91Ó°ÊÓ

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 $\mathrm{θ}$ of the $mth$order diffraction pattern produced from a grating having line separation $d$ is

role="math" localid="1663073298458" $\mathit{d}\mathit{s}\mathit{i}\mathit{n}\mathit{θ}\mathbf{=}\mathit{m}\mathit{λ}$ …(¾±)

Here, $\mathrm{λ}$ 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),

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

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 $\sin \mathrm{θ}$ is 1. Thus, from equation (i),

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

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