Interference: double slit and diffraction grating

What you are trying to do

Measure the wavelength of laser light twice over: from the fringe spacing of a double-slit pattern, and from the diffraction angles of a grating. The two answers should agree, and comparing them is the point.

Apparatus

• A laser pen or lab laser (class 2), on a stand so it cannot wander
• A double-slit slide of known slit separation, and a diffraction grating of known lines per millimetre
• A plain screen or wall, several metres away
• Metre rule for the fringes; tape measure for the longer distances
• A darkened room

Variables

• Independent: the slit-to-screen distance D (double slit); the order number n (grating)
• Dependent: the fringe spacing w; the position of each diffracted spot
• Control: the light source and its wavelength, and the slide in use — same slit separation or grating throughout

Method

1. Part one, the double slit. Mount the laser so its beam passes through both slits and falls on the screen a measured distance D away; start with D at least two metres, and darken the room until the fringes are crisp.

1. Never measure one fringe. Mark the centres of ten spacings with a pencil, measure across them all with a millimetre rule, and divide by ten; a tenth of a millimetre ruler error becomes a hundredth on w.
2. Repeat the measurement at several different values of D, moving the screen back in steps of half a metre.
3. Part two, the grating. Replace the slide with the grating, set it square to the beam, and let the pattern of sharp spots fall on the wall a measured distance L away.

1. For each order n, measure the distance from the central spot to the nth spot on both sides, and average the pair: any small misalignment of the grating then cancels.

Analysis

1. Double slit: plot w against D. The points should form a straight line through the origin with gradient λ/s, so the wavelength is the gradient multiplied by the slit separation.
2. Grating: for each order, θ comes from tan θ = x/L, and the wavelength from λ = d sin θ/n. Use sin of the measured angle, never x/L itself: the grating angles are far too large for the small-angle shortcut.
3. Compare the two wavelengths. They were measured with different apparatus and different equations, so their agreement is real evidence, not circularity.

Worked readings: the double slit

A 650 nm laser through slits 0.40 mm apart, fringes measured ten spacings at a time:

Each w is a ten-spacing measurement divided by ten. Plotting w against D gives a straight line through the origin with gradient (5.69 − 2.44) mm/(3.50 − 1.50) m = 1.63 mm per metre. Then λ = gradient × s = 1.63 × 10⁻³ × 4.00 × 10⁻⁴ = 6.50 × 10⁻⁷ m, which is 650 nm.

Worked readings: the grating

The same laser through a 300 lines-per-millimetre grating, spots measured on a wall:

With 300 lines per millimetre the slit spacing is d = 1/300 mm = 3.33 × 10⁻⁶ m, and the wall is L = 2.00 m away. For each order, θ comes from tan θ = x/L and the wavelength from d sin θ/n; all three orders return 650 nm, agreeing with the double-slit value. As a final check, n ≤ d/λ = 5.13, so only 5 orders can exist each side, however big the screen.

Where the uncertainty comes from

• Fringe spacing: The fringes sit a few millimetres apart, so a single spacing cannot be measured well with a rule. Measuring across ten and dividing is the standard cure, and the reason it works belongs in your evaluation.
• Slit separation: Taken from the manufacturer's label, so its uncertainty is fixed before you start; it usually dominates the double-slit result.
• Distances D and L: Metres long and measured with a tape, so their percentage uncertainty is small; longer is better on both counts.
• Grating angles: Higher orders sit at larger angles, where the same centimetre of position error moves sin θ less: quote the higher orders with more confidence, and average both sides.

What earns the marks

Safety