Diffraction grating

diffraction grating is when we have a lot of tiny slits (openings) close to each other in a material, and we send waves through those slits.

When the waves pass through the slits, they diffract (spread out) and overlap with each other

This creates a pattern of bright and dark spots on a screen placed behind the grating. This is because of the interference of the waves - where the waves overlap, they can either add together (constructive) or cancel out (destructive).

Formula

The formula for diffraction grating is:

n \lambda = d \sin \theta

Where:

• n is the order of the spectrum (or maximum).
• \lambda is the wavelength of the wave.
• d is the distance between the slits (also called the grating spacing).
• \theta is the angle between the central maximum and the nth maximum.

Order of diffraction fringes

The order tells you how far the fringe we’re looking at is from the centre line. For example, a fringe 3 away from the centre line has an order of 3.

Converting ‘lines per millimetre’ to ‘distance between slits’

If we know the number of lines that are in one millimetre of the diffraction grating, we can find the distance between the slits by doing:

\text{distance}=\frac1{\text{lines per millimetre}}

That gives you an answer in millimetres, so

Finding the highest order

Just make \theta=90\degree, and then rearrange and solve for n!

n=\frac{d\sin\theta}{\lambda}

Given a grating with 400 lines per millimetre, how many orders can be produced on each side with an EM wave of wavelength 700nm?

• d=\frac1{400}=2.5\times10^{-3}mm
• d=2.5\times10^{-6}m
• \lambda=7\times10^{-7}m
• Let \theta=90\degree:
  • \sin\theta=1
• Rearrange the normal equation to find n on its own:
  • n=\frac{d\sin\theta}{\lambda}
• Substitute the values and solve!
  • n=\frac{2.5\times10^{-6}\times1}{700}
  • n\approx3.57\times10^{-9}