Topic 7 of 9
Diffraction gratings and wavelength
A grating's regularly spaced slits reinforce light in particular directions. Use the spacing between slits, the order and the actual angle from the normal.
A transmission grating has many parallel slits. With monochromatic light normally incident, adjacent slits start in phase. Towards an outgoing angle θ, their path difference is a sin θ, where a is the slit spacing.
Locate principal maxima
When adjacent contributions differ by an integer number of wavelengths, they reinforce to form a principal maximum:
The central direction has n = 0. Matching positive and negative orders lie on opposite sides of the normal. Use the angle from that normal, not from the grating surface. The spacing a is a length; it is not the quoted number of lines per millimetre.
Estimate the optical scales
A few hundred lines per millimetre means a slit spacing of order micrometres. Visible wavelength is of order half a micrometre, so a/λ is only a few. Expect a few possible orders and first-order angles of order a few tenths of a radian. The exact sine bound, not this estimate, decides the highest integer order.
Worked orders and angles
Convert line density before using the equation
A grating has 600 lines/mm and is illuminated normally by light of wavelength 500 nm.
a = 1/600 000 m
≈ 1.67 × 10-6 m
sin θ = nλ/a = 0.300n
| Order n | sin θ | θ / ° |
|---|---|---|
| 0 | 0 | 0 |
| 1 | 0.300 | 17.46 |
| 2 | 0.600 | 36.87 |
| 3 | 0.900 | 64.16 |
Keep the unrounded spacing in the calculation. Since |sin θ| cannot exceed 1:
Highest integer order = 3
The allowed directions are 0, ±1, ±2 and ±3: seven in total. Order 4 would require sin θ = 1.2 and has no real direction. These are geometrically allowed maxima, not a promise that every spot has equal brightness or can be detected.
The third-order angle is about 64°. Using sin θ ≈ θ there would be inappropriate; use the inverse sine.
Use path difference for maxima and exact angles for large orders
Adjacent-slit path difference
Seven geometrically allowed directions
Measure a first-order screen displacement
Use θ = arctan(0.3145/1.000), then λ = a sin θ for first order. Measurements on both sides help locate the centre and reveal alignment problems.
Optional check Normally incident 500 nm light falls on a grating with 600 lines/mm. What is the highest geometrically allowed principal order and the total number of allowed order directions?
Determine wavelength from a screen measurement
Use a monochromatic source and a grating of known line density. Direct the beam normally onto the grating. Establish the zero-order direction, then measure a chosen order's angle or obtain it from a calibrated perpendicular screen.
For a screen distance D = 1.000 m and a first-order displacement y = 0.3145 m from the zero-order centre:
= arctan(0.3145/1.000) ≈ 17.46°
λ = a sin θ/n
≈ 5.00 × 10-7 m = 500 nm
Here y/D is the tangent, not the exact sine. First find the angle, then use its sine in the grating equation.
Record the order, line density, screen geometry and spot-centre positions with units. Measure matching spots on both sides to help establish the centre and check alignment. Repeated centre readings assess variation, but averaging cannot automatically correct a tilted grating. In a school laser experiment, use the approved equipment and beam-control procedure.