Diffraction Gratings

A diffraction grating is the natural extension of the double-slit experiment to many slits. Instead of two slits, we use hundreds or thousands of closely-spaced parallel slits — typically several hundred per millimetre. The result is a much sharper, more intense and more precisely-located fringe pattern that allows extremely accurate measurement of wavelengths. Diffraction gratings are the standard tool of spectroscopy, used to analyse the light from stars, the composition of flames and the chemistry of laboratory samples.

OCR A-Level Physics A requires you to derive and apply the grating equation:

d sin θ = n λ

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Physical Principle

Like Young's experiment, a diffraction grating works by superposition of coherent waves from multiple sources. Each slit in the grating produces its own secondary wave (Huygens). Because the same primary wavefront reaches all slits at the same time, the secondary waves are all coherent with each other.

At an angle θ from the straight-through direction, all the secondary waves travel extra path lengths that depend linearly on slit position. Specifically, if the slit spacing is d, the path difference between neighbouring slits is d sin θ.

For a maximum, all the waves must arrive in phase, which requires:

d sin θ = n λ

where n is an integer (0, 1, 2, 3, …). This is the diffraction grating equation.

What the Variables Mean

Symbol	Meaning
d	Slit spacing — the distance between the centres of adjacent slits (m)
θ	Angle of the maximum, measured from the straight-through direction
n	Order of the maximum (integer: 0 for central, 1 for first order, 2 for second order, …)
λ	Wavelength of the light

Why Gratings Are Sharper Than Double-Slits

With only two slits, the transition from "in phase" to "out of phase" as you move off the central maximum is gradual — the fringes are broad sinusoids. With N slits, all N waves must be simultaneously in phase for a maximum, which is a much more restrictive condition. The consequence is that the bright fringes become very narrow and very intense, separated by wide regions of near-zero intensity. For a grating with N = 600 lines per mm, the fringes are essentially sharp lines.

This sharpness is what makes gratings useful for spectroscopy — they can resolve wavelengths that differ by less than a nanometre, which a double-slit cannot.

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From Lines-per-Millimetre to d

The slit spacing d and the "number of lines per unit length" are related. A grating labelled "300 lines per mm" has:

d = 1 mm / 300 = 1/300 mm = 3.33 × 10⁻⁶ m

Similarly:

600 lines/mm: d = 1.67 × 10⁻⁶ m 1200 lines/mm: d = 8.33 × 10⁻⁷ m

You must be fluent in this conversion. Very fine gratings (with d close to the wavelength of light) produce only a few orders of maximum and at wide angles; coarser gratings produce many orders at smaller angles.

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Worked Example 1 — First-Order Angle

Q. Light of wavelength 600 nm is incident normally on a diffraction grating with 300 lines per mm. Calculate the angle of the first-order maximum.

A. Slit spacing: d = 1/300 mm = 3.33 × 10⁻⁶ m.

For n = 1:

sin θ = n λ / d = (1)(6.00 × 10⁻⁷) / (3.33 × 10⁻⁶) = 0.180 θ = arcsin(0.180) = 10.4°

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Worked Example 2 — Higher Orders

Q. For the same grating and wavelength as above, how many orders can be seen?

A. The highest order corresponds to sin θ ≤ 1. At the limit sin θ = 1, θ = 90°:

n_max = d / λ = (3.33 × 10⁻⁶) / (6.00 × 10⁻⁷) = 5.55

Since n must be an integer, n_max = 5. So you will see the central (n = 0) maximum plus five orders on each side, for a total of eleven bright fringes.

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Worked Example 3 — Measuring Wavelength

Q. When illuminated with monochromatic light, a grating with 500 lines per mm shows its first-order maximum at 18.2°. Calculate the wavelength.

A. d = 1/500 mm = 2.00 × 10⁻⁶ m.

λ = d sin θ / n = (2.00 × 10⁻⁶)(sin 18.2°) / 1 λ = (2.00 × 10⁻⁶)(0.3123) λ = 6.25 × 10⁻⁷ m = 625 nm

This is in the orange-red part of the visible spectrum.

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Worked Example 4 — Resolving Two Wavelengths

Q. A grating with 600 lines per mm is illuminated with a sodium lamp emitting two close wavelengths: 589.0 nm and 589.6 nm. Calculate the angular separation of the first-order maxima.

A. d = 1/600 mm = 1.667 × 10⁻⁶ m.