Question

What is the proof for a diffraction grating formula `dsintheta=nlamda`?

Answer 1

Let parallel rays of a monochromatic light of wavelength `lambda` be incident on a diffraction grating having slit separation `d`. If the grating has `N` lines per meter, the grating spacing is given by

`d=1/N`

Consider two parallel rays of light diffracted from two adjacent slits `A` and `B` as shown in the enlarged part of the figure. `theta` is the angle refracted ray makes with the the normal or angle of diffraction.

Path difference between the two rays is `AC`. Which is `=dsin theta`.

Light from `A` must be in phase with light from `B` for constructive interference. This happens when the path difference between the two is a whole number of wavelength `lambda`. Therefore,

`dsintheta=nlambda`
where `n` is an integer `=0,1,2,3...`

1. `n` is called the spectrum order. For `n = 1`, we have the first diffraction maximum.
2. `sintheta` can never be greater than `1`. Therefore, there is limit to the number of spectra that can be obtained.