Investigate the wave model of light, including diffraction and constructive and destructive interference (Young's double-slit experiment), and apply $\Delta x = \lambda L / d$ for fringe spacing in the small-angle limit

What this dot point is asking

VCAA wants you to describe the wave model of light through its key signatures (interference and diffraction), explain Young's double-slit experiment, apply the path-difference conditions for constructive and destructive interference, and use the small-angle fringe-spacing formula $\Delta x = \lambda L / d$. The dot point is the wave-side of the wave-particle duality story in Unit 4.

The wave model of light

Light is a transverse electromagnetic wave. Like all waves, it can:

• Reflect off surfaces.
• Refract when passing between media of different refractive index.
• Diffract around obstacles or through narrow openings.
• Interfere with other waves, producing constructive and destructive patterns.

The interference and diffraction phenomena are the diagnostic wave behaviours. They cannot be explained by a purely particle model of light, so their observation is direct evidence for the wave nature of light.

The wave equation $v = f \lambda$ connects wave speed, frequency and wavelength. For light in vacuum, $v = c \approx 3.0 \times 10^8$ m/s, so:

Cross-link: see the wavelength-frequency calculator for fast conversions.

Coherent light and the double-slit experiment

Young's double-slit experiment (1801) is the canonical demonstration of light interference.

Setup. Monochromatic (single-wavelength), coherent light passes through two narrow slits separated by distance $d$. The light reaching the screen at distance $L$ from the slits is the superposition of two waves, one from each slit.

Coherence requirement. The two sources must have a fixed phase relationship. In practice, both slits are illuminated by the same monochromatic source (e.g., a laser, or a single slit illuminated first), ensuring coherence.

Observation. A regular pattern of bright and dark fringes appears on the screen. Bright fringes correspond to constructive interference (the waves arrive in phase); dark fringes correspond to destructive interference (the waves arrive out of phase by $\pi$).

Path difference conditions

The phase relationship between the two waves at a point on the screen depends on the path difference: the difference in the distance each wave has travelled.

where $\theta$ is the angle from the centreline to the point on the screen.

Constructive interference (bright fringe):

Destructive interference (dark fringe):

The integer $m$ is the order of the fringe. $m = 0$ is the central bright fringe (directly opposite the midpoint of the slits, equal path lengths from both slits). $m = \pm 1$ are the first-order bright fringes on either side, and so on.

Fringe spacing in the small-angle limit

For small $\theta$ (the typical Young's setup where $L \gg d$), $\sin \theta \approx \tan \theta = y / L$ where $y$ is the distance from the centreline on the screen.

Substituting into the constructive condition: $d \cdot y / L = m \lambda$, so $y = m \lambda L / d$.

The fringe spacing $\Delta x$ (distance between adjacent bright fringes, or between adjacent dark fringes) is:

This formula is the working tool for Young's double-slit problems. Three takeaways:

1. Longer wavelength gives wider spacing. Red light (700 nm) makes wider fringes than blue (450 nm).
2. Greater screen distance gives wider spacing. Moving the screen further from the slits spreads the pattern.
3. Wider slit separation gives narrower spacing. Slits further apart cram the fringes closer.

The formula is approximate; it assumes the small-angle limit (typically valid when $\theta < 5$ degrees) and that the slits are very narrow compared to the wavelength (diffraction from each slit is wide compared to the separation).

Single-slit diffraction

When light passes through a single narrow slit of width $w$, it spreads out and produces a diffraction pattern on a distant screen.

Pattern. A central bright band, flanked by progressively dimmer side bands separated by dark fringes. The central band is twice as wide as the side bands.

Dark fringe condition. Dark minima occur at angles where:

(Note: $m = 0$ corresponds to the centre of the bright central maximum, not a dark fringe.)

Width of the central maximum. From the first dark fringes on either side: $\sin \theta_1 \approx \lambda / w$, so the angular half-width is $\theta_1 \approx \lambda / w$. The angular full-width is $2 \lambda / w$.

This means narrow slits diffract more (wide pattern) and wide slits diffract less (narrow pattern). The same principle explains why low-frequency (long-wavelength) sound waves diffract around corners more than high-frequency (short-wavelength) sound waves.

Diffraction grating

A diffraction grating is many parallel slits (typically thousands per mm). The constructive interference condition is the same as for two slits:

where $d$ is the spacing between adjacent slits. The diffraction pattern produced by a grating has very sharp bright maxima (because of constructive interference between many sources) separated by dark regions.

Diffraction gratings are used to disperse light into its spectrum (different wavelengths diffract at different angles) and underpin spectroscopy.

What the interference and diffraction patterns tell us

The wave model predicts both the position and the spacing of the fringes correctly. Specifically:

• Existence of the pattern. A purely particle (corpuscular) model of light predicts a single bright band where the particles pass through, not a fringe pattern.
• Wavelength dependence of fringe spacing. Different wavelengths give different spacings, exactly as $\Delta x = \lambda L / d$ predicts.
• Wavelength dependence of single-slit diffraction. Narrower slits and longer wavelengths give wider diffraction, again consistent with the wave model.

The wave model is therefore the working model for low-intensity classical optics: refraction, reflection, interference, diffraction, polarisation. The photon (particle) model takes over for high-energy interactions with matter (photoelectric effect, atomic transitions), and the matter-wave model unifies both.

Worked examples

Example 1. Calculate fringe spacing

Wavelength 600 nm, slit separation 0.20 mm, screen distance 1.5 m.

$\Delta x = (600 \times 10^{-9}) \times 1.5 / (0.20 \times 10^{-3}) = 9.0 \times 10^{-7} / 0.20 \times 10^{-3} = 4.5 \times 10^{-3}$ m = 4.5 mm.

Example 2. Wavelength from fringe spacing

Fringe spacing 2.0 mm, slit separation 0.50 mm, screen distance 1.2 m. Find $\lambda$.

Rearrange: $\lambda = \Delta x \cdot d / L = (2.0 \times 10^{-3}) \times (0.50 \times 10^{-3}) / 1.2 = 1.0 \times 10^{-6} / 1.2 \approx 8.3 \times 10^{-7}$ m $= 830$ nm (in the infrared / red region).

Example 3. Path difference and fringe identification

In a setup with $\lambda = 500$ nm, the path difference at a point on the screen is 1500 nm.

Express in wavelengths: $1500 / 500 = 3$. The path difference is exactly $3 \lambda$ (integer multiple), so constructive interference: this is the third-order bright fringe ($m = 3$).

Common errors

Unit conversion forgotten. Wavelengths in nm, slit separation in mm, screen distance in m. Convert all to metres before substituting.

Confusing $d$ and $w$. $d$ is the slit separation in Young's double-slit. $w$ is the slit width in single-slit diffraction. Different quantities, different formulas.

Applying small-angle formula at large angles. $\Delta x = \lambda L / d$ assumes small $\theta$. For large $\theta$ (close to the slits, or first-order maxima with very short $L$), use $d \sin \theta = m \lambda$ directly.

Treating the $m = 0$ position as a dark fringe in single-slit diffraction. The single-slit minimum condition has $m \neq 0$; $m = 0$ is the bright centre.

Forgetting coherence. Two independent light sources are not coherent, so they do not produce a stable interference pattern. Young used one source illuminating both slits to ensure coherence.

Confusing path difference with phase difference. Path difference (in metres) becomes phase difference (in radians) via $\phi = 2 \pi \Delta / \lambda$. Constructive: $\Delta = m \lambda$ (so $\phi = 2 \pi m$). Destructive: $\Delta = (m + 0.5) \lambda$ (so $\phi = (2m + 1) \pi$).

In one sentence

The wave model of light is supported by the diffraction and interference patterns produced when coherent monochromatic light passes through one or more slits; for Young's double-slit, constructive fringes occur where the path difference is $m \lambda$ and destructive fringes where it is $(m + \frac{1}{2}) \lambda$, with fringe spacing on a distant screen given by $\Delta x = \lambda L / d$ in the small-angle limit.