Resolving Power of Grating

Resolving Power Definition:

The separation between two objects which are very close to each other is called resolution and the ability of an optical instrument to produce their separate patterns is known as resolving power.

According to Rayleigh’s criterion of the resolution, the two spectral lines (or two-point sources) are said to be just resolved when the principal maximum in the diffraction pattern of one falls over the 1^st minimum in the diffraction pattern of the other and vice versa.

Let λ₁ and λ₂ be the wavelengths of two spectral lines resolved by the grating. Then if their wavelength difference is large, their principal maxima are separately visible and lines are well resolved (Fig. a).

Now when (λ₂ – λ₁) is small in such a way that the central maximum of λ, coincides with 1^st minimum of λ₂ and vice-versa, then the lines are said to be just resolved (Fig. b).

Resolving Power of Grating:

Resolving Power of Grating is its ability to form two separate maxima of two wavelengths which are very close to each other. It is defined as the ratio of the wavelength of one of the spectral line to the difference in the wavelengths between them such that the two lines appear to be just resolved.

Thus, R.P. = λ/dλ

(1) Consider incident light is having two spectral lines of wavelengths ‘λ’ and ‘λ + dλ’.

(2) Let ‘θ’ and ‘θ + dθ’ be their angles of diffraction respectively for the n^th order as shown in the figure.

(3) Condition for n^th principal maximum for λ is-

(a + b) sin θ = nλ ……….(i)

(4) Similarly n^th principal maximum of (λ + dλ) is expressed as-

(a + b) sin (θ + dθ) = n (λ + dλ) ……….(ii)

(5) Now according to Rayleigh’s criterion, if n^th principal maximum due to (λ + dλ) coincides on 1^st minimum after n^th principal maximum due to λ, then only two objects appear as separate.

(6) We know that condition for minima is-

(a + b) sin θ = mλ/N ……….(iii)
But m = 1, as it is 1^st minimum.

(7) Thus at an angle ‘θ + dθ’ there is 1^st minimum (λ/N) after n^th principal maximum due to λ (i.e. nλ).

Therefore, we can write-

(a + b) sin (θ + dθ) = nλ + λ/N ……….(iv)
⇒ n (λ + dλ) = nλ + λ/N [using (ii) and (iv)]
⇒ n dλ = λ/N
⇒ λ/dλ = Resolving Power = nN ……….(v) which is required equation.

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