AS 91523 · 2023 paper · diffraction glasses

Question 3(b) Merit to Excellence

At a fair, children buy “Rainbow Glasses” made of diffraction gratings in a cardboard frame. Steve shines a laser pointer of wavelength 643 nm (6.43 × 10⁻⁷ m) through one of the gratings onto a wall. The pattern is a bright red spot at the centre, with slightly dimmer red spots either side. The wall is 1.43 m from the grating, and the distance from the central bright spot to the second dimmer spot is 1.75 m.

λ	643 nm (6.43 × 10⁻⁷ m)
grating to wall	1.43 m
centre to 2nd spot	1.75 m

Give an in-depth explanation of why this pattern is seen: explain how diffraction and interference cause the bright spots, and why there are large sections with no light between them.

The 2023 paper, page 6 ↗ Marking schedule, page 5 ↗

See it move: Two sources, and the path difference between them

Work it out

Try it on paper first. Stuck? Take one hint at a time.

1 Why does the grating need diffraction to make a pattern at all?
Each slit is narrow, so light spreads out of it in every direction. Without that spreading, the beams from neighbouring slits would never overlap, and light that does not overlap cannot interfere.
2 Where the spread-out light overlaps, when do the waves from neighbouring slits reinforce?
When one path is longer than its neighbour’s by a whole number of wavelengths, nλ: they arrive in phase and interfere constructively. In those directions, d sin θ = nλ, and there is a bright spot.
3 Just off one of those directions, waves from neighbouring slits are only slightly out of phase. Why is it still dark there?
The small phase difference between neighbours adds up across the grating: the 2nd slit is a little behind the 1st, the 3rd twice as far, and so on. With hundreds of slits, every wave has another arriving 180° out of phase with it, and they cancel. So the light dies away very close to each bright direction.
4 So what does the wall show?
Bright spots only at the few angles where every slit’s light arrives in phase, and wide dark stretches everywhere between.

Answer Each narrow slit diffracts the light, so light from neighbouring slits overlaps and can interfere. Where each slit’s path is longer than its neighbour’s by a whole number of wavelengths (d sin θ = nλ), every wave arrives in phase and they interfere constructively: a bright spot. At any other angle, even a small phase difference between neighbours builds up across the many slits until each wave is cancelled by another arriving half a cycle out of step, so the wall is dark over wide stretches between the spots.

What the marker wanted

Achieved One idea: the waves must spread out (diffract) to overlap and interfere; a whole-wavelength path difference gives a bright spot; waves in phase give a bright spot; or the dark stretches are waves arriving out of phase and cancelling.
Merit Explains why there are bright spots (or dark gaps): a whole-wavelength path difference puts the waves in phase. Explaining the gaps with half-wavelength path differences, as for two slits, does not count.
Excellence The full answer: the diffracted waves overlap and interfere, whole-wavelength path differences give the bright spots, and the many slits make the wide dark regions.

In our words, from NZQA’s 2023 marking schedule.

Common mistake

Explaining the dark stretches as half-wavelength path differences, as for two slits. The schedule turns it down: with many slits, nearly every angle between the bright spots is dark, not only the half-wavelength ones.

How did it go?

Got it moves it further out. Review brings it back tomorrow.