The diffraction grating
Section: Superposition | Syllabus: Cambridge AS Level Physics (9702)
What is a Diffraction Grating?
A diffraction grating consists of a large number of equally spaced parallel slits (or lines). When light passes through, it produces sharp, bright maxima at specific angles.
Diffraction Grating A plate with many parallel, equally spaced slits (typically hundreds to thousands per millimetre) used to produce interference patterns and measure wavelengths with high precision. Grating Spacing:If grating has N lines per mm: d = (1/N) mm = (1/N × 10^3) m
Example: 500 lines/mm → d = (1/500 × 10^3) m = 2.0 × 10^-6 m
Difference Between Young's Slits and Diffraction Gratings
FIG 8.20: Comparison of Double-Slit and Grating PatternsShow side-by-side: (a) Double-slit pattern with broad, overlapping fringes of similar intensity, gradual intensity variation. (b) Diffraction grating pattern with sharp, narrow, intense maxima separated by wide dark regions. Label: central maximum (n=0), first order (n=1), second order (n=2). Show intensity vs angle graphs below each.
| Feature | Young's Double Slit | Diffraction Grating |
|---|---|---|
| Number of slits | 2 | Many (hundreds to thousands) |
| Bright maxima | Broad, gradually varying | Sharp, narrow, intense |
| Dark regions | Gradual transition | Wide and dark |
| Measurement precision | Lower | Higher (sharp maxima easier to locate) |
Why More Slits Give Sharper Maxima
With two slits: gradual transition between constructive and destructive interference → broad fringes.
With many slits:
- At maxima: all slits in phase → very bright, intense spots
- Slightly away from maxima: waves from different slits have many different phases → destructive interference
- More slits = sharper maxima (small deviations cause significant cancellation)
More slits does NOT mean more bright maxima. It means sharper maxima, wider dark regions, and fewer visible spots (secondary maxima suppressed). Students sometimes think incorrectly that more slits = more light = more maxima.
The Diffraction Grating Equation
FIG 8.21: Diffraction Grating GeometryShow: parallel light rays incident on grating (multiple slits with spacing d). Rays diffracted at angle θ to the normal. Show path difference between rays from adjacent slits = d sin θ. Mark the central maximum (n = 0, θ = 0), first order (n = 1), second order (n = 2) on both sides. Show how angle increases with order number.
The Diffraction Grating Equation:d sinθ = nλ
where:
- d = grating spacing (m)
- θ = diffraction angle from normal (° or rad)
- n = order number (0, 1, 2, 3, ...)
- λ = wavelength (m)
Order Numbers
- n = 0: Central maximum (θ = 0°), all wavelengths superimposed
- n = 1: First-order maxima (path difference = 1λ between adjacent slits)
- n = 2: Second-order maxima (path difference = 2λ)
- Higher orders at larger angles; symmetrical on both sides
Since sinθ ≤ 1: n_max = floor((d/λ))
If calculation gives sinθ > 1, that order does not exist.
Using a Diffraction Grating to Determine Wavelength
Diffraction gratings provide a highly accurate method for measuring the wavelength of light.
Procedure (Conceptual)
- Direct monochromatic light (e.g., from a laser or discharge lamp) at the grating
- Observe the diffraction pattern-a series of bright spots at specific angles
- Measure the angle θ to a particular order maximum (e.g., first order)
- Use the grating equation dsinθ = nλ to calculate λ
Why Gratings are More Accurate than Young's Slits
- Sharp maxima: The bright spots are narrow and well-defined, making it easier to locate their exact positions
- Large angles: Higher order maxima appear at larger angles, which can be measured more precisely
- Multiple orders: Measurements from different orders can be averaged to improve accuracy
- Young's slits produce broad, overlapping fringes that are harder to measure precisely
For the Cambridge AS Level syllabus, knowledge of the structure and use of the spectrometer is not required. The focus is on understanding the principles of the diffraction grating.
Worked Examples
Worked Example 1: Finding WavelengthQuestion: A diffraction grating has 600 lines per mm. Monochromatic light produces a first-order maximum at an angle of 20.5° to the normal. Calculate the wavelength of the light.
SolutionFirst, find the grating spacing:
d = (1/600) mm = (1/600 × 10^3) m = 1.67 × 10^-6 m
Using dsinθ = nλ with n = 1:
λ = (dsinθ/n) = 1.67 × 10^-6 × sin(20.5°)1
λ = 1.67 × 10^-6 × 0.350 = 5.85 × 10^-7 m
λ = 585 nm (yellow light)
Worked Example 2: Finding the AngleQuestion: Light of wavelength 650 nm passes through a diffraction grating with spacing 2.0 μm. At what angle does the second-order maximum appear?
SolutionGiven: λ = 650 × 10^-9 m, d = 2.0 × 10^-6 m, n = 2
Using dsinθ = nλ:
sinθ = (nλ/d) = 2 × 650 × 10^-92.0 × 10^-6
sinθ = 1.3 × 10^-62.0 × 10^-6 = 0.65
θ = sin^-1(0.65) = 40.5°
Worked Example 3: Maximum Possible OrderQuestion: A diffraction grating has 400 lines per mm. What is the maximum order that can be observed for light of wavelength 700 nm?
SolutionGrating spacing: d = (1/400 × 10^3) m = 2.5 × 10^-6 m
For the maximum to be visible, sinθ ≤ 1
From dsinθ = nλ: n ≤ (d/λ)
n ≤ 2.5 × 10^-6700 × 10^-9 = 2.5 × 10^-67.0 × 10^-7 = 3.57
Maximum order n_max = 3 (taking the integer part)
Orders 0, 1, 2, and 3 can be observed (on each side of centre).
Worked Example 4: Grating with Lines per mmQuestion: A grating has 750 lines per mm. Calculate the angle at which the first-order maximum for light of wavelength 450 nm appears.
SolutionGrating spacing: d = (1/750 × 10^3) m = 1.33 × 10^-6 m
Using sinθ = (nλ/d):
sinθ = 1 × 450 × 10^-91.33 × 10^-6 = 0.338
θ = sin^-1(0.338) = 19.7°
Common Exam Mistakes
Avoid These Errors- Confusing lines per mm with grating spacing: If there are N lines per mm, then d = (1/N) mm. Remember to convert to metres.
- Forgetting that sinθ ≤ 1: If your calculation gives sinθ > 1, that order does not exist.
- Using degrees vs radians: Ensure your calculator is in the correct mode. The grating equation works with either, but be consistent.
- Misidentifying the order: n = 0 is the central maximum (straight through); n = 1 is the first-order maximum on either side.
- Thinking more slits = more maxima: More slits give sharper, brighter maxima and wider dark regions-not more maxima.
- Unit errors: Wavelength is often given in nm (10^-9 m) and grating spacing derived from lines/mm. Convert carefully.
Concept Check Questions
Interactive revision notes, videos and practice questions load below.