A diffraction grating has 200 lines/mm. Light consisting of a continuous range of wavelengths between 550 nm and 700 nm s incident perpendicularly on the grating.

(a) What is the lowest order that is overlapped by another order?

(b) What is the highest order for which the complete spectrum is present?

Number of lines per mm of the grating = 200

Lower limit of wavelength in the incident spectrum = 550 nm

Upper limit of wavelength in the incident spectrum = 700 nm

An optical element with a periodic structure that divides light into a number of beams that move in diverse directions is known as a diffraction grating.

The angular distance $\theta$ of the $mth$order diffraction pattern produced from a grating having line separation $d$ is

role="math" localid="1663073298458" $\mathit{d}\mathit{s}\mathit{i}\mathit{n}\mathit{\theta }\mathbf{=}\mathit{m}\mathit{\lambda }$ …(i)

Here, $\lambda$ is the wavelength of the incident light.

For overlap of order the $m_{\text{th}}$ order produced from the 700nm wave should match with the ${\left(m+1\right)}_{\text{th}}$ order produced from the 550 nm wave.

Thus from equation (i),

$$\begin{gathered} m\times 700\text{nm}=\left(m+1\right)\times 550\text{nm} \\ m\left(700-550\right)=550 \\ m=\frac{550}{150} \\ =3.67 \end{gathered}$$

Thus the lowest order where overlap can occur is the 4th order

The line separation of the grating is

$$\begin{gathered} d=\frac{1}{200}\text{mm} \\ =0.005\text{mm} \\ =5000\text{nm} \end{gathered}$$

Maximum order can be obtained from the 700 nm wave. Maximum value of $\sin \theta$ is 1. Thus, from equation (i),

$$\begin{gathered} 5000\text{nm}=m_{\max }\times 700\text{nm} \\ {m}_{\max }=\frac{5000}{700} \\ =7.14 \end{gathered}$$

Thus, the maximum order for which the full spectrum can be obtained is the 7th order.