Diffraction and Gratings

Overview

Diffraction and gratings combine two closely related wave phenomena:

• diffraction: spreading of waves after passing through gaps or around edges;
• diffraction gratings: many closely spaced slits producing sharp interference maxima.

Both arise from Superposition of Waves and are central to Light Waves and other Waves.

Definition

Diffraction is the spreading of waves when they pass through an aperture or move around the edge of an obstacle.

A diffraction grating is a surface containing a very large number of equally spaced parallel slits or lines. Each slit diffracts the wave, and waves from many slits interfere to form sharp maxima.

Why It Matters

Diffraction confirms wave behaviour and explains why waves spread around obstacles. Gratings turn this spreading into precise wavelength measurement, making them fundamental tools in optics and spectroscopy.

Key Representations

Dependence on Gap Size

Let $a$ be aperture width and $\lambda$ be wavelength.

Strong diffraction occurs when:

$$a\sim \lambda$$

Weak diffraction occurs when:

$$a\gg \lambda$$

Examples of Diffraction

Water waves in a ripple tank spread strongly through narrow gaps.

Sound waves have relatively large wavelengths, so they diffract around doors and corners. See Sound Waves.

Light has a very small wavelength, so everyday diffraction is less obvious. Noticeable diffraction requires narrow slits or fine structures. See Light Waves.

Diffraction Grating Spacing

If $N$ is the number of lines per metre, then slit spacing is:

$$d=\frac{1}{N}$$

where $d$ is the distance between adjacent slits.

Grating Equation

Constructive interference occurs when:

$$d\sin \theta =n\lambda$$

where:

• $d$ is slit spacing;
• $\theta$ is angle from central direction;
• $n$ is order number;
• $\lambda$ is wavelength.

Orders of Maxima

Zero order has:

$$n=0$$

so:

$$\theta =0$$

First, second, and higher orders occur for:

$$n=1,2,3,\dots$$

Maximum Possible Order

Since:

$$\sin \theta \le 1$$

then:

$$n\lambda \le d$$

so:

$$n\le \frac{d}{\lambda }$$

The greatest whole number satisfying this is the maximum observable order.

Measuring Wavelength

If $d$ is known and $\theta$ is measured:

$$\lambda =\frac{d\sin \theta }{n}$$

White Light and Spectra

White light contains many wavelengths. From:

$$d\sin \theta =n\lambda$$

different wavelengths diffract at different angles.

The zero-order image is white because all wavelengths overlap at $\theta =0$.

In first and higher orders, longer wavelengths such as red appear at larger angles than shorter wavelengths such as violet.

Why Maxima Are Sharper with Many Slits

With many slits, constructive interference occurs strongly only at precise angles, while destructive interference occurs almost everywhere else. The peaks become narrow and intense.

Double Slit versus Grating

Feature	Young Double Slit	Diffraction Grating
Number of slits	2	Many
Bright maxima	Moderate	Very bright
Width of maxima	Broad	Narrow
Best for wavelength measurement	Good	Excellent

See Young Double Slit.

Formula Summary

Grating spacing:

$$d=\frac{1}{N}$$

Maxima condition:

$$d\sin \theta =n\lambda$$

Wavelength:

$$\lambda =\frac{d\sin \theta }{n}$$

Maximum order:

$$n_{\max }=\text{largest integer}\le \frac{d}{\lambda }$$

Links

• Core hub: Superposition of Waves
• Prerequisite: Waves
• Related: Interference
• Related: Young Double Slit
• Related: Polarisation
• Related: Light Waves
• Related: Superposition Common Exam Traps