A diffraction grating is made up of slits of width 260 nm with separation 810 nm. The grating is illuminated by monochromatic plane waves of wavelength λ = 550 nm at normal incidence.

a. How many maxima are there in the full diffraction pattern?
b. What is the angular width of a spectral line observed in the first order if the grating has 1300 slits?

Explanation:

Given that,

Grafting width is 260nm

Separation is d= 810nm

Wavelength is λ = 550 nm

A. The condition for maximum grating is given as

d Sinθ = mλ

where,

λ is the wavelength

d is the operation between the two slit.

The range is Sinθ is -1 and 1

But since θ Is between 0-180°

Then, 0≤Sinθ≤1

Then, we must have d ≥ mλ

So, d/λ ≤m

Then, m ≥ d/λ

m ≥ 810/550

m ≥1.47

So, we can take m = 1

There are three maxima m = -1,0,1

b. When N = 1300.

For first order maxima m = 1

Then, d Sinθ = mλ

d Sinθ = λ

The angular width is given as.

∆θ = λ/NdCosθ, where d Sinθ = λ

∆θ = dSinθ/NdCosθ

∆θ = tanθ/N, equation 1

d Sinθ = λ

Sinθ = λ / d

θ = ArcSin(λ/d)

θ = ArcSin(550/810)

θ = 42.77°

Substituting theta into equation 1

∆θ = tanθ/N

∆θ = tan(42.77)/1000

∆θ = 0.000925rad

To degree 2πrad = 360°

∆θ = 0.000925rad × 360°/2πrad

∆θ = 0.053°

The angular width of the spectral line is observed to be 0.053°