Physics 9702 · Paper 2

The Diffraction Grating

With two slits you get broad, blurry bright fringes. A has thousands of equally spaced slits instead, and its bright lines come out very thin and much brighter.
The reason they are so sharp: with so many slits, if the angle is even slightly wrong, some pair of beams cancels. Only at a few exact angles do all the beams add, and each of those angles gives a thin, bright line.

Many slits, sharp maxima

All the beams add up only at certain exact angles. There they add together, giving a sharp, bright band called a .
The straight-through band is the zeroth , $n = 0$ . Then the first order $n = 1$ on each side, then the second order, and so on.
The bands are sharper, brighter and further apart than the double-slit fringes, because many beams add, not just two.

A laser beam passing through a diffraction grating and splitting into five beams that land as five red dots on a screen — One laser beam in, a fan of sharp beams out: one bright dot for each order, brightest at n = 0.

The grating formula

Take two slits next to each other. Look at the angle where a bright band forms. Along that angle, one beam travels a little further than its neighbour. That extra distance is $d\sin\theta$ .
For a bright band, that extra path must be a whole number of wavelengths. That gives the grating formula.

d\sin\theta = n\lambda

Symbols

$d$ = grating spacing, the distance between neighbouring lines (m)
$\theta$ = angle of the nth maximum from the straight-through direction (degrees)
$n$ = order, a whole number 0, 1, 2, … (no unit)
$\lambda$ = wavelength of the light (m)

Two neighbouring slits, a distance d apart, send light off at the same angle θ. The lower ray travels one extra bit, d sinθ. For a bright line this must be a whole number of wavelengths: d sinθ = nλ.

Common mistake

Gratings are quoted as lines per metre (or per cm), not as a spacing. Turn that into the spacing first:

d = \dfrac{1}{\text{lines per metre}}

. This one step is where most marks are lost.

Worked example

Wavelength from a grating

A grating has 3000 lines per cm. The first order ( $n = 1$ ) sits at $\theta = 10^\circ$ . Find the wavelength.

Find the spacing. There are 3000 lines per cm, and 1 cm is $10^{-2}$ m, so there are $3000 \times 100 = 3.0 \times 10^{5}$ lines per metre.
Spacing: $d = \dfrac{1}{3.0 \times 10^{5}} = 3.33 \times 10^{-6}$ m.
Substitute into $d\sin\theta = n\lambda$ with $n = 1$ : $\lambda = d\sin\theta = (3.33 \times 10^{-6})\sin 10^\circ$ .
This gives $\lambda = 5.8 \times 10^{-7}$ m.

Answer

\lambda = 5.8 \times 10^{-7}\ \text{m} = 580\ \text{nm}

Your turn— tap to reveal the worked answer (9702/11/M/J/25 Q30)

Light of wavelength 680 nm hits a grating with 450 lines per mm. Find the angle of the second-order maximum.

Answer: spacing $d = \dfrac{1}{450 \times 1000} = 2.22 \times 10^{-6}$ m. Then $\sin\theta = \dfrac{n\lambda}{d} = \dfrac{2 \times 680 \times 10^{-9}}{2.22 \times 10^{-6}} = 0.612$ , so $\theta \approx 38^\circ$ . Check the unit on d, and remember d is the line spacing, not the grating-to-screen distance. (9702/11/M/J/25 Q30)

The highest order you can see

Two more things about a grating: there is a limit to how many orders it can make, and in white light it splits the colours into a . Take the order limit first.
A beam can bend at most until it travels along the grating surface. That is 90°, and it cannot go past that. So $\sin\theta$ can never be bigger than 1.
Put the biggest case, $\sin\theta = 1$ , into $d\sin\theta = n\lambda$ . That gives $n \le \dfrac{d}{\lambda}$ . So the highest order is the whole-number part of $d/\lambda$ .

Worked example

How many orders fit

Green light, wavelength 550 nm, through a grating of 400 lines per mm. What is the highest order?

Spacing: 400 lines per mm is $400 \times 1000 = 4.0 \times 10^{5}$ lines per metre, so $d = \dfrac{1}{4.0 \times 10^{5}} = 2.5 \times 10^{-6}$ m.
Divide: $\dfrac{d}{\lambda} = \dfrac{2.5 \times 10^{-6}}{550 \times 10^{-9}} = 4.55$ .
n must be a whole number and cannot go above 4.55, so the highest order is 4.

Answer

n_{\text{max}} = 4

Your turn— tap to reveal the worked answer (9702/11/O/N/25 Q30)

For a given grating and wavelength, $d/\lambda = 4.55$ . What is the highest order maximum?

Answer: take the whole-number part of 4.55, so the highest order is 4. You cannot round up to 5, because that would need $\sin\theta$ above 1. (9702/11/O/N/25 Q30)

White light through a grating

The zeroth order stays white. Every colour goes straight through with zero path difference, so they all add together.
Every higher order spreads into a spectrum. A longer wavelength bends more, so each colour comes out at its own angle.
Violet has the shortest wavelength, so it bends the least and is on the inside. Red bends the most and is on the outside.

White light splits into a spectrum: violet bends least (inner), red bends most (outer). Higher orders spread even wider.

Your turn— tap to reveal the worked answer (9702/12/O/N/22 Q30)

Red laser light through a grating gives a row of separate red dots on a screen. The red light is swapped for white light. What changes?

Answer: the central dot stays a single white dot (zero path difference for every colour), and each side dot spreads into a short spectrum, violet on the inside, red on the outside. (9702/12/O/N/22 Q30)

Your turn— tap to reveal the worked answer (9702/11/O/N/20 Q29)

White light passes through a grating. Orange light of wavelength 600 nm in the 4th order lands at the same angle as blue light in the 5th order. Find the wavelength of the blue light.

Answer: same grating and same angle means $d\sin\theta$ is the same, so $n\lambda$ is the same: $4 \times 600 = 5 \times \lambda_{\text{blue}}$ , giving $\lambda_{\text{blue}} = 480$ nm. (9702/11/O/N/20 Q29)

A grating gives many sharp, bright, unevenly spaced maxima, far cleaner than two slits. The highest order is the whole-number part of $d/\lambda$ , and white light spreads each higher order into a violet-to-red spectrum.

PreviousThe Double-Slit Experiment NextHow a Stationary Wave Forms

The Diffraction Grating

With two slits you get broad, blurry bright fringes. A has thousands of equally spaced slits instead, and its bright lines come out very thin and much brighter.
The reason they are so sharp: with so many slits, if the angle is even slightly wrong, some pair of beams cancels. Only at a few exact angles do all the beams add, and each of those angles gives a thin, bright line.

Many slits, sharp maxima

All the beams add up only at certain exact angles. There they add together, giving a sharp, bright band called a .
The straight-through band is the zeroth , $n = 0$ . Then the first order $n = 1$ on each side, then the second order, and so on.
The bands are sharper, brighter and further apart than the double-slit fringes, because many beams add, not just two.

The grating formula

Take two slits next to each other. Look at the angle where a bright band forms. Along that angle, one beam travels a little further than its neighbour. That extra distance is $d\sin\theta$ .
For a bright band, that extra path must be a whole number of wavelengths. That gives the grating formula.

d\sin\theta = n\lambda

Symbols

$d$ = grating spacing, the distance between neighbouring lines (m)
$\theta$ = angle of the nth maximum from the straight-through direction (degrees)
$n$ = order, a whole number 0, 1, 2, … (no unit)
$\lambda$ = wavelength of the light (m)

Common mistake

Gratings are quoted as lines per metre (or per cm), not as a spacing. Turn that into the spacing first:

d = \dfrac{1}{\text{lines per metre}}

. This one step is where most marks are lost.

Worked example

Wavelength from a grating

A grating has 3000 lines per cm. The first order ( $n = 1$ ) sits at $\theta = 10^\circ$ . Find the wavelength.

Find the spacing. There are 3000 lines per cm, and 1 cm is $10^{-2}$ m, so there are $3000 \times 100 = 3.0 \times 10^{5}$ lines per metre.
Spacing: $d = \dfrac{1}{3.0 \times 10^{5}} = 3.33 \times 10^{-6}$ m.
Substitute into $d\sin\theta = n\lambda$ with $n = 1$ : $\lambda = d\sin\theta = (3.33 \times 10^{-6})\sin 10^\circ$ .
This gives $\lambda = 5.8 \times 10^{-7}$ m.

Answer

\lambda = 5.8 \times 10^{-7}\ \text{m} = 580\ \text{nm}

Your turn— tap to reveal the worked answer (9702/11/M/J/25 Q30)

Light of wavelength 680 nm hits a grating with 450 lines per mm. Find the angle of the second-order maximum.

The highest order you can see

Two more things about a grating: there is a limit to how many orders it can make, and in white light it splits the colours into a . Take the order limit first.
A beam can bend at most until it travels along the grating surface. That is 90°, and it cannot go past that. So $\sin\theta$ can never be bigger than 1.
Put the biggest case, $\sin\theta = 1$ , into $d\sin\theta = n\lambda$ . That gives $n \le \dfrac{d}{\lambda}$ . So the highest order is the whole-number part of $d/\lambda$ .

Worked example

How many orders fit

Green light, wavelength 550 nm, through a grating of 400 lines per mm. What is the highest order?

Spacing: 400 lines per mm is $400 \times 1000 = 4.0 \times 10^{5}$ lines per metre, so $d = \dfrac{1}{4.0 \times 10^{5}} = 2.5 \times 10^{-6}$ m.
Divide: $\dfrac{d}{\lambda} = \dfrac{2.5 \times 10^{-6}}{550 \times 10^{-9}} = 4.55$ .
n must be a whole number and cannot go above 4.55, so the highest order is 4.

Answer

n_{\text{max}} = 4

Your turn— tap to reveal the worked answer (9702/11/O/N/25 Q30)

For a given grating and wavelength, $d/\lambda = 4.55$ . What is the highest order maximum?

Answer: take the whole-number part of 4.55, so the highest order is 4. You cannot round up to 5, because that would need $\sin\theta$ above 1. (9702/11/O/N/25 Q30)

White light through a grating

The zeroth order stays white. Every colour goes straight through with zero path difference, so they all add together.
Every higher order spreads into a spectrum. A longer wavelength bends more, so each colour comes out at its own angle.
Violet has the shortest wavelength, so it bends the least and is on the inside. Red bends the most and is on the outside.

White light splits into a spectrum: violet bends least (inner), red bends most (outer). Higher orders spread even wider.

Your turn— tap to reveal the worked answer (9702/12/O/N/22 Q30)

Red laser light through a grating gives a row of separate red dots on a screen. The red light is swapped for white light. What changes?

Your turn— tap to reveal the worked answer (9702/11/O/N/20 Q29)

White light passes through a grating. Orange light of wavelength 600 nm in the 4th order lands at the same angle as blue light in the 5th order. Find the wavelength of the blue light.