Diffraction Gratings — Question 7

Question 7

Different spectral orders can place different wavelengths at the same angle

A normally illuminated grating has spacing $d=2.00\,\mu\mathrm m$. Its source includes wavelengths from 400 to $700\,\mathrm{nm}$. Orders m and n overlap when $m\lambda_m=n\lambda_n$.

1. Find the shared angle of $400\,\mathrm{nm}$ in third order and $600\,\mathrm{nm}$ in second order.

2. Find the second-order wavelength interval whose directions overlap some third-order wavelength in the stated source band.

Question 7 - Solution

Different spectral orders can place different wavelengths at the same angle

(a) Method and calculation

$$3(400)=2(600)=1200\,\mathrm{nm}\Longrightarrow\sin\theta=1200/2000.$$$$\boxed{\theta=36.8699^\circ}.$$

(b) Interpretation and further calculation

$$2\lambda_2=3\lambda_3,\quad400\leq\lambda_3\leq700\Longrightarrow600\leq\lambda_2\leq1050\,\mathrm{nm}.$$$$\lambda_2\in[400,700]\Longrightarrow\boxed{\lambda_2\in[600,700]\,\mathrm{nm},\quad\lambda_3\in[400,466.667]\,\mathrm{nm}}.$$

Physical check

A measured angle alone does not identify wavelength unless the order is known or unwanted orders are filtered.