IB Physics HLTopic 3 — Oscillations & WavesPaper 1 & 2d sin θ = nλ~15 min read
Diffraction Gratings
Two slits gave you fringes you could just about measure. Now cut thousands of slits into a piece of glass, all the same distance apart. The bright maxima become razor-thin, blindingly sharp lines separated by broad darkness — precise enough to fingerprint the light from a distant star. Everything about a grating follows from a single equation, and you already know why it’s true.
📘 What you need to know
A grating is a large number of thin, equally spaced parallel slits
The grating equation: d sin θ = nλ
d = slit spacing, θ = angle of the nth maximum from the normal, n = order (0, 1, 2…)
From lines per metre: d = 1/N
More slits → maxima get sharper and brighter, dark regions wider
Highest visible order: n = d/λ, always rounded down to a whole number
White light: a thin white central maximum, then a spectrum in every other order — violet nearest the centre, red furthest
What a grating does
A diffraction grating is a plate with a huge number of very thin, equally spaced parallel slits ruled into it — typically hundreds per millimetre. Light diffracts at every slit, and all those diffracted beams interfere.
The result looks like a double-slit pattern with the contrast turned up to maximum:
Bright fringes become narrower and much brighter
The dark regions between them become wider and darker
The maxima sit at exactly the same angles a double slit of the same spacing would give
The angles are drawn to scale: sin θ2 = 2 sin θ1, so θ1 = 15° gives θ2 = 31.2°. Higher orders sit at larger angles, and the gaps between them grow.
The grating equation
Look at two adjacent slits. Light leaving them at angle θ to the normal travels along parallel paths, and the ray from one slit lags the ray from its neighbour by exactly d sin θ.
Now the key step. Because every slit is the same distance d from the next, that same lag repeats all the way across the grating. If the lag equals a whole number of wavelengths, then every single slit arrives in phase with every other one — thousands of waves adding together at once.
The lag builds up in equal steps across the grating. When one step is a whole number of wavelengths, so is every step — and all the slits reinforce at once.
The grating equationd sin θ = nλ
d = distance between adjacent slits (m)
θ = angle of diffraction of the nth order, measured from the normal
n = order of the maximum: 0, 1, 2, 3… (n = 0 is the central maximum)
λ = wavelength of the light (m)
Notice the shape of this equation. For a single slit, b sin θ = nλ located the dark fringes. For a grating, d sin θ = nλ locates the bright ones. Different letter, opposite meaning. Check which one you’re using before you touch the calculator: b is a width, d is a spacing.
Slit spacing from lines per millimetre
Gratings are sold by how many lines they have per millimetre — 100, 300, 600 lines/mm. Call that N. The spacing is simply its reciprocal:
Slit spacingd = 1 / N
If N is in lines per metre, d comes out in metres. If N is per millimetre, d is in millimetres — convert before substituting.
More slits, sharper lines
Why bother with thousands of slits when two already gave interference? Because more slits sharpen the maxima. The bright lines stay at the same angles, but get narrower and far more intense, with wide dark gaps in between. That sharpness is what makes a grating a precision instrument.
All three curves use the same slit spacing, so the maxima land in identical places. With twenty slits the peaks are already twenty times narrower than the two-slit fringes — a real grating has thousands.
Angular separation and the highest order
Rearranging the grating equation gives the angle of any order:
Angle of the nth order
sin θ = nλ / d
Because θ is measured from the centre, higher orders sit at bigger angles. The angular separation between two orders is just the difference: θ2 − θ1.
How many orders can you see?
The furthest a beam could possibly be diffracted is 90°, straight along the grating. Setting sin θ = 1 in the grating equation gives the largest possible n:
Highest visible ordern = d / λ
Since n must be a whole number, you always round down. If d/λ comes out as 2.7, the highest order you can see is n = 2 — there is no such thing as a 2.7th maximum.
Lines per mm N
d = 1/N
Spacing d
sin θ = nλ/d
Angle of each order
White light through a grating
Send white light through a grating and each wavelength obeys the same equation — but with a different λ, so each is sent to a different angle. Every order except the centre becomes a full spectrum.
The central maximum (n = 0) is a thin white line: at θ = 0 every wavelength interferes constructively together
Violet has the shortest λ, so it is diffracted least — it appears nearest the centre
Red has the longest λ, so it is diffracted most — it appears furthest out
At high orders the spectra get wider and eventually overlap, which is why the first-order spectrum is used for analysis
Drawn to scale for a 300 lines/mm grating: the bar positions really are proportional to sin θ = nλ/d. The third-order violet (sin θ = 0.36) genuinely falls inside the second-order red (sin θ = 0.42) — the orders really do overlap.
🧮 Working with the grating equation
Find d first. If given lines per mm, convert to lines per metre, then d = 1/N.
Write sin θ = nλ/d and substitute. Keep everything in metres.
Check sin θ ≤ 1. If it exceeds 1, that order does not exist.
For the highest order, work out d/λ and round down.
θ is from the centre — the angle between the two nth-order beams on either side is 2θ.
WE 1
Light of wavelength 600 nm is shone normally at a diffraction grating with 300 lines per millimetre. Calculate the angle of the first-order maximum.
Step 1 — find the slit spacing
N = 300 lines/mm = 3.00 × 10⁵ lines/m
d = 1/N = 3.33 × 10⁻⁶ mStep 2 — rearrange the grating equation
sin θ = nλ / d
Step 3 — substitute with n = 1sin θ = (1 × 6.00 × 10⁻⁷) / (3.33 × 10⁻⁶) = 0.180Step 4 — inverse sineθ = 10.4°The two first-order beams sit 10.4° either side of the centre, so 20.8° apart from each other.
WE 2
For the same grating and wavelength, determine the highest order of maximum that can be observed.
Step 1 — the largest possible angle is 90°, so sin θ = 1
d × 1 = nλ
Step 2 — rearrange for n
n = d / λ
n = (3.33 × 10⁻⁶) / (6.00 × 10⁻⁷) = 5.56Step 3 — n must be a whole number, so round DOWNn = 5Check: n = 5 needs sin θ = 0.900 (fine), but n = 6 would need sin θ = 1.08 — impossible. Rounding up is the classic error.
WE 3
White light (400 nm to 700 nm) is shone at the same 300 lines/mm grating. Calculate the angular width of the first-order spectrum.
Step 1 — violet is diffracted least, so find its angle firstsin θ = (4.00 × 10⁻⁷)/(3.33 × 10⁻⁶) = 0.120 → θ = 6.89°Step 2 — red is diffracted mostsin θ = (7.00 × 10⁻⁷)/(3.33 × 10⁻⁶) = 0.210 → θ = 12.12°Step 3 — the spectrum spans the gap between them12.12 − 6.89angular width = 5.2°Violet on the inside, red on the outside — the opposite way round to a glass prism, where violet bends most.
Grating vs prism: both split white light, but they disagree about which colour bends furthest. A prism refracts violet the most (higher n for short λ). A grating diffracts red the most (larger λ in d sin θ = nλ). Examiners love that contrast.
Feature
Double slit
Diffraction grating
Number of slits
2
Thousands
Bright fringes
Broad, fairly dim
Very narrow and bright
Dark regions
Narrow
Wide and dark
Equation
s = λD/d
d sin θ = nλ
Best used for
Demonstrating interference
Measuring wavelength precisely
💡 Top tips
d sin θ = nλ gives the BRIGHT maxima. The single-slit look-alike gives the dark ones.
Convert lines per mm to lines per metre before using d = 1/N.
The highest order is d/λ, rounded down. Never round up.
If sin θ comes out greater than 1, that order simply doesn’t exist — say so.
θ is measured from the normal, not between two orders. The two nth-order beams are 2θ apart.
In a grating spectrum, violet is nearest the centre.
⚠ Common mistakes
Rounding the highest order up — you cannot see a fraction of a maximum
Using θ as the angle between two orders instead of from the normal
Forgetting to convert lines per mm into lines per metre
Confusing d (slit spacing) with b (slit width) from single-slit diffraction
Saying red is nearest the centre — that’s a prism, not a grating
Thinking more slits moves the maxima — it only makes them sharper
Quick recap: A grating is thousands of equally spaced slits. Adjacent slits differ in path by d sin θ, so all of them reinforce when d sin θ = nλ. Get d from d = 1/N. More slits means sharper, brighter maxima at the same angles. The highest order is d/λ rounded down, and white light produces a spectrum in every order with violet nearest the centre.
That completes Wave Phenomena. Look back at what one idea did: superposition gave interference, interference gave Young’s fringes, adding slit width gave the diffraction envelope, and adding thousands of slits gave an instrument that reads the chemistry of stars from their light. Every equation on these pages came from the same sentence — where waves overlap, add the displacements.
Gratings and orders got you tangled?
Book a free meeting and we’ll drill d sin θ = nλ, highest orders and past-paper grating questions together.
