C.3.14 (HL)—Diffraction gratings

Core idea

A diffraction grating has many equally spaced parallel slits. Bright principal maxima occur when the path difference between adjacent slits is an integer number of wavelengths:

$$n\lambda=d\sin\theta$$

Here, $n$ is the order number $0,1,2,\ldots$, $\lambda$ is the wavelength, $d$ is the spacing between adjacent slits, and $\theta$ is measured from the central maximum to the chosen maximum.

Build the model

If a grating has $N$ lines per metre, the slit spacing is $d=1/N$. For a selected maximum, identify its order $n$, convert $d$ and $\lambda$ to consistent units, and solve for the unknown angle, wavelength or spacing. The central maximum is $n=0$; the first maxima on either side are $n=1$.

Check the allowed orders

Because $\lvert\sin\theta\rvert\le 1$, a wavelength can only produce orders satisfying $n\lambda\le d$. The largest possible order is therefore the greatest integer not exceeding $d/\lambda$. For overlapping wavelengths, equate their path-difference conditions: if the second-order maximum of $\lambda_1$ coincides with the third-order maximum of $\lambda_2$, then $2\lambda_1=3\lambda_2$.

Common trap

Do not use the number of lines per metre as $d$; invert it first. Do not count the central maximum as first order, and do not replace the grating equation with the small-angle approximation unless the question explicitly permits that approximation.