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 = fraction mm = fraction m Example: 500 lines/mm → d = fraction m = 2.0 × 10^-6 m Difference Between Young's Slits and Diffraction Gratings FIG 8.20: Comparison of Double-Slit and Grating Patterns Show 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) Counter-Intuitive Result 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 Geometry Show: 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 θ = 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 Maximum Observable Order: Since θ 1: n_max = floor(fraction) If calculation gives θ > 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 dθ = 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 Spectrometer Not Required 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 Wavelength Question: 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. Solution First, find the grating spacing: d = fraction mm = fraction m = 1.67 × 10^-6 m Using dθ = nλ with n = 1: λ = fraction = 1.67 × 10^-6 × (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 Angle Question: Light of wavelength 650 nm passes through a diffraction grating with spacing 2.0 μm.
At what angle does the second-order maximum appear? Solution Given: λ = 650 × 10^-9 m, d = 2.0 × 10^-6 m, n = 2 Using dθ = nλ: θ = fraction = 2 × 650 × 10^-92.0 × 10^-6 θ = 1.3 × 10^-62.0 × 10^-6 = 0.65 θ = ^-1(0.65) = 40.5° Worked Example 3: Maximum Possible Order Question: A diffraction grating has 400 lines per mm.
What is the maximum order that can be observed for light of wavelength 700 nm? Solution Grating spacing: d = fraction m = 2.5 × 10^-6 m For the maximum to be…
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