Diffraction Grating Spectra: Orders, Formula and Uses

A diffraction grating is a surface with thousands of very fine parallel lines or slits close together. When light passes through it, each slit makes its own diffraction pattern, and the waves from all slits interfere. The result is very bright, sharp lines in different directions called orders.

Gratings matter because they separate light into its colours much better than a prism. Scientists use them to study the light from stars and hot gases. Each element gives its own pattern of bright lines, acting like a fingerprint.

Key Concepts

Grating element (d): Distance between the centres of two neighbouring slits, in metres. If a grating has 500 lines per mm, then d = 1/500 mm.

Orders (n): The bright directions are counted as zero order (straight through), first order, second order, and so on. Higher orders spread the colours more but are dimmer.

Spectrum: The set of colours or wavelengths separated by the grating. White light gives a rainbow in each order except the zero order, which stays white.

Formulas

Grating equation: d sin θ = n λ, where d is the grating spacing in metres, θ is the angle of the bright line, n is the order (0, 1, 2, …) and λ is the wavelength in metres. This holds for light incident normally (straight on) and requires n λ / d ≤ 1, otherwise that order does not exist.

Maximum order: n_max is the largest whole number less than d / λ. Orders beyond this are impossible because sin θ cannot exceed 1.

Worked Example

A grating has 600 lines per mm. Light of wavelength 500 nm falls on it. At what angle is the first-order line seen?

Step 1: Find d. 600 lines per mm means d = 1/600 mm = 1.667 × 10⁻⁶ m.

Step 2: Use d sin θ = n λ with n = 1: sin θ = λ / d = 500 × 10⁻⁹ / 1.667 × 10⁻⁶ = 0.30.

Step 3: θ = sin⁻¹(0.30) ≈ 17.5 degrees.

The first-order bright line appears at about 17.5° from the centre. The second order would be at sin θ = 0.60, about 36.9°.

Common Misconceptions

• Misconception: A grating with more lines per mm always gives a wider pattern. Correct: More lines per mm means smaller d, so angles are larger for a given order, but the sharpness also increases.
• Misconception: Zero order shows colours. Correct: In zero order all wavelengths overlap at θ = 0, so white light stays white.
• Misconception: Orders go on forever. Correct: Orders stop when n λ exceeds d.

Real-World Applications

Gratings are used in spectrometers in labs, in astronomy to analyse starlight, and in CD and DVD surfaces where the fine tracks act as a grating and make rainbow colours. They are also used in laser systems and in instruments that check water and air quality by their spectra.

Remember, a diffraction grating uses many slits to make sharp, bright spectra. The angle of each colour is given by d sin θ = n λ.