Diffraction Gratings

A diffraction grating is an optical component with many equally spaced parallel slits. It produces sharp, well-defined maxima and is far more useful than a double slit for precise wavelength measurements and spectroscopy.

Spec mapping: This lesson sits under AQA 7408 section 3.3.2 and develops the diffraction grating: the grating equation d sin θ = nλ, the relationship d = 1/N between line density and slit spacing, the maximum-order constraint d sin θ ≤ d (hence n_max = floor(d/λ)), and the qualitative comparison with double-slit interference (sharper maxima from many-slit superposition). Spectroscopic applications across atomic and stellar physics are also part of the spec. (Refer to the official AQA specification document for exact wording.)

Synoptic links:

• Section 3.3.2 (Young's double slits): the grating equation d sin θ = nλ is the wide-angle generalisation of w = λD/s; the conceptual prerequisite is the double-slit derivation.
• Section 3.2.2 (atomic line spectra): diffraction gratings are the standard tool for resolving emission and absorption lines in laboratory and stellar spectroscopy; the hydrogen Balmer series is a classic Paper 2 synoptic example.
• Section 3.9 (astrophysics, optional unit): stellar classification and Doppler-shift measurements both rely on grating spectroscopy; even non-astrophysics candidates may meet stellar absorption-line questions in Paper 3.

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Structure of a Diffraction Grating

A typical diffraction grating consists of a glass or plastic slide with thousands of equally spaced parallel lines ruled on its surface. The unruled gaps between the lines act as slits.

Gratings are specified by the number of lines per unit length, N. For example:

• A grating with 300 lines per mm has N = 300 mm⁻¹ = 3.0 × 10⁵ m⁻¹
• A grating with 600 lines per mm has N = 600 mm⁻¹ = 6.0 × 10⁵ m⁻¹

The slit spacing (also called the grating spacing or grating element), d, is:

d = 1/N

For a grating with 300 lines per mm: d = 1/(3.0 × 10⁵) = 3.33 × 10⁻⁶ m = 3.33 μm

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The Grating Equation

When monochromatic light passes through a diffraction grating, sharp bright maxima (principal maxima) are observed at specific angles. The condition for a maximum is:

d sin θ = nλ

where:

• d = slit spacing (m)
• θ = angle of diffraction measured from the normal to the grating
• n = order number (n = 0, 1, 2, 3, ...)
• λ = wavelength (m)

Derivation

Adjacent slits are separated by distance d. For constructive interference at angle θ, the path difference between light from adjacent slits must equal a whole number of wavelengths:

Path difference = d sin θ = nλ

This is the same principle as for a double slit, but with many more slits producing much sharper maxima.

The Zero Order

At n = 0: d sin θ = 0, so θ = 0°. The zero-order maximum is always straight ahead, regardless of wavelength. This is simply the undeviated beam.

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Why Are Grating Maxima Sharp?

With a double slit, the maxima are broad because only two waves are interfering. With a grating containing thousands of slits, the waves from all the slits must arrive in phase for a maximum. Even a slight departure from the exact angle causes significant cancellation across the many slit pairs. The result is extremely narrow, intense maxima separated by wide dark regions.

The more slits, the sharper the maxima.

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Worked Examples

Worked Example 1 — Light of wavelength 550 nm is incident on a diffraction grating with 400 lines per mm. Calculate the angle of the first-order maximum.

d = 1/N = 1/(4.00 × 10⁵) = 2.50 × 10⁻⁶ m

For n = 1: d sin θ = λ

sin θ = λ/d = (550 × 10⁻⁹)/(2.50 × 10⁻⁶) = 0.220

θ = arcsin(0.220) = 12.7°

Worked Example 2 — Using the same grating as above, calculate the angle of the second-order maximum.

For n = 2: sin θ = 2λ/d = 2 × 0.220 = 0.440

θ = arcsin(0.440) = 26.1°

Worked Example 3 — Determine the maximum order visible for this grating and wavelength.

The maximum possible value of sin θ is 1. So:

d sin θ = nλ → n_max = d/λ = (2.50 × 10⁻⁶)/(550 × 10⁻⁹) = 4.55

Since n must be an integer, the maximum order is n = 4.

Check: sin θ = 4 × 550 × 10⁻⁹/(2.50 × 10⁻⁶) = 0.880, so θ = 61.6° — this is valid.

For n = 5: sin θ = 5 × 0.220 = 1.10 > 1 — impossible. So n = 4 is confirmed as the highest order.

Worked Example 4 — A diffraction grating is used to analyse light from a sodium lamp. Two closely spaced yellow lines are observed at wavelengths 589.0 nm and 589.6 nm. In the second order, calculate the angular separation of these two lines using a grating with 600 lines per mm.

d = 1/(6.00 × 10⁵) = 1.667 × 10⁻⁶ m

For λ₁ = 589.0 nm, n = 2: sin θ₁ = 2 × 589.0 × 10⁻⁹/(1.667 × 10⁻⁶) = 0.7068 θ₁ = 44.94°

For λ₂ = 589.6 nm, n = 2: sin θ₂ = 2 × 589.6 × 10⁻⁹/(1.667 × 10⁻⁶) = 0.7075 θ₂ = 45.00°

Angular separation = θ₂ − θ₁ = 45.00° − 44.94° = 0.06°

This is small but measurable with a good spectrometer. Higher orders give greater angular separation.

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Angular Separation and Resolving Power

Angular Dispersion

The angular separation between two wavelengths increases with:

• Higher order n — the factor n appears in d sin θ = nλ
• Smaller grating spacing d (more lines per mm)

Resolving Power

A grating can resolve (distinguish) two closely spaced wavelengths if they produce separable maxima. The resolving power is defined as:

R = λ/Δλ = nN_total

where N_total is the total number of slits illuminated and n is the order. A higher resolving power means the grating can distinguish smaller wavelength differences.

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Grating Spectra with White Light

When white light passes through a diffraction grating:

1. The zero order (n = 0) is white — all wavelengths constructively interfere at θ = 0°.
2. Each higher order produces a spectrum — violet is closest to the centre (smallest θ because shortest λ) and red is furthest from the centre (largest θ because longest λ).
3. Higher orders are spread over wider angles and may overlap with adjacent orders.

Exam Tip: Compare grating spectra with double slit fringes: the grating produces much sharper, more widely spaced maxima. The grating is far superior for measuring wavelengths accurately.

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Applications in Spectroscopy

Diffraction gratings are used in:

Atomic Spectroscopy

Gases can be excited (heated or placed in an electric discharge) to emit light at characteristic wavelengths. A diffraction grating separates these wavelengths into a line spectrum (discrete bright lines on a dark background), allowing identification of elements.

Stellar Spectroscopy

Analysing the light from stars reveals:

• The chemical composition (from absorption or emission lines)
• The surface temperature (from the peak wavelength)
• The radial velocity (from Doppler shifts of known spectral lines)

CD/DVD Tracks

The closely spaced tracks on a CD or DVD act as a reflection diffraction grating. White light reflecting from the surface is split into its component colours, producing the rainbow pattern you see on the disc surface.

Worked Example 5 — The track spacing on a DVD is 0.74 μm. White light is reflected from the surface. Calculate the angle at which the first-order maximum for red light (λ = 650 nm) is observed.

d = 0.74 μm = 7.4 × 10⁻⁷ m

sin θ = nλ/d = (1 × 650 × 10⁻⁹)/(7.4 × 10⁻⁷) = 0.878

θ = arcsin(0.878) = 61.4°

Worked Example 6 — Is a second-order red maximum possible for this DVD?

sin θ = 2 × 650 × 10⁻⁹/(7.4 × 10⁻⁷) = 1.757

Since sin θ > 1, this is impossible. Only the first-order maximum exists for red light with this grating.

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Reflection Gratings and Modern Spectroscopy

The gratings discussed so far are transmission gratings — light passes through the slits. Most modern spectrometers use reflection gratings: light is reflected from a precisely ruled metallic surface, and the same condition d sin θ = nλ governs the constructive maxima (with both angles measured from the grating normal). Reflection gratings are easier to manufacture at high line densities (~2400 lines/mm achievable), don't suffer from material absorption, and can be coated to optimise efficiency at particular wavelengths.

Blazed Gratings

A standard grating spreads its diffracted intensity across many orders, with no single order dominating. A blazed grating has its rulings cut at a specific angle (the blaze angle) such that one chosen order receives the bulk of the diffracted intensity — typically the first order. This is the standard configuration in laboratory and astronomical spectrometers, achieving up to 80% efficiency in the chosen order.

Worked Example 7 — A blazed reflection grating with 1200 lines/mm is used at first-order in the visible. Calculate the angular range over which it disperses visible light (400 nm to 700 nm).

d = 1/(1.2 × 10⁶) = 8.33 × 10⁻⁷ m

For λ = 400 nm: sin θ = (400 × 10⁻⁹) / (8.33 × 10⁻⁷) = 0.480; θ = 28.7°

For λ = 700 nm: sin θ = (700 × 10⁻⁹) / (8.33 × 10⁻⁷) = 0.840; θ = 57.1°

Angular range = 57.1° − 28.7° = 28.4° — a substantial fanning out of the visible spectrum across nearly 30°, allowing easy measurement of wavelength.

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The Resolving Power of a Grating

The minimum wavelength separation that a grating can resolve in the nth order is:

Δλ_min = λ/(n × N_illuminated)

where N_illuminated is the total number of slits illuminated by the beam (not the line density per unit length). This is the standard Rayleigh-criterion resolution analogous to single-slit diffraction.

Worked Example 8 — A grating with 600 lines/mm is illuminated over a width of 25 mm. Calculate the resolving power at first order, second order, and the minimum resolvable wavelength separation at λ = 589 nm in each order.

N_illuminated = (600 lines/mm) × (25 mm) = 15000 slits

R₁ = n × N = 1 × 15000 = 15000 at first order R₂ = n × N = 2 × 15000 = 30000 at second order

Δλ_min at first order: λ/R = 589/15000 = 0.039 nm Δλ_min at second order: λ/R = 589/30000 = 0.020 nm

The sodium D-line doublet (Δλ = 0.6 nm) is comfortably resolved at both orders, but the much finer hyperfine structure (Δλ ≈ 0.002 nm) is not resolved even at second order — finer gratings or higher orders would be needed.

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Crystal Diffraction (Bragg's Law)

The diffraction-grating equation has a remarkable extension to crystal structures. When X-rays of wavelength λ are reflected from successive crystal planes separated by distance d, constructive interference occurs at:

2d sin θ = nλ (Bragg's law)

The factor of 2 arises because the X-rays must travel an extra path length 2d sin θ in reaching the second plane — this is the round-trip path difference, not the one-way grating equation difference. Bragg's law underpins X-ray crystallography, the technique that revealed the structure of DNA (Watson, Crick, Wilkins, Franklin 1953), the atomic structure of countless minerals and proteins, and remains a central tool in materials science.

Worked Example 9 — Copper K-α X-rays of wavelength 1.54 × 10⁻¹⁰ m are diffracted from a sodium chloride crystal. The first-order diffraction maximum is observed at θ = 14.0°. Calculate the spacing between successive Na-Cl planes.

2d sin θ = nλ → d = nλ/(2 sin θ) = (1 × 1.54 × 10⁻¹⁰) / (2 × sin 14.0°)

d = (1.54 × 10⁻¹⁰) / (2 × 0.2419) = (1.54 × 10⁻¹⁰) / 0.4838 = 3.18 × 10⁻¹⁰ m = 3.18 Å

The known Na-Cl plane spacing in rock salt is approximately 2.82 Å (depending on which family of planes is sampled). The 13% discrepancy reflects either the choice of plane family or experimental error.

Worked Example 10 — DNA was solved (Watson, Crick, Wilkins, Franklin, 1953) using X-ray diffraction of fibres. A characteristic diffraction spot for DNA appears at angle 1.7° from the central beam when X-rays of wavelength 1.54 × 10⁻¹⁰ m are used. Estimate the corresponding plane spacing in the DNA structure.

Using Bragg's law with n = 1:

d = nλ / (2 sin θ) = (1 × 1.54 × 10⁻¹⁰) / (2 × sin 1.7°) = (1.54 × 10⁻¹⁰) / (2 × 0.0297) = (1.54 × 10⁻¹⁰) / 0.0593 = 2.6 × 10⁻⁹ m = 26 Å