White light through a grating
The zero order is white, because all wavelengths have zero path difference there. Every higher order is a continuous spectrum, with violet closest to the centre and red furthest out, since d sin θ = nλ means longer wavelength deviates more. Higher orders can overlap, for example the red end of the second order can fall beyond the violet end of the third order.
sin θ = nλ/d, so θ increases with λ at fixed nWhat this actually means
This is the opposite way round from a glass prism, where violet is deviated most. A grating disperses by path difference, a prism by refractive index, so the colour order reverses. Examiners love this contrast.
Explain the white centre properly: at n = 0 the path difference is zero for every wavelength, so all colours interfere constructively at the same place and recombine to white.
For the overlap question, compute the extreme angles. With d = 1.67 µm, the second order red (700 nm) is at 57.1° while the third order violet (450 nm) is at 54.1°, so the third order violet appears before the second order red has finished. That is a genuine overlap.
Each spectrum also gets wider with order, because the angular gap between violet and red grows with n. The first order might span 11°, the second 24°.
Draw it as a fan, symmetric about the centre: white line in the middle, then violet-to-red spectra spreading outwards on both sides, getting broader and fainter.
Putting red nearest the centre by copying the prism spectrum. On a grating the longest wavelength deviates most.
Prove it — watch it be true
- Switch the source to white light and observe the fanned spectra.
- Confirm the central maximum is white and that in every higher order violet sits closest to the centre.
- Increase the order and confirm the second and third order spectra begin to overlap.