Difficulty: Intermediate | Prerequisites: Wave basics (wavelength, frequency, superposition), Ch. 16 on wave optics fundamentals
When light passes through two or more narrow slits, waves from each slit interfere to produce bright and dark fringes on a screen. A diffraction grating uses hundreds or thousands of slits to make these fringes much sharper and brighter. The key equation, d sin θ = mλ, predicts exactly where each bright fringe lands.
Path-length difference (Δr)
The difference in distance that two waves travel from their respective slits to a given point on the screen. This quantity determines whether the waves arrive in phase (constructive) or out of phase (destructive). In simple terms, it is the extra distance one wave has to travel compared to its neighbour.
Slit spacing (d)
The centre-to-centre distance between adjacent slits in a double-slit setup or a diffraction grating. For a grating with N lines per millimetre, d = 1 mm / N. Think of it as the repeat distance of the pattern of openings.
Order number (m)
An integer (0, ±1, ±2, ...) that labels each bright fringe. m = 0 is the central maximum, m = ±1 the first-order fringes on either side, and so on. Each order corresponds to a path-length difference of exactly m wavelengths.
Fringe spacing (Δy)
The distance between consecutive bright fringes on the screen, given by Δy = λL / d (for small angles). In simple terms, this is how far apart the bright bands appear on your detector or screen.
Diffraction grating
An optical element with a very large number of equally spaced parallel slits. It works on the same principle as a double slit but produces much narrower, brighter fringes because hundreds or thousands of waves must all arrive in phase simultaneously.
Light from a single source passes through two narrow slits separated by distance d. Each slit acts as a source of circular (spherical) wavefronts. Where crest meets crest on the far screen, a bright fringe appears (constructive interference). Where crest meets trough, darkness (destructive interference).
Constructive interference occurs when the path-length difference equals a whole number of wavelengths:
\Delta r = d \sin\theta_m = m\lambdaRearranging for the angle of the m-th bright fringe:
\sin\theta_m = m \frac{\lambda}{d}For a screen at distance L from the slits, the position of the m-th bright fringe is:
y_m = L \tan\theta_mWhen angles are small (which they usually are in practice), tan θ ≈ sin θ, so this simplifies to:
y_m = \frac{m\lambda L}{d}The distance between adjacent bright fringes is constant (for small angles):
\Delta y = \frac{\lambda L}{d}Larger wavelength or larger screen distance spreads fringes out. Larger slit spacing pushes them closer together.
A diffraction grating is the same physics as double-slit interference, scaled up. Instead of two slits, a grating has N slits (often hundreds or thousands per millimetre). The bright-fringe condition is identical:
d \sin\theta_m = m\lambdaThe slit spacing d is found from the line density: a grating with N lines per mm has d = (1 mm) / N.
With only two slits, the constructive-interference condition is easy to satisfy approximately, so bright fringes are broad and soft. With many slits, all N waves must arrive in phase simultaneously. Even a small deviation from the exact angle causes most of the waves to cancel. The result: very narrow, very bright peaks at the exact angles predicted by the equation, with near-darkness between them.
Increasing the number of slits produces narrower and brighter fringes, with no change to their positions.
The position of the m-th order bright fringe on a screen at distance L:
y_m = L \tan\theta_mBecause the fringes are so sharp, a grating can separate light of slightly different wavelengths into distinct lines. This is the basis of grating spectroscopy: measure the angle, know d, and solve for λ.
Formula | Meaning | When to use |
|---|---|---|
d sin θ_m = mλ | Bright fringe condition | Finding the angle of the m-th order maximum (double slit or grating) |
sin θ_m = m(λ/d) | Rearranged for angle | Same, solving for angle directly |
y_m = L tan θ_m | Fringe position on screen | Finding where a fringe lands (exact) |
y_m = mλL / d | Fringe position (small angle) | Quick calculation when θ is small |
Δy = λL / d | Fringe spacing | Distance between consecutive bright fringes |
d = 1 mm / N | Slit spacing from line density | Converting "lines per mm" to slit spacing |
Δr = d sin θ | Path-length difference | The extra distance one wave travels relative to its neighbour |
Diffraction gratings are the working principle behind spectrometers used in chemistry labs, astronomy, and forensic science. When white light hits a grating, each wavelength diffracts to a different angle, spreading the light into a spectrum, which is how astronomers determine the chemical composition of distant stars.
The rainbow patterns you see on the surface of a CD or DVD are diffraction patterns. The tiny, regularly spaced tracks on the disc act as a reflection grating.
Students often think that increasing the number of slits changes the positions of the bright fringes. It does not. More slits make the same fringes narrower and brighter, but the angles stay the same.
It is common to confuse slit spacing (d) with slit width (a). Slit spacing is the distance between the centres of adjacent slits and controls where interference maxima appear. Slit width controls the single-slit diffraction envelope (covered in Section 17.5).
Students sometimes assume that m can take any integer value. In reality, sin θ cannot exceed 1, so the maximum observable order is limited to m_max = d / λ (rounded down).
Forgetting to convert line density to slit spacing is a frequent source of errors. If a grating has 500 lines/mm, d = 1/500 mm = 2 × 10⁻⁶ m, not 500 m.
⚠️ The equation d sin θ = mλ is the single most tested formula from this section. Be ready to use it in both directions: given λ and d, find θ; given θ and d, find λ.
⚠️ Exam questions frequently give slit density (lines/mm) rather than slit spacing. Always convert to d in metres before substituting.
⚠️ Expect to be asked what happens when the number of slits increases. The answer: fringes get narrower and brighter, but their positions do not change.
⚠️ Be prepared to sketch the pattern for a double slit vs. a grating, showing broader fringes for the double slit and sharp peaks for the grating, both at the same angles.
True or false: A diffraction grating with more slits produces fringes at different angles than a double slit with the same spacing. (False: same angles, just sharper peaks.)
Fill in the blank: The path-length difference for the m-th order bright fringe is Δr = ______. (mλ)
True or false: Doubling the slit spacing d doubles the fringe spacing Δy on the screen. (False: Δy = λL/d, so doubling d halves Δy.)
Fill in the blank: A grating with 600 lines/mm has a slit spacing of d = ______ m. (1.67 × 10⁻⁶ m)
True or false: The central maximum (m = 0) is always at θ = 0 regardless of wavelength. (True.)
Q: A diffraction grating has 800 lines/mm. What is the angle of the first-order maximum for light of wavelength 550 nm?
A: d = 1/800 mm = 1.25 × 10⁻⁶ m. sin θ₁ = (1)(550 × 10⁻⁹) / (1.25 × 10⁻⁶) = 0.44. θ₁ = arcsin(0.44) ≈ 26.1°.
Q: In a double-slit experiment, d = 0.25 mm, λ = 632 nm, and L = 1.5 m. What is the fringe spacing on the screen?
A: Δy = λL/d = (632 × 10⁻⁹)(1.5) / (0.25 × 10⁻³) = 3.79 × 10⁻³ m ≈ 3.8 mm.
Q: What is the highest order visible for a grating with d = 2.0 μm illuminated by 700 nm light?
A: m_max = d/λ = 2.0 × 10⁻⁶ / 700 × 10⁻⁹ = 2.86. Since m must be an integer, m_max = 2.
Q: Explain why increasing the number of slits in a grating makes the fringes narrower but does not change their positions.
A: The positions depend only on d sin θ = mλ, which involves slit spacing, not the number of slits. However, with more slits, even a slight deviation from the exact angle causes destructive interference among the many waves, so only the precise angle survives. The peaks become very narrow as a result.
This material builds directly on the superposition principle from earlier wave chapters. If two-source interference (e.g. two speakers, two water-wave sources) made sense, this is the same idea with light.
Diffraction gratings connect forward to thin-film interference (Ch. 17.4), which also uses path-length differences but with reflected rather than transmitted waves. Single-slit diffraction (Ch. 17.5) governs the envelope that modulates the multi-slit pattern, so these topics are often combined in exam questions.
In later courses, this same mathematics appears in X-ray crystallography (Bragg diffraction), where the atomic planes of a crystal act as a natural diffraction grating.
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