large; may exceed $90°$	Very wide; fills half-space	Strong diffraction
Moderate ( $a \sim 10\lambda$ )	A few degrees	Easily measurable	Classic textbook pattern
Wide ( $a \sim 100\lambda$ )	Sub-degree	Very narrow	Faint secondary maxima
Very wide ( $a \gg 100\lambda$ )	Sub-arcminute	Effectively zero	Sharp shadow of slit

Narrower slits give wider central maxima — the less localised the wave is as it passes through the slit, the more spread out the diffraction pattern. This trade-off — between confinement in real space and spread in angle — is a fundamental consequence of wave behaviour. In quantum mechanics it appears as the Heisenberg uncertainty principle applied to a photon's transverse position and momentum: $\Delta x \, \Delta p_x \gtrsim \hbar$ .

Effect of Wavelength on the Pattern

For a fixed slit width $a$ , longer wavelengths give wider diffraction patterns. Since the first minimum is at $\sin\theta = \lambda/a$ :

Red light ( $\lambda \approx 700\text{ nm}$ ) produces a wider central maximum than
Blue light ( $\lambda \approx 450\text{ nm}$ ).

Doubling $\lambda$ doubles the angular width of the central maximum.

If a single slit is illuminated with white light, the central maximum is white (all colours superpose constructively along the axis), but the secondary maxima are coloured — red on the outside, violet on the inside of each — because red diffracts more widely than violet.

Worked Example 1 — First Minimum Angle

Q. Light of wavelength $\lambda = 600\text{ nm}$ passes through a single slit of width $a = 0.10\text{ mm}$ . Calculate the angle of the first minimum and the linear width of the central maximum on a screen $2.0\text{ m}$ away.

A. Using $\sin\theta = \lambda/a$ :

$\sin\theta = \frac{6.00\times 10^{-7}}{1.00\times 10^{-4}} = 6.00\times 10^{-3}$

For such a small value, $\sin\theta \approx \theta$ in radians:

$\theta \approx 6.00\times 10^{-3}\text{ rad} \approx 0.34°$

Linear half-width on the screen: $D\tan\theta \approx D\theta = 2.0 \times 6.0\times 10^{-3} = 1.2\times 10^{-2}\text{ m} = 1.2\text{ cm}$ .

Full width of central maximum $= 2 \times 1.2\text{ cm} = 2.4\text{ cm}$ .

Worked Example 2 — Comparison of Two Wavelengths

Q. The same slit ( $a = 0.10\text{ mm}$ ) is illuminated alternately with red light ( $\lambda = 700\text{ nm}$ ) and blue light ( $\lambda = 450\text{ nm}$ ). Compare the linear widths of the central maxima on a screen $1.50\text{ m}$ away.

A. For each colour, central-max half-width $= D\lambda/a$ :

Red: $1.50 \times (7.00\times 10^{-7})/(1.00\times 10^{-4}) = 1.05\times 10^{-2}\text{ m} = 10.5\text{ mm}$ . Full width $= 21.0\text{ mm}$ .
Blue: $1.50 \times (4.50\times 10^{-7})/(1.00\times 10^{-4}) = 6.75\times 10^{-3}\text{ m} = 6.75\text{ mm}$ . Full width $= 13.5\text{ mm}$ .

Ratio: $w_{\text{red}}/w_{\text{blue}} = 21.0/13.5 = 1.56 = 700/450$ — consistent with the linear scaling $w \propto \lambda$ .

Worked Example 3 — Width of Side-Maxima

Q. For the slit in Worked Example 1, calculate the linear width of the first side-maximum on the screen.

A. Side-maxima sit between consecutive minima. The first side-maximum sits between $\sin\theta = \lambda/a$ and $\sin\theta = 2\lambda/a$ , a span of $\lambda/a$ in $\sin\theta$ .

For small angles, this corresponds to a linear width $D\lambda/a = 1.2\text{ cm}$ — half the central maximum's $2.4\text{ cm}$ . This is the "twice as wide" claim made quantitative.

Worked Example 4 — Acoustic Diffraction Through a Doorway

Q. A speaker emitting sound of frequency $f = 100\text{ Hz}$ in air ( $v = 340\text{ m s}^{-1}$ ) plays through an open doorway of width $a = 1.00\text{ m}$ . At what angle does the first diffraction minimum occur?

A. Wavelength: $\lambda = v/f = 340/100 = 3.40\text{ m}$ .

This is larger than the slit width! In the formula $\sin\theta = \lambda/a = 3.40$ — which is unphysical ( $\sin\theta > 1$ has no solution). The interpretation: the wave diffracts so strongly that there is no first minimum at all within the half-space behind the doorway. The sound spreads essentially uniformly through the full angular range.

This is why low-frequency bass sounds carry around corners and through doorways far more effectively than high-frequency treble. For $f = 5\text{ kHz}$ (treble) the wavelength is $\lambda = 6.8\text{ cm}$ , $\sin\theta = 0.068$ , $\theta = 3.9°$ — the first minimum is only a few degrees off-axis, so most of the treble propagates roughly straight ahead and is heard only by listeners directly in line with the doorway.

The everyday experience of "the bass is louder around the corner" is exactly this single-slit diffraction effect in real life. Concert venues and recording studios use diffuser panels (a "single slit pattern" generator at acoustic wavelengths) deliberately to mix the sound.

Diffraction at Other Apertures

Single-slit-like diffraction occurs at any narrow aperture, but the exact pattern depends on the shape:

Circular aperture (telescope objective, camera lens, eye pupil): produces an Airy pattern — a bright central disc (the Airy disc) surrounded by concentric dark and bright rings. The first dark ring sits at $\sin\theta = 1.22\lambda/D$ , where $D$ is the aperture diameter. Two stars closer together than this angle cannot be resolved by the optical system — the Rayleigh resolution criterion.
Narrow rectangular slit: single-slit diffraction in one dimension, very little spreading in the perpendicular dimension. The pattern is a thin bright line with side-maxima along it.
Square aperture: Cartesian product of two single-slit patterns — a central spot surrounded by side-spots along both axes (and weaker spots in the diagonals).

Diffraction and the Double-Slit Pattern

The double-slit pattern studied in Lesson 8 is actually a combination of two effects:

Single-Slit Diffraction

Single-Slit Diffraction

Diffraction at a Single Slit

The Shape of the Pattern

Why the Central Maximum Is So Wide

The First Minimum

Higher Minima

Effect of Slit Width on the Pattern

Effect of Wavelength on the Pattern

Worked Example 1 — First Minimum Angle

Worked Example 2 — Comparison of Two Wavelengths

Worked Example 3 — Width of Side-Maxima

Worked Example 4 — Acoustic Diffraction Through a Doorway

Diffraction at Other Apertures

Diffraction and the Double-Slit Pattern

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