Dispersive and resolving power of grating

Diffraction of light is the bending of light around the edges of an obstacle or aperture whose size is comparable to the wavelength, so that light enters the geometrical shadow and bright and dark fringes form.

A plane transmission grating has many equal parallel slits of width a separated by opaque spaces b; d = a + b is the grating element. Maxima of order n occur at

d sin θ = n λ        (n = 0, 1, 2 ...)         (1)

Dispersive power

It is the rate of change of the diffraction angle with wavelength, dθ/dλ.

1. Differentiate (1) with respect to λ: d cos θ · dθ = n · dλ
2. Therefore dθ/dλ = n / (d cos θ)

So the dispersion is larger for a higher order, a smaller grating element (more lines per cm) and a larger θ.

Resolving power

It is R = λ/dλ, where dλ is the smallest wavelength difference between two lines (λ and λ + dλ) that are just resolved by the Rayleigh criterion (the principal maximum of one falls on the first minimum of the other).

1. Let the grating have N lines. The nth maximum of λ + dλ is at: d sin θ = n(λ + dλ)
2. The first minimum next to the nth maximum of λ occurs when the path difference across the whole grating is (nN + 1)λ: N d sin θ = (nN + 1)λ
3. Multiply step 1 by N: N d sin θ = nN(λ + dλ)
4. Equate: nN(λ + dλ) = nNλ + λ, so nN dλ = λ
5. Therefore R = λ/dλ = nN

Relation between them

Using N d = total width W of the ruled part of the grating:

R = nN = (n / d cos θ) × (N d cos θ) = (dθ/dλ) × W cos θ

So resolving power = (dispersive power) × (width of the diffracted beam, W cos θ). Also R = nN: the resolving power equals the total number of rulings times the order. A wide grating with many lines, used in a higher order, resolves closer wavelengths. For example, in the first order (n = 1) a grating with N = 15000 lines has R = λ/dλ = 15000.