Wize University Physics Textbook (Master) > Wave Optics

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.

$Image illustrating Diffraction Grating$

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

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

$n=0,\pm1,\pm2, \ ...$

$d$ is the distance between two adjacent slits
$\theta$ is the angle between each ray and the horizontal line

For dark points, the path difference $\Delta l$ is given by:

\boxed{ \ d\sin\theta=\left(n+\frac{1}{2}\right)\lambda \ }

$n=0,\pm1,\pm2, \ ...$

If you're given number of lines per unit length ( $N$ ) , the distance between each two slits could be found from:

\boxed{ \ d=\frac{1}{N} \ }

Example: Diffraction Grating

We are supposed to design a spectrometer that can map the maximum of the first spectral order of the orange light $1.45$ cm away from the center of the zero spectral order ( $\lambda=590$ nm). The screen is at the distance of $2$ m from the source. How many lines per centimeter should the grating have?

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

$\dfrac{y}{D}=\tan\theta\approx\sin\theta=\dfrac{m\lambda}{d}$

Let's isolate $d$ to get:

$d=\dfrac{m\lambda D}y{}$

$=\dfrac{590\times10^{-9}\times2}{0.0145}$

$=8.1379\times10^{-5}$ (m)

From this we can get the number of lines:

$N=\dfrac{1}{d}=\dfrac{1}{8.1379\times10^{-5}}=12288$ lines/m

$=123$ lines/mm

Practice: Highest Possible Order

What is the highest order complete visible spectrum that can be produced by a grating with $680$ lines per mm?

$m=1$

$m=2$

$m=3$

$m=4$

I don't know