Diffraction Grating with Overlapping Orders
A student uses a diffraction grating with 600 lines per millimetre and a red laser of wavelength 632.8 nm. A screen is placed 1.50 m away. The student observes a bright central spot and several orders on each side, but the second-order red line appears to overlap with a faint line the student believes is a different wavelength from the same laser. The student also notices that the third-order line is missing entirely. The laser is known to emit only the 632.8 nm line.
1. Calculate the grating spacing and the angle of the first three orders, and determine whether a third order should exist.
Read the worked answer
Grating spacing: d = 1/(600 × 10³) m = 1.667 × 10⁻⁶ m = 1.667 μm.
Grating equation: d sin θ = mλ, so sin θ = mλ/d = m × 632.8 × 10⁻⁹/1.667 × 10⁻⁶ = m × 0.3797.
- m = 1: sin θ = 0.3797, θ = 22.3°
- m = 2: sin θ = 0.7594, θ = 49.4°
- m = 3: sin θ = 1.139 > 1, impossible.
Conclusion: A third order cannot exist because sin θ would exceed 1. The 'missing' third order is simply not physically possible for this grating and wavelength. The student's expectation of a third order is the error.
2. Explain the faint line overlapping the second order and the absence of a third order, using physical reasoning rather than calculation alone.
Read the worked answer
Overlap: The faint line near the second order is almost certainly a reflection or ghost from the grating substrate, or light from a room source diffracting through the same grating. Since the laser emits only 632.8 nm, a genuine second wavelength is ruled out. The student should test by blocking the laser: if the faint line persists, it is ambient light.
Missing third order: The maximum order is m_max = floor(d/λ) = floor(1.667/0.6328) = floor(2.63) = 2. So only orders 0, ±1, ±2 are possible. There is no 'missing' order; the geometry forbids it.
Judgement: The student must distinguish between an absent order (physically impossible) and a spurious line (experimental artefact). Checking m_max first prevents misidentification.