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Diffraction gratings

Swap two slits for thousands and the fuzzy fringes sharpen into thin, brilliant lines at exactly predictable angles. One short equation locates every line, and the geometry of a sine caps how many lines can exist.

Year 12AQA 3.3.2.2

Builds on Interference and Young's double slit.

IN THIS TOPIC

  • Explain why many slits produce sharper, brighter maxima than two.
  • Use d sin θ = nλ, finding d from the number of lines per millimetre.
  • Determine the highest order visible and give uses of diffraction gratings.

WHAT YOU PROBABLY THINK

Adding thousands more slits should smear the pattern into a blur.

From two slits to thousands

A diffraction grating is a plate ruled with hundreds of slits per millimetre. Light leaving all of them overlaps, and the many-slit sum is far stricter than the two-slit one. At most angles, the thousands of contributions arrive with a scatter of phases and cancel almost perfectly. Only at a few special angles does every slit's light arrive exactly in step, and there the maxima are brighter, fed by every slit at once, and much sharper, because even a tiny step away from the exact angle restores the cancellation.

The grating equation

The special angles come from the path difference between neighbouring slits, whose centres sit a distance d apart, the grating spacing. Light leaving adjacent slits at angle θ to the normal differs in path by dsin θ, and every slit stays in step with every other only when that difference is a whole number of wavelengths:

d sin θ = nλON YOUR DATA SHEET
Adjacent grating slits: the path difference d sin theta must be a whole number of wavelengthsincident lightdθd sin θin step only when d sin θ = nλ
FIG. 1Adjacent slits, and the extra distance dsin θ the lower ray travels. When it equals nλ, light from every slit on the grating arrives in phase.

The whole number n is the order of the maximum: n = 0 is the straight-through beam, n = 1 the first order either side, and so on. Gratings are labelled in lines per millimetre, and d is one millimetre shared between N lines:

d = 1NNOT ON THE DATA SHEET — LEARN IT

with N in lines per metre giving d in metres. A 600 lines per mm grating, for instance, has d = 1.67 × 10− 6 m.

Orders, and the one that cannot exist

Rearranged, sin θ = nλ/d, and a sine can never exceed 1. That single fact caps the pattern: orders exist only while nλ/d ≤ 1, so the highest order is the whole-number part of d/λ. Beyond it, the geometry simply has no angle to offer.

Grating orders: sharp beams at the angles where d sin theta equals n lambdan = 0n = 1n = 1n = 2n = 2grating
FIG. 2The full pattern for d = 2.4λ: a zero order and two orders either side, each a sharp beam. A third order would need sin θ = 1.25, which no angle can supply.

WORKED EXAMPLE

Monochromatic light of wavelength 550 nm falls on a grating with 600 lines per mm. Find the angle of the first-order maximum, and the highest order visible.

d = 1/N = (1.0 × 10− 3) / 600 = 1.67 × 10− 6 m.

First order: sin θ = λ/d = (550 × 10− 9) / (1.67 × 10− 6) = 0.330, so θ = 19.3°.

Highest order: d/λ = 3.03, so n = 3 works (sin θ = 0.990) and n = 4 would need sin θ = 1.32. The third order is the last one visible, at a steep 81.9°.

Because the maxima are so sharp, their angles can be measured precisely, and the grating equation then delivers λ to matching precision. That is the instrument's real job: gratings split light into line spectra, letting chemists identify elements by the wavelengths they emit and letting astronomers read the composition of a star from its light.

THE EXAM BIT

  • Finding d is where marks leak: convert the millimetre to metres first, then divide by the number of lines. For 600 lines per mm, d = (1.0 × 10− 3)/600, and the answer should land near 10− 6 m.
  • For the highest order, compute d/λ and take the whole-number part. Writing n = 3.03 as the answer, or rounding it up to 4, both lose the mark.
  • If sin θ comes out above 1, the order does not exist; say so rather than forcing an angle from the calculator.
  • Comparison wording: grating maxima are sharper and brighter than double-slit fringes, and the sharpness is what makes wavelength measurements precise.
  • Check the mode. An angle of 19.3° emerging as 0.337 usually means the calculator is in radians.

CHECK YOURSELF

Light of wavelength 550 nm falls on a grating with 300 lines per mm. Find (a) the angle of the first-order maximum and (b) the highest order that can be seen.

Show a hint

Find d first, and remember that sin θ can never pass 1.

Show the answer

(a) d = 1/N = (1.0 × 10− 3) / 300 = 3.33 × 10− 6 m. Then sin θ = λ/d = (550 × 10− 9) / (3.33 × 10− 6) = 0.165, so θ = 9.5°.

(b) d/λ = 6.06, so the highest whole n with sin θ ≤ 1 is n = 6, and that sixth order sits out at 81.9°.

d sin θ = nλ.

sin θ can never pass 1, and that caps n.

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