Grating Thermals and Laser Wavelength Drift
A student uses a transmission diffraction grating with a nominal 600 lines per millimetre to measure the wavelength of a laser pointer. The grating is mounted in an aluminium holder in a laboratory whose temperature rises by 8.0 K during the experiment. The first-order spot is measured at 22.30° from the straight-through direction. The laser beam diameter is 1.2 mm and the grating has a ruled width of 25 mm. The glass grating substrate has a linear expansion coefficient of 8.0×10−6 K−1 and the aluminium holder 23×10−6 K−1. A second measurement later in the afternoon gives 22.32°. The student’s partner says the shift is due to thermal expansion of the grating and that the wavelength has therefore changed; the student thinks it is a misalignment.
1. Calculate the grating spacing, the wavelength implied by 22.30°, and the angular shift expected from thermal expansion of the glass over 8.0 K.
Read the worked answer
d = 1/(600 mm−1) = 1.667×10−6 m.
λ = d sinθ = 1.667×10−6 × sin 22.30° = 1.667×10−6 × 0.3794 ≈ 6.32×10−7 m = 632 nm.
For small fractional change, d sinθ = λ gives Δθ ≈ (Δd/d) tanθ. Thermal expansion of the glass gives Δd/d = αΔT = 8.0×10−6 × 8.0 = 6.4×10−5. Thus Δθ ≈ 6.4×10−5 × tan 22.30° = 2.6×10−5 rad ≈ 0.0015°. The observed 0.02° is more than ten times larger, so thermal expansion of the grating cannot by itself explain the shift.
2. Estimate the number of grating lines illuminated by the beam and the resolving power at first and second order. Can the setup resolve the sodium D lines at 589.0 and 589.6 nm if the grating is fully illuminated? Does the narrow beam change the answer?
Read the worked answer
Lines illuminated: N = 600 mm−1 × 1.2 mm = 720. Resolving power R = mN: 720 at m = 1, 1440 at m = 2.
Required R for the sodium doublet: λ/Δλ = 589.0/0.6 ≈ 982. With the narrow 1.2 mm beam, first order (720) is insufficient, but second order (1440) is sufficient. If the full 25 mm ruled width were illuminated, N = 600 × 25 = 15 000, giving R = 15 000 and 30 000, so the doublet would be easily resolved at both orders. The narrow beam, not the grating’s full width, limits the first-order resolution.
3. The student wants to decide whether the 0.02° shift is thermal or misalignment. Describe one experimental check and one calculation that would help, and state what result would support each explanation.
Read the worked answer
A calculation: 0.02° corresponds to Δsinθ = cosθ Δθ = 0.925 × 3.49×10−4 = 3.23×10−4, so Δλ/λ = Δsinθ/sinθ ≈ 8.5×10−4, or Δλ ≈ 0.54 nm. That is far too large for 8 K of glass expansion, but it is plausible for a diode laser whose wavelength drifts by about 0.05–0.07 nm K−1.
An experimental check: measure the zero-order position and the +1 and −1 orders. A real grating-temperature change should shift the diffraction pattern symmetrically about the zero order; a mechanical rotation or holder expansion may move the zero order or shift one side differently. Also measure the grating temperature directly and repeat with a stabilised He–Ne laser. If the zero order moves, suspect misalignment; if the pattern stays centred but the angle changes with laser temperature, suspect laser wavelength drift.