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Wize University Physics Textbook (Master) > Wave Optics

Diffraction Grating

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Diffraction Grating


A diffraction grating has multiple slits located the same distance apart. But same principle and same formulas work as of the double-slit.



For bright points, the path difference Δl\Delta l is given by:

 dsin⁡θ=nλ \boxed { \ d\sin\theta=n\lambda \ }
n=0,±1,±2, ...n=0,\pm1,\pm2, \ ...

  • dd is the distance between two adjacent slits
  • θ\theta is the angle between each ray and the horizontal line




For dark points, the path difference Δl\Delta l is given by:
 dsin⁡θ=(n+12)λ \boxed{ \ d\sin\theta=\left(n+\frac{1}{2}\right)\lambda \ }
n=0,±1,±2, ...n=0,\pm1,\pm2, \ ...


PAGE BREAK


If you're given number of lines per unit length (NN) , the distance between each two slits could be found from:
 d=1N \boxed{ \ d=\frac{1}{N} \ }


Exam Tip
Just remember that NN and dd are reciprocals of each other.



Watch Out!
Often you're given the number of lines per mmmm or cmcm. This means the reciprocal will give you dd in those units as well, so you'll have to convert it to meters.

Example: Diffraction Grating


We are supposed to design a spectrometer that can map the maximum of the first spectral order of the orange light 1.451.45 cm away from the center of the zero spectral order (λ=590\lambda=590nm). The screen is at the distance of 22 m from the source. How many lines per centimeter should the grating have?

Here the distance to the screen D=2D=2 is much larger than the location of the first max (m=1m=1) which is at y=0.0145y=0.0145. Therefore we can use the small angle approximation where:

yD=tan⁡θ≈sin⁡θ=mλd\dfrac{y}{D}=\tan\theta\approx\sin\theta=\dfrac{m\lambda}{d}

Let's isolate dd to get:

d=mλDyd=\dfrac{m\lambda D}y{}

=590×10−9×20.0145=\dfrac{590\times10^{-9}\times2}{0.0145}

=8.1379×10−5=8.1379\times10^{-5} (m)

From this we can get the number of lines:

N=1d=18.1379×10−5=12288N=\dfrac{1}{d}=\dfrac{1}{8.1379\times10^{-5}}=12288 lines/m
=123=123 lines/mm

Practice: Highest Possible Order


What is the highest order complete visible spectrum that can be produced by a grating with 680680 lines per mm?