Diffraction gratings: dispersion and resolving power

The diffraction grating

A grating is a periodic array of N parallel slits, separated by a pitch d (the grating pitch). When illuminated by a monochromatic plane wave, it produces very narrow principal maxima in directions θ_k that satisfy the fundamental diffraction relation:

dsin(θ_k) = kλ, where k is a relative integer (diffraction order)

Effect of the number of slits N

The larger N is, the narrower and more intense the principal maxima are (their angular width decreases by a factor of 1/N), which allows closely spaced wavelengths to be clearly separated. This is why diffraction gratings used in spectroscopy often have several hundred lines per millimetre.

Dispersion and resolving power

• Angular dispersion: d(theta)/d(lambda) = k/(d*cos(theta)) -> the higher the order k, the more spread out the spectrum is.
• Resolving power: R = λ/δλ = k*N -> the ability to separate two neighbouring wavelengths, λ and λ + δλ.

Order k	Dispersion	Possible overlap
1	low	rare
2	medium	possible with order 1 in white light
3	high	frequent

Classical trap

The resolving power R = k*N does not depend directly on the pitch d, but on the total number of illuminated lines N and the order k used: a wider diffraction grating (more illuminated lines) provides better resolution, even at the same pitch d.