## Diffraction gratings and patterns

Posted in AQA AS Unit 2, AS Unit 2: Waves by Mr A on 21 Feb 2010

Diffraction grating equation

Important: The following derivation assumes that all rays incident on each part of the screen are parallel. This is a fair assumption, provided the distance from the slits to the screen is much larger than the slit separation.

Thus, for the central fringe, the rays travel exactly the same distance as one another.

For the first order fringe, each successive ray travels an extra path difference of $d \sin{\theta}$. Incidentally, this extra path difference must also be equal to $\lambda$, for constructive interference to occur.

If we proceed to the second order fringe, each successive ray must travel an extra $n \lambda$. It, therefore, follows that

$\boxed{n \lambda = d \sin{\theta}}$

Worked example (class demo)

If a red laser is shone through a diffraction grating with ? lines per mm at a screen ? m away, and the first order fringe makes an anlge of ?, what is the wavelength of the light?