Diffraction gratings

What a grating is

• A diffraction grating is a piece of glass or plastic ruled with a very large number of equally spaced parallel slits — typically hundreds or thousands per millimetre.
• Light passing through is diffracted at every slit, and the waves from all the slits superpose.
• The result is the same family of maxima as two slits, but far sharper and brighter.

$$d\sin\theta = n\lambda$$

• $d$ — the grating spacing, the distance between adjacent slits (m)
• $\theta$ — the angle of the $n$th maximum from the straight-through direction
• $n$ — the order of the maximum ($n = 0, 1, 2, \dots$)
• $\lambda$ — wavelength (m)

Finding $d$ from the grating's rating

Gratings are labelled by lines (or slits) per millimetre, not by their spacing, so the first step in almost every question is a conversion:

$$d = \frac{1}{N}$$

• $N$ — the number of lines per metre

• A grating marked "$600$ lines per mm" has $N = 600 \times 10^{3} = 6.00 \times 10^{5}$ lines per metre, so:

$$d = \frac{1}{6.00 \times 10^{5}} = 1.67 \times 10^{-6} \text{ m}$$

• Convert lines per millimetre to lines per metre before taking the reciprocal. Forgetting this factor of $1000$ is the most common error on this page.

Why the maxima are so sharp

• With two slits, moving slightly away from a maximum only slightly reduces the reinforcement, so the fringes are broad and fade gradually.
• With thousands of slits, a small change in angle puts the many waves progressively out of step with one another, and they cancel almost completely.
• The result is that light appears only at very precise angles, with near-total darkness in between:
  • the maxima are very narrow and sharp,
  • they are much brighter, because the light from thousands of slits is concentrated into them,
  • they are more widely separated than two-slit fringes, because $d$ is very small.
• This sharpness is why gratings, not double slits, are used for measuring wavelengths accurately.

How many orders are possible

• Since $\sin\theta$ cannot exceed $1$, the equation $d\sin\theta = n\lambda$ places a hard limit on the order:

$$n_{\text{max}} \le \frac{d}{\lambda}$$

• Work out $d/\lambda$ and round down to the nearest whole number — that is the highest order visible.
• Include the zero order and both sides when a question asks for the total number of maxima: orders $0, \pm1, \pm2, \dots, \pm n_{\text{max}}$, giving $2n_{\text{max}} + 1$ in total.

Grating compared with two slits

	Two slits	Diffraction grating
Number of slits	2	thousands
Maxima	broad, gradually fading	very sharp and narrow
Brightness	dim	much brighter
Separation of orders	small (mm on a screen)	large (measured as angles)
Best measured with	a ruler on a screen	a protractor / spectrometer
Used for	demonstrating interference	measuring wavelengths precisely

• Note the change of method: two-slit work measures distances on a screen and uses $n\lambda = dx/L$; grating work measures angles and uses $d\sin\theta = n\lambda$.